| {"accepted_answer": {"answer_html": "<p>Rather than fitting separate models, you can handle both age groups in a single <code>gls()</code> model with <code>age</code> interactions. This has two advantages over separate <code>auto.arima()</code> models: you get formal tests of whether intervention effects differ between groups via the interaction terms, and you can account for the fact that the two series have very different variances.</p>\n<p><code>age * (time + step1 + ramp1 + ...)</code> gives each group its own intercept, pre-intervention slope, and intervention effects, plus tests of whether they differ. Then, <code>corAR1(form = ~ 1 | age)</code> fits AR(1) autocorrelation within each group and <code>varIdent(form = ~ 1 | age)</code> allows different residual variances per group, which matters here since children's prevalence is an order of magnitude larger.</p>\n<pre class=\"lang-r prettyprint-override\"><code>library(nlme)\nlibrary(dplyr)\n\ninterventions <- as.Date(c("2021-09-13", "2022-06-02", "2023-06-29"))\n\nd_its <- d |>\n arrange(age, month) |>\n mutate(.by = age,\n time = row_number() - 1,\n step1 = as.numeric(month >= interventions[1]),\n ramp1 = ifelse(step1 == 1, cumsum(step1) - 1, 0),\n step2 = as.numeric(month >= interventions[2]),\n ramp2 = ifelse(step2 == 1, cumsum(step2) - 1, 0),\n step3 = as.numeric(month >= interventions[3]),\n ramp3 = ifelse(step3 == 1, cumsum(step3) - 1, 0),\n sin12 = sin(2 * pi * time / 12),\n cos12 = cos(2 * pi * time / 12),\n sin6 = sin(2 * pi * time / 6),\n cos6 = cos(2 * pi * time / 6),\n sin4 = sin(2 * pi * time / 4),\n cos4 = cos(2 * pi * time / 4)\n ) |>\n mutate(age = factor(age))\n\nfit <- gls(\n prevalence ~ age * (time + step1 + ramp1 + step2 + ramp2 + step3 + ramp3 +\n sin12 + cos12 + sin6 + cos6 + sin4 + cos4),\n data = d_its,\n correlation = corAR1(form = ~ 1 | age),\n weights = varIdent(form = ~ 1 | age)\n)\n</code></pre>\n<p>The Fourier terms (<code>sin12</code>, <code>cos12</code>, etc.) handle the recurring seasonal pattern, which is more stable than trying to absorb it through ARMA error terms (see <a href=\"https://otexts.com/fpp3/dhr.html\" rel=\"nofollow noreferrer\">https://otexts.com/fpp3/dhr.html</a>).</p>\n<p>The reference level is Adults, so coefficients are interpreted as:</p>\n<ul>\n<li><code>time</code>: pre-intervention slope for adults (prevalence per month)</li>\n<li><code>ageChildren:time</code>: <em>additional</em> slope for children;</li>\n</ul>\n<p>The same logic applies for all <code>ramp</code> and <code>step</code> terms.</p>\n<pre class=\"lang-r prettyprint-override\"><code>coef(fit) |>\n (\\(e) data.frame(\n Segment = c("Pre", "Post-1", "Post-2", "Post-3"),\n Adults = round(cumsum(c(e["time"],\n e["ramp1"],\n e["ramp2"],\n e["ramp3"])), 3),\n Children = round(cumsum(c(e["time"] + e["ageChildren:time"],\n e["ramp1"] + e["ageChildren:ramp1"],\n e["ramp2"] + e["ageChildren:ramp2"],\n e["ramp3"] + e["ageChildren:ramp3"])), 3)\n))()\n\n#> Segment Adults Children\n#> time Pre 0.152 1.445\n#> ramp1 Post-1 0.919 9.276\n#> ramp2 Post-2 1.265 6.667\n#> ramp3 Post-3 2.387 8.418\n</code></pre>\n<pre class=\"lang-r prettyprint-override\"><code>par(mfrow = c(1, 2))\nacf(residuals(fit)[d_its<span class=\"math-container\">$age == \"Adults\"], main = \"Adults\")\nacf(residuals(fit)[d_its$</span>age == "Children"], main = "Children")\ndev.off()\n</code></pre>\n<p><img src=\"https://i.sstatic.net/2fnDCjiM.png\" alt=\"\" /></p>\n<pre class=\"lang-r prettyprint-override\"><code>summary(fit)\n\n#> Generalized least squares fit by REML\n#> Model: prevalence ~ age * (time + step1 + ramp1 + step2 + ramp2 + step3 + ramp3 + sin12 + cos12 + sin6 + cos6 + sin4 + cos4) \n#> Data: d_its \n#> AIC BIC logLik\n#> 1665.172 1769.226 -801.5858\n#> \n#> Correlation Structure: AR(1)\n#> Formula: ~1 | age \n#> Parameter estimate(s):\n#> Phi \n#> 0.1231768 \n#> Variance function:\n#> Structure: Different standard deviations per stratum\n#> Formula: ~1 | age \n#> Parameter estimates:\n#> Adults Children \n#> 1.000000 9.435418 \n#> \n#> Coefficients:\n#> Value Std.Error t-value p-value\n#> (Intercept) 10.25450 0.701853 14.610597 0.0000\n#> ageChildren 177.16318 6.659369 26.603599 0.0000\n#> time 0.15162 0.017799 8.518225 0.0000\n#> step1 1.83019 1.934456 0.946100 0.3452\n#> ramp1 0.76689 0.379858 2.018892 0.0448\n#> step2 -0.98456 2.681497 -0.367167 0.7139\n#> ramp2 0.34620 0.449034 0.770992 0.4416\n#> step3 -4.38832 2.094825 -2.094838 0.0374\n#> ramp3 1.12257 0.257109 4.366133 0.0000\n#> sin12 0.01845 0.399784 0.046145 0.9632\n#> cos12 1.16269 0.380402 3.056467 0.0025\n#> sin6 -0.70555 0.358820 -1.966309 0.0506\n#> cos6 -0.97079 0.355213 -2.732980 0.0068\n#> sin4 0.78337 0.333359 2.349941 0.0197\n#> cos4 0.05479 0.333865 0.164117 0.8698\n#> ageChildren:time 1.29381 0.168883 7.661002 0.0000\n#> ageChildren:step1 -1.02662 18.354630 -0.055933 0.9554\n#> ageChildren:ramp1 7.06407 3.604192 1.959960 0.0513\n#> ageChildren:step2 -35.82645 25.442751 -1.408120 0.1606\n#> ageChildren:ramp2 -2.95530 4.260553 -0.693643 0.4887\n#> ageChildren:step3 -29.49930 19.876247 -1.484148 0.1393\n#> ageChildren:ramp3 0.62770 2.439516 0.257306 0.7972\n#> ageChildren:sin12 4.22521 3.793256 1.113874 0.2666\n#> ageChildren:cos12 36.54005 3.609352 10.123715 0.0000\n#> ageChildren:sin6 -9.76953 3.404578 -2.869528 0.0045\n#> ageChildren:cos6 -25.08251 3.370352 -7.442104 0.0000\n#> ageChildren:sin4 13.68702 3.162997 4.327230 0.0000\n#> ageChildren:cos4 14.66062 3.167797 4.628016 0.0000\n#> \n#> Correlation: \n#> (Intr) agChld time step1 ramp1 step2 ramp2 step3 \n#> ageChildren -0.105 \n#> time -0.862 0.091 \n#> step1 0.173 -0.018 -0.309 \n#> ramp1 0.057 -0.006 -0.063 -0.763 \n#> step2 -0.021 0.002 0.020 0.366 -0.711 \n#> ramp2 -0.004 0.000 0.005 0.655 -0.831 0.306 \n#> step3 -0.015 0.002 0.015 0.007 -0.031 0.300 -0.403 \n#> ramp3 -0.016 0.002 0.016 0.008 -0.023 0.514 -0.520 0.649\n#> sin12 -0.056 0.006 0.043 0.155 -0.247 0.317 0.074 0.209\n#> cos12 -0.050 0.005 0.069 -0.188 0.137 -0.028 -0.155 0.065\n#> sin6 -0.008 0.001 -0.006 0.034 0.018 -0.103 0.052 -0.126\n#> cos6 -0.010 0.001 0.002 0.047 -0.026 -0.035 0.060 -0.071\n#> sin4 -0.024 0.002 0.028 -0.033 -0.004 0.059 -0.040 0.066\n#> cos4 -0.011 0.001 0.002 0.063 -0.078 0.095 0.024 0.069\n#> ageChildren:time 0.091 -0.862 -0.105 0.033 0.007 -0.002 0.000 -0.002\n#> ageChildren:step1 -0.018 0.173 0.033 -0.105 0.080 -0.039 -0.069 -0.001\n#> ageChildren:ramp1 -0.006 0.057 0.007 0.080 -0.105 0.075 0.088 0.003\n#> ageChildren:step2 0.002 -0.021 -0.002 -0.039 0.075 -0.105 -0.032 -0.032\n#> ageChildren:ramp2 0.000 -0.004 0.000 -0.069 0.088 -0.032 -0.105 0.043\n#> ageChildren:step3 0.002 -0.015 -0.002 -0.001 0.003 -0.032 0.043 -0.105\n#> ageChildren:ramp3 0.002 -0.016 -0.002 -0.001 0.002 -0.054 0.055 -0.068\n#> ageChildren:sin12 0.006 -0.056 -0.005 -0.016 0.026 -0.033 -0.008 -0.022\n#> ageChildren:cos12 0.005 -0.050 -0.007 0.020 -0.014 0.003 0.016 -0.007\n#> ageChildren:sin6 0.001 -0.008 0.001 -0.004 -0.002 0.011 -0.005 0.013\n#> ageChildren:cos6 0.001 -0.010 0.000 -0.005 0.003 0.004 -0.006 0.007\n#> ageChildren:sin4 0.002 -0.024 -0.003 0.003 0.000 -0.006 0.004 -0.007\n#> ageChildren:cos4 0.001 -0.011 0.000 -0.007 0.008 -0.010 -0.002 -0.007\n#> ramp3 sin12 cos12 sin6 cos6 sin4 cos4 agChl:\n#> ageChildren \n#> time \n#> step1 \n#> ramp1 \n#> step2 \n#> ramp2 \n#> step3 \n#> ramp3 \n#> sin12 0.235 \n#> cos12 0.057 -0.008 \n#> sin6 -0.099 -0.036 -0.015 \n#> cos6 -0.058 -0.011 -0.017 0.017 \n#> sin4 0.074 0.025 0.014 -0.013 -0.006 \n#> cos4 0.073 0.041 -0.007 -0.009 -0.004 0.009 \n#> ageChildren:time -0.002 -0.005 -0.007 0.001 0.000 -0.003 0.000 \n#> ageChildren:step1 -0.001 -0.016 0.020 -0.004 -0.005 0.003 -0.007 -0.309\n#> ageChildren:ramp1 0.002 0.026 -0.014 -0.002 0.003 0.000 0.008 -0.063\n#> ageChildren:step2 -0.054 -0.033 0.003 0.011 0.004 -0.006 -0.010 0.020\n#> ageChildren:ramp2 0.055 -0.008 0.016 -0.005 -0.006 0.004 -0.002 0.005\n#> ageChildren:step3 -0.068 -0.022 -0.007 0.013 0.007 -0.007 -0.007 0.015\n#> ageChildren:ramp3 -0.105 -0.025 -0.006 0.010 0.006 -0.008 -0.008 0.016\n#> ageChildren:sin12 -0.025 -0.105 0.001 0.004 0.001 -0.003 -0.004 0.043\n#> ageChildren:cos12 -0.006 0.001 -0.105 0.002 0.002 -0.002 0.001 0.069\n#> ageChildren:sin6 0.010 0.004 0.002 -0.105 -0.002 0.001 0.001 -0.006\n#> ageChildren:cos6 0.006 0.001 0.002 -0.002 -0.105 0.001 0.000 0.002\n#> ageChildren:sin4 -0.008 -0.003 -0.002 0.001 0.001 -0.105 -0.001 0.028\n#> ageChildren:cos4 -0.008 -0.004 0.001 0.001 0.000 -0.001 -0.105 0.002\n#> agChldrn:s1 agChldrn:r1 agChldrn:s2 agChldrn:r2 agChldrn:s3\n#> ageChildren \n#> time \n#> step1 \n#> ramp1 \n#> step2 \n#> ramp2 \n#> step3 \n#> ramp3 \n#> sin12 \n#> cos12 \n#> sin6 \n#> cos6 \n#> sin4 \n#> cos4 \n#> ageChildren:time \n#> ageChildren:step1 \n#> ageChildren:ramp1 -0.763 \n#> ageChildren:step2 0.366 -0.711 \n#> ageChildren:ramp2 0.655 -0.831 0.306 \n#> ageChildren:step3 0.007 -0.031 0.300 -0.403 \n#> ageChildren:ramp3 0.008 -0.023 0.514 -0.520 0.649 \n#> ageChildren:sin12 0.155 -0.247 0.317 0.074 0.209 \n#> ageChildren:cos12 -0.188 0.137 -0.028 -0.155 0.065 \n#> ageChildren:sin6 0.034 0.018 -0.103 0.052 -0.126 \n#> ageChildren:cos6 0.047 -0.026 -0.035 0.060 -0.071 \n#> ageChildren:sin4 -0.033 -0.004 0.059 -0.040 0.066 \n#> ageChildren:cos4 0.063 -0.078 0.095 0.024 0.069 \n#> agChldrn:r3 agChldrn:s12 agChldrn:c12 agChldrn:s6 agChldrn:c6\n#> ageChildren \n#> time \n#> step1 \n#> ramp1 \n#> step2 \n#> ramp2 \n#> step3 \n#> ramp3 \n#> sin12 \n#> cos12 \n#> sin6 \n#> cos6 \n#> sin4 \n#> cos4 \n#> ageChildren:time \n#> ageChildren:step1 \n#> ageChildren:ramp1 \n#> ageChildren:step2 \n#> ageChildren:ramp2 \n#> ageChildren:step3 \n#> ageChildren:ramp3 \n#> ageChildren:sin12 0.235 \n#> ageChildren:cos12 0.057 -0.008 \n#> ageChildren:sin6 -0.099 -0.036 -0.015 \n#> ageChildren:cos6 -0.058 -0.011 -0.017 0.017 \n#> ageChildren:sin4 0.074 0.025 0.014 -0.013 -0.006 \n#> ageChildren:cos4 0.073 0.041 -0.007 -0.009 -0.004 \n#> agChldrn:s4\n#> ageChildren \n#> time \n#> step1 \n#> ramp1 \n#> step2 \n#> ramp2 \n#> step3 \n#> ramp3 \n#> sin12 \n#> cos12 \n#> sin6 \n#> cos6 \n#> sin4 \n#> cos4 \n#> ageChildren:time \n#> ageChildren:step1 \n#> ageChildren:ramp1 \n#> ageChildren:step2 \n#> ageChildren:ramp2 \n#> ageChildren:step3 \n#> ageChildren:ramp3 \n#> ageChildren:sin12 \n#> ageChildren:cos12 \n#> ageChildren:sin6 \n#> ageChildren:cos6 \n#> ageChildren:sin4 \n#> ageChildren:cos4 0.009 \n#> \n#> Standardized residuals:\n#> Min Q1 Med Q3 Max \n#> -5.25291094 -0.42240774 0.04331099 0.45224365 3.26323939 \n#> \n#> Residual standard error: 2.608572 \n#> Degrees of freedom: 240 total; 212 residual\n</code></pre>\n<p><sup>Created on 2026-08-29 with <a href=\"https://reprex.tidyverse.org\" rel=\"nofollow noreferrer\">reprex v2.1.1</a></sup></p>\n", "answer_id": 676996, "answer_text": "Rather than fitting separate models, you can handle both age groups in a single gls() model with age interactions. This has two advantages over separate auto.arima() models: you get formal tests of whether intervention effects differ between groups via the interaction terms, and you can account for the fact that the two series have very different variances.\n\n\n\n\nage * (time + step1 + ramp1 + ...) gives each group its own intercept, pre-intervention slope, and intervention effects, plus tests of whether they differ. Then, corAR1(form = ~ 1 | age) fits AR(1) autocorrelation within each group and varIdent(form = ~ 1 | age) allows different residual variances per group, which matters here since children's prevalence is an order of magnitude larger.\n\n\n\n\nlibrary(nlme)\nlibrary(dplyr)\n\ninterventions <- as.Date(c(\"2021-09-13\", \"2022-06-02\", \"2023-06-29\"))\n\nd_its <- d |>\n arrange(age, month) |>\n mutate(.by = age,\n time = row_number() - 1,\n step1 = as.numeric(month >= interventions[1]),\n ramp1 = ifelse(step1 == 1, cumsum(step1) - 1, 0),\n step2 = as.numeric(month >= interventions[2]),\n ramp2 = ifelse(step2 == 1, cumsum(step2) - 1, 0),\n step3 = as.numeric(month >= interventions[3]),\n ramp3 = ifelse(step3 == 1, cumsum(step3) - 1, 0),\n sin12 = sin(2 * pi * time / 12),\n cos12 = cos(2 * pi * time / 12),\n sin6 = sin(2 * pi * time / 6),\n cos6 = cos(2 * pi * time / 6),\n sin4 = sin(2 * pi * time / 4),\n cos4 = cos(2 * pi * time / 4)\n ) |>\n mutate(age = factor(age))\n\nfit <- gls(\n prevalence ~ age * (time + step1 + ramp1 + step2 + ramp2 + step3 + ramp3 +\n sin12 + cos12 + sin6 + cos6 + sin4 + cos4),\n data = d_its,\n correlation = corAR1(form = ~ 1 | age),\n weights = varIdent(form = ~ 1 | age)\n)\n\n\n\n\n\nThe Fourier terms (sin12, cos12, etc.) handle the recurring seasonal pattern, which is more stable than trying to absorb it through ARMA error terms (see https://otexts.com/fpp3/dhr.html (https://otexts.com/fpp3/dhr.html)).\n\n\n\n\nThe reference level is Adults, so coefficients are interpreted as:\n\n\n\n\n\ntime: pre-intervention slope for adults (prevalence per month)\n\n\n\n\nageChildren:time: additional slope for children;\n\n\n\n\n\nThe same logic applies for all ramp and step terms.\n\n\n\n\ncoef(fit) |>\n (\\(e) data.frame(\n Segment = c(\"Pre\", \"Post-1\", \"Post-2\", \"Post-3\"),\n Adults = round(cumsum(c(e[\"time\"],\n e[\"ramp1\"],\n e[\"ramp2\"],\n e[\"ramp3\"])), 3),\n Children = round(cumsum(c(e[\"time\"] + e[\"ageChildren:time\"],\n e[\"ramp1\"] + e[\"ageChildren:ramp1\"],\n e[\"ramp2\"] + e[\"ageChildren:ramp2\"],\n e[\"ramp3\"] + e[\"ageChildren:ramp3\"])), 3)\n))()\n\n#> Segment Adults Children\n#> time Pre 0.152 1.445\n#> ramp1 Post-1 0.919 9.276\n#> ramp2 Post-2 1.265 6.667\n#> ramp3 Post-3 2.387 8.418\n\n\n\n\n\npar(mfrow = c(1, 2))\nacf(residuals(fit)[d_its$age == \"Adults\"], main = \"Adults\")\nacf(residuals(fit)[d_its$age == \"Children\"], main = \"Children\")\ndev.off()\n\n\n\n\n\n[image: ; source: https://i.sstatic.net/2fnDCjiM.png]\n\n\n\n\nsummary(fit)\n\n#> Generalized least squares fit by REML\n#> Model: prevalence ~ age * (time + step1 + ramp1 + step2 + ramp2 + step3 + ramp3 + sin12 + cos12 + sin6 + cos6 + sin4 + cos4) \n#> Data: d_its \n#> AIC BIC logLik\n#> 1665.172 1769.226 -801.5858\n#> \n#> Correlation Structure: AR(1)\n#> Formula: ~1 | age \n#> Parameter estimate(s):\n#> Phi \n#> 0.1231768 \n#> Variance function:\n#> Structure: Different standard deviations per stratum\n#> Formula: ~1 | age \n#> Parameter estimates:\n#> Adults Children \n#> 1.000000 9.435418 \n#> \n#> Coefficients:\n#> Value Std.Error t-value p-value\n#> (Intercept) 10.25450 0.701853 14.610597 0.0000\n#> ageChildren 177.16318 6.659369 26.603599 0.0000\n#> time 0.15162 0.017799 8.518225 0.0000\n#> step1 1.83019 1.934456 0.946100 0.3452\n#> ramp1 0.76689 0.379858 2.018892 0.0448\n#> step2 -0.98456 2.681497 -0.367167 0.7139\n#> ramp2 0.34620 0.449034 0.770992 0.4416\n#> step3 -4.38832 2.094825 -2.094838 0.0374\n#> ramp3 1.12257 0.257109 4.366133 0.0000\n#> sin12 0.01845 0.399784 0.046145 0.9632\n#> cos12 1.16269 0.380402 3.056467 0.0025\n#> sin6 -0.70555 0.358820 -1.966309 0.0506\n#> cos6 -0.97079 0.355213 -2.732980 0.0068\n#> sin4 0.78337 0.333359 2.349941 0.0197\n#> cos4 0.05479 0.333865 0.164117 0.8698\n#> ageChildren:time 1.29381 0.168883 7.661002 0.0000\n#> ageChildren:step1 -1.02662 18.354630 -0.055933 0.9554\n#> ageChildren:ramp1 7.06407 3.604192 1.959960 0.0513\n#> ageChildren:step2 -35.82645 25.442751 -1.408120 0.1606\n#> ageChildren:ramp2 -2.95530 4.260553 -0.693643 0.4887\n#> ageChildren:step3 -29.49930 19.876247 -1.484148 0.1393\n#> ageChildren:ramp3 0.62770 2.439516 0.257306 0.7972\n#> ageChildren:sin12 4.22521 3.793256 1.113874 0.2666\n#> ageChildren:cos12 36.54005 3.609352 10.123715 0.0000\n#> ageChildren:sin6 -9.76953 3.404578 -2.869528 0.0045\n#> ageChildren:cos6 -25.08251 3.370352 -7.442104 0.0000\n#> ageChildren:sin4 13.68702 3.162997 4.327230 0.0000\n#> ageChildren:cos4 14.66062 3.167797 4.628016 0.0000\n#> \n#> Correlation: \n#> (Intr) agChld time step1 ramp1 step2 ramp2 step3 \n#> ageChildren -0.105 \n#> time -0.862 0.091 \n#> step1 0.173 -0.018 -0.309 \n#> ramp1 0.057 -0.006 -0.063 -0.763 \n#> step2 -0.021 0.002 0.020 0.366 -0.711 \n#> ramp2 -0.004 0.000 0.005 0.655 -0.831 0.306 \n#> step3 -0.015 0.002 0.015 0.007 -0.031 0.300 -0.403 \n#> ramp3 -0.016 0.002 0.016 0.008 -0.023 0.514 -0.520 0.649\n#> sin12 -0.056 0.006 0.043 0.155 -0.247 0.317 0.074 0.209\n#> cos12 -0.050 0.005 0.069 -0.188 0.137 -0.028 -0.155 0.065\n#> sin6 -0.008 0.001 -0.006 0.034 0.018 -0.103 0.052 -0.126\n#> cos6 -0.010 0.001 0.002 0.047 -0.026 -0.035 0.060 -0.071\n#> sin4 -0.024 0.002 0.028 -0.033 -0.004 0.059 -0.040 0.066\n#> cos4 -0.011 0.001 0.002 0.063 -0.078 0.095 0.024 0.069\n#> ageChildren:time 0.091 -0.862 -0.105 0.033 0.007 -0.002 0.000 -0.002\n#> ageChildren:step1 -0.018 0.173 0.033 -0.105 0.080 -0.039 -0.069 -0.001\n#> ageChildren:ramp1 -0.006 0.057 0.007 0.080 -0.105 0.075 0.088 0.003\n#> ageChildren:step2 0.002 -0.021 -0.002 -0.039 0.075 -0.105 -0.032 -0.032\n#> ageChildren:ramp2 0.000 -0.004 0.000 -0.069 0.088 -0.032 -0.105 0.043\n#> ageChildren:step3 0.002 -0.015 -0.002 -0.001 0.003 -0.032 0.043 -0.105\n#> ageChildren:ramp3 0.002 -0.016 -0.002 -0.001 0.002 -0.054 0.055 -0.068\n#> ageChildren:sin12 0.006 -0.056 -0.005 -0.016 0.026 -0.033 -0.008 -0.022\n#> ageChildren:cos12 0.005 -0.050 -0.007 0.020 -0.014 0.003 0.016 -0.007\n#> ageChildren:sin6 0.001 -0.008 0.001 -0.004 -0.002 0.011 -0.005 0.013\n#> ageChildren:cos6 0.001 -0.010 0.000 -0.005 0.003 0.004 -0.006 0.007\n#> ageChildren:sin4 0.002 -0.024 -0.003 0.003 0.000 -0.006 0.004 -0.007\n#> ageChildren:cos4 0.001 -0.011 0.000 -0.007 0.008 -0.010 -0.002 -0.007\n#> ramp3 sin12 cos12 sin6 cos6 sin4 cos4 agChl:\n#> ageChildren \n#> time \n#> step1 \n#> ramp1 \n#> step2 \n#> ramp2 \n#> step3 \n#> ramp3 \n#> sin12 0.235 \n#> cos12 0.057 -0.008 \n#> sin6 -0.099 -0.036 -0.015 \n#> cos6 -0.058 -0.011 -0.017 0.017 \n#> sin4 0.074 0.025 0.014 -0.013 -0.006 \n#> cos4 0.073 0.041 -0.007 -0.009 -0.004 0.009 \n#> ageChildren:time -0.002 -0.005 -0.007 0.001 0.000 -0.003 0.000 \n#> ageChildren:step1 -0.001 -0.016 0.020 -0.004 -0.005 0.003 -0.007 -0.309\n#> ageChildren:ramp1 0.002 0.026 -0.014 -0.002 0.003 0.000 0.008 -0.063\n#> ageChildren:step2 -0.054 -0.033 0.003 0.011 0.004 -0.006 -0.010 0.020\n#> ageChildren:ramp2 0.055 -0.008 0.016 -0.005 -0.006 0.004 -0.002 0.005\n#> ageChildren:step3 -0.068 -0.022 -0.007 0.013 0.007 -0.007 -0.007 0.015\n#> ageChildren:ramp3 -0.105 -0.025 -0.006 0.010 0.006 -0.008 -0.008 0.016\n#> ageChildren:sin12 -0.025 -0.105 0.001 0.004 0.001 -0.003 -0.004 0.043\n#> ageChildren:cos12 -0.006 0.001 -0.105 0.002 0.002 -0.002 0.001 0.069\n#> ageChildren:sin6 0.010 0.004 0.002 -0.105 -0.002 0.001 0.001 -0.006\n#> ageChildren:cos6 0.006 0.001 0.002 -0.002 -0.105 0.001 0.000 0.002\n#> ageChildren:sin4 -0.008 -0.003 -0.002 0.001 0.001 -0.105 -0.001 0.028\n#> ageChildren:cos4 -0.008 -0.004 0.001 0.001 0.000 -0.001 -0.105 0.002\n#> agChldrn:s1 agChldrn:r1 agChldrn:s2 agChldrn:r2 agChldrn:s3\n#> ageChildren \n#> time \n#> step1 \n#> ramp1 \n#> step2 \n#> ramp2 \n#> step3 \n#> ramp3 \n#> sin12 \n#> cos12 \n#> sin6 \n#> cos6 \n#> sin4 \n#> cos4 \n#> ageChildren:time \n#> ageChildren:step1 \n#> ageChildren:ramp1 -0.763 \n#> ageChildren:step2 0.366 -0.711 \n#> ageChildren:ramp2 0.655 -0.831 0.306 \n#> ageChildren:step3 0.007 -0.031 0.300 -0.403 \n#> ageChildren:ramp3 0.008 -0.023 0.514 -0.520 0.649 \n#> ageChildren:sin12 0.155 -0.247 0.317 0.074 0.209 \n#> ageChildren:cos12 -0.188 0.137 -0.028 -0.155 0.065 \n#> ageChildren:sin6 0.034 0.018 -0.103 0.052 -0.126 \n#> ageChildren:cos6 0.047 -0.026 -0.035 0.060 -0.071 \n#> ageChildren:sin4 -0.033 -0.004 0.059 -0.040 0.066 \n#> ageChildren:cos4 0.063 -0.078 0.095 0.024 0.069 \n#> agChldrn:r3 agChldrn:s12 agChldrn:c12 agChldrn:s6 agChldrn:c6\n#> ageChildren \n#> time \n#> step1 \n#> ramp1 \n#> step2 \n#> ramp2 \n#> step3 \n#> ramp3 \n#> sin12 \n#> cos12 \n#> sin6 \n#> cos6 \n#> sin4 \n#> cos4 \n#> ageChildren:time \n#> ageChildren:step1 \n#> ageChildren:ramp1 \n#> ageChildren:step2 \n#> ageChildren:ramp2 \n#> ageChildren:step3 \n#> ageChildren:ramp3 \n#> ageChildren:sin12 0.235 \n#> ageChildren:cos12 0.057 -0.008 \n#> ageChildren:sin6 -0.099 -0.036 -0.015 \n#> ageChildren:cos6 -0.058 -0.011 -0.017 0.017 \n#> ageChildren:sin4 0.074 0.025 0.014 -0.013 -0.006 \n#> ageChildren:cos4 0.073 0.041 -0.007 -0.009 -0.004 \n#> agChldrn:s4\n#> ageChildren \n#> time \n#> step1 \n#> ramp1 \n#> step2 \n#> ramp2 \n#> step3 \n#> ramp3 \n#> sin12 \n#> cos12 \n#> sin6 \n#> cos6 \n#> sin4 \n#> cos4 \n#> ageChildren:time \n#> ageChildren:step1 \n#> ageChildren:ramp1 \n#> ageChildren:step2 \n#> ageChildren:ramp2 \n#> ageChildren:step3 \n#> ageChildren:ramp3 \n#> ageChildren:sin12 \n#> ageChildren:cos12 \n#> ageChildren:sin6 \n#> ageChildren:cos6 \n#> ageChildren:sin4 \n#> ageChildren:cos4 0.009 \n#> \n#> Standardized residuals:\n#> Min Q1 Med Q3 Max \n#> -5.25291094 -0.42240774 0.04331099 0.45224365 3.26323939 \n#> \n#> Residual standard error: 2.608572 \n#> Degrees of freedom: 240 total; 212 residual\n\n\n\n\n\nCreated on 2026-08-29 with reprex v2.1.1 (https://reprex.tidyverse.org)", "answer_url": "https://stats.stackexchange.com/a/676996", "author": "M--", "author_url": "https://stats.stackexchange.com/users/154449/m", "author_user_type": "registered", "content_license": "CC BY-SA 4.0", "created_at": "2026-08-29T05:35:19+00:00", "is_accepted": true, "provenance": {"attribution_required": true, "collected_at": "2026-10-01T03:27:06.169765+00:00", "license": "CC BY-SA 4.0", "license_url": "https://creativecommons.org/licenses/by-sa/4.0/", "raw_file": "raw/codex_api_v1/9155f81b0f1dac4b0f83fad58469033af46ad34ee8d731e569d063b41b3df6b6_1790825226563497500_0.json", "raw_sha256": "cbac4b1b24e2159f1c17ea9902eaf89a0a5260344e6dba6f75790ba955f0d04b", "source_api": "Stack Exchange API 2.3", "source_url": "https://api.stackexchange.com/2.3/questions/677298;677295;677280;677279;677278;677275;677269;677266;677262;677256;677245;677243;677242;677226;677225;677217;677211;677201;677194;677193;677186;677183;677179;677177;677171;677169;677168;677151;677149;677147;677137;677135;677131;677129;677115;677110;677109;677101;677099;677098;677096;677095;677094;677085;677083;677079;677078;677075;677071;677066;677065;677062;677058;677045;677041;677035;677033;677023;677021;676999;676997;676991;676982;676981;676979;676977;676975;676971;676966;676961;676958;676956;676952;676947;676945;676937;676935;676933;676924;676922;676899;676898;676893;676889;676888;676879;676874;676873;676870;676867;676865;676858;676855;676850;676842;676832;676830;676824;676823;676821/answers?filter=withbody&order=asc&page=1&pagesize=100&site=stats&sort=creation", "transformation": "API HTML retained; mechanical HTML-to-text; no LLM rewriting"}, "question_id": 676979, "revision_attribution": [{"content_license": "CC BY-SA 4.0", "contributor": {"display_name": "M--", "profile_url": "https://stats.stackexchange.com/users/154449/m", "user_type": "registered"}, "created_at": "2026-08-29T05:35:19+00:00", "raw_file": "raw/codex_api_v1/84bb317e707465a80f008019404ef1f770e152d74ef692d3b81c48903ec855cc_1790825232001984700_0.json", "raw_sha256": "75778d86d5325e15f4b491a748dc70ae6ac6ee98840d584e01ed28416cb71fbe", "revision_guid": "4F437721-3A00-4B4D-BA06-A2F50516CC59", "revision_number": 1, "revision_type": "single_user", "revision_url": "https://stats.stackexchange.com/revisions/4F437721-3A00-4B4D-BA06-A2F50516CC59/view-source"}], "score": 3, "updated_at": "2026-08-29T05:35:19+00:00"}, "code_blocks": [{"block_index": 0, "code_text": "month <- seq(\n from = as.Date(\"2016-01-01\"),\n to = as.Date(\"2025-12-31\"),\n by = \"month\"\n)\nmonth <- rep(month, each = 2)\nage <- rep(c(\"Children\", \"Adults\"), 120)\nprevalence <- c(\n 214.1, 11.3, 194.9, 11.1, 220.3, 11.8, 200.7, 11.7, 213.5, \n 11.7, 197.6, 11.5, 128.2, 10.3, 138.6, 9.9, 215, 12.1, 201.9, \n 11.9, 223.8, 12.2, 204.9, 12.4, 234.1, 12.7, 202, 11.9, 247, \n 13.6, 209.8, 13, 236.7, 13.4, 216.9, 13.1, 145.1, 12, 144.3, \n 11, 235.1, 13.7, 227.5, 13.9, 242.5, 13.9, 229.5, 13.8, 255.3, \n 14.5, 226.3, 13.6, 260.8, 15.2, 229.7, 14.2, 245.5, 14.7, 232.6, \n 14.4, 158.8, 13.4, 165, 12.5, 244.3, 14.8, 249.4, 15.7, 259.4, \n 15.5, 244, 15.3, 275.7, 16.2, 240.7, 14.9, 274.4, 16.2, 256.8, \n 16.3, 274.3, 16.4, 240.3, 15.5, 178.2, 15.5, 180.3, 13.6, 264, \n 16.3, 271.3, 17.4, 277.8, 16.9, 261.5, 16.8, 296.9, 17.8, 265.8, \n 17.2, 270.4, 17.9, 218.8, 16, 214.7, 16, 221.8, 16.7, 183.6, \n 16.8, 185.4, 14.9, 273.6, 18.7, 276.7, 19.3, 285, 19.4, 282.8, \n 19.9, 308.3, 20.2, 279.2, 19.3, 331.1, 22, 289.3, 21.2, 303, \n 20.9, 307.5, 21.8, 221.2, 20.9, 215.4, 18.9, 324, 23.4, 323.3, \n 24.3, 328.3, 24.7, 326.5, 25.8, 349.2, 26.1, 319.1, 25.5, 379.3, \n 29, 341.7, 28.3, 365.1, 28.6, 351.7, 30.1, 253.7, 28.6, 266.9, \n 27, 375, 33.4, 373.5, 34.4, 385.8, 35.5, 368.1, 36, 413.7, 38.8, \n 376, 37.4, 440.1, 42.5, 390.3, 40.8, 409.5, 42.6, 402, 44.9, \n 297.9, 43.4, 297.1, 40.2, 430.9, 49.3, 433.4, 51.2, 454.1, 53.1, \n 441.4, 53.4, 472.9, 57, 455.8, 57, 496.3, 62.6, 475, 63.1, 483.4, \n 63.3, 466, 65.1, 367.3, 66.2, 368.4, 58.6, 499.4, 70.8, 528.3, \n 76.1, 533.8, 76, 525.6, 77.2, 588.3, 84.8, 532.4, 81.1, 599.6, \n 90.9, 580.3, 90.8, 601.5, 93.8, 560.6, 95.1, 455.2, 96.9, 440.6, \n 83.7, 629.2, 107.7, 655.7, 112.8, 653.8, 113.2, 674.9, 118.7\n)\nd <- data.frame(month, age, prevalence)\n\npar(mfrow = c(1, 2))\nplot(prevalence ~ month, d, type = \"l\", main = \"Children\", subset = age == \"Children\")\nplot(prevalence ~ month, d, type = \"l\", main = \"Adults\", subset = age == \"Adults\")\ndev.off()\n", "language": "unspecified", "language_note": "Syntax-screened scientific code; language not asserted; not executed", "post_id": 676979, "sha256": "9cace690276c17ab9dec1c8efa1588d9d0fdb834a516aefaad95181c002107bf", "source_url": "https://stats.stackexchange.com/questions/676979/strategy-for-interrupted-time-series-of-correlated-stratified-time-series"}, {"block_index": 1, "code_text": "d2 <- subset(d, age == \"Children\")\nd2$prevalence <- ts(d2$prevalence, start = 2016, frequency = 12)\n\ninterventions <- as.Date(c(\"2021-09-13\", \"2022-06-02\", \"2023-06-29\"))\n\nstep1 <- as.numeric(d2$month >= interventions[1])\nramp1 <- ifelse(step1 == 1, cumsum(step1) - 1, 0)\n\nstep2 <- as.numeric(d2$month >= interventions[2])\nramp2 <- ifelse(step2 == 1, cumsum(step2) - 1, 0)\n\nstep3 <- as.numeric(d2$month >= interventions[3])\nramp3 <- ifelse(step3 == 1, cumsum(step3) - 1, 0)\n\ntime <- seq(from = 0, along.with = d2$month)\n\nxreg <- cbind(time, step1, ramp1, step2, ramp2, step3, ramp3)\n\ncbind(d2, xreg)\n# month age prevalence time step1 ramp1 step2 ramp2 step3 ramp3\n# 1 2016-01-01 Children 214.1 0 0 0 0 0 0 0\n# 3 2016-02-01 Children 194.9 1 0 0 0 0 0 0\n# 5 2016-03-01 Children 220.3 2 0 0 0 0 0 0\n# 7 2016-04-01 Children 200.7 3 0 0 0 0 0 0\n# 9 2016-05-01 Children 213.5 4 0 0 0 0 0 0\n# 11 2016-06-01 Children 197.6 5 0 0 0 0 0 0\n# 13 2016-07-01 Children 128.2 6 0 0 0 0 0 0\n# 15 2016-08-01 Children 138.6 7 0 0 0 0 0 0\n# 17 2016-09-01 Children 215.0 8 0 0 0 0 0 0\n# 19 2016-10-01 Children 201.9 9 0 0 0 0 0 0\n# 21 2016-11-01 Children 223.8 10 0 0 0 0 0 0\n# 23 2016-12-01 Children 204.9 11 0 0 0 0 0 0\n# 25 2017-01-01 Children 234.1 12 0 0 0 0 0 0\n# 27 2017-02-01 Children 202.0 13 0 0 0 0 0 0\n# 29 2017-03-01 Children 247.0 14 0 0 0 0 0 0\n# 31 2017-04-01 Children 209.8 15 0 0 0 0 0 0\n# 33 2017-05-01 Children 236.7 16 0 0 0 0 0 0\n# 35 2017-06-01 Children 216.9 17 0 0 0 0 0 0\n# 37 2017-07-01 Children 145.1 18 0 0 0 0 0 0\n# 39 2017-08-01 Children 144.3 19 0 0 0 0 0 0\n# 41 2017-09-01 Children 235.1 20 0 0 0 0 0 0\n# 43 2017-10-01 Children 227.5 21 0 0 0 0 0 0\n# 45 2017-11-01 Children 242.5 22 0 0 0 0 0 0\n# 47 2017-12-01 Children 229.5 23 0 0 0 0 0 0\n# 49 2018-01-01 Children 255.3 24 0 0 0 0 0 0\n# 51 2018-02-01 Children 226.3 25 0 0 0 0 0 0\n# 53 2018-03-01 Children 260.8 26 0 0 0 0 0 0\n# 55 2018-04-01 Children 229.7 27 0 0 0 0 0 0\n# 57 2018-05-01 Children 245.5 28 0 0 0 0 0 0\n# 59 2018-06-01 Children 232.6 29 0 0 0 0 0 0\n# 61 2018-07-01 Children 158.8 30 0 0 0 0 0 0\n# 63 2018-08-01 Children 165.0 31 0 0 0 0 0 0\n# 65 2018-09-01 Children 244.3 32 0 0 0 0 0 0\n# 67 2018-10-01 Children 249.4 33 0 0 0 0 0 0\n# 69 2018-11-01 Children 259.4 34 0 0 0 0 0 0\n# 71 2018-12-01 Children 244.0 35 0 0 0 0 0 0\n# 73 2019-01-01 Children 275.7 36 0 0 0 0 0 0\n# 75 2019-02-01 Children 240.7 37 0 0 0 0 0 0\n# 77 2019-03-01 Children 274.4 38 0 0 0 0 0 0\n# 79 2019-04-01 Children 256.8 39 0 0 0 0 0 0\n# 81 2019-05-01 Children 274.3 40 0 0 0 0 0 0\n# 83 2019-06-01 Children 240.3 41 0 0 0 0 0 0\n# 85 2019-07-01 Children 178.2 42 0 0 0 0 0 0\n# 87 2019-08-01 Children 180.3 43 0 0 0 0 0 0\n# 89 2019-09-01 Children 264.0 44 0 0 0 0 0 0\n# 91 2019-10-01 Children 271.3 45 0 0 0 0 0 0\n# 93 2019-11-01 Children 277.8 46 0 0 0 0 0 0\n# 95 2019-12-01 Children 261.5 47 0 0 0 0 0 0\n# 97 2020-01-01 Children 296.9 48 0 0 0 0 0 0\n# 99 2020-02-01 Children 265.8 49 0 0 0 0 0 0\n# 101 2020-03-01 Children 270.4 50 0 0 0 0 0 0\n# 103 2020-04-01 Children 218.8 51 0 0 0 0 0 0\n# 105 2020-05-01 Children 214.7 52 0 0 0 0 0 0\n# 107 2020-06-01 Children 221.8 53 0 0 0 0 0 0\n# 109 2020-07-01 Children 183.6 54 0 0 0 0 0 0\n# 111 2020-08-01 Children 185.4 55 0 0 0 0 0 0\n# 113 2020-09-01 Children 273.6 56 0 0 0 0 0 0\n# 115 2020-10-01 Children 276.7 57 0 0 0 0 0 0\n# 117 2020-11-01 Children 285.0 58 0 0 0 0 0 0\n# 119 2020-12-01 Children 282.8 59 0 0 0 0 0 0\n# 121 2021-01-01 Children 308.3 60 0 0 0 0 0 0\n# 123 2021-02-01 Children 279.2 61 0 0 0 0 0 0\n# 125 2021-03-01 Children 331.1 62 0 0 0 0 0 0\n# 127 2021-04-01 Children 289.3 63 0 0 0 0 0 0\n# 129 2021-05-01 Children 303.0 64 0 0 0 0 0 0\n# 131 2021-06-01 Children 307.5 65 0 0 0 0 0 0\n# 133 2021-07-01 Children 221.2 66 0 0 0 0 0 0\n# 135 2021-08-01 Children 215.4 67 0 0 0 0 0 0\n# 137 2021-09-01 Children 324.0 68 0 0 0 0 0 0\n# 139 2021-10-01 Children 323.3 69 1 0 0 0 0 0\n# 141 2021-11-01 Children 328.3 70 1 1 0 0 0 0\n# 143 2021-12-01 Children 326.5 71 1 2 0 0 0 0\n# 145 2022-01-01 Children 349.2 72 1 3 0 0 0 0\n# 147 2022-02-01 Children 319.1 73 1 4 0 0 0 0\n# 149 2022-03-01 Children 379.3 74 1 5 0 0 0 0\n# 151 2022-04-01 Children 341.7 75 1 6 0 0 0 0\n# 153 2022-05-01 Children 365.1 76 1 7 0 0 0 0\n# 155 2022-06-01 Children 351.7 77 1 8 0 0 0 0\n# 157 2022-07-01 Children 253.7 78 1 9 1 0 0 0\n# 159 2022-08-01 Children 266.9 79 1 10 1 1 0 0\n# 161 2022-09-01 Children 375.0 80 1 11 1 2 0 0\n# 163 2022-10-01 Children 373.5 81 1 12 1 3 0 0\n# 165 2022-11-01 Children 385.8 82 1 13 1 4 0 0\n# 167 2022-12-01 Children 368.1 83 1 14 1 5 0 0\n# 169 2023-01-01 Children 413.7 84 1 15 1 6 0 0\n# 171 2023-02-01 Children 376.0 85 1 16 1 7 0 0\n# 173 2023-03-01 Children 440.1 86 1 17 1 8 0 0\n# 175 2023-04-01 Children 390.3 87 1 18 1 9 0 0\n# 177 2023-05-01 Children 409.5 88 1 19 1 10 0 0\n# 179 2023-06-01 Children 402.0 89 1 20 1 11 0 0\n# 181 2023-07-01 Children 297.9 90 1 21 1 12 1 0\n# 183 2023-08-01 Children 297.1 91 1 22 1 13 1 1\n# 185 2023-09-01 Children 430.9 92 1 23 1 14 1 2\n# 187 2023-10-01 Children 433.4 93 1 24 1 15 1 3\n# 189 2023-11-01 Children 454.1 94 1 25 1 16 1 4\n# 191 2023-12-01 Children 441.4 95 1 26 1 17 1 5\n# 193 2024-01-01 Children 472.9 96 1 27 1 18 1 6\n# 195 2024-02-01 Children 455.8 97 1 28 1 19 1 7\n# 197 2024-03-01 Children 496.3 98 1 29 1 20 1 8\n# 199 2024-04-01 Children 475.0 99 1 30 1 21 1 9\n# 201 2024-05-01 Children 483.4 100 1 31 1 22 1 10\n# 203 2024-06-01 Children 466.0 101 1 32 1 23 1 11\n# 205 2024-07-01 Children 367.3 102 1 33 1 24 1 12\n# 207 2024-08-01 Children 368.4 103 1 34 1 25 1 13\n# 209 2024-09-01 Children 499.4 104 1 35 1 26 1 14\n# 211 2024-10-01 Children 528.3 105 1 36 1 27 1 15\n# 213 2024-11-01 Children 533.8 106 1 37 1 28 1 16\n# 215 2024-12-01 Children 525.6 107 1 38 1 29 1 17\n# 217 2025-01-01 Children 588.3 108 1 39 1 30 1 18\n# 219 2025-02-01 Children 532.4 109 1 40 1 31 1 19\n# 221 2025-03-01 Children 599.6 110 1 41 1 32 1 20\n# 223 2025-04-01 Children 580.3 111 1 42 1 33 1 21\n# 225 2025-05-01 Children 601.5 112 1 43 1 34 1 22\n# 227 2025-06-01 Children 560.6 113 1 44 1 35 1 23\n# 229 2025-07-01 Children 455.2 114 1 45 1 36 1 24\n# 231 2025-08-01 Children 440.6 115 1 46 1 37 1 25\n# 233 2025-09-01 Children 629.2 116 1 47 1 38 1 26\n# 235 2025-10-01 Children 655.7 117 1 48 1 39 1 27\n# 237 2025-11-01 Children 653.8 118 1 49 1 40 1 28\n# 239 2025-12-01 Children 674.9 119 1 50 1 41 1 29\n", "language": "unspecified", "language_note": "Syntax-screened scientific code; language not asserted; not executed", "post_id": 676979, "sha256": "4c753734c9398639a80aeac59ebd68bf3f63a83e5732ab45ae175d77017a8eda", "source_url": "https://stats.stackexchange.com/questions/676979/strategy-for-interrupted-time-series-of-correlated-stratified-time-series"}, {"block_index": 2, "code_text": "fit <- forecast::auto.arima(\n d2$prevalence,\n d = 0,\n D = 0,\n stepwise = FALSE,\n trace = TRUE,\n approximation = FALSE,\n xreg = xreg\n)\n# \n# Regression with ARIMA(0,0,0) errors : 1422.701\n# Regression with ARIMA(0,0,0) errors : 1256.821\n# Regression with ARIMA(0,0,0)(0,0,1)[12] errors : Inf\n# Regression with ARIMA(0,0,0)(0,0,1)[12] errors : 1186.586\n# Regression with ARIMA(0,0,0)(0,0,2)[12] errors : Inf\n# Regression with ARIMA(0,0,0)(0,0,2)[12] errors : Inf\n# ARIMA(0,0,0)(1,0,0)[12] with zero mean : Inf\n# ARIMA(0,0,0)(1,0,0)[12] with non-zero mean : Inf\n# ARIMA(0,0,0)(1,0,1)[12] with zero mean : Inf\n# ARIMA(0,0,0)(1,0,1)[12] with non-zero mean : Inf\n# ARIMA(0,0,0)(1,0,2)[12] with zero mean : Inf\n# ARIMA(0,0,0)(1,0,2)[12] with non-zero mean : Inf\n# ARIMA(0,0,0)(2,0,0)[12] with zero mean : Inf\n# ARIMA(0,0,0)(2,0,0)[12] with non-zero mean : Inf\n# ARIMA(0,0,0)(2,0,1)[12] with zero mean : Inf\n# ARIMA(0,0,0)(2,0,1)[12] with non-zero mean : Inf\n# ARIMA(0,0,0)(2,0,2)[12] with zero mean : Inf\n# ARIMA(0,0,0)(2,0,2)[12] with non-zero mean : Inf\n# Regression with ARIMA(0,0,1) errors : 1348.712\n# Regression with ARIMA(0,0,1) errors : 1237.018\n# Regression with ARIMA(0,0,1)(0,0,1)[12] errors : Inf\n# Regression with ARIMA(0,0,1)(0,0,1)[12] errors : 1167.011\n# Regression with ARIMA(0,0,1)(0,0,2)[12] errors : Inf\n# Regression with ARIMA(0,0,1)(0,0,2)[12] errors : Inf\n# ARIMA(0,0,1)(1,0,0)[12] with zero mean : Inf\n# ARIMA(0,0,1)(1,0,0)[12] with non-zero mean : Inf\n# ARIMA(0,0,1)(1,0,1)[12] with zero mean : Inf\n# ARIMA(0,0,1)(1,0,1)[12] with non-zero mean : Inf\n# ARIMA(0,0,1)(1,0,2)[12] with zero mean : Inf\n# ARIMA(0,0,1)(1,0,2)[12] with non-zero mean : Inf\n# ARIMA(0,0,1)(2,0,0)[12] with zero mean : Inf\n# ARIMA(0,0,1)(2,0,0)[12] with non-zero mean : Inf\n# ARIMA(0,0,1)(2,0,1)[12] with zero mean : Inf\n# ARIMA(0,0,1)(2,0,1)[12] with non-zero mean : Inf\n# ARIMA(0,0,1)(2,0,2)[12] with zero mean : Inf\n# Regression with ARIMA(0,0,1)(2,0,2)[12] errors : Inf\n# Regression with ARIMA(0,0,2) errors : 1304.633\n# Regression with ARIMA(0,0,2) errors : 1235.236\n# Regression with ARIMA(0,0,2)(0,0,1)[12] errors : 1226.35\n# Regression with ARIMA(0,0,2)(0,0,1)[12] errors : 1164.06\n# Regression with ARIMA(0,0,2)(0,0,2)[12] errors : Inf\n# Regression with ARIMA(0,0,2)(0,0,2)[12] errors : 1122.153\n# ARIMA(0,0,2)(1,0,0)[12] with zero mean : Inf\n# Regression with ARIMA(0,0,2)(1,0,0)[12] errors : Inf\n# ARIMA(0,0,2)(1,0,1)[12] with zero mean : Inf\n# ARIMA(0,0,2)(1,0,1)[12] with non-zero mean : Inf\n# ARIMA(0,0,2)(1,0,2)[12] with zero mean : Inf\n# ARIMA(0,0,2)(1,0,2)[12] with non-zero mean : Inf\n# ARIMA(0,0,2)(2,0,0)[12] with zero mean : Inf\n# ARIMA(0,0,2)(2,0,0)[12] with non-zero mean : Inf\n# ARIMA(0,0,2)(2,0,1)[12] with zero mean : Inf\n# ARIMA(0,0,2)(2,0,1)[12] with non-zero mean : Inf\n# Regression with ARIMA(0,0,3) errors : 1301.979\n# Regression with ARIMA(0,0,3) errors : Inf\n# Regression with ARIMA(0,0,3)(0,0,1)[12] errors : Inf\n# Regression with ARIMA(0,0,3)(0,0,1)[12] errors : 1166.444\n# Regression with ARIMA(0,0,3)(0,0,2)[12] errors : Inf\n# Regression with ARIMA(0,0,3)(0,0,2)[12] errors : Inf\n# ARIMA(0,0,3)(1,0,0)[12] with zero mean : Inf\n# Regression with ARIMA(0,0,3)(1,0,0)[12] errors : Inf\n# ARIMA(0,0,3)(1,0,1)[12] with zero mean : Inf\n# ARIMA(0,0,3)(1,0,1)[12] with non-zero mean : Inf\n# ARIMA(0,0,3)(2,0,0)[12] with zero mean : Inf\n# ARIMA(0,0,3)(2,0,0)[12] with non-zero mean : Inf\n# Regression with ARIMA(0,0,4) errors : Inf\n# Regression with ARIMA(0,0,4) errors : Inf\n# Regression with ARIMA(0,0,4)(0,0,1)[12] errors : Inf\n# Regression with ARIMA(0,0,4)(0,0,1)[12] errors : 1167.795\n# ARIMA(0,0,4)(1,0,0)[12] with zero mean : Inf\n# Regression with ARIMA(0,0,4)(1,0,0)[12] errors : Inf\n# Regression with ARIMA(0,0,5) errors : Inf\n# Regression with ARIMA(0,0,5) errors : Inf\n# Regression with ARIMA(1,0,0) errors : 1269.039\n# Regression with ARIMA(1,0,0) errors : 1236.551\n# Regression with ARIMA(1,0,0)(0,0,1)[12] errors : 1191.463\n# Regression with ARIMA(1,0,0)(0,0,1)[12] errors : 1163.925\n# Regression with ARIMA(1,0,0)(0,0,2)[12] errors : 1144.206\n# Regression with ARIMA(1,0,0)(0,0,2)[12] errors : 1122.139\n# ARIMA(1,0,0)(1,0,0)[12] with zero mean : Inf\n# ARIMA(1,0,0)(1,0,0)[12] with non-zero mean : Inf\n# ARIMA(1,0,0)(1,0,1)[12] with zero mean : Inf\n# ARIMA(1,0,0)(1,0,1)[12] with non-zero mean : Inf\n# ARIMA(1,0,0)(1,0,2)[12] with zero mean : Inf\n# ARIMA(1,0,0)(1,0,2)[12] with non-zero mean : Inf\n# ARIMA(1,0,0)(2,0,0)[12] with zero mean : Inf\n# Regression with ARIMA(1,0,0)(2,0,0)[12] errors : Inf\n# ARIMA(1,0,0)(2,0,1)[12] with zero mean : Inf\n# ARIMA(1,0,0)(2,0,1)[12] with non-zero mean : Inf\n# ARIMA(1,0,0)(2,0,2)[12] with zero mean : Inf\n# ARIMA(1,0,0)(2,0,2)[12] with non-zero mean : Inf\n# Regression with ARIMA(1,0,1) errors : 1269.741\n# Regression with ARIMA(1,0,1) errors : 1237.127\n# Regression with ARIMA(1,0,1)(0,0,1)[12] errors : 1191.287\n# Regression with ARIMA(1,0,1)(0,0,1)[12] errors : 1165.649\n# Regression with ARIMA(1,0,1)(0,0,2)[12] errors : 1145.611\n# Regression with ARIMA(1,0,1)(0,0,2)[12] errors : 1123.56\n# ARIMA(1,0,1)(1,0,0)[12] with zero mean : Inf\n# ARIMA(1,0,1)(1,0,0)[12] with non-zero mean : Inf\n# ARIMA(1,0,1)(1,0,1)[12] with zero mean : Inf\n# ARIMA(1,0,1)(1,0,1)[12] with non-zero mean : Inf\n# ARIMA(1,0,1)(1,0,2)[12] with zero mean : Inf\n# ARIMA(1,0,1)(1,0,2)[12] with non-zero mean : Inf\n# ARIMA(1,0,1)(2,0,0)[12] with zero mean : Inf\n# Regression with ARIMA(1,0,1)(2,0,0)[12] errors : Inf\n# ARIMA(1,0,1)(2,0,1)[12] with zero mean : Inf\n# ARIMA(1,0,1)(2,0,1)[12] with non-zero mean : Inf\n# Regression with ARIMA(1,0,2) errors : Inf\n# ARIMA(1,0,2) with non-zero mean : Inf\n# Regression with ARIMA(1,0,2)(0,0,1)[12] errors : Inf\n# ARIMA(1,0,1)(2,0,1)[12] with non-zero mean : Inf\n# Regression with ARIMA(1,0,2) errors : Inf\n# ARIMA(1,0,2) with non-zero mean : Inf\n# Regression with ARIMA(1,0,2)(0,0,1)[12] errors : Inf\n# Regression with ARIMA(1,0,2)(0,0,1)[12] errors : Inf\n# Regression with ARIMA(1,0,2)(0,0,2)[12] errors : Inf\n# Regression with ARIMA(1,0,2)(0,0,2)[12] errors : Inf\n# ARIMA(1,0,2)(1,0,0)[12] with zero mean : Inf\n# ARIMA(1,0,2)(1,0,0)[12] with non-zero mean : Inf\n# ARIMA(1,0,2)(1,0,1)[12] with zero mean : Inf\n# ARIMA(1,0,2)(1,0,1)[12] with non-zero mean : Inf\n# ARIMA(1,0,2)(2,0,0)[12] with zero mean : Inf\n# Regression with ARIMA(1,0,2)(2,0,0)[12] errors : Inf\n# Regression with ARIMA(1,0,3) errors : Inf\n# Regression with ARIMA(1,0,3) errors : Inf\n# Regression with ARIMA(1,0,3)(0,0,1)[12] errors : 1196.785\n# Regression with ARIMA(1,0,3)(0,0,1)[12] errors : Inf\n# ARIMA(1,0,3)(1,0,0)[12] with zero mean : Inf\n# ARIMA(1,0,3)(1,0,0)[12] with non-zero mean : Inf\n# Regression with ARIMA(1,0,4) errors : Inf\n# Regression with ARIMA(1,0,4) errors : Inf\n# Regression with ARIMA(2,0,0) errors : 1270.227\n# Regression with ARIMA(2,0,0) errors : 1235.453\n# Regression with ARIMA(2,0,0)(0,0,1)[12] errors : 1191.855\n# Regression with ARIMA(2,0,0)(0,0,1)[12] errors : 1165.096\n# Regression with ARIMA(2,0,0)(0,0,2)[12] errors : 1145.887\n# Regression with ARIMA(2,0,0)(0,0,2)[12] errors : 1123.031\n# ARIMA(2,0,0)(1,0,0)[12] with zero mean : Inf\n# ARIMA(2,0,0)(1,0,0)[12] with non-zero mean : Inf\n# ARIMA(2,0,0)(1,0,1)[12] with zero mean : Inf\n# ARIMA(2,0,0)(1,0,1)[12] with non-zero mean : Inf\n# ARIMA(2,0,0)(1,0,2)[12] with zero mean : Inf\n# ARIMA(2,0,0)(1,0,2)[12] with non-zero mean : Inf\n# ARIMA(2,0,0)(2,0,0)[12] with zero mean : Inf\n# ARIMA(2,0,0)(2,0,0)[12] with non-zero mean : Inf\n# ARIMA(2,0,0)(2,0,1)[12] with zero mean : Inf\n# Regression with ARIMA(2,0,0)(2,0,1)[12] errors : Inf\n# Regression with ARIMA(2,0,1) errors : Inf\n# Regression with ARIMA(2,0,1) errors : Inf\n# Regression with ARIMA(2,0,1)(0,0,1)[12] errors : Inf\n# Regression with ARIMA(2,0,1)(0,0,1)[12] errors : Inf\n# Regression with ARIMA(2,0,1)(0,0,2)[12] errors : Inf\n# Regression with ARIMA(2,0,1)(0,0,2)[12] errors : Inf\n# ARIMA(2,0,1)(1,0,0)[12] with zero mean : Inf\n# ARIMA(2,0,1)(1,0,0)[12] with non-zero mean : Inf\n# Regression with ARIMA(2,0,1)(1,0,1)[12] errors : Inf\n# ARIMA(2,0,1)(1,0,1)[12] with non-zero mean : Inf\n# Regression with ARIMA(2,0,1)(2,0,0)[12] errors : Inf\n# Regression with ARIMA(2,0,1)(2,0,0)[12] errors : Inf\n# Regression with ARIMA(2,0,2) errors : Inf\n# Regression with ARIMA(2,0,2) errors : Inf\n# ARIMA(2,0,2)(0,0,1)[12] with zero mean : Inf\n# Regression with ARIMA(2,0,2)(0,0,1)[12] errors : Inf\n# ARIMA(2,0,2)(1,0,0)[12] with zero mean : Inf\n# ARIMA(2,0,2)(1,0,0)[12] with non-zero mean : Inf\n# ARIMA(2,0,3) with zero mean : Inf\n# Regression with ARIMA(2,0,3) errors : Inf\n# Regression with ARIMA(3,0,0) errors : 1271.552\n# Regression with ARIMA(3,0,0) errors : 1234.442\n# Regression with ARIMA(3,0,0)(0,0,1)[12] errors : 1193.944\n# Regression with ARIMA(3,0,0)(0,0,1)[12] errors : 1165.004\n# Regression with ARIMA(3,0,0)(0,0,2)[12] errors : 1147.337\n# Regression with ARIMA(3,0,0)(0,0,2)[12] errors : 1124.69\n# ARIMA(3,0,0)(1,0,0)[12] with zero mean : Inf\n# ARIMA(3,0,0)(1,0,0)[12] with non-zero mean : Inf\n# ARIMA(3,0,0)(1,0,1)[12] with zero mean : Inf\n# ARIMA(3,0,0)(1,0,1)[12] with non-zero mean : Inf\n# ARIMA(3,0,0)(2,0,0)[12] with zero mean : Inf\n# ARIMA(3,0,0)(2,0,0)[12] with non-zero mean : Inf\n# Regression with ARIMA(3,0,1) errors : Inf\n# Regression with ARIMA(3,0,1) errors : Inf\n# Regression with ARIMA(3,0,1)(0,0,1)[12] errors : Inf\n# Regression with ARIMA(3,0,1)(0,0,1)[12] errors : Inf\n# ARIMA(3,0,1)(1,0,0)[12] with zero mean : Inf\n# ARIMA(3,0,1)(1,0,0)[12] with non-zero mean : Inf\n# Regression with ARIMA(3,0,2) errors : Inf\n# Regression with ARIMA(3,0,2) errors : Inf\n# Regression with ARIMA(4,0,0) errors : 1268.176\n# Regression with ARIMA(4,0,0) errors : 1236.501\n# Regression with ARIMA(4,0,0)(0,0,1)[12] errors : 1191.278\n# Regression with ARIMA(4,0,0)(0,0,1)[12] errors : 1167.39\n# ARIMA(4,0,0)(1,0,0)[12] with zero mean : Inf\n# ARIMA(4,0,0)(1,0,0)[12] with non-zero mean : Inf\n# Regression with ARIMA(4,0,1) errors : Inf\n# Regression with ARIMA(4,0,1) errors : Inf\n# Regression with ARIMA(5,0,0) errors : 1270.388\n# Regression with ARIMA(5,0,0) errors : 1226.449\n# \n# \n# \n# Best model: Regression with ARIMA(1,0,0)(0,0,2)[12] errors \n# \n\nsummary(fit)\n# Series: d2$prevalence \n# Regression with ARIMA(1,0,0)(0,0,2)[12] errors \n# \n# Coefficients:\n# ar1 sma1 sma2 intercept time step1 ramp1 step2 ramp2 step3 ramp3\n# 0.5256 1.0413 0.6987 188.8179 1.5717 3.3284 2.1966 -5.6181 0.4640 0.4378 4.6447\n# s.e. 0.0800 0.1536 0.1312 20.6895 0.4724 13.0492 3.5219 23.9955 3.8197 19.1790 2.5724\n# \n# sigma^2 = 494.9: log likelihood = -547.61\n# AIC=1119.22 AICc=1122.14 BIC=1152.67\n# \n# Training set error measures:\n# ME RMSE MAE MPE MAPE MASE ACF1\n# Training set -0.2760281 21.20232 16.21419 -0.9807653 5.578825 0.3604042 0.05861382\n", "language": "unspecified", "language_note": "Syntax-screened scientific code; language not asserted; not executed", "post_id": 676979, "sha256": "91f70a0cc2d89034fa701fed928aa731ddc1fd142c8e82f207b970b96a31654f", "source_url": "https://stats.stackexchange.com/questions/676979/strategy-for-interrupted-time-series-of-correlated-stratified-time-series"}, {"block_index": 3, "code_text": "res <- residuals(fit)\nl <- 36\n\npar(mfrow = c(3, 1))\nplot(res)\nacf(res, lag.max = l)\npacf(res, lag.max = l)\ndev.off()\n\nBox.test(res, lag = 24, type = \"Ljung-Box\", fitdf = 3)\n# \n# Box-Ljung test\n# \n# data: res\n# X-squared = 49.814, df = 21, p-value = 0.0003872\n# \n", "language": "unspecified", "language_note": "Syntax-screened scientific code; language not asserted; not executed", "post_id": 676979, "sha256": "eceaa543efc075433b65e321fb99244a1259e5c10bf8ecf2c60a1a24e576db3f", "source_url": "https://stats.stackexchange.com/questions/676979/strategy-for-interrupted-time-series-of-correlated-stratified-time-series"}, {"block_index": 0, "code_text": "library(nlme)\nlibrary(dplyr)\n\ninterventions <- as.Date(c(\"2021-09-13\", \"2022-06-02\", \"2023-06-29\"))\n\nd_its <- d |>\n arrange(age, month) |>\n mutate(.by = age,\n time = row_number() - 1,\n step1 = as.numeric(month >= interventions[1]),\n ramp1 = ifelse(step1 == 1, cumsum(step1) - 1, 0),\n step2 = as.numeric(month >= interventions[2]),\n ramp2 = ifelse(step2 == 1, cumsum(step2) - 1, 0),\n step3 = as.numeric(month >= interventions[3]),\n ramp3 = ifelse(step3 == 1, cumsum(step3) - 1, 0),\n sin12 = sin(2 * pi * time / 12),\n cos12 = cos(2 * pi * time / 12),\n sin6 = sin(2 * pi * time / 6),\n cos6 = cos(2 * pi * time / 6),\n sin4 = sin(2 * pi * time / 4),\n cos4 = cos(2 * pi * time / 4)\n ) |>\n mutate(age = factor(age))\n\nfit <- gls(\n prevalence ~ age * (time + step1 + ramp1 + step2 + ramp2 + step3 + ramp3 +\n sin12 + cos12 + sin6 + cos6 + sin4 + cos4),\n data = d_its,\n correlation = corAR1(form = ~ 1 | age),\n weights = varIdent(form = ~ 1 | age)\n)\n", "language": "unspecified", "language_note": "Syntax-screened scientific code; language not asserted; not executed", "post_id": 676996, "sha256": "62139218828fa06f701a06c8cf502690ac9f06dd8b87b97cf97522170fb9a176", "source_url": "https://stats.stackexchange.com/a/676996"}, {"block_index": 3, "code_text": "summary(fit)\n\n#> Generalized least squares fit by REML\n#> Model: prevalence ~ age * (time + step1 + ramp1 + step2 + ramp2 + step3 + ramp3 + sin12 + cos12 + sin6 + cos6 + sin4 + cos4) \n#> Data: d_its \n#> AIC BIC logLik\n#> 1665.172 1769.226 -801.5858\n#> \n#> Correlation Structure: AR(1)\n#> Formula: ~1 | age \n#> Parameter estimate(s):\n#> Phi \n#> 0.1231768 \n#> Variance function:\n#> Structure: Different standard deviations per stratum\n#> Formula: ~1 | age \n#> Parameter estimates:\n#> Adults Children \n#> 1.000000 9.435418 \n#> \n#> Coefficients:\n#> Value Std.Error t-value p-value\n#> (Intercept) 10.25450 0.701853 14.610597 0.0000\n#> ageChildren 177.16318 6.659369 26.603599 0.0000\n#> time 0.15162 0.017799 8.518225 0.0000\n#> step1 1.83019 1.934456 0.946100 0.3452\n#> ramp1 0.76689 0.379858 2.018892 0.0448\n#> step2 -0.98456 2.681497 -0.367167 0.7139\n#> ramp2 0.34620 0.449034 0.770992 0.4416\n#> step3 -4.38832 2.094825 -2.094838 0.0374\n#> ramp3 1.12257 0.257109 4.366133 0.0000\n#> sin12 0.01845 0.399784 0.046145 0.9632\n#> cos12 1.16269 0.380402 3.056467 0.0025\n#> sin6 -0.70555 0.358820 -1.966309 0.0506\n#> cos6 -0.97079 0.355213 -2.732980 0.0068\n#> sin4 0.78337 0.333359 2.349941 0.0197\n#> cos4 0.05479 0.333865 0.164117 0.8698\n#> ageChildren:time 1.29381 0.168883 7.661002 0.0000\n#> ageChildren:step1 -1.02662 18.354630 -0.055933 0.9554\n#> ageChildren:ramp1 7.06407 3.604192 1.959960 0.0513\n#> ageChildren:step2 -35.82645 25.442751 -1.408120 0.1606\n#> ageChildren:ramp2 -2.95530 4.260553 -0.693643 0.4887\n#> ageChildren:step3 -29.49930 19.876247 -1.484148 0.1393\n#> ageChildren:ramp3 0.62770 2.439516 0.257306 0.7972\n#> ageChildren:sin12 4.22521 3.793256 1.113874 0.2666\n#> ageChildren:cos12 36.54005 3.609352 10.123715 0.0000\n#> ageChildren:sin6 -9.76953 3.404578 -2.869528 0.0045\n#> ageChildren:cos6 -25.08251 3.370352 -7.442104 0.0000\n#> ageChildren:sin4 13.68702 3.162997 4.327230 0.0000\n#> ageChildren:cos4 14.66062 3.167797 4.628016 0.0000\n#> \n#> Correlation: \n#> (Intr) agChld time step1 ramp1 step2 ramp2 step3 \n#> ageChildren -0.105 \n#> time -0.862 0.091 \n#> step1 0.173 -0.018 -0.309 \n#> ramp1 0.057 -0.006 -0.063 -0.763 \n#> step2 -0.021 0.002 0.020 0.366 -0.711 \n#> ramp2 -0.004 0.000 0.005 0.655 -0.831 0.306 \n#> step3 -0.015 0.002 0.015 0.007 -0.031 0.300 -0.403 \n#> ramp3 -0.016 0.002 0.016 0.008 -0.023 0.514 -0.520 0.649\n#> sin12 -0.056 0.006 0.043 0.155 -0.247 0.317 0.074 0.209\n#> cos12 -0.050 0.005 0.069 -0.188 0.137 -0.028 -0.155 0.065\n#> sin6 -0.008 0.001 -0.006 0.034 0.018 -0.103 0.052 -0.126\n#> cos6 -0.010 0.001 0.002 0.047 -0.026 -0.035 0.060 -0.071\n#> sin4 -0.024 0.002 0.028 -0.033 -0.004 0.059 -0.040 0.066\n#> cos4 -0.011 0.001 0.002 0.063 -0.078 0.095 0.024 0.069\n#> ageChildren:time 0.091 -0.862 -0.105 0.033 0.007 -0.002 0.000 -0.002\n#> ageChildren:step1 -0.018 0.173 0.033 -0.105 0.080 -0.039 -0.069 -0.001\n#> ageChildren:ramp1 -0.006 0.057 0.007 0.080 -0.105 0.075 0.088 0.003\n#> ageChildren:step2 0.002 -0.021 -0.002 -0.039 0.075 -0.105 -0.032 -0.032\n#> ageChildren:ramp2 0.000 -0.004 0.000 -0.069 0.088 -0.032 -0.105 0.043\n#> ageChildren:step3 0.002 -0.015 -0.002 -0.001 0.003 -0.032 0.043 -0.105\n#> ageChildren:ramp3 0.002 -0.016 -0.002 -0.001 0.002 -0.054 0.055 -0.068\n#> ageChildren:sin12 0.006 -0.056 -0.005 -0.016 0.026 -0.033 -0.008 -0.022\n#> ageChildren:cos12 0.005 -0.050 -0.007 0.020 -0.014 0.003 0.016 -0.007\n#> ageChildren:sin6 0.001 -0.008 0.001 -0.004 -0.002 0.011 -0.005 0.013\n#> ageChildren:cos6 0.001 -0.010 0.000 -0.005 0.003 0.004 -0.006 0.007\n#> ageChildren:sin4 0.002 -0.024 -0.003 0.003 0.000 -0.006 0.004 -0.007\n#> ageChildren:cos4 0.001 -0.011 0.000 -0.007 0.008 -0.010 -0.002 -0.007\n#> ramp3 sin12 cos12 sin6 cos6 sin4 cos4 agChl:\n#> ageChildren \n#> time \n#> step1 \n#> ramp1 \n#> step2 \n#> ramp2 \n#> step3 \n#> ramp3 \n#> sin12 0.235 \n#> cos12 0.057 -0.008 \n#> sin6 -0.099 -0.036 -0.015 \n#> cos6 -0.058 -0.011 -0.017 0.017 \n#> sin4 0.074 0.025 0.014 -0.013 -0.006 \n#> cos4 0.073 0.041 -0.007 -0.009 -0.004 0.009 \n#> ageChildren:time -0.002 -0.005 -0.007 0.001 0.000 -0.003 0.000 \n#> ageChildren:step1 -0.001 -0.016 0.020 -0.004 -0.005 0.003 -0.007 -0.309\n#> ageChildren:ramp1 0.002 0.026 -0.014 -0.002 0.003 0.000 0.008 -0.063\n#> ageChildren:step2 -0.054 -0.033 0.003 0.011 0.004 -0.006 -0.010 0.020\n#> ageChildren:ramp2 0.055 -0.008 0.016 -0.005 -0.006 0.004 -0.002 0.005\n#> ageChildren:step3 -0.068 -0.022 -0.007 0.013 0.007 -0.007 -0.007 0.015\n#> ageChildren:ramp3 -0.105 -0.025 -0.006 0.010 0.006 -0.008 -0.008 0.016\n#> ageChildren:sin12 -0.025 -0.105 0.001 0.004 0.001 -0.003 -0.004 0.043\n#> ageChildren:cos12 -0.006 0.001 -0.105 0.002 0.002 -0.002 0.001 0.069\n#> ageChildren:sin6 0.010 0.004 0.002 -0.105 -0.002 0.001 0.001 -0.006\n#> ageChildren:cos6 0.006 0.001 0.002 -0.002 -0.105 0.001 0.000 0.002\n#> ageChildren:sin4 -0.008 -0.003 -0.002 0.001 0.001 -0.105 -0.001 0.028\n#> ageChildren:cos4 -0.008 -0.004 0.001 0.001 0.000 -0.001 -0.105 0.002\n#> agChldrn:s1 agChldrn:r1 agChldrn:s2 agChldrn:r2 agChldrn:s3\n#> ageChildren \n#> time \n#> step1 \n#> ramp1 \n#> step2 \n#> ramp2 \n#> step3 \n#> ramp3 \n#> sin12 \n#> cos12 \n#> sin6 \n#> cos6 \n#> sin4 \n#> cos4 \n#> ageChildren:time \n#> ageChildren:step1 \n#> ageChildren:ramp1 -0.763 \n#> ageChildren:step2 0.366 -0.711 \n#> ageChildren:ramp2 0.655 -0.831 0.306 \n#> ageChildren:step3 0.007 -0.031 0.300 -0.403 \n#> ageChildren:ramp3 0.008 -0.023 0.514 -0.520 0.649 \n#> ageChildren:sin12 0.155 -0.247 0.317 0.074 0.209 \n#> ageChildren:cos12 -0.188 0.137 -0.028 -0.155 0.065 \n#> ageChildren:sin6 0.034 0.018 -0.103 0.052 -0.126 \n#> ageChildren:cos6 0.047 -0.026 -0.035 0.060 -0.071 \n#> ageChildren:sin4 -0.033 -0.004 0.059 -0.040 0.066 \n#> ageChildren:cos4 0.063 -0.078 0.095 0.024 0.069 \n#> agChldrn:r3 agChldrn:s12 agChldrn:c12 agChldrn:s6 agChldrn:c6\n#> ageChildren \n#> time \n#> step1 \n#> ramp1 \n#> step2 \n#> ramp2 \n#> step3 \n#> ramp3 \n#> sin12 \n#> cos12 \n#> sin6 \n#> cos6 \n#> sin4 \n#> cos4 \n#> ageChildren:time \n#> ageChildren:step1 \n#> ageChildren:ramp1 \n#> ageChildren:step2 \n#> ageChildren:ramp2 \n#> ageChildren:step3 \n#> ageChildren:ramp3 \n#> ageChildren:sin12 0.235 \n#> ageChildren:cos12 0.057 -0.008 \n#> ageChildren:sin6 -0.099 -0.036 -0.015 \n#> ageChildren:cos6 -0.058 -0.011 -0.017 0.017 \n#> ageChildren:sin4 0.074 0.025 0.014 -0.013 -0.006 \n#> ageChildren:cos4 0.073 0.041 -0.007 -0.009 -0.004 \n#> agChldrn:s4\n#> ageChildren \n#> time \n#> step1 \n#> ramp1 \n#> step2 \n#> ramp2 \n#> step3 \n#> ramp3 \n#> sin12 \n#> cos12 \n#> sin6 \n#> cos6 \n#> sin4 \n#> cos4 \n#> ageChildren:time \n#> ageChildren:step1 \n#> ageChildren:ramp1 \n#> ageChildren:step2 \n#> ageChildren:ramp2 \n#> ageChildren:step3 \n#> ageChildren:ramp3 \n#> ageChildren:sin12 \n#> ageChildren:cos12 \n#> ageChildren:sin6 \n#> ageChildren:cos6 \n#> ageChildren:sin4 \n#> ageChildren:cos4 0.009 \n#> \n#> Standardized residuals:\n#> Min Q1 Med Q3 Max \n#> -5.25291094 -0.42240774 0.04331099 0.45224365 3.26323939 \n#> \n#> Residual standard error: 2.608572 \n#> Degrees of freedom: 240 total; 212 residual\n", "language": "unspecified", "language_note": "Syntax-screened scientific code; language not asserted; not executed", "post_id": 676996, "sha256": "2efb6dfee26f9f445e3475854cc386d4e974fbf2a8a395357cc7ed6d71558eda", "source_url": "https://stats.stackexchange.com/a/676996"}], "other_answers": [], "product": "code_qa", "question": "Background\n\n\n\n\n\nData: time series of the monthly prevalence of patients receiving a given medication by age group (children or adults) from January 2016 to December 2026\n\n\n\n\nInterventions: three successive regulatory interventions implemented on September 2021, June 2022, and June 2023 to facilitate prescription of the medication (dotted lines on the figure below)\n\n\n\n\nAim: to estimate pre-intervention slope, post-intervention 1 slope, post-intervention 2 slope, and post-intervention 3 slope depending on age group to assess the impact of the interventions (i.e., interrupted time series analysis, ITS)\n\n\n\n\n\nmonth <- seq(\n from = as.Date(\"2016-01-01\"),\n to = as.Date(\"2025-12-31\"),\n by = \"month\"\n)\nmonth <- rep(month, each = 2)\nage <- rep(c(\"Children\", \"Adults\"), 120)\nprevalence <- c(\n 214.1, 11.3, 194.9, 11.1, 220.3, 11.8, 200.7, 11.7, 213.5, \n 11.7, 197.6, 11.5, 128.2, 10.3, 138.6, 9.9, 215, 12.1, 201.9, \n 11.9, 223.8, 12.2, 204.9, 12.4, 234.1, 12.7, 202, 11.9, 247, \n 13.6, 209.8, 13, 236.7, 13.4, 216.9, 13.1, 145.1, 12, 144.3, \n 11, 235.1, 13.7, 227.5, 13.9, 242.5, 13.9, 229.5, 13.8, 255.3, \n 14.5, 226.3, 13.6, 260.8, 15.2, 229.7, 14.2, 245.5, 14.7, 232.6, \n 14.4, 158.8, 13.4, 165, 12.5, 244.3, 14.8, 249.4, 15.7, 259.4, \n 15.5, 244, 15.3, 275.7, 16.2, 240.7, 14.9, 274.4, 16.2, 256.8, \n 16.3, 274.3, 16.4, 240.3, 15.5, 178.2, 15.5, 180.3, 13.6, 264, \n 16.3, 271.3, 17.4, 277.8, 16.9, 261.5, 16.8, 296.9, 17.8, 265.8, \n 17.2, 270.4, 17.9, 218.8, 16, 214.7, 16, 221.8, 16.7, 183.6, \n 16.8, 185.4, 14.9, 273.6, 18.7, 276.7, 19.3, 285, 19.4, 282.8, \n 19.9, 308.3, 20.2, 279.2, 19.3, 331.1, 22, 289.3, 21.2, 303, \n 20.9, 307.5, 21.8, 221.2, 20.9, 215.4, 18.9, 324, 23.4, 323.3, \n 24.3, 328.3, 24.7, 326.5, 25.8, 349.2, 26.1, 319.1, 25.5, 379.3, \n 29, 341.7, 28.3, 365.1, 28.6, 351.7, 30.1, 253.7, 28.6, 266.9, \n 27, 375, 33.4, 373.5, 34.4, 385.8, 35.5, 368.1, 36, 413.7, 38.8, \n 376, 37.4, 440.1, 42.5, 390.3, 40.8, 409.5, 42.6, 402, 44.9, \n 297.9, 43.4, 297.1, 40.2, 430.9, 49.3, 433.4, 51.2, 454.1, 53.1, \n 441.4, 53.4, 472.9, 57, 455.8, 57, 496.3, 62.6, 475, 63.1, 483.4, \n 63.3, 466, 65.1, 367.3, 66.2, 368.4, 58.6, 499.4, 70.8, 528.3, \n 76.1, 533.8, 76, 525.6, 77.2, 588.3, 84.8, 532.4, 81.1, 599.6, \n 90.9, 580.3, 90.8, 601.5, 93.8, 560.6, 95.1, 455.2, 96.9, 440.6, \n 83.7, 629.2, 107.7, 655.7, 112.8, 653.8, 113.2, 674.9, 118.7\n)\nd <- data.frame(month, age, prevalence)\n\npar(mfrow = c(1, 2))\nplot(prevalence ~ month, d, type = \"l\", main = \"Children\", subset = age == \"Children\")\nplot(prevalence ~ month, d, type = \"l\", main = \"Adults\", subset = age == \"Adults\")\ndev.off()\n\n\n\n\n\n[image: time series; source: https://i.sstatic.net/4a0pdydL.png] (https://i.sstatic.net/4a0pdydL.png)\n\n\n\n\nWhat I did\n\n\n\n\nI started with children.\n\n\n\n\nI constructed the ITS variables for the 3 interventions (step1, ramp1, step2, ramp2, step3, ramp3) following Schaffer et al. (https://doi.org/10.1186/s12874-021-01235-8) and Xiao et al. (https://doi.org/10.1093/ije/dyaa148)\n\n\n\n\nd2 <- subset(d, age == \"Children\")\nd2$prevalence <- ts(d2$prevalence, start = 2016, frequency = 12)\n\ninterventions <- as.Date(c(\"2021-09-13\", \"2022-06-02\", \"2023-06-29\"))\n\nstep1 <- as.numeric(d2$month >= interventions[1])\nramp1 <- ifelse(step1 == 1, cumsum(step1) - 1, 0)\n\nstep2 <- as.numeric(d2$month >= interventions[2])\nramp2 <- ifelse(step2 == 1, cumsum(step2) - 1, 0)\n\nstep3 <- as.numeric(d2$month >= interventions[3])\nramp3 <- ifelse(step3 == 1, cumsum(step3) - 1, 0)\n\ntime <- seq(from = 0, along.with = d2$month)\n\nxreg <- cbind(time, step1, ramp1, step2, ramp2, step3, ramp3)\n\ncbind(d2, xreg)\n# month age prevalence time step1 ramp1 step2 ramp2 step3 ramp3\n# 1 2016-01-01 Children 214.1 0 0 0 0 0 0 0\n# 3 2016-02-01 Children 194.9 1 0 0 0 0 0 0\n# 5 2016-03-01 Children 220.3 2 0 0 0 0 0 0\n# 7 2016-04-01 Children 200.7 3 0 0 0 0 0 0\n# 9 2016-05-01 Children 213.5 4 0 0 0 0 0 0\n# 11 2016-06-01 Children 197.6 5 0 0 0 0 0 0\n# 13 2016-07-01 Children 128.2 6 0 0 0 0 0 0\n# 15 2016-08-01 Children 138.6 7 0 0 0 0 0 0\n# 17 2016-09-01 Children 215.0 8 0 0 0 0 0 0\n# 19 2016-10-01 Children 201.9 9 0 0 0 0 0 0\n# 21 2016-11-01 Children 223.8 10 0 0 0 0 0 0\n# 23 2016-12-01 Children 204.9 11 0 0 0 0 0 0\n# 25 2017-01-01 Children 234.1 12 0 0 0 0 0 0\n# 27 2017-02-01 Children 202.0 13 0 0 0 0 0 0\n# 29 2017-03-01 Children 247.0 14 0 0 0 0 0 0\n# 31 2017-04-01 Children 209.8 15 0 0 0 0 0 0\n# 33 2017-05-01 Children 236.7 16 0 0 0 0 0 0\n# 35 2017-06-01 Children 216.9 17 0 0 0 0 0 0\n# 37 2017-07-01 Children 145.1 18 0 0 0 0 0 0\n# 39 2017-08-01 Children 144.3 19 0 0 0 0 0 0\n# 41 2017-09-01 Children 235.1 20 0 0 0 0 0 0\n# 43 2017-10-01 Children 227.5 21 0 0 0 0 0 0\n# 45 2017-11-01 Children 242.5 22 0 0 0 0 0 0\n# 47 2017-12-01 Children 229.5 23 0 0 0 0 0 0\n# 49 2018-01-01 Children 255.3 24 0 0 0 0 0 0\n# 51 2018-02-01 Children 226.3 25 0 0 0 0 0 0\n# 53 2018-03-01 Children 260.8 26 0 0 0 0 0 0\n# 55 2018-04-01 Children 229.7 27 0 0 0 0 0 0\n# 57 2018-05-01 Children 245.5 28 0 0 0 0 0 0\n# 59 2018-06-01 Children 232.6 29 0 0 0 0 0 0\n# 61 2018-07-01 Children 158.8 30 0 0 0 0 0 0\n# 63 2018-08-01 Children 165.0 31 0 0 0 0 0 0\n# 65 2018-09-01 Children 244.3 32 0 0 0 0 0 0\n# 67 2018-10-01 Children 249.4 33 0 0 0 0 0 0\n# 69 2018-11-01 Children 259.4 34 0 0 0 0 0 0\n# 71 2018-12-01 Children 244.0 35 0 0 0 0 0 0\n# 73 2019-01-01 Children 275.7 36 0 0 0 0 0 0\n# 75 2019-02-01 Children 240.7 37 0 0 0 0 0 0\n# 77 2019-03-01 Children 274.4 38 0 0 0 0 0 0\n# 79 2019-04-01 Children 256.8 39 0 0 0 0 0 0\n# 81 2019-05-01 Children 274.3 40 0 0 0 0 0 0\n# 83 2019-06-01 Children 240.3 41 0 0 0 0 0 0\n# 85 2019-07-01 Children 178.2 42 0 0 0 0 0 0\n# 87 2019-08-01 Children 180.3 43 0 0 0 0 0 0\n# 89 2019-09-01 Children 264.0 44 0 0 0 0 0 0\n# 91 2019-10-01 Children 271.3 45 0 0 0 0 0 0\n# 93 2019-11-01 Children 277.8 46 0 0 0 0 0 0\n# 95 2019-12-01 Children 261.5 47 0 0 0 0 0 0\n# 97 2020-01-01 Children 296.9 48 0 0 0 0 0 0\n# 99 2020-02-01 Children 265.8 49 0 0 0 0 0 0\n# 101 2020-03-01 Children 270.4 50 0 0 0 0 0 0\n# 103 2020-04-01 Children 218.8 51 0 0 0 0 0 0\n# 105 2020-05-01 Children 214.7 52 0 0 0 0 0 0\n# 107 2020-06-01 Children 221.8 53 0 0 0 0 0 0\n# 109 2020-07-01 Children 183.6 54 0 0 0 0 0 0\n# 111 2020-08-01 Children 185.4 55 0 0 0 0 0 0\n# 113 2020-09-01 Children 273.6 56 0 0 0 0 0 0\n# 115 2020-10-01 Children 276.7 57 0 0 0 0 0 0\n# 117 2020-11-01 Children 285.0 58 0 0 0 0 0 0\n# 119 2020-12-01 Children 282.8 59 0 0 0 0 0 0\n# 121 2021-01-01 Children 308.3 60 0 0 0 0 0 0\n# 123 2021-02-01 Children 279.2 61 0 0 0 0 0 0\n# 125 2021-03-01 Children 331.1 62 0 0 0 0 0 0\n# 127 2021-04-01 Children 289.3 63 0 0 0 0 0 0\n# 129 2021-05-01 Children 303.0 64 0 0 0 0 0 0\n# 131 2021-06-01 Children 307.5 65 0 0 0 0 0 0\n# 133 2021-07-01 Children 221.2 66 0 0 0 0 0 0\n# 135 2021-08-01 Children 215.4 67 0 0 0 0 0 0\n# 137 2021-09-01 Children 324.0 68 0 0 0 0 0 0\n# 139 2021-10-01 Children 323.3 69 1 0 0 0 0 0\n# 141 2021-11-01 Children 328.3 70 1 1 0 0 0 0\n# 143 2021-12-01 Children 326.5 71 1 2 0 0 0 0\n# 145 2022-01-01 Children 349.2 72 1 3 0 0 0 0\n# 147 2022-02-01 Children 319.1 73 1 4 0 0 0 0\n# 149 2022-03-01 Children 379.3 74 1 5 0 0 0 0\n# 151 2022-04-01 Children 341.7 75 1 6 0 0 0 0\n# 153 2022-05-01 Children 365.1 76 1 7 0 0 0 0\n# 155 2022-06-01 Children 351.7 77 1 8 0 0 0 0\n# 157 2022-07-01 Children 253.7 78 1 9 1 0 0 0\n# 159 2022-08-01 Children 266.9 79 1 10 1 1 0 0\n# 161 2022-09-01 Children 375.0 80 1 11 1 2 0 0\n# 163 2022-10-01 Children 373.5 81 1 12 1 3 0 0\n# 165 2022-11-01 Children 385.8 82 1 13 1 4 0 0\n# 167 2022-12-01 Children 368.1 83 1 14 1 5 0 0\n# 169 2023-01-01 Children 413.7 84 1 15 1 6 0 0\n# 171 2023-02-01 Children 376.0 85 1 16 1 7 0 0\n# 173 2023-03-01 Children 440.1 86 1 17 1 8 0 0\n# 175 2023-04-01 Children 390.3 87 1 18 1 9 0 0\n# 177 2023-05-01 Children 409.5 88 1 19 1 10 0 0\n# 179 2023-06-01 Children 402.0 89 1 20 1 11 0 0\n# 181 2023-07-01 Children 297.9 90 1 21 1 12 1 0\n# 183 2023-08-01 Children 297.1 91 1 22 1 13 1 1\n# 185 2023-09-01 Children 430.9 92 1 23 1 14 1 2\n# 187 2023-10-01 Children 433.4 93 1 24 1 15 1 3\n# 189 2023-11-01 Children 454.1 94 1 25 1 16 1 4\n# 191 2023-12-01 Children 441.4 95 1 26 1 17 1 5\n# 193 2024-01-01 Children 472.9 96 1 27 1 18 1 6\n# 195 2024-02-01 Children 455.8 97 1 28 1 19 1 7\n# 197 2024-03-01 Children 496.3 98 1 29 1 20 1 8\n# 199 2024-04-01 Children 475.0 99 1 30 1 21 1 9\n# 201 2024-05-01 Children 483.4 100 1 31 1 22 1 10\n# 203 2024-06-01 Children 466.0 101 1 32 1 23 1 11\n# 205 2024-07-01 Children 367.3 102 1 33 1 24 1 12\n# 207 2024-08-01 Children 368.4 103 1 34 1 25 1 13\n# 209 2024-09-01 Children 499.4 104 1 35 1 26 1 14\n# 211 2024-10-01 Children 528.3 105 1 36 1 27 1 15\n# 213 2024-11-01 Children 533.8 106 1 37 1 28 1 16\n# 215 2024-12-01 Children 525.6 107 1 38 1 29 1 17\n# 217 2025-01-01 Children 588.3 108 1 39 1 30 1 18\n# 219 2025-02-01 Children 532.4 109 1 40 1 31 1 19\n# 221 2025-03-01 Children 599.6 110 1 41 1 32 1 20\n# 223 2025-04-01 Children 580.3 111 1 42 1 33 1 21\n# 225 2025-05-01 Children 601.5 112 1 43 1 34 1 22\n# 227 2025-06-01 Children 560.6 113 1 44 1 35 1 23\n# 229 2025-07-01 Children 455.2 114 1 45 1 36 1 24\n# 231 2025-08-01 Children 440.6 115 1 46 1 37 1 25\n# 233 2025-09-01 Children 629.2 116 1 47 1 38 1 26\n# 235 2025-10-01 Children 655.7 117 1 48 1 39 1 27\n# 237 2025-11-01 Children 653.8 118 1 49 1 40 1 28\n# 239 2025-12-01 Children 674.9 119 1 50 1 41 1 29\n\n\n\n\n\nAccording to this thread (https://stats.stackexchange.com/q/665216/), I included the ITS variables for the 3 interventions in a single model. I used forecast::auto.arima() to select (p, q) and (P, Q) parameters automatically.\n\n\n\n\nAccording to Schaffer et al. (https://doi.org/10.1186/s12874-021-01235-8), I included a time variable and set $d = 0$ and $D = 0$ (but I am not sure that I need both $d = 0$ and $D = 0$): (emphasis mine)\n\n\n\n\n\n\n\nIn ITS analysis, ARIMA forecasts $Y_t$ in the absence of the intervention (the “counterfactual”) and determines how the observed diverges from this forecast. Unlike segmented regression, including time or seasonal dummy variables in the ARIMA model is not necessary, as ARIMA can eliminate trends and seasonality through differencing. If the trend is eliminated via differencing then the pre- and post-intervention trends cannot be estimated from the model. However, if estimation of the pre- and/or post-intervention slope is desired, this can be accommodated by including time as a covariate and incorporating AR and MA terms to address autocorrelation (e.g. ARMA models) [21 (https://doi.org/10.1186/s12889-017-4998-9), 22 (https://doi.org/10.3111/13696998.2011.626097)].\n\n\n\n\n\n\n\nfit <- forecast::auto.arima(\n d2$prevalence,\n d = 0,\n D = 0,\n stepwise = FALSE,\n trace = TRUE,\n approximation = FALSE,\n xreg = xreg\n)\n# \n# Regression with ARIMA(0,0,0) errors : 1422.701\n# Regression with ARIMA(0,0,0) errors : 1256.821\n# Regression with ARIMA(0,0,0)(0,0,1)[12] errors : Inf\n# Regression with ARIMA(0,0,0)(0,0,1)[12] errors : 1186.586\n# Regression with ARIMA(0,0,0)(0,0,2)[12] errors : Inf\n# Regression with ARIMA(0,0,0)(0,0,2)[12] errors : Inf\n# ARIMA(0,0,0)(1,0,0)[12] with zero mean : Inf\n# ARIMA(0,0,0)(1,0,0)[12] with non-zero mean : Inf\n# ARIMA(0,0,0)(1,0,1)[12] with zero mean : Inf\n# ARIMA(0,0,0)(1,0,1)[12] with non-zero mean : Inf\n# ARIMA(0,0,0)(1,0,2)[12] with zero mean : Inf\n# ARIMA(0,0,0)(1,0,2)[12] with non-zero mean : Inf\n# ARIMA(0,0,0)(2,0,0)[12] with zero mean : Inf\n# ARIMA(0,0,0)(2,0,0)[12] with non-zero mean : Inf\n# ARIMA(0,0,0)(2,0,1)[12] with zero mean : Inf\n# ARIMA(0,0,0)(2,0,1)[12] with non-zero mean : Inf\n# ARIMA(0,0,0)(2,0,2)[12] with zero mean : Inf\n# ARIMA(0,0,0)(2,0,2)[12] with non-zero mean : Inf\n# Regression with ARIMA(0,0,1) errors : 1348.712\n# Regression with ARIMA(0,0,1) errors : 1237.018\n# Regression with ARIMA(0,0,1)(0,0,1)[12] errors : Inf\n# Regression with ARIMA(0,0,1)(0,0,1)[12] errors : 1167.011\n# Regression with ARIMA(0,0,1)(0,0,2)[12] errors : Inf\n# Regression with ARIMA(0,0,1)(0,0,2)[12] errors : Inf\n# ARIMA(0,0,1)(1,0,0)[12] with zero mean : Inf\n# ARIMA(0,0,1)(1,0,0)[12] with non-zero mean : Inf\n# ARIMA(0,0,1)(1,0,1)[12] with zero mean : Inf\n# ARIMA(0,0,1)(1,0,1)[12] with non-zero mean : Inf\n# ARIMA(0,0,1)(1,0,2)[12] with zero mean : Inf\n# ARIMA(0,0,1)(1,0,2)[12] with non-zero mean : Inf\n# ARIMA(0,0,1)(2,0,0)[12] with zero mean : Inf\n# ARIMA(0,0,1)(2,0,0)[12] with non-zero mean : Inf\n# ARIMA(0,0,1)(2,0,1)[12] with zero mean : Inf\n# ARIMA(0,0,1)(2,0,1)[12] with non-zero mean : Inf\n# ARIMA(0,0,1)(2,0,2)[12] with zero mean : Inf\n# Regression with ARIMA(0,0,1)(2,0,2)[12] errors : Inf\n# Regression with ARIMA(0,0,2) errors : 1304.633\n# Regression with ARIMA(0,0,2) errors : 1235.236\n# Regression with ARIMA(0,0,2)(0,0,1)[12] errors : 1226.35\n# Regression with ARIMA(0,0,2)(0,0,1)[12] errors : 1164.06\n# Regression with ARIMA(0,0,2)(0,0,2)[12] errors : Inf\n# Regression with ARIMA(0,0,2)(0,0,2)[12] errors : 1122.153\n# ARIMA(0,0,2)(1,0,0)[12] with zero mean : Inf\n# Regression with ARIMA(0,0,2)(1,0,0)[12] errors : Inf\n# ARIMA(0,0,2)(1,0,1)[12] with zero mean : Inf\n# ARIMA(0,0,2)(1,0,1)[12] with non-zero mean : Inf\n# ARIMA(0,0,2)(1,0,2)[12] with zero mean : Inf\n# ARIMA(0,0,2)(1,0,2)[12] with non-zero mean : Inf\n# ARIMA(0,0,2)(2,0,0)[12] with zero mean : Inf\n# ARIMA(0,0,2)(2,0,0)[12] with non-zero mean : Inf\n# ARIMA(0,0,2)(2,0,1)[12] with zero mean : Inf\n# ARIMA(0,0,2)(2,0,1)[12] with non-zero mean : Inf\n# Regression with ARIMA(0,0,3) errors : 1301.979\n# Regression with ARIMA(0,0,3) errors : Inf\n# Regression with ARIMA(0,0,3)(0,0,1)[12] errors : Inf\n# Regression with ARIMA(0,0,3)(0,0,1)[12] errors : 1166.444\n# Regression with ARIMA(0,0,3)(0,0,2)[12] errors : Inf\n# Regression with ARIMA(0,0,3)(0,0,2)[12] errors : Inf\n# ARIMA(0,0,3)(1,0,0)[12] with zero mean : Inf\n# Regression with ARIMA(0,0,3)(1,0,0)[12] errors : Inf\n# ARIMA(0,0,3)(1,0,1)[12] with zero mean : Inf\n# ARIMA(0,0,3)(1,0,1)[12] with non-zero mean : Inf\n# ARIMA(0,0,3)(2,0,0)[12] with zero mean : Inf\n# ARIMA(0,0,3)(2,0,0)[12] with non-zero mean : Inf\n# Regression with ARIMA(0,0,4) errors : Inf\n# Regression with ARIMA(0,0,4) errors : Inf\n# Regression with ARIMA(0,0,4)(0,0,1)[12] errors : Inf\n# Regression with ARIMA(0,0,4)(0,0,1)[12] errors : 1167.795\n# ARIMA(0,0,4)(1,0,0)[12] with zero mean : Inf\n# Regression with ARIMA(0,0,4)(1,0,0)[12] errors : Inf\n# Regression with ARIMA(0,0,5) errors : Inf\n# Regression with ARIMA(0,0,5) errors : Inf\n# Regression with ARIMA(1,0,0) errors : 1269.039\n# Regression with ARIMA(1,0,0) errors : 1236.551\n# Regression with ARIMA(1,0,0)(0,0,1)[12] errors : 1191.463\n# Regression with ARIMA(1,0,0)(0,0,1)[12] errors : 1163.925\n# Regression with ARIMA(1,0,0)(0,0,2)[12] errors : 1144.206\n# Regression with ARIMA(1,0,0)(0,0,2)[12] errors : 1122.139\n# ARIMA(1,0,0)(1,0,0)[12] with zero mean : Inf\n# ARIMA(1,0,0)(1,0,0)[12] with non-zero mean : Inf\n# ARIMA(1,0,0)(1,0,1)[12] with zero mean : Inf\n# ARIMA(1,0,0)(1,0,1)[12] with non-zero mean : Inf\n# ARIMA(1,0,0)(1,0,2)[12] with zero mean : Inf\n# ARIMA(1,0,0)(1,0,2)[12] with non-zero mean : Inf\n# ARIMA(1,0,0)(2,0,0)[12] with zero mean : Inf\n# Regression with ARIMA(1,0,0)(2,0,0)[12] errors : Inf\n# ARIMA(1,0,0)(2,0,1)[12] with zero mean : Inf\n# ARIMA(1,0,0)(2,0,1)[12] with non-zero mean : Inf\n# ARIMA(1,0,0)(2,0,2)[12] with zero mean : Inf\n# ARIMA(1,0,0)(2,0,2)[12] with non-zero mean : Inf\n# Regression with ARIMA(1,0,1) errors : 1269.741\n# Regression with ARIMA(1,0,1) errors : 1237.127\n# Regression with ARIMA(1,0,1)(0,0,1)[12] errors : 1191.287\n# Regression with ARIMA(1,0,1)(0,0,1)[12] errors : 1165.649\n# Regression with ARIMA(1,0,1)(0,0,2)[12] errors : 1145.611\n# Regression with ARIMA(1,0,1)(0,0,2)[12] errors : 1123.56\n# ARIMA(1,0,1)(1,0,0)[12] with zero mean : Inf\n# ARIMA(1,0,1)(1,0,0)[12] with non-zero mean : Inf\n# ARIMA(1,0,1)(1,0,1)[12] with zero mean : Inf\n# ARIMA(1,0,1)(1,0,1)[12] with non-zero mean : Inf\n# ARIMA(1,0,1)(1,0,2)[12] with zero mean : Inf\n# ARIMA(1,0,1)(1,0,2)[12] with non-zero mean : Inf\n# ARIMA(1,0,1)(2,0,0)[12] with zero mean : Inf\n# Regression with ARIMA(1,0,1)(2,0,0)[12] errors : Inf\n# ARIMA(1,0,1)(2,0,1)[12] with zero mean : Inf\n# ARIMA(1,0,1)(2,0,1)[12] with non-zero mean : Inf\n# Regression with ARIMA(1,0,2) errors : Inf\n# ARIMA(1,0,2) with non-zero mean : Inf\n# Regression with ARIMA(1,0,2)(0,0,1)[12] errors : Inf\n# ARIMA(1,0,1)(2,0,1)[12] with non-zero mean : Inf\n# Regression with ARIMA(1,0,2) errors : Inf\n# ARIMA(1,0,2) with non-zero mean : Inf\n# Regression with ARIMA(1,0,2)(0,0,1)[12] errors : Inf\n# Regression with ARIMA(1,0,2)(0,0,1)[12] errors : Inf\n# Regression with ARIMA(1,0,2)(0,0,2)[12] errors : Inf\n# Regression with ARIMA(1,0,2)(0,0,2)[12] errors : Inf\n# ARIMA(1,0,2)(1,0,0)[12] with zero mean : Inf\n# ARIMA(1,0,2)(1,0,0)[12] with non-zero mean : Inf\n# ARIMA(1,0,2)(1,0,1)[12] with zero mean : Inf\n# ARIMA(1,0,2)(1,0,1)[12] with non-zero mean : Inf\n# ARIMA(1,0,2)(2,0,0)[12] with zero mean : Inf\n# Regression with ARIMA(1,0,2)(2,0,0)[12] errors : Inf\n# Regression with ARIMA(1,0,3) errors : Inf\n# Regression with ARIMA(1,0,3) errors : Inf\n# Regression with ARIMA(1,0,3)(0,0,1)[12] errors : 1196.785\n# Regression with ARIMA(1,0,3)(0,0,1)[12] errors : Inf\n# ARIMA(1,0,3)(1,0,0)[12] with zero mean : Inf\n# ARIMA(1,0,3)(1,0,0)[12] with non-zero mean : Inf\n# Regression with ARIMA(1,0,4) errors : Inf\n# Regression with ARIMA(1,0,4) errors : Inf\n# Regression with ARIMA(2,0,0) errors : 1270.227\n# Regression with ARIMA(2,0,0) errors : 1235.453\n# Regression with ARIMA(2,0,0)(0,0,1)[12] errors : 1191.855\n# Regression with ARIMA(2,0,0)(0,0,1)[12] errors : 1165.096\n# Regression with ARIMA(2,0,0)(0,0,2)[12] errors : 1145.887\n# Regression with ARIMA(2,0,0)(0,0,2)[12] errors : 1123.031\n# ARIMA(2,0,0)(1,0,0)[12] with zero mean : Inf\n# ARIMA(2,0,0)(1,0,0)[12] with non-zero mean : Inf\n# ARIMA(2,0,0)(1,0,1)[12] with zero mean : Inf\n# ARIMA(2,0,0)(1,0,1)[12] with non-zero mean : Inf\n# ARIMA(2,0,0)(1,0,2)[12] with zero mean : Inf\n# ARIMA(2,0,0)(1,0,2)[12] with non-zero mean : Inf\n# ARIMA(2,0,0)(2,0,0)[12] with zero mean : Inf\n# ARIMA(2,0,0)(2,0,0)[12] with non-zero mean : Inf\n# ARIMA(2,0,0)(2,0,1)[12] with zero mean : Inf\n# Regression with ARIMA(2,0,0)(2,0,1)[12] errors : Inf\n# Regression with ARIMA(2,0,1) errors : Inf\n# Regression with ARIMA(2,0,1) errors : Inf\n# Regression with ARIMA(2,0,1)(0,0,1)[12] errors : Inf\n# Regression with ARIMA(2,0,1)(0,0,1)[12] errors : Inf\n# Regression with ARIMA(2,0,1)(0,0,2)[12] errors : Inf\n# Regression with ARIMA(2,0,1)(0,0,2)[12] errors : Inf\n# ARIMA(2,0,1)(1,0,0)[12] with zero mean : Inf\n# ARIMA(2,0,1)(1,0,0)[12] with non-zero mean : Inf\n# Regression with ARIMA(2,0,1)(1,0,1)[12] errors : Inf\n# ARIMA(2,0,1)(1,0,1)[12] with non-zero mean : Inf\n# Regression with ARIMA(2,0,1)(2,0,0)[12] errors : Inf\n# Regression with ARIMA(2,0,1)(2,0,0)[12] errors : Inf\n# Regression with ARIMA(2,0,2) errors : Inf\n# Regression with ARIMA(2,0,2) errors : Inf\n# ARIMA(2,0,2)(0,0,1)[12] with zero mean : Inf\n# Regression with ARIMA(2,0,2)(0,0,1)[12] errors : Inf\n# ARIMA(2,0,2)(1,0,0)[12] with zero mean : Inf\n# ARIMA(2,0,2)(1,0,0)[12] with non-zero mean : Inf\n# ARIMA(2,0,3) with zero mean : Inf\n# Regression with ARIMA(2,0,3) errors : Inf\n# Regression with ARIMA(3,0,0) errors : 1271.552\n# Regression with ARIMA(3,0,0) errors : 1234.442\n# Regression with ARIMA(3,0,0)(0,0,1)[12] errors : 1193.944\n# Regression with ARIMA(3,0,0)(0,0,1)[12] errors : 1165.004\n# Regression with ARIMA(3,0,0)(0,0,2)[12] errors : 1147.337\n# Regression with ARIMA(3,0,0)(0,0,2)[12] errors : 1124.69\n# ARIMA(3,0,0)(1,0,0)[12] with zero mean : Inf\n# ARIMA(3,0,0)(1,0,0)[12] with non-zero mean : Inf\n# ARIMA(3,0,0)(1,0,1)[12] with zero mean : Inf\n# ARIMA(3,0,0)(1,0,1)[12] with non-zero mean : Inf\n# ARIMA(3,0,0)(2,0,0)[12] with zero mean : Inf\n# ARIMA(3,0,0)(2,0,0)[12] with non-zero mean : Inf\n# Regression with ARIMA(3,0,1) errors : Inf\n# Regression with ARIMA(3,0,1) errors : Inf\n# Regression with ARIMA(3,0,1)(0,0,1)[12] errors : Inf\n# Regression with ARIMA(3,0,1)(0,0,1)[12] errors : Inf\n# ARIMA(3,0,1)(1,0,0)[12] with zero mean : Inf\n# ARIMA(3,0,1)(1,0,0)[12] with non-zero mean : Inf\n# Regression with ARIMA(3,0,2) errors : Inf\n# Regression with ARIMA(3,0,2) errors : Inf\n# Regression with ARIMA(4,0,0) errors : 1268.176\n# Regression with ARIMA(4,0,0) errors : 1236.501\n# Regression with ARIMA(4,0,0)(0,0,1)[12] errors : 1191.278\n# Regression with ARIMA(4,0,0)(0,0,1)[12] errors : 1167.39\n# ARIMA(4,0,0)(1,0,0)[12] with zero mean : Inf\n# ARIMA(4,0,0)(1,0,0)[12] with non-zero mean : Inf\n# Regression with ARIMA(4,0,1) errors : Inf\n# Regression with ARIMA(4,0,1) errors : Inf\n# Regression with ARIMA(5,0,0) errors : 1270.388\n# Regression with ARIMA(5,0,0) errors : 1226.449\n# \n# \n# \n# Best model: Regression with ARIMA(1,0,0)(0,0,2)[12] errors \n# \n\nsummary(fit)\n# Series: d2$prevalence \n# Regression with ARIMA(1,0,0)(0,0,2)[12] errors \n# \n# Coefficients:\n# ar1 sma1 sma2 intercept time step1 ramp1 step2 ramp2 step3 ramp3\n# 0.5256 1.0413 0.6987 188.8179 1.5717 3.3284 2.1966 -5.6181 0.4640 0.4378 4.6447\n# s.e. 0.0800 0.1536 0.1312 20.6895 0.4724 13.0492 3.5219 23.9955 3.8197 19.1790 2.5724\n# \n# sigma^2 = 494.9: log likelihood = -547.61\n# AIC=1119.22 AICc=1122.14 BIC=1152.67\n# \n# Training set error measures:\n# ME RMSE MAE MPE MAPE MASE ACF1\n# Training set -0.2760281 21.20232 16.21419 -0.9807653 5.578825 0.3604042 0.05861382\n\n\n\n\n\nQuestion\n\n\n\n\nShould I just apply the same strategy to adults? Or can such models handle correlated/stratified time series?\n\n\n\n\nI guess that there is a better strategy than fitting separate models within each subgroup. However, I suspect that I cannot simply stack the time series and include age in xreg. Additionally, I probably need an interaction terms (e.g. time * age).\n\n\n\n\nI have also read about nlme::gls() (https://stat.ethz.ch/R-manual/R-devel/library/nlme/html/gls.html) but I would like guidance/confirmation whether it is more appropriate.\n\n\n\n\n\n\n\nBTW, I am not satisfied with the fit, but I will explore that once I have decided the modelling strategy.\n\n\n\n\nres <- residuals(fit)\nl <- 36\n\npar(mfrow = c(3, 1))\nplot(res)\nacf(res, lag.max = l)\npacf(res, lag.max = l)\ndev.off()\n\nBox.test(res, lag = 24, type = \"Ljung-Box\", fitdf = 3)\n# \n# Box-Ljung test\n# \n# data: res\n# X-squared = 49.814, df = 21, p-value = 0.0003872\n# \n\n\n\n\n\n[image: residuals; source: https://i.sstatic.net/pzs11NTf.png] (https://i.sstatic.net/pzs11NTf.png)", "record_id": "Scientific-Code-and-Analysis-QA:stats:676979", "selected_answer": {"answer_html": "<p>Rather than fitting separate models, you can handle both age groups in a single <code>gls()</code> model with <code>age</code> interactions. This has two advantages over separate <code>auto.arima()</code> models: you get formal tests of whether intervention effects differ between groups via the interaction terms, and you can account for the fact that the two series have very different variances.</p>\n<p><code>age * (time + step1 + ramp1 + ...)</code> gives each group its own intercept, pre-intervention slope, and intervention effects, plus tests of whether they differ. Then, <code>corAR1(form = ~ 1 | age)</code> fits AR(1) autocorrelation within each group and <code>varIdent(form = ~ 1 | age)</code> allows different residual variances per group, which matters here since children's prevalence is an order of magnitude larger.</p>\n<pre class=\"lang-r prettyprint-override\"><code>library(nlme)\nlibrary(dplyr)\n\ninterventions <- as.Date(c("2021-09-13", "2022-06-02", "2023-06-29"))\n\nd_its <- d |>\n arrange(age, month) |>\n mutate(.by = age,\n time = row_number() - 1,\n step1 = as.numeric(month >= interventions[1]),\n ramp1 = ifelse(step1 == 1, cumsum(step1) - 1, 0),\n step2 = as.numeric(month >= interventions[2]),\n ramp2 = ifelse(step2 == 1, cumsum(step2) - 1, 0),\n step3 = as.numeric(month >= interventions[3]),\n ramp3 = ifelse(step3 == 1, cumsum(step3) - 1, 0),\n sin12 = sin(2 * pi * time / 12),\n cos12 = cos(2 * pi * time / 12),\n sin6 = sin(2 * pi * time / 6),\n cos6 = cos(2 * pi * time / 6),\n sin4 = sin(2 * pi * time / 4),\n cos4 = cos(2 * pi * time / 4)\n ) |>\n mutate(age = factor(age))\n\nfit <- gls(\n prevalence ~ age * (time + step1 + ramp1 + step2 + ramp2 + step3 + ramp3 +\n sin12 + cos12 + sin6 + cos6 + sin4 + cos4),\n data = d_its,\n correlation = corAR1(form = ~ 1 | age),\n weights = varIdent(form = ~ 1 | age)\n)\n</code></pre>\n<p>The Fourier terms (<code>sin12</code>, <code>cos12</code>, etc.) handle the recurring seasonal pattern, which is more stable than trying to absorb it through ARMA error terms (see <a href=\"https://otexts.com/fpp3/dhr.html\" rel=\"nofollow noreferrer\">https://otexts.com/fpp3/dhr.html</a>).</p>\n<p>The reference level is Adults, so coefficients are interpreted as:</p>\n<ul>\n<li><code>time</code>: pre-intervention slope for adults (prevalence per month)</li>\n<li><code>ageChildren:time</code>: <em>additional</em> slope for children;</li>\n</ul>\n<p>The same logic applies for all <code>ramp</code> and <code>step</code> terms.</p>\n<pre class=\"lang-r prettyprint-override\"><code>coef(fit) |>\n (\\(e) data.frame(\n Segment = c("Pre", "Post-1", "Post-2", "Post-3"),\n Adults = round(cumsum(c(e["time"],\n e["ramp1"],\n e["ramp2"],\n e["ramp3"])), 3),\n Children = round(cumsum(c(e["time"] + e["ageChildren:time"],\n e["ramp1"] + e["ageChildren:ramp1"],\n e["ramp2"] + e["ageChildren:ramp2"],\n e["ramp3"] + e["ageChildren:ramp3"])), 3)\n))()\n\n#> Segment Adults Children\n#> time Pre 0.152 1.445\n#> ramp1 Post-1 0.919 9.276\n#> ramp2 Post-2 1.265 6.667\n#> ramp3 Post-3 2.387 8.418\n</code></pre>\n<pre class=\"lang-r prettyprint-override\"><code>par(mfrow = c(1, 2))\nacf(residuals(fit)[d_its<span class=\"math-container\">$age == \"Adults\"], main = \"Adults\")\nacf(residuals(fit)[d_its$</span>age == "Children"], main = "Children")\ndev.off()\n</code></pre>\n<p><img src=\"https://i.sstatic.net/2fnDCjiM.png\" alt=\"\" /></p>\n<pre class=\"lang-r prettyprint-override\"><code>summary(fit)\n\n#> Generalized least squares fit by REML\n#> Model: prevalence ~ age * (time + step1 + ramp1 + step2 + ramp2 + step3 + ramp3 + sin12 + cos12 + sin6 + cos6 + sin4 + cos4) \n#> Data: d_its \n#> AIC BIC logLik\n#> 1665.172 1769.226 -801.5858\n#> \n#> Correlation Structure: AR(1)\n#> Formula: ~1 | age \n#> Parameter estimate(s):\n#> Phi \n#> 0.1231768 \n#> Variance function:\n#> Structure: Different standard deviations per stratum\n#> Formula: ~1 | age \n#> Parameter estimates:\n#> Adults Children \n#> 1.000000 9.435418 \n#> \n#> Coefficients:\n#> Value Std.Error t-value p-value\n#> (Intercept) 10.25450 0.701853 14.610597 0.0000\n#> ageChildren 177.16318 6.659369 26.603599 0.0000\n#> time 0.15162 0.017799 8.518225 0.0000\n#> step1 1.83019 1.934456 0.946100 0.3452\n#> ramp1 0.76689 0.379858 2.018892 0.0448\n#> step2 -0.98456 2.681497 -0.367167 0.7139\n#> ramp2 0.34620 0.449034 0.770992 0.4416\n#> step3 -4.38832 2.094825 -2.094838 0.0374\n#> ramp3 1.12257 0.257109 4.366133 0.0000\n#> sin12 0.01845 0.399784 0.046145 0.9632\n#> cos12 1.16269 0.380402 3.056467 0.0025\n#> sin6 -0.70555 0.358820 -1.966309 0.0506\n#> cos6 -0.97079 0.355213 -2.732980 0.0068\n#> sin4 0.78337 0.333359 2.349941 0.0197\n#> cos4 0.05479 0.333865 0.164117 0.8698\n#> ageChildren:time 1.29381 0.168883 7.661002 0.0000\n#> ageChildren:step1 -1.02662 18.354630 -0.055933 0.9554\n#> ageChildren:ramp1 7.06407 3.604192 1.959960 0.0513\n#> ageChildren:step2 -35.82645 25.442751 -1.408120 0.1606\n#> ageChildren:ramp2 -2.95530 4.260553 -0.693643 0.4887\n#> ageChildren:step3 -29.49930 19.876247 -1.484148 0.1393\n#> ageChildren:ramp3 0.62770 2.439516 0.257306 0.7972\n#> ageChildren:sin12 4.22521 3.793256 1.113874 0.2666\n#> ageChildren:cos12 36.54005 3.609352 10.123715 0.0000\n#> ageChildren:sin6 -9.76953 3.404578 -2.869528 0.0045\n#> ageChildren:cos6 -25.08251 3.370352 -7.442104 0.0000\n#> ageChildren:sin4 13.68702 3.162997 4.327230 0.0000\n#> ageChildren:cos4 14.66062 3.167797 4.628016 0.0000\n#> \n#> Correlation: \n#> (Intr) agChld time step1 ramp1 step2 ramp2 step3 \n#> ageChildren -0.105 \n#> time -0.862 0.091 \n#> step1 0.173 -0.018 -0.309 \n#> ramp1 0.057 -0.006 -0.063 -0.763 \n#> step2 -0.021 0.002 0.020 0.366 -0.711 \n#> ramp2 -0.004 0.000 0.005 0.655 -0.831 0.306 \n#> step3 -0.015 0.002 0.015 0.007 -0.031 0.300 -0.403 \n#> ramp3 -0.016 0.002 0.016 0.008 -0.023 0.514 -0.520 0.649\n#> sin12 -0.056 0.006 0.043 0.155 -0.247 0.317 0.074 0.209\n#> cos12 -0.050 0.005 0.069 -0.188 0.137 -0.028 -0.155 0.065\n#> sin6 -0.008 0.001 -0.006 0.034 0.018 -0.103 0.052 -0.126\n#> cos6 -0.010 0.001 0.002 0.047 -0.026 -0.035 0.060 -0.071\n#> sin4 -0.024 0.002 0.028 -0.033 -0.004 0.059 -0.040 0.066\n#> cos4 -0.011 0.001 0.002 0.063 -0.078 0.095 0.024 0.069\n#> ageChildren:time 0.091 -0.862 -0.105 0.033 0.007 -0.002 0.000 -0.002\n#> ageChildren:step1 -0.018 0.173 0.033 -0.105 0.080 -0.039 -0.069 -0.001\n#> ageChildren:ramp1 -0.006 0.057 0.007 0.080 -0.105 0.075 0.088 0.003\n#> ageChildren:step2 0.002 -0.021 -0.002 -0.039 0.075 -0.105 -0.032 -0.032\n#> ageChildren:ramp2 0.000 -0.004 0.000 -0.069 0.088 -0.032 -0.105 0.043\n#> ageChildren:step3 0.002 -0.015 -0.002 -0.001 0.003 -0.032 0.043 -0.105\n#> ageChildren:ramp3 0.002 -0.016 -0.002 -0.001 0.002 -0.054 0.055 -0.068\n#> ageChildren:sin12 0.006 -0.056 -0.005 -0.016 0.026 -0.033 -0.008 -0.022\n#> ageChildren:cos12 0.005 -0.050 -0.007 0.020 -0.014 0.003 0.016 -0.007\n#> ageChildren:sin6 0.001 -0.008 0.001 -0.004 -0.002 0.011 -0.005 0.013\n#> ageChildren:cos6 0.001 -0.010 0.000 -0.005 0.003 0.004 -0.006 0.007\n#> ageChildren:sin4 0.002 -0.024 -0.003 0.003 0.000 -0.006 0.004 -0.007\n#> ageChildren:cos4 0.001 -0.011 0.000 -0.007 0.008 -0.010 -0.002 -0.007\n#> ramp3 sin12 cos12 sin6 cos6 sin4 cos4 agChl:\n#> ageChildren \n#> time \n#> step1 \n#> ramp1 \n#> step2 \n#> ramp2 \n#> step3 \n#> ramp3 \n#> sin12 0.235 \n#> cos12 0.057 -0.008 \n#> sin6 -0.099 -0.036 -0.015 \n#> cos6 -0.058 -0.011 -0.017 0.017 \n#> sin4 0.074 0.025 0.014 -0.013 -0.006 \n#> cos4 0.073 0.041 -0.007 -0.009 -0.004 0.009 \n#> ageChildren:time -0.002 -0.005 -0.007 0.001 0.000 -0.003 0.000 \n#> ageChildren:step1 -0.001 -0.016 0.020 -0.004 -0.005 0.003 -0.007 -0.309\n#> ageChildren:ramp1 0.002 0.026 -0.014 -0.002 0.003 0.000 0.008 -0.063\n#> ageChildren:step2 -0.054 -0.033 0.003 0.011 0.004 -0.006 -0.010 0.020\n#> ageChildren:ramp2 0.055 -0.008 0.016 -0.005 -0.006 0.004 -0.002 0.005\n#> ageChildren:step3 -0.068 -0.022 -0.007 0.013 0.007 -0.007 -0.007 0.015\n#> ageChildren:ramp3 -0.105 -0.025 -0.006 0.010 0.006 -0.008 -0.008 0.016\n#> ageChildren:sin12 -0.025 -0.105 0.001 0.004 0.001 -0.003 -0.004 0.043\n#> ageChildren:cos12 -0.006 0.001 -0.105 0.002 0.002 -0.002 0.001 0.069\n#> ageChildren:sin6 0.010 0.004 0.002 -0.105 -0.002 0.001 0.001 -0.006\n#> ageChildren:cos6 0.006 0.001 0.002 -0.002 -0.105 0.001 0.000 0.002\n#> ageChildren:sin4 -0.008 -0.003 -0.002 0.001 0.001 -0.105 -0.001 0.028\n#> ageChildren:cos4 -0.008 -0.004 0.001 0.001 0.000 -0.001 -0.105 0.002\n#> agChldrn:s1 agChldrn:r1 agChldrn:s2 agChldrn:r2 agChldrn:s3\n#> ageChildren \n#> time \n#> step1 \n#> ramp1 \n#> step2 \n#> ramp2 \n#> step3 \n#> ramp3 \n#> sin12 \n#> cos12 \n#> sin6 \n#> cos6 \n#> sin4 \n#> cos4 \n#> ageChildren:time \n#> ageChildren:step1 \n#> ageChildren:ramp1 -0.763 \n#> ageChildren:step2 0.366 -0.711 \n#> ageChildren:ramp2 0.655 -0.831 0.306 \n#> ageChildren:step3 0.007 -0.031 0.300 -0.403 \n#> ageChildren:ramp3 0.008 -0.023 0.514 -0.520 0.649 \n#> ageChildren:sin12 0.155 -0.247 0.317 0.074 0.209 \n#> ageChildren:cos12 -0.188 0.137 -0.028 -0.155 0.065 \n#> ageChildren:sin6 0.034 0.018 -0.103 0.052 -0.126 \n#> ageChildren:cos6 0.047 -0.026 -0.035 0.060 -0.071 \n#> ageChildren:sin4 -0.033 -0.004 0.059 -0.040 0.066 \n#> ageChildren:cos4 0.063 -0.078 0.095 0.024 0.069 \n#> agChldrn:r3 agChldrn:s12 agChldrn:c12 agChldrn:s6 agChldrn:c6\n#> ageChildren \n#> time \n#> step1 \n#> ramp1 \n#> step2 \n#> ramp2 \n#> step3 \n#> ramp3 \n#> sin12 \n#> cos12 \n#> sin6 \n#> cos6 \n#> sin4 \n#> cos4 \n#> ageChildren:time \n#> ageChildren:step1 \n#> ageChildren:ramp1 \n#> ageChildren:step2 \n#> ageChildren:ramp2 \n#> ageChildren:step3 \n#> ageChildren:ramp3 \n#> ageChildren:sin12 0.235 \n#> ageChildren:cos12 0.057 -0.008 \n#> ageChildren:sin6 -0.099 -0.036 -0.015 \n#> ageChildren:cos6 -0.058 -0.011 -0.017 0.017 \n#> ageChildren:sin4 0.074 0.025 0.014 -0.013 -0.006 \n#> ageChildren:cos4 0.073 0.041 -0.007 -0.009 -0.004 \n#> agChldrn:s4\n#> ageChildren \n#> time \n#> step1 \n#> ramp1 \n#> step2 \n#> ramp2 \n#> step3 \n#> ramp3 \n#> sin12 \n#> cos12 \n#> sin6 \n#> cos6 \n#> sin4 \n#> cos4 \n#> ageChildren:time \n#> ageChildren:step1 \n#> ageChildren:ramp1 \n#> ageChildren:step2 \n#> ageChildren:ramp2 \n#> ageChildren:step3 \n#> ageChildren:ramp3 \n#> ageChildren:sin12 \n#> ageChildren:cos12 \n#> ageChildren:sin6 \n#> ageChildren:cos6 \n#> ageChildren:sin4 \n#> ageChildren:cos4 0.009 \n#> \n#> Standardized residuals:\n#> Min Q1 Med Q3 Max \n#> -5.25291094 -0.42240774 0.04331099 0.45224365 3.26323939 \n#> \n#> Residual standard error: 2.608572 \n#> Degrees of freedom: 240 total; 212 residual\n</code></pre>\n<p><sup>Created on 2026-08-29 with <a href=\"https://reprex.tidyverse.org\" rel=\"nofollow noreferrer\">reprex v2.1.1</a></sup></p>\n", "answer_id": 676996, "answer_text": "Rather than fitting separate models, you can handle both age groups in a single gls() model with age interactions. This has two advantages over separate auto.arima() models: you get formal tests of whether intervention effects differ between groups via the interaction terms, and you can account for the fact that the two series have very different variances.\n\n\n\n\nage * (time + step1 + ramp1 + ...) gives each group its own intercept, pre-intervention slope, and intervention effects, plus tests of whether they differ. Then, corAR1(form = ~ 1 | age) fits AR(1) autocorrelation within each group and varIdent(form = ~ 1 | age) allows different residual variances per group, which matters here since children's prevalence is an order of magnitude larger.\n\n\n\n\nlibrary(nlme)\nlibrary(dplyr)\n\ninterventions <- as.Date(c(\"2021-09-13\", \"2022-06-02\", \"2023-06-29\"))\n\nd_its <- d |>\n arrange(age, month) |>\n mutate(.by = age,\n time = row_number() - 1,\n step1 = as.numeric(month >= interventions[1]),\n ramp1 = ifelse(step1 == 1, cumsum(step1) - 1, 0),\n step2 = as.numeric(month >= interventions[2]),\n ramp2 = ifelse(step2 == 1, cumsum(step2) - 1, 0),\n step3 = as.numeric(month >= interventions[3]),\n ramp3 = ifelse(step3 == 1, cumsum(step3) - 1, 0),\n sin12 = sin(2 * pi * time / 12),\n cos12 = cos(2 * pi * time / 12),\n sin6 = sin(2 * pi * time / 6),\n cos6 = cos(2 * pi * time / 6),\n sin4 = sin(2 * pi * time / 4),\n cos4 = cos(2 * pi * time / 4)\n ) |>\n mutate(age = factor(age))\n\nfit <- gls(\n prevalence ~ age * (time + step1 + ramp1 + step2 + ramp2 + step3 + ramp3 +\n sin12 + cos12 + sin6 + cos6 + sin4 + cos4),\n data = d_its,\n correlation = corAR1(form = ~ 1 | age),\n weights = varIdent(form = ~ 1 | age)\n)\n\n\n\n\n\nThe Fourier terms (sin12, cos12, etc.) handle the recurring seasonal pattern, which is more stable than trying to absorb it through ARMA error terms (see https://otexts.com/fpp3/dhr.html (https://otexts.com/fpp3/dhr.html)).\n\n\n\n\nThe reference level is Adults, so coefficients are interpreted as:\n\n\n\n\n\ntime: pre-intervention slope for adults (prevalence per month)\n\n\n\n\nageChildren:time: additional slope for children;\n\n\n\n\n\nThe same logic applies for all ramp and step terms.\n\n\n\n\ncoef(fit) |>\n (\\(e) data.frame(\n Segment = c(\"Pre\", \"Post-1\", \"Post-2\", \"Post-3\"),\n Adults = round(cumsum(c(e[\"time\"],\n e[\"ramp1\"],\n e[\"ramp2\"],\n e[\"ramp3\"])), 3),\n Children = round(cumsum(c(e[\"time\"] + e[\"ageChildren:time\"],\n e[\"ramp1\"] + e[\"ageChildren:ramp1\"],\n e[\"ramp2\"] + e[\"ageChildren:ramp2\"],\n e[\"ramp3\"] + e[\"ageChildren:ramp3\"])), 3)\n))()\n\n#> Segment Adults Children\n#> time Pre 0.152 1.445\n#> ramp1 Post-1 0.919 9.276\n#> ramp2 Post-2 1.265 6.667\n#> ramp3 Post-3 2.387 8.418\n\n\n\n\n\npar(mfrow = c(1, 2))\nacf(residuals(fit)[d_its$age == \"Adults\"], main = \"Adults\")\nacf(residuals(fit)[d_its$age == \"Children\"], main = \"Children\")\ndev.off()\n\n\n\n\n\n[image: ; source: https://i.sstatic.net/2fnDCjiM.png]\n\n\n\n\nsummary(fit)\n\n#> Generalized least squares fit by REML\n#> Model: prevalence ~ age * (time + step1 + ramp1 + step2 + ramp2 + step3 + ramp3 + sin12 + cos12 + sin6 + cos6 + sin4 + cos4) \n#> Data: d_its \n#> AIC BIC logLik\n#> 1665.172 1769.226 -801.5858\n#> \n#> Correlation Structure: AR(1)\n#> Formula: ~1 | age \n#> Parameter estimate(s):\n#> Phi \n#> 0.1231768 \n#> Variance function:\n#> Structure: Different standard deviations per stratum\n#> Formula: ~1 | age \n#> Parameter estimates:\n#> Adults Children \n#> 1.000000 9.435418 \n#> \n#> Coefficients:\n#> Value Std.Error t-value p-value\n#> (Intercept) 10.25450 0.701853 14.610597 0.0000\n#> ageChildren 177.16318 6.659369 26.603599 0.0000\n#> time 0.15162 0.017799 8.518225 0.0000\n#> step1 1.83019 1.934456 0.946100 0.3452\n#> ramp1 0.76689 0.379858 2.018892 0.0448\n#> step2 -0.98456 2.681497 -0.367167 0.7139\n#> ramp2 0.34620 0.449034 0.770992 0.4416\n#> step3 -4.38832 2.094825 -2.094838 0.0374\n#> ramp3 1.12257 0.257109 4.366133 0.0000\n#> sin12 0.01845 0.399784 0.046145 0.9632\n#> cos12 1.16269 0.380402 3.056467 0.0025\n#> sin6 -0.70555 0.358820 -1.966309 0.0506\n#> cos6 -0.97079 0.355213 -2.732980 0.0068\n#> sin4 0.78337 0.333359 2.349941 0.0197\n#> cos4 0.05479 0.333865 0.164117 0.8698\n#> ageChildren:time 1.29381 0.168883 7.661002 0.0000\n#> ageChildren:step1 -1.02662 18.354630 -0.055933 0.9554\n#> ageChildren:ramp1 7.06407 3.604192 1.959960 0.0513\n#> ageChildren:step2 -35.82645 25.442751 -1.408120 0.1606\n#> ageChildren:ramp2 -2.95530 4.260553 -0.693643 0.4887\n#> ageChildren:step3 -29.49930 19.876247 -1.484148 0.1393\n#> ageChildren:ramp3 0.62770 2.439516 0.257306 0.7972\n#> ageChildren:sin12 4.22521 3.793256 1.113874 0.2666\n#> ageChildren:cos12 36.54005 3.609352 10.123715 0.0000\n#> ageChildren:sin6 -9.76953 3.404578 -2.869528 0.0045\n#> ageChildren:cos6 -25.08251 3.370352 -7.442104 0.0000\n#> ageChildren:sin4 13.68702 3.162997 4.327230 0.0000\n#> ageChildren:cos4 14.66062 3.167797 4.628016 0.0000\n#> \n#> Correlation: \n#> (Intr) agChld time step1 ramp1 step2 ramp2 step3 \n#> ageChildren -0.105 \n#> time -0.862 0.091 \n#> step1 0.173 -0.018 -0.309 \n#> ramp1 0.057 -0.006 -0.063 -0.763 \n#> step2 -0.021 0.002 0.020 0.366 -0.711 \n#> ramp2 -0.004 0.000 0.005 0.655 -0.831 0.306 \n#> step3 -0.015 0.002 0.015 0.007 -0.031 0.300 -0.403 \n#> ramp3 -0.016 0.002 0.016 0.008 -0.023 0.514 -0.520 0.649\n#> sin12 -0.056 0.006 0.043 0.155 -0.247 0.317 0.074 0.209\n#> cos12 -0.050 0.005 0.069 -0.188 0.137 -0.028 -0.155 0.065\n#> sin6 -0.008 0.001 -0.006 0.034 0.018 -0.103 0.052 -0.126\n#> cos6 -0.010 0.001 0.002 0.047 -0.026 -0.035 0.060 -0.071\n#> sin4 -0.024 0.002 0.028 -0.033 -0.004 0.059 -0.040 0.066\n#> cos4 -0.011 0.001 0.002 0.063 -0.078 0.095 0.024 0.069\n#> ageChildren:time 0.091 -0.862 -0.105 0.033 0.007 -0.002 0.000 -0.002\n#> ageChildren:step1 -0.018 0.173 0.033 -0.105 0.080 -0.039 -0.069 -0.001\n#> ageChildren:ramp1 -0.006 0.057 0.007 0.080 -0.105 0.075 0.088 0.003\n#> ageChildren:step2 0.002 -0.021 -0.002 -0.039 0.075 -0.105 -0.032 -0.032\n#> ageChildren:ramp2 0.000 -0.004 0.000 -0.069 0.088 -0.032 -0.105 0.043\n#> ageChildren:step3 0.002 -0.015 -0.002 -0.001 0.003 -0.032 0.043 -0.105\n#> ageChildren:ramp3 0.002 -0.016 -0.002 -0.001 0.002 -0.054 0.055 -0.068\n#> ageChildren:sin12 0.006 -0.056 -0.005 -0.016 0.026 -0.033 -0.008 -0.022\n#> ageChildren:cos12 0.005 -0.050 -0.007 0.020 -0.014 0.003 0.016 -0.007\n#> ageChildren:sin6 0.001 -0.008 0.001 -0.004 -0.002 0.011 -0.005 0.013\n#> ageChildren:cos6 0.001 -0.010 0.000 -0.005 0.003 0.004 -0.006 0.007\n#> ageChildren:sin4 0.002 -0.024 -0.003 0.003 0.000 -0.006 0.004 -0.007\n#> ageChildren:cos4 0.001 -0.011 0.000 -0.007 0.008 -0.010 -0.002 -0.007\n#> ramp3 sin12 cos12 sin6 cos6 sin4 cos4 agChl:\n#> ageChildren \n#> time \n#> step1 \n#> ramp1 \n#> step2 \n#> ramp2 \n#> step3 \n#> ramp3 \n#> sin12 0.235 \n#> cos12 0.057 -0.008 \n#> sin6 -0.099 -0.036 -0.015 \n#> cos6 -0.058 -0.011 -0.017 0.017 \n#> sin4 0.074 0.025 0.014 -0.013 -0.006 \n#> cos4 0.073 0.041 -0.007 -0.009 -0.004 0.009 \n#> ageChildren:time -0.002 -0.005 -0.007 0.001 0.000 -0.003 0.000 \n#> ageChildren:step1 -0.001 -0.016 0.020 -0.004 -0.005 0.003 -0.007 -0.309\n#> ageChildren:ramp1 0.002 0.026 -0.014 -0.002 0.003 0.000 0.008 -0.063\n#> ageChildren:step2 -0.054 -0.033 0.003 0.011 0.004 -0.006 -0.010 0.020\n#> ageChildren:ramp2 0.055 -0.008 0.016 -0.005 -0.006 0.004 -0.002 0.005\n#> ageChildren:step3 -0.068 -0.022 -0.007 0.013 0.007 -0.007 -0.007 0.015\n#> ageChildren:ramp3 -0.105 -0.025 -0.006 0.010 0.006 -0.008 -0.008 0.016\n#> ageChildren:sin12 -0.025 -0.105 0.001 0.004 0.001 -0.003 -0.004 0.043\n#> ageChildren:cos12 -0.006 0.001 -0.105 0.002 0.002 -0.002 0.001 0.069\n#> ageChildren:sin6 0.010 0.004 0.002 -0.105 -0.002 0.001 0.001 -0.006\n#> ageChildren:cos6 0.006 0.001 0.002 -0.002 -0.105 0.001 0.000 0.002\n#> ageChildren:sin4 -0.008 -0.003 -0.002 0.001 0.001 -0.105 -0.001 0.028\n#> ageChildren:cos4 -0.008 -0.004 0.001 0.001 0.000 -0.001 -0.105 0.002\n#> agChldrn:s1 agChldrn:r1 agChldrn:s2 agChldrn:r2 agChldrn:s3\n#> ageChildren \n#> time \n#> step1 \n#> ramp1 \n#> step2 \n#> ramp2 \n#> step3 \n#> ramp3 \n#> sin12 \n#> cos12 \n#> sin6 \n#> cos6 \n#> sin4 \n#> cos4 \n#> ageChildren:time \n#> ageChildren:step1 \n#> ageChildren:ramp1 -0.763 \n#> ageChildren:step2 0.366 -0.711 \n#> ageChildren:ramp2 0.655 -0.831 0.306 \n#> ageChildren:step3 0.007 -0.031 0.300 -0.403 \n#> ageChildren:ramp3 0.008 -0.023 0.514 -0.520 0.649 \n#> ageChildren:sin12 0.155 -0.247 0.317 0.074 0.209 \n#> ageChildren:cos12 -0.188 0.137 -0.028 -0.155 0.065 \n#> ageChildren:sin6 0.034 0.018 -0.103 0.052 -0.126 \n#> ageChildren:cos6 0.047 -0.026 -0.035 0.060 -0.071 \n#> ageChildren:sin4 -0.033 -0.004 0.059 -0.040 0.066 \n#> ageChildren:cos4 0.063 -0.078 0.095 0.024 0.069 \n#> agChldrn:r3 agChldrn:s12 agChldrn:c12 agChldrn:s6 agChldrn:c6\n#> ageChildren \n#> time \n#> step1 \n#> ramp1 \n#> step2 \n#> ramp2 \n#> step3 \n#> ramp3 \n#> sin12 \n#> cos12 \n#> sin6 \n#> cos6 \n#> sin4 \n#> cos4 \n#> ageChildren:time \n#> ageChildren:step1 \n#> ageChildren:ramp1 \n#> ageChildren:step2 \n#> ageChildren:ramp2 \n#> ageChildren:step3 \n#> ageChildren:ramp3 \n#> ageChildren:sin12 0.235 \n#> ageChildren:cos12 0.057 -0.008 \n#> ageChildren:sin6 -0.099 -0.036 -0.015 \n#> ageChildren:cos6 -0.058 -0.011 -0.017 0.017 \n#> ageChildren:sin4 0.074 0.025 0.014 -0.013 -0.006 \n#> ageChildren:cos4 0.073 0.041 -0.007 -0.009 -0.004 \n#> agChldrn:s4\n#> ageChildren \n#> time \n#> step1 \n#> ramp1 \n#> step2 \n#> ramp2 \n#> step3 \n#> ramp3 \n#> sin12 \n#> cos12 \n#> sin6 \n#> cos6 \n#> sin4 \n#> cos4 \n#> ageChildren:time \n#> ageChildren:step1 \n#> ageChildren:ramp1 \n#> ageChildren:step2 \n#> ageChildren:ramp2 \n#> ageChildren:step3 \n#> ageChildren:ramp3 \n#> ageChildren:sin12 \n#> ageChildren:cos12 \n#> ageChildren:sin6 \n#> ageChildren:cos6 \n#> ageChildren:sin4 \n#> ageChildren:cos4 0.009 \n#> \n#> Standardized residuals:\n#> Min Q1 Med Q3 Max \n#> -5.25291094 -0.42240774 0.04331099 0.45224365 3.26323939 \n#> \n#> Residual standard error: 2.608572 \n#> Degrees of freedom: 240 total; 212 residual\n\n\n\n\n\nCreated on 2026-08-29 with reprex v2.1.1 (https://reprex.tidyverse.org)", "answer_url": "https://stats.stackexchange.com/a/676996", "author": "M--", "author_url": "https://stats.stackexchange.com/users/154449/m", "author_user_type": "registered", "content_license": "CC BY-SA 4.0", "created_at": "2026-08-29T05:35:19+00:00", "is_accepted": true, "provenance": {"attribution_required": true, "collected_at": "2026-10-01T03:27:06.169765+00:00", "license": "CC BY-SA 4.0", "license_url": "https://creativecommons.org/licenses/by-sa/4.0/", "raw_file": "raw/codex_api_v1/9155f81b0f1dac4b0f83fad58469033af46ad34ee8d731e569d063b41b3df6b6_1790825226563497500_0.json", "raw_sha256": "cbac4b1b24e2159f1c17ea9902eaf89a0a5260344e6dba6f75790ba955f0d04b", "source_api": "Stack Exchange API 2.3", "source_url": "https://api.stackexchange.com/2.3/questions/677298;677295;677280;677279;677278;677275;677269;677266;677262;677256;677245;677243;677242;677226;677225;677217;677211;677201;677194;677193;677186;677183;677179;677177;677171;677169;677168;677151;677149;677147;677137;677135;677131;677129;677115;677110;677109;677101;677099;677098;677096;677095;677094;677085;677083;677079;677078;677075;677071;677066;677065;677062;677058;677045;677041;677035;677033;677023;677021;676999;676997;676991;676982;676981;676979;676977;676975;676971;676966;676961;676958;676956;676952;676947;676945;676937;676935;676933;676924;676922;676899;676898;676893;676889;676888;676879;676874;676873;676870;676867;676865;676858;676855;676850;676842;676832;676830;676824;676823;676821/answers?filter=withbody&order=asc&page=1&pagesize=100&site=stats&sort=creation", "transformation": "API HTML retained; mechanical HTML-to-text; no LLM rewriting"}, "question_id": 676979, "revision_attribution": [{"content_license": "CC BY-SA 4.0", "contributor": {"display_name": "M--", "profile_url": "https://stats.stackexchange.com/users/154449/m", "user_type": "registered"}, "created_at": "2026-08-29T05:35:19+00:00", "raw_file": "raw/codex_api_v1/84bb317e707465a80f008019404ef1f770e152d74ef692d3b81c48903ec855cc_1790825232001984700_0.json", "raw_sha256": "75778d86d5325e15f4b491a748dc70ae6ac6ee98840d584e01ed28416cb71fbe", "revision_guid": "4F437721-3A00-4B4D-BA06-A2F50516CC59", "revision_number": 1, "revision_type": "single_user", "revision_url": "https://stats.stackexchange.com/revisions/4F437721-3A00-4B4D-BA06-A2F50516CC59/view-source"}], "score": 3, "updated_at": "2026-08-29T05:35:19+00:00"}, "selection_rule": "accepted; otherwise maximum score >= 1, tie lowest ID; no correctness label", "split": "validation", "thread": {"accepted_answer_id": 676996, "answers": [{"answer_html": "<p>Rather than fitting separate models, you can handle both age groups in a single <code>gls()</code> model with <code>age</code> interactions. This has two advantages over separate <code>auto.arima()</code> models: you get formal tests of whether intervention effects differ between groups via the interaction terms, and you can account for the fact that the two series have very different variances.</p>\n<p><code>age * (time + step1 + ramp1 + ...)</code> gives each group its own intercept, pre-intervention slope, and intervention effects, plus tests of whether they differ. Then, <code>corAR1(form = ~ 1 | age)</code> fits AR(1) autocorrelation within each group and <code>varIdent(form = ~ 1 | age)</code> allows different residual variances per group, which matters here since children's prevalence is an order of magnitude larger.</p>\n<pre class=\"lang-r prettyprint-override\"><code>library(nlme)\nlibrary(dplyr)\n\ninterventions <- as.Date(c("2021-09-13", "2022-06-02", "2023-06-29"))\n\nd_its <- d |>\n arrange(age, month) |>\n mutate(.by = age,\n time = row_number() - 1,\n step1 = as.numeric(month >= interventions[1]),\n ramp1 = ifelse(step1 == 1, cumsum(step1) - 1, 0),\n step2 = as.numeric(month >= interventions[2]),\n ramp2 = ifelse(step2 == 1, cumsum(step2) - 1, 0),\n step3 = as.numeric(month >= interventions[3]),\n ramp3 = ifelse(step3 == 1, cumsum(step3) - 1, 0),\n sin12 = sin(2 * pi * time / 12),\n cos12 = cos(2 * pi * time / 12),\n sin6 = sin(2 * pi * time / 6),\n cos6 = cos(2 * pi * time / 6),\n sin4 = sin(2 * pi * time / 4),\n cos4 = cos(2 * pi * time / 4)\n ) |>\n mutate(age = factor(age))\n\nfit <- gls(\n prevalence ~ age * (time + step1 + ramp1 + step2 + ramp2 + step3 + ramp3 +\n sin12 + cos12 + sin6 + cos6 + sin4 + cos4),\n data = d_its,\n correlation = corAR1(form = ~ 1 | age),\n weights = varIdent(form = ~ 1 | age)\n)\n</code></pre>\n<p>The Fourier terms (<code>sin12</code>, <code>cos12</code>, etc.) handle the recurring seasonal pattern, which is more stable than trying to absorb it through ARMA error terms (see <a href=\"https://otexts.com/fpp3/dhr.html\" rel=\"nofollow noreferrer\">https://otexts.com/fpp3/dhr.html</a>).</p>\n<p>The reference level is Adults, so coefficients are interpreted as:</p>\n<ul>\n<li><code>time</code>: pre-intervention slope for adults (prevalence per month)</li>\n<li><code>ageChildren:time</code>: <em>additional</em> slope for children;</li>\n</ul>\n<p>The same logic applies for all <code>ramp</code> and <code>step</code> terms.</p>\n<pre class=\"lang-r prettyprint-override\"><code>coef(fit) |>\n (\\(e) data.frame(\n Segment = c("Pre", "Post-1", "Post-2", "Post-3"),\n Adults = round(cumsum(c(e["time"],\n e["ramp1"],\n e["ramp2"],\n e["ramp3"])), 3),\n Children = round(cumsum(c(e["time"] + e["ageChildren:time"],\n e["ramp1"] + e["ageChildren:ramp1"],\n e["ramp2"] + e["ageChildren:ramp2"],\n e["ramp3"] + e["ageChildren:ramp3"])), 3)\n))()\n\n#> Segment Adults Children\n#> time Pre 0.152 1.445\n#> ramp1 Post-1 0.919 9.276\n#> ramp2 Post-2 1.265 6.667\n#> ramp3 Post-3 2.387 8.418\n</code></pre>\n<pre class=\"lang-r prettyprint-override\"><code>par(mfrow = c(1, 2))\nacf(residuals(fit)[d_its<span class=\"math-container\">$age == \"Adults\"], main = \"Adults\")\nacf(residuals(fit)[d_its$</span>age == "Children"], main = "Children")\ndev.off()\n</code></pre>\n<p><img src=\"https://i.sstatic.net/2fnDCjiM.png\" alt=\"\" /></p>\n<pre class=\"lang-r prettyprint-override\"><code>summary(fit)\n\n#> Generalized least squares fit by REML\n#> Model: prevalence ~ age * (time + step1 + ramp1 + step2 + ramp2 + step3 + ramp3 + sin12 + cos12 + sin6 + cos6 + sin4 + cos4) \n#> Data: d_its \n#> AIC BIC logLik\n#> 1665.172 1769.226 -801.5858\n#> \n#> Correlation Structure: AR(1)\n#> Formula: ~1 | age \n#> Parameter estimate(s):\n#> Phi \n#> 0.1231768 \n#> Variance function:\n#> Structure: Different standard deviations per stratum\n#> Formula: ~1 | age \n#> Parameter estimates:\n#> Adults Children \n#> 1.000000 9.435418 \n#> \n#> Coefficients:\n#> Value Std.Error t-value p-value\n#> (Intercept) 10.25450 0.701853 14.610597 0.0000\n#> ageChildren 177.16318 6.659369 26.603599 0.0000\n#> time 0.15162 0.017799 8.518225 0.0000\n#> step1 1.83019 1.934456 0.946100 0.3452\n#> ramp1 0.76689 0.379858 2.018892 0.0448\n#> step2 -0.98456 2.681497 -0.367167 0.7139\n#> ramp2 0.34620 0.449034 0.770992 0.4416\n#> step3 -4.38832 2.094825 -2.094838 0.0374\n#> ramp3 1.12257 0.257109 4.366133 0.0000\n#> sin12 0.01845 0.399784 0.046145 0.9632\n#> cos12 1.16269 0.380402 3.056467 0.0025\n#> sin6 -0.70555 0.358820 -1.966309 0.0506\n#> cos6 -0.97079 0.355213 -2.732980 0.0068\n#> sin4 0.78337 0.333359 2.349941 0.0197\n#> cos4 0.05479 0.333865 0.164117 0.8698\n#> ageChildren:time 1.29381 0.168883 7.661002 0.0000\n#> ageChildren:step1 -1.02662 18.354630 -0.055933 0.9554\n#> ageChildren:ramp1 7.06407 3.604192 1.959960 0.0513\n#> ageChildren:step2 -35.82645 25.442751 -1.408120 0.1606\n#> ageChildren:ramp2 -2.95530 4.260553 -0.693643 0.4887\n#> ageChildren:step3 -29.49930 19.876247 -1.484148 0.1393\n#> ageChildren:ramp3 0.62770 2.439516 0.257306 0.7972\n#> ageChildren:sin12 4.22521 3.793256 1.113874 0.2666\n#> ageChildren:cos12 36.54005 3.609352 10.123715 0.0000\n#> ageChildren:sin6 -9.76953 3.404578 -2.869528 0.0045\n#> ageChildren:cos6 -25.08251 3.370352 -7.442104 0.0000\n#> ageChildren:sin4 13.68702 3.162997 4.327230 0.0000\n#> ageChildren:cos4 14.66062 3.167797 4.628016 0.0000\n#> \n#> Correlation: \n#> (Intr) agChld time step1 ramp1 step2 ramp2 step3 \n#> ageChildren -0.105 \n#> time -0.862 0.091 \n#> step1 0.173 -0.018 -0.309 \n#> ramp1 0.057 -0.006 -0.063 -0.763 \n#> step2 -0.021 0.002 0.020 0.366 -0.711 \n#> ramp2 -0.004 0.000 0.005 0.655 -0.831 0.306 \n#> step3 -0.015 0.002 0.015 0.007 -0.031 0.300 -0.403 \n#> ramp3 -0.016 0.002 0.016 0.008 -0.023 0.514 -0.520 0.649\n#> sin12 -0.056 0.006 0.043 0.155 -0.247 0.317 0.074 0.209\n#> cos12 -0.050 0.005 0.069 -0.188 0.137 -0.028 -0.155 0.065\n#> sin6 -0.008 0.001 -0.006 0.034 0.018 -0.103 0.052 -0.126\n#> cos6 -0.010 0.001 0.002 0.047 -0.026 -0.035 0.060 -0.071\n#> sin4 -0.024 0.002 0.028 -0.033 -0.004 0.059 -0.040 0.066\n#> cos4 -0.011 0.001 0.002 0.063 -0.078 0.095 0.024 0.069\n#> ageChildren:time 0.091 -0.862 -0.105 0.033 0.007 -0.002 0.000 -0.002\n#> ageChildren:step1 -0.018 0.173 0.033 -0.105 0.080 -0.039 -0.069 -0.001\n#> ageChildren:ramp1 -0.006 0.057 0.007 0.080 -0.105 0.075 0.088 0.003\n#> ageChildren:step2 0.002 -0.021 -0.002 -0.039 0.075 -0.105 -0.032 -0.032\n#> ageChildren:ramp2 0.000 -0.004 0.000 -0.069 0.088 -0.032 -0.105 0.043\n#> ageChildren:step3 0.002 -0.015 -0.002 -0.001 0.003 -0.032 0.043 -0.105\n#> ageChildren:ramp3 0.002 -0.016 -0.002 -0.001 0.002 -0.054 0.055 -0.068\n#> ageChildren:sin12 0.006 -0.056 -0.005 -0.016 0.026 -0.033 -0.008 -0.022\n#> ageChildren:cos12 0.005 -0.050 -0.007 0.020 -0.014 0.003 0.016 -0.007\n#> ageChildren:sin6 0.001 -0.008 0.001 -0.004 -0.002 0.011 -0.005 0.013\n#> ageChildren:cos6 0.001 -0.010 0.000 -0.005 0.003 0.004 -0.006 0.007\n#> ageChildren:sin4 0.002 -0.024 -0.003 0.003 0.000 -0.006 0.004 -0.007\n#> ageChildren:cos4 0.001 -0.011 0.000 -0.007 0.008 -0.010 -0.002 -0.007\n#> ramp3 sin12 cos12 sin6 cos6 sin4 cos4 agChl:\n#> ageChildren \n#> time \n#> step1 \n#> ramp1 \n#> step2 \n#> ramp2 \n#> step3 \n#> ramp3 \n#> sin12 0.235 \n#> cos12 0.057 -0.008 \n#> sin6 -0.099 -0.036 -0.015 \n#> cos6 -0.058 -0.011 -0.017 0.017 \n#> sin4 0.074 0.025 0.014 -0.013 -0.006 \n#> cos4 0.073 0.041 -0.007 -0.009 -0.004 0.009 \n#> ageChildren:time -0.002 -0.005 -0.007 0.001 0.000 -0.003 0.000 \n#> ageChildren:step1 -0.001 -0.016 0.020 -0.004 -0.005 0.003 -0.007 -0.309\n#> ageChildren:ramp1 0.002 0.026 -0.014 -0.002 0.003 0.000 0.008 -0.063\n#> ageChildren:step2 -0.054 -0.033 0.003 0.011 0.004 -0.006 -0.010 0.020\n#> ageChildren:ramp2 0.055 -0.008 0.016 -0.005 -0.006 0.004 -0.002 0.005\n#> ageChildren:step3 -0.068 -0.022 -0.007 0.013 0.007 -0.007 -0.007 0.015\n#> ageChildren:ramp3 -0.105 -0.025 -0.006 0.010 0.006 -0.008 -0.008 0.016\n#> ageChildren:sin12 -0.025 -0.105 0.001 0.004 0.001 -0.003 -0.004 0.043\n#> ageChildren:cos12 -0.006 0.001 -0.105 0.002 0.002 -0.002 0.001 0.069\n#> ageChildren:sin6 0.010 0.004 0.002 -0.105 -0.002 0.001 0.001 -0.006\n#> ageChildren:cos6 0.006 0.001 0.002 -0.002 -0.105 0.001 0.000 0.002\n#> ageChildren:sin4 -0.008 -0.003 -0.002 0.001 0.001 -0.105 -0.001 0.028\n#> ageChildren:cos4 -0.008 -0.004 0.001 0.001 0.000 -0.001 -0.105 0.002\n#> agChldrn:s1 agChldrn:r1 agChldrn:s2 agChldrn:r2 agChldrn:s3\n#> ageChildren \n#> time \n#> step1 \n#> ramp1 \n#> step2 \n#> ramp2 \n#> step3 \n#> ramp3 \n#> sin12 \n#> cos12 \n#> sin6 \n#> cos6 \n#> sin4 \n#> cos4 \n#> ageChildren:time \n#> ageChildren:step1 \n#> ageChildren:ramp1 -0.763 \n#> ageChildren:step2 0.366 -0.711 \n#> ageChildren:ramp2 0.655 -0.831 0.306 \n#> ageChildren:step3 0.007 -0.031 0.300 -0.403 \n#> ageChildren:ramp3 0.008 -0.023 0.514 -0.520 0.649 \n#> ageChildren:sin12 0.155 -0.247 0.317 0.074 0.209 \n#> ageChildren:cos12 -0.188 0.137 -0.028 -0.155 0.065 \n#> ageChildren:sin6 0.034 0.018 -0.103 0.052 -0.126 \n#> ageChildren:cos6 0.047 -0.026 -0.035 0.060 -0.071 \n#> ageChildren:sin4 -0.033 -0.004 0.059 -0.040 0.066 \n#> ageChildren:cos4 0.063 -0.078 0.095 0.024 0.069 \n#> agChldrn:r3 agChldrn:s12 agChldrn:c12 agChldrn:s6 agChldrn:c6\n#> ageChildren \n#> time \n#> step1 \n#> ramp1 \n#> step2 \n#> ramp2 \n#> step3 \n#> ramp3 \n#> sin12 \n#> cos12 \n#> sin6 \n#> cos6 \n#> sin4 \n#> cos4 \n#> ageChildren:time \n#> ageChildren:step1 \n#> ageChildren:ramp1 \n#> ageChildren:step2 \n#> ageChildren:ramp2 \n#> ageChildren:step3 \n#> ageChildren:ramp3 \n#> ageChildren:sin12 0.235 \n#> ageChildren:cos12 0.057 -0.008 \n#> ageChildren:sin6 -0.099 -0.036 -0.015 \n#> ageChildren:cos6 -0.058 -0.011 -0.017 0.017 \n#> ageChildren:sin4 0.074 0.025 0.014 -0.013 -0.006 \n#> ageChildren:cos4 0.073 0.041 -0.007 -0.009 -0.004 \n#> agChldrn:s4\n#> ageChildren \n#> time \n#> step1 \n#> ramp1 \n#> step2 \n#> ramp2 \n#> step3 \n#> ramp3 \n#> sin12 \n#> cos12 \n#> sin6 \n#> cos6 \n#> sin4 \n#> cos4 \n#> ageChildren:time \n#> ageChildren:step1 \n#> ageChildren:ramp1 \n#> ageChildren:step2 \n#> ageChildren:ramp2 \n#> ageChildren:step3 \n#> ageChildren:ramp3 \n#> ageChildren:sin12 \n#> ageChildren:cos12 \n#> ageChildren:sin6 \n#> ageChildren:cos6 \n#> ageChildren:sin4 \n#> ageChildren:cos4 0.009 \n#> \n#> Standardized residuals:\n#> Min Q1 Med Q3 Max \n#> -5.25291094 -0.42240774 0.04331099 0.45224365 3.26323939 \n#> \n#> Residual standard error: 2.608572 \n#> Degrees of freedom: 240 total; 212 residual\n</code></pre>\n<p><sup>Created on 2026-08-29 with <a href=\"https://reprex.tidyverse.org\" rel=\"nofollow noreferrer\">reprex v2.1.1</a></sup></p>\n", "answer_id": 676996, "answer_text": "Rather than fitting separate models, you can handle both age groups in a single gls() model with age interactions. This has two advantages over separate auto.arima() models: you get formal tests of whether intervention effects differ between groups via the interaction terms, and you can account for the fact that the two series have very different variances.\n\n\n\n\nage * (time + step1 + ramp1 + ...) gives each group its own intercept, pre-intervention slope, and intervention effects, plus tests of whether they differ. Then, corAR1(form = ~ 1 | age) fits AR(1) autocorrelation within each group and varIdent(form = ~ 1 | age) allows different residual variances per group, which matters here since children's prevalence is an order of magnitude larger.\n\n\n\n\nlibrary(nlme)\nlibrary(dplyr)\n\ninterventions <- as.Date(c(\"2021-09-13\", \"2022-06-02\", \"2023-06-29\"))\n\nd_its <- d |>\n arrange(age, month) |>\n mutate(.by = age,\n time = row_number() - 1,\n step1 = as.numeric(month >= interventions[1]),\n ramp1 = ifelse(step1 == 1, cumsum(step1) - 1, 0),\n step2 = as.numeric(month >= interventions[2]),\n ramp2 = ifelse(step2 == 1, cumsum(step2) - 1, 0),\n step3 = as.numeric(month >= interventions[3]),\n ramp3 = ifelse(step3 == 1, cumsum(step3) - 1, 0),\n sin12 = sin(2 * pi * time / 12),\n cos12 = cos(2 * pi * time / 12),\n sin6 = sin(2 * pi * time / 6),\n cos6 = cos(2 * pi * time / 6),\n sin4 = sin(2 * pi * time / 4),\n cos4 = cos(2 * pi * time / 4)\n ) |>\n mutate(age = factor(age))\n\nfit <- gls(\n prevalence ~ age * (time + step1 + ramp1 + step2 + ramp2 + step3 + ramp3 +\n sin12 + cos12 + sin6 + cos6 + sin4 + cos4),\n data = d_its,\n correlation = corAR1(form = ~ 1 | age),\n weights = varIdent(form = ~ 1 | age)\n)\n\n\n\n\n\nThe Fourier terms (sin12, cos12, etc.) handle the recurring seasonal pattern, which is more stable than trying to absorb it through ARMA error terms (see https://otexts.com/fpp3/dhr.html (https://otexts.com/fpp3/dhr.html)).\n\n\n\n\nThe reference level is Adults, so coefficients are interpreted as:\n\n\n\n\n\ntime: pre-intervention slope for adults (prevalence per month)\n\n\n\n\nageChildren:time: additional slope for children;\n\n\n\n\n\nThe same logic applies for all ramp and step terms.\n\n\n\n\ncoef(fit) |>\n (\\(e) data.frame(\n Segment = c(\"Pre\", \"Post-1\", \"Post-2\", \"Post-3\"),\n Adults = round(cumsum(c(e[\"time\"],\n e[\"ramp1\"],\n e[\"ramp2\"],\n e[\"ramp3\"])), 3),\n Children = round(cumsum(c(e[\"time\"] + e[\"ageChildren:time\"],\n e[\"ramp1\"] + e[\"ageChildren:ramp1\"],\n e[\"ramp2\"] + e[\"ageChildren:ramp2\"],\n e[\"ramp3\"] + e[\"ageChildren:ramp3\"])), 3)\n))()\n\n#> Segment Adults Children\n#> time Pre 0.152 1.445\n#> ramp1 Post-1 0.919 9.276\n#> ramp2 Post-2 1.265 6.667\n#> ramp3 Post-3 2.387 8.418\n\n\n\n\n\npar(mfrow = c(1, 2))\nacf(residuals(fit)[d_its$age == \"Adults\"], main = \"Adults\")\nacf(residuals(fit)[d_its$age == \"Children\"], main = \"Children\")\ndev.off()\n\n\n\n\n\n[image: ; source: https://i.sstatic.net/2fnDCjiM.png]\n\n\n\n\nsummary(fit)\n\n#> Generalized least squares fit by REML\n#> Model: prevalence ~ age * (time + step1 + ramp1 + step2 + ramp2 + step3 + ramp3 + sin12 + cos12 + sin6 + cos6 + sin4 + cos4) \n#> Data: d_its \n#> AIC BIC logLik\n#> 1665.172 1769.226 -801.5858\n#> \n#> Correlation Structure: AR(1)\n#> Formula: ~1 | age \n#> Parameter estimate(s):\n#> Phi \n#> 0.1231768 \n#> Variance function:\n#> Structure: Different standard deviations per stratum\n#> Formula: ~1 | age \n#> Parameter estimates:\n#> Adults Children \n#> 1.000000 9.435418 \n#> \n#> Coefficients:\n#> Value Std.Error t-value p-value\n#> (Intercept) 10.25450 0.701853 14.610597 0.0000\n#> ageChildren 177.16318 6.659369 26.603599 0.0000\n#> time 0.15162 0.017799 8.518225 0.0000\n#> step1 1.83019 1.934456 0.946100 0.3452\n#> ramp1 0.76689 0.379858 2.018892 0.0448\n#> step2 -0.98456 2.681497 -0.367167 0.7139\n#> ramp2 0.34620 0.449034 0.770992 0.4416\n#> step3 -4.38832 2.094825 -2.094838 0.0374\n#> ramp3 1.12257 0.257109 4.366133 0.0000\n#> sin12 0.01845 0.399784 0.046145 0.9632\n#> cos12 1.16269 0.380402 3.056467 0.0025\n#> sin6 -0.70555 0.358820 -1.966309 0.0506\n#> cos6 -0.97079 0.355213 -2.732980 0.0068\n#> sin4 0.78337 0.333359 2.349941 0.0197\n#> cos4 0.05479 0.333865 0.164117 0.8698\n#> ageChildren:time 1.29381 0.168883 7.661002 0.0000\n#> ageChildren:step1 -1.02662 18.354630 -0.055933 0.9554\n#> ageChildren:ramp1 7.06407 3.604192 1.959960 0.0513\n#> ageChildren:step2 -35.82645 25.442751 -1.408120 0.1606\n#> ageChildren:ramp2 -2.95530 4.260553 -0.693643 0.4887\n#> ageChildren:step3 -29.49930 19.876247 -1.484148 0.1393\n#> ageChildren:ramp3 0.62770 2.439516 0.257306 0.7972\n#> ageChildren:sin12 4.22521 3.793256 1.113874 0.2666\n#> ageChildren:cos12 36.54005 3.609352 10.123715 0.0000\n#> ageChildren:sin6 -9.76953 3.404578 -2.869528 0.0045\n#> ageChildren:cos6 -25.08251 3.370352 -7.442104 0.0000\n#> ageChildren:sin4 13.68702 3.162997 4.327230 0.0000\n#> ageChildren:cos4 14.66062 3.167797 4.628016 0.0000\n#> \n#> Correlation: \n#> (Intr) agChld time step1 ramp1 step2 ramp2 step3 \n#> ageChildren -0.105 \n#> time -0.862 0.091 \n#> step1 0.173 -0.018 -0.309 \n#> ramp1 0.057 -0.006 -0.063 -0.763 \n#> step2 -0.021 0.002 0.020 0.366 -0.711 \n#> ramp2 -0.004 0.000 0.005 0.655 -0.831 0.306 \n#> step3 -0.015 0.002 0.015 0.007 -0.031 0.300 -0.403 \n#> ramp3 -0.016 0.002 0.016 0.008 -0.023 0.514 -0.520 0.649\n#> sin12 -0.056 0.006 0.043 0.155 -0.247 0.317 0.074 0.209\n#> cos12 -0.050 0.005 0.069 -0.188 0.137 -0.028 -0.155 0.065\n#> sin6 -0.008 0.001 -0.006 0.034 0.018 -0.103 0.052 -0.126\n#> cos6 -0.010 0.001 0.002 0.047 -0.026 -0.035 0.060 -0.071\n#> sin4 -0.024 0.002 0.028 -0.033 -0.004 0.059 -0.040 0.066\n#> cos4 -0.011 0.001 0.002 0.063 -0.078 0.095 0.024 0.069\n#> ageChildren:time 0.091 -0.862 -0.105 0.033 0.007 -0.002 0.000 -0.002\n#> ageChildren:step1 -0.018 0.173 0.033 -0.105 0.080 -0.039 -0.069 -0.001\n#> ageChildren:ramp1 -0.006 0.057 0.007 0.080 -0.105 0.075 0.088 0.003\n#> ageChildren:step2 0.002 -0.021 -0.002 -0.039 0.075 -0.105 -0.032 -0.032\n#> ageChildren:ramp2 0.000 -0.004 0.000 -0.069 0.088 -0.032 -0.105 0.043\n#> ageChildren:step3 0.002 -0.015 -0.002 -0.001 0.003 -0.032 0.043 -0.105\n#> ageChildren:ramp3 0.002 -0.016 -0.002 -0.001 0.002 -0.054 0.055 -0.068\n#> ageChildren:sin12 0.006 -0.056 -0.005 -0.016 0.026 -0.033 -0.008 -0.022\n#> ageChildren:cos12 0.005 -0.050 -0.007 0.020 -0.014 0.003 0.016 -0.007\n#> ageChildren:sin6 0.001 -0.008 0.001 -0.004 -0.002 0.011 -0.005 0.013\n#> ageChildren:cos6 0.001 -0.010 0.000 -0.005 0.003 0.004 -0.006 0.007\n#> ageChildren:sin4 0.002 -0.024 -0.003 0.003 0.000 -0.006 0.004 -0.007\n#> ageChildren:cos4 0.001 -0.011 0.000 -0.007 0.008 -0.010 -0.002 -0.007\n#> ramp3 sin12 cos12 sin6 cos6 sin4 cos4 agChl:\n#> ageChildren \n#> time \n#> step1 \n#> ramp1 \n#> step2 \n#> ramp2 \n#> step3 \n#> ramp3 \n#> sin12 0.235 \n#> cos12 0.057 -0.008 \n#> sin6 -0.099 -0.036 -0.015 \n#> cos6 -0.058 -0.011 -0.017 0.017 \n#> sin4 0.074 0.025 0.014 -0.013 -0.006 \n#> cos4 0.073 0.041 -0.007 -0.009 -0.004 0.009 \n#> ageChildren:time -0.002 -0.005 -0.007 0.001 0.000 -0.003 0.000 \n#> ageChildren:step1 -0.001 -0.016 0.020 -0.004 -0.005 0.003 -0.007 -0.309\n#> ageChildren:ramp1 0.002 0.026 -0.014 -0.002 0.003 0.000 0.008 -0.063\n#> ageChildren:step2 -0.054 -0.033 0.003 0.011 0.004 -0.006 -0.010 0.020\n#> ageChildren:ramp2 0.055 -0.008 0.016 -0.005 -0.006 0.004 -0.002 0.005\n#> ageChildren:step3 -0.068 -0.022 -0.007 0.013 0.007 -0.007 -0.007 0.015\n#> ageChildren:ramp3 -0.105 -0.025 -0.006 0.010 0.006 -0.008 -0.008 0.016\n#> ageChildren:sin12 -0.025 -0.105 0.001 0.004 0.001 -0.003 -0.004 0.043\n#> ageChildren:cos12 -0.006 0.001 -0.105 0.002 0.002 -0.002 0.001 0.069\n#> ageChildren:sin6 0.010 0.004 0.002 -0.105 -0.002 0.001 0.001 -0.006\n#> ageChildren:cos6 0.006 0.001 0.002 -0.002 -0.105 0.001 0.000 0.002\n#> ageChildren:sin4 -0.008 -0.003 -0.002 0.001 0.001 -0.105 -0.001 0.028\n#> ageChildren:cos4 -0.008 -0.004 0.001 0.001 0.000 -0.001 -0.105 0.002\n#> agChldrn:s1 agChldrn:r1 agChldrn:s2 agChldrn:r2 agChldrn:s3\n#> ageChildren \n#> time \n#> step1 \n#> ramp1 \n#> step2 \n#> ramp2 \n#> step3 \n#> ramp3 \n#> sin12 \n#> cos12 \n#> sin6 \n#> cos6 \n#> sin4 \n#> cos4 \n#> ageChildren:time \n#> ageChildren:step1 \n#> ageChildren:ramp1 -0.763 \n#> ageChildren:step2 0.366 -0.711 \n#> ageChildren:ramp2 0.655 -0.831 0.306 \n#> ageChildren:step3 0.007 -0.031 0.300 -0.403 \n#> ageChildren:ramp3 0.008 -0.023 0.514 -0.520 0.649 \n#> ageChildren:sin12 0.155 -0.247 0.317 0.074 0.209 \n#> ageChildren:cos12 -0.188 0.137 -0.028 -0.155 0.065 \n#> ageChildren:sin6 0.034 0.018 -0.103 0.052 -0.126 \n#> ageChildren:cos6 0.047 -0.026 -0.035 0.060 -0.071 \n#> ageChildren:sin4 -0.033 -0.004 0.059 -0.040 0.066 \n#> ageChildren:cos4 0.063 -0.078 0.095 0.024 0.069 \n#> agChldrn:r3 agChldrn:s12 agChldrn:c12 agChldrn:s6 agChldrn:c6\n#> ageChildren \n#> time \n#> step1 \n#> ramp1 \n#> step2 \n#> ramp2 \n#> step3 \n#> ramp3 \n#> sin12 \n#> cos12 \n#> sin6 \n#> cos6 \n#> sin4 \n#> cos4 \n#> ageChildren:time \n#> ageChildren:step1 \n#> ageChildren:ramp1 \n#> ageChildren:step2 \n#> ageChildren:ramp2 \n#> ageChildren:step3 \n#> ageChildren:ramp3 \n#> ageChildren:sin12 0.235 \n#> ageChildren:cos12 0.057 -0.008 \n#> ageChildren:sin6 -0.099 -0.036 -0.015 \n#> ageChildren:cos6 -0.058 -0.011 -0.017 0.017 \n#> ageChildren:sin4 0.074 0.025 0.014 -0.013 -0.006 \n#> ageChildren:cos4 0.073 0.041 -0.007 -0.009 -0.004 \n#> agChldrn:s4\n#> ageChildren \n#> time \n#> step1 \n#> ramp1 \n#> step2 \n#> ramp2 \n#> step3 \n#> ramp3 \n#> sin12 \n#> cos12 \n#> sin6 \n#> cos6 \n#> sin4 \n#> cos4 \n#> ageChildren:time \n#> ageChildren:step1 \n#> ageChildren:ramp1 \n#> ageChildren:step2 \n#> ageChildren:ramp2 \n#> ageChildren:step3 \n#> ageChildren:ramp3 \n#> ageChildren:sin12 \n#> ageChildren:cos12 \n#> ageChildren:sin6 \n#> ageChildren:cos6 \n#> ageChildren:sin4 \n#> ageChildren:cos4 0.009 \n#> \n#> Standardized residuals:\n#> Min Q1 Med Q3 Max \n#> -5.25291094 -0.42240774 0.04331099 0.45224365 3.26323939 \n#> \n#> Residual standard error: 2.608572 \n#> Degrees of freedom: 240 total; 212 residual\n\n\n\n\n\nCreated on 2026-08-29 with reprex v2.1.1 (https://reprex.tidyverse.org)", "answer_url": "https://stats.stackexchange.com/a/676996", "author": "M--", "author_url": "https://stats.stackexchange.com/users/154449/m", "author_user_type": "registered", "content_license": "CC BY-SA 4.0", "created_at": 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"https://otexts.com/fpp3/dhr.html", "https://reprex.tidyverse.org", "https://stat.ethz.ch/R-manual/R-devel/library/nlme/html/gls.html"], "medical_sensitive": true, "patient_specific": false, "provenance": {"attribution_required": true, "collected_at": "2026-10-01T03:27:03.428408+00:00", "license": "CC BY-SA 4.0", "license_url": "https://creativecommons.org/licenses/by-sa/4.0/", "raw_file": "raw/codex_api_v1/774052c18cd9e8fa951e893347bfa3c1a85e743fc46ec4667572002cadb10cbf_1790825224163423600_0.json", "raw_sha256": "fe4dd06d3de6c0b1bb33ba88aee0ed4284118e01f76a58dcdce220a4946329b2", "source_api": "Stack Exchange API 2.3", "source_url": "https://api.stackexchange.com/2.3/search/advanced?answers=1&filter=withbody&order=desc&page=1&pagesize=100&site=stats&sort=creation", "transformation": "API HTML retained; mechanical HTML-to-text; no LLM rewriting"}, "question_author": "Thomas", "question_author_url": "https://stats.stackexchange.com/users/263046/thomas", "question_author_user_type": "registered", "question_created_at": "2026-08-27T20:48:18+00:00", "question_html": "<h2>Background</h2>\n<ul>\n<li>Data: time series of the monthly prevalence of patients receiving a given medication by age group (children or adults) from January 2016 to December 2026</li>\n<li>Interventions: three successive regulatory interventions implemented on September 2021, June 2022, and June 2023 to facilitate prescription of the medication (dotted lines on the figure below)</li>\n<li>Aim: to estimate pre-intervention slope, post-intervention 1 slope, post-intervention 2 slope, and post-intervention 3 slope depending on age group to assess the impact of the interventions (i.e., interrupted time series analysis, ITS)</li>\n</ul>\n<pre class=\"lang-r prettyprint-override\"><code>month <- seq(\n from = as.Date("2016-01-01"),\n to = as.Date("2025-12-31"),\n by = "month"\n)\nmonth <- rep(month, each = 2)\nage <- rep(c("Children", "Adults"), 120)\nprevalence <- c(\n 214.1, 11.3, 194.9, 11.1, 220.3, 11.8, 200.7, 11.7, 213.5, \n 11.7, 197.6, 11.5, 128.2, 10.3, 138.6, 9.9, 215, 12.1, 201.9, \n 11.9, 223.8, 12.2, 204.9, 12.4, 234.1, 12.7, 202, 11.9, 247, \n 13.6, 209.8, 13, 236.7, 13.4, 216.9, 13.1, 145.1, 12, 144.3, \n 11, 235.1, 13.7, 227.5, 13.9, 242.5, 13.9, 229.5, 13.8, 255.3, \n 14.5, 226.3, 13.6, 260.8, 15.2, 229.7, 14.2, 245.5, 14.7, 232.6, \n 14.4, 158.8, 13.4, 165, 12.5, 244.3, 14.8, 249.4, 15.7, 259.4, \n 15.5, 244, 15.3, 275.7, 16.2, 240.7, 14.9, 274.4, 16.2, 256.8, \n 16.3, 274.3, 16.4, 240.3, 15.5, 178.2, 15.5, 180.3, 13.6, 264, \n 16.3, 271.3, 17.4, 277.8, 16.9, 261.5, 16.8, 296.9, 17.8, 265.8, \n 17.2, 270.4, 17.9, 218.8, 16, 214.7, 16, 221.8, 16.7, 183.6, \n 16.8, 185.4, 14.9, 273.6, 18.7, 276.7, 19.3, 285, 19.4, 282.8, \n 19.9, 308.3, 20.2, 279.2, 19.3, 331.1, 22, 289.3, 21.2, 303, \n 20.9, 307.5, 21.8, 221.2, 20.9, 215.4, 18.9, 324, 23.4, 323.3, \n 24.3, 328.3, 24.7, 326.5, 25.8, 349.2, 26.1, 319.1, 25.5, 379.3, \n 29, 341.7, 28.3, 365.1, 28.6, 351.7, 30.1, 253.7, 28.6, 266.9, \n 27, 375, 33.4, 373.5, 34.4, 385.8, 35.5, 368.1, 36, 413.7, 38.8, \n 376, 37.4, 440.1, 42.5, 390.3, 40.8, 409.5, 42.6, 402, 44.9, \n 297.9, 43.4, 297.1, 40.2, 430.9, 49.3, 433.4, 51.2, 454.1, 53.1, \n 441.4, 53.4, 472.9, 57, 455.8, 57, 496.3, 62.6, 475, 63.1, 483.4, \n 63.3, 466, 65.1, 367.3, 66.2, 368.4, 58.6, 499.4, 70.8, 528.3, \n 76.1, 533.8, 76, 525.6, 77.2, 588.3, 84.8, 532.4, 81.1, 599.6, \n 90.9, 580.3, 90.8, 601.5, 93.8, 560.6, 95.1, 455.2, 96.9, 440.6, \n 83.7, 629.2, 107.7, 655.7, 112.8, 653.8, 113.2, 674.9, 118.7\n)\nd <- data.frame(month, age, prevalence)\n\npar(mfrow = c(1, 2))\nplot(prevalence ~ month, d, type = "l", main = "Children", subset = age == "Children")\nplot(prevalence ~ month, d, type = "l", main = "Adults", subset = age == "Adults")\ndev.off()\n</code></pre>\n<p><a href=\"https://i.sstatic.net/4a0pdydL.png\" rel=\"nofollow noreferrer\"><img src=\"https://i.sstatic.net/4a0pdydL.png\" alt=\"time series\" /></a></p>\n<h2>What I did</h2>\n<p>I started with children.</p>\n<p>I constructed the ITS variables for the 3 interventions (<code>step1</code>, <code>ramp1</code>, <code>step2</code>, <code>ramp2</code>, <code>step3</code>, <code>ramp3</code>) following <a href=\"https://doi.org/10.1186/s12874-021-01235-8\" rel=\"nofollow noreferrer\">Schaffer et al.</a> and <a href=\"https://doi.org/10.1093/ije/dyaa148\" rel=\"nofollow noreferrer\">Xiao et al.</a></p>\n<pre class=\"lang-r prettyprint-override\"><code>d2 <- subset(d, age == "Children")\nd2<span class=\"math-container\">$prevalence <- ts(d2$</span>prevalence, start = 2016, frequency = 12)\n\ninterventions <- as.Date(c("2021-09-13", "2022-06-02", "2023-06-29"))\n\nstep1 <- as.numeric(d2$month >= interventions[1])\nramp1 <- ifelse(step1 == 1, cumsum(step1) - 1, 0)\n\nstep2 <- as.numeric(d2$month >= interventions[2])\nramp2 <- ifelse(step2 == 1, cumsum(step2) - 1, 0)\n\nstep3 <- as.numeric(d2$month >= interventions[3])\nramp3 <- ifelse(step3 == 1, cumsum(step3) - 1, 0)\n\ntime <- seq(from = 0, along.with = d2$month)\n\nxreg <- cbind(time, step1, ramp1, step2, ramp2, step3, ramp3)\n\ncbind(d2, xreg)\n# month age prevalence time step1 ramp1 step2 ramp2 step3 ramp3\n# 1 2016-01-01 Children 214.1 0 0 0 0 0 0 0\n# 3 2016-02-01 Children 194.9 1 0 0 0 0 0 0\n# 5 2016-03-01 Children 220.3 2 0 0 0 0 0 0\n# 7 2016-04-01 Children 200.7 3 0 0 0 0 0 0\n# 9 2016-05-01 Children 213.5 4 0 0 0 0 0 0\n# 11 2016-06-01 Children 197.6 5 0 0 0 0 0 0\n# 13 2016-07-01 Children 128.2 6 0 0 0 0 0 0\n# 15 2016-08-01 Children 138.6 7 0 0 0 0 0 0\n# 17 2016-09-01 Children 215.0 8 0 0 0 0 0 0\n# 19 2016-10-01 Children 201.9 9 0 0 0 0 0 0\n# 21 2016-11-01 Children 223.8 10 0 0 0 0 0 0\n# 23 2016-12-01 Children 204.9 11 0 0 0 0 0 0\n# 25 2017-01-01 Children 234.1 12 0 0 0 0 0 0\n# 27 2017-02-01 Children 202.0 13 0 0 0 0 0 0\n# 29 2017-03-01 Children 247.0 14 0 0 0 0 0 0\n# 31 2017-04-01 Children 209.8 15 0 0 0 0 0 0\n# 33 2017-05-01 Children 236.7 16 0 0 0 0 0 0\n# 35 2017-06-01 Children 216.9 17 0 0 0 0 0 0\n# 37 2017-07-01 Children 145.1 18 0 0 0 0 0 0\n# 39 2017-08-01 Children 144.3 19 0 0 0 0 0 0\n# 41 2017-09-01 Children 235.1 20 0 0 0 0 0 0\n# 43 2017-10-01 Children 227.5 21 0 0 0 0 0 0\n# 45 2017-11-01 Children 242.5 22 0 0 0 0 0 0\n# 47 2017-12-01 Children 229.5 23 0 0 0 0 0 0\n# 49 2018-01-01 Children 255.3 24 0 0 0 0 0 0\n# 51 2018-02-01 Children 226.3 25 0 0 0 0 0 0\n# 53 2018-03-01 Children 260.8 26 0 0 0 0 0 0\n# 55 2018-04-01 Children 229.7 27 0 0 0 0 0 0\n# 57 2018-05-01 Children 245.5 28 0 0 0 0 0 0\n# 59 2018-06-01 Children 232.6 29 0 0 0 0 0 0\n# 61 2018-07-01 Children 158.8 30 0 0 0 0 0 0\n# 63 2018-08-01 Children 165.0 31 0 0 0 0 0 0\n# 65 2018-09-01 Children 244.3 32 0 0 0 0 0 0\n# 67 2018-10-01 Children 249.4 33 0 0 0 0 0 0\n# 69 2018-11-01 Children 259.4 34 0 0 0 0 0 0\n# 71 2018-12-01 Children 244.0 35 0 0 0 0 0 0\n# 73 2019-01-01 Children 275.7 36 0 0 0 0 0 0\n# 75 2019-02-01 Children 240.7 37 0 0 0 0 0 0\n# 77 2019-03-01 Children 274.4 38 0 0 0 0 0 0\n# 79 2019-04-01 Children 256.8 39 0 0 0 0 0 0\n# 81 2019-05-01 Children 274.3 40 0 0 0 0 0 0\n# 83 2019-06-01 Children 240.3 41 0 0 0 0 0 0\n# 85 2019-07-01 Children 178.2 42 0 0 0 0 0 0\n# 87 2019-08-01 Children 180.3 43 0 0 0 0 0 0\n# 89 2019-09-01 Children 264.0 44 0 0 0 0 0 0\n# 91 2019-10-01 Children 271.3 45 0 0 0 0 0 0\n# 93 2019-11-01 Children 277.8 46 0 0 0 0 0 0\n# 95 2019-12-01 Children 261.5 47 0 0 0 0 0 0\n# 97 2020-01-01 Children 296.9 48 0 0 0 0 0 0\n# 99 2020-02-01 Children 265.8 49 0 0 0 0 0 0\n# 101 2020-03-01 Children 270.4 50 0 0 0 0 0 0\n# 103 2020-04-01 Children 218.8 51 0 0 0 0 0 0\n# 105 2020-05-01 Children 214.7 52 0 0 0 0 0 0\n# 107 2020-06-01 Children 221.8 53 0 0 0 0 0 0\n# 109 2020-07-01 Children 183.6 54 0 0 0 0 0 0\n# 111 2020-08-01 Children 185.4 55 0 0 0 0 0 0\n# 113 2020-09-01 Children 273.6 56 0 0 0 0 0 0\n# 115 2020-10-01 Children 276.7 57 0 0 0 0 0 0\n# 117 2020-11-01 Children 285.0 58 0 0 0 0 0 0\n# 119 2020-12-01 Children 282.8 59 0 0 0 0 0 0\n# 121 2021-01-01 Children 308.3 60 0 0 0 0 0 0\n# 123 2021-02-01 Children 279.2 61 0 0 0 0 0 0\n# 125 2021-03-01 Children 331.1 62 0 0 0 0 0 0\n# 127 2021-04-01 Children 289.3 63 0 0 0 0 0 0\n# 129 2021-05-01 Children 303.0 64 0 0 0 0 0 0\n# 131 2021-06-01 Children 307.5 65 0 0 0 0 0 0\n# 133 2021-07-01 Children 221.2 66 0 0 0 0 0 0\n# 135 2021-08-01 Children 215.4 67 0 0 0 0 0 0\n# 137 2021-09-01 Children 324.0 68 0 0 0 0 0 0\n# 139 2021-10-01 Children 323.3 69 1 0 0 0 0 0\n# 141 2021-11-01 Children 328.3 70 1 1 0 0 0 0\n# 143 2021-12-01 Children 326.5 71 1 2 0 0 0 0\n# 145 2022-01-01 Children 349.2 72 1 3 0 0 0 0\n# 147 2022-02-01 Children 319.1 73 1 4 0 0 0 0\n# 149 2022-03-01 Children 379.3 74 1 5 0 0 0 0\n# 151 2022-04-01 Children 341.7 75 1 6 0 0 0 0\n# 153 2022-05-01 Children 365.1 76 1 7 0 0 0 0\n# 155 2022-06-01 Children 351.7 77 1 8 0 0 0 0\n# 157 2022-07-01 Children 253.7 78 1 9 1 0 0 0\n# 159 2022-08-01 Children 266.9 79 1 10 1 1 0 0\n# 161 2022-09-01 Children 375.0 80 1 11 1 2 0 0\n# 163 2022-10-01 Children 373.5 81 1 12 1 3 0 0\n# 165 2022-11-01 Children 385.8 82 1 13 1 4 0 0\n# 167 2022-12-01 Children 368.1 83 1 14 1 5 0 0\n# 169 2023-01-01 Children 413.7 84 1 15 1 6 0 0\n# 171 2023-02-01 Children 376.0 85 1 16 1 7 0 0\n# 173 2023-03-01 Children 440.1 86 1 17 1 8 0 0\n# 175 2023-04-01 Children 390.3 87 1 18 1 9 0 0\n# 177 2023-05-01 Children 409.5 88 1 19 1 10 0 0\n# 179 2023-06-01 Children 402.0 89 1 20 1 11 0 0\n# 181 2023-07-01 Children 297.9 90 1 21 1 12 1 0\n# 183 2023-08-01 Children 297.1 91 1 22 1 13 1 1\n# 185 2023-09-01 Children 430.9 92 1 23 1 14 1 2\n# 187 2023-10-01 Children 433.4 93 1 24 1 15 1 3\n# 189 2023-11-01 Children 454.1 94 1 25 1 16 1 4\n# 191 2023-12-01 Children 441.4 95 1 26 1 17 1 5\n# 193 2024-01-01 Children 472.9 96 1 27 1 18 1 6\n# 195 2024-02-01 Children 455.8 97 1 28 1 19 1 7\n# 197 2024-03-01 Children 496.3 98 1 29 1 20 1 8\n# 199 2024-04-01 Children 475.0 99 1 30 1 21 1 9\n# 201 2024-05-01 Children 483.4 100 1 31 1 22 1 10\n# 203 2024-06-01 Children 466.0 101 1 32 1 23 1 11\n# 205 2024-07-01 Children 367.3 102 1 33 1 24 1 12\n# 207 2024-08-01 Children 368.4 103 1 34 1 25 1 13\n# 209 2024-09-01 Children 499.4 104 1 35 1 26 1 14\n# 211 2024-10-01 Children 528.3 105 1 36 1 27 1 15\n# 213 2024-11-01 Children 533.8 106 1 37 1 28 1 16\n# 215 2024-12-01 Children 525.6 107 1 38 1 29 1 17\n# 217 2025-01-01 Children 588.3 108 1 39 1 30 1 18\n# 219 2025-02-01 Children 532.4 109 1 40 1 31 1 19\n# 221 2025-03-01 Children 599.6 110 1 41 1 32 1 20\n# 223 2025-04-01 Children 580.3 111 1 42 1 33 1 21\n# 225 2025-05-01 Children 601.5 112 1 43 1 34 1 22\n# 227 2025-06-01 Children 560.6 113 1 44 1 35 1 23\n# 229 2025-07-01 Children 455.2 114 1 45 1 36 1 24\n# 231 2025-08-01 Children 440.6 115 1 46 1 37 1 25\n# 233 2025-09-01 Children 629.2 116 1 47 1 38 1 26\n# 235 2025-10-01 Children 655.7 117 1 48 1 39 1 27\n# 237 2025-11-01 Children 653.8 118 1 49 1 40 1 28\n# 239 2025-12-01 Children 674.9 119 1 50 1 41 1 29\n</code></pre>\n<p>According to <a href=\"https://stats.stackexchange.com/q/665216/\">this thread</a>, I included the ITS variables for the 3 interventions in a single model. I used <code>forecast::auto.arima()</code> to select (p, q) and (P, Q) parameters automatically.</p>\n<p>According to <a href=\"https://doi.org/10.1186/s12874-021-01235-8\" rel=\"nofollow noreferrer\">Schaffer et al.</a>, I included a <code>time</code> variable and set <span class=\"math-container\">$d = 0$</span> and <span class=\"math-container\">$D = 0$</span> (but I am not sure that I need both <span class=\"math-container\">$d = 0$</span> and <span class=\"math-container\">$D = 0$</span>): (emphasis mine)</p>\n<blockquote>\n<p>In ITS analysis, ARIMA forecasts <span class=\"math-container\">$Y_t$</span> in the absence of the intervention (the “counterfactual”) and determines how the observed diverges from this forecast. Unlike segmented regression, including time or seasonal dummy variables in the ARIMA model is not necessary, as ARIMA can eliminate trends and seasonality through differencing. <em>If the trend is eliminated via differencing then the pre- and post-intervention trends cannot be estimated from the model. However, if estimation of the pre- and/or post-intervention slope is desired, this can be accommodated by including time as a covariate and incorporating AR and MA terms to address autocorrelation (e.g. ARMA models)</em> [<a href=\"https://doi.org/10.1186/s12889-017-4998-9\" rel=\"nofollow noreferrer\">21</a>, <a href=\"https://doi.org/10.3111/13696998.2011.626097\" rel=\"nofollow noreferrer\">22</a>].</p>\n</blockquote>\n<pre class=\"lang-r prettyprint-override\"><code>fit <- forecast::auto.arima(\n d2$prevalence,\n d = 0,\n D = 0,\n stepwise = FALSE,\n trace = TRUE,\n approximation = FALSE,\n xreg = xreg\n)\n# \n# Regression with ARIMA(0,0,0) errors : 1422.701\n# Regression with ARIMA(0,0,0) errors : 1256.821\n# Regression with ARIMA(0,0,0)(0,0,1)[12] errors : Inf\n# Regression with ARIMA(0,0,0)(0,0,1)[12] errors : 1186.586\n# Regression with ARIMA(0,0,0)(0,0,2)[12] errors : Inf\n# Regression with ARIMA(0,0,0)(0,0,2)[12] errors : Inf\n# ARIMA(0,0,0)(1,0,0)[12] with zero mean : Inf\n# ARIMA(0,0,0)(1,0,0)[12] with non-zero mean : Inf\n# ARIMA(0,0,0)(1,0,1)[12] with zero mean : Inf\n# ARIMA(0,0,0)(1,0,1)[12] with non-zero mean : Inf\n# ARIMA(0,0,0)(1,0,2)[12] with zero mean : Inf\n# ARIMA(0,0,0)(1,0,2)[12] with non-zero mean : Inf\n# ARIMA(0,0,0)(2,0,0)[12] with zero mean : Inf\n# ARIMA(0,0,0)(2,0,0)[12] with non-zero mean : Inf\n# ARIMA(0,0,0)(2,0,1)[12] with zero mean : Inf\n# ARIMA(0,0,0)(2,0,1)[12] with non-zero mean : Inf\n# ARIMA(0,0,0)(2,0,2)[12] with zero mean : Inf\n# ARIMA(0,0,0)(2,0,2)[12] with non-zero mean : Inf\n# Regression with ARIMA(0,0,1) errors : 1348.712\n# Regression with ARIMA(0,0,1) errors : 1237.018\n# Regression with ARIMA(0,0,1)(0,0,1)[12] errors : Inf\n# Regression with ARIMA(0,0,1)(0,0,1)[12] errors : 1167.011\n# Regression with ARIMA(0,0,1)(0,0,2)[12] errors : Inf\n# Regression with ARIMA(0,0,1)(0,0,2)[12] errors : Inf\n# ARIMA(0,0,1)(1,0,0)[12] with zero mean : Inf\n# ARIMA(0,0,1)(1,0,0)[12] with non-zero mean : Inf\n# ARIMA(0,0,1)(1,0,1)[12] with zero mean : Inf\n# ARIMA(0,0,1)(1,0,1)[12] with non-zero mean : Inf\n# ARIMA(0,0,1)(1,0,2)[12] with zero mean : Inf\n# ARIMA(0,0,1)(1,0,2)[12] with non-zero mean : Inf\n# ARIMA(0,0,1)(2,0,0)[12] with zero mean : Inf\n# ARIMA(0,0,1)(2,0,0)[12] with non-zero mean : Inf\n# ARIMA(0,0,1)(2,0,1)[12] with zero mean : Inf\n# ARIMA(0,0,1)(2,0,1)[12] with non-zero mean : Inf\n# ARIMA(0,0,1)(2,0,2)[12] with zero mean : Inf\n# Regression with ARIMA(0,0,1)(2,0,2)[12] errors : Inf\n# Regression with ARIMA(0,0,2) errors : 1304.633\n# Regression with ARIMA(0,0,2) errors : 1235.236\n# Regression with ARIMA(0,0,2)(0,0,1)[12] errors : 1226.35\n# Regression with ARIMA(0,0,2)(0,0,1)[12] errors : 1164.06\n# Regression with ARIMA(0,0,2)(0,0,2)[12] errors : Inf\n# Regression with ARIMA(0,0,2)(0,0,2)[12] errors : 1122.153\n# ARIMA(0,0,2)(1,0,0)[12] with zero mean : Inf\n# Regression with ARIMA(0,0,2)(1,0,0)[12] errors : Inf\n# ARIMA(0,0,2)(1,0,1)[12] with zero mean : Inf\n# ARIMA(0,0,2)(1,0,1)[12] with non-zero mean : Inf\n# ARIMA(0,0,2)(1,0,2)[12] with zero mean : Inf\n# ARIMA(0,0,2)(1,0,2)[12] with non-zero mean : Inf\n# ARIMA(0,0,2)(2,0,0)[12] with zero mean : Inf\n# ARIMA(0,0,2)(2,0,0)[12] with non-zero mean : Inf\n# ARIMA(0,0,2)(2,0,1)[12] with zero mean : Inf\n# ARIMA(0,0,2)(2,0,1)[12] with non-zero mean : Inf\n# Regression with ARIMA(0,0,3) errors : 1301.979\n# Regression with ARIMA(0,0,3) errors : Inf\n# Regression with ARIMA(0,0,3)(0,0,1)[12] errors : Inf\n# Regression with ARIMA(0,0,3)(0,0,1)[12] errors : 1166.444\n# Regression with ARIMA(0,0,3)(0,0,2)[12] errors : Inf\n# Regression with ARIMA(0,0,3)(0,0,2)[12] errors : Inf\n# ARIMA(0,0,3)(1,0,0)[12] with zero mean : Inf\n# Regression with ARIMA(0,0,3)(1,0,0)[12] errors : Inf\n# ARIMA(0,0,3)(1,0,1)[12] with zero mean : Inf\n# ARIMA(0,0,3)(1,0,1)[12] with non-zero mean : Inf\n# ARIMA(0,0,3)(2,0,0)[12] with zero mean : Inf\n# ARIMA(0,0,3)(2,0,0)[12] with non-zero mean : Inf\n# Regression with ARIMA(0,0,4) errors : Inf\n# Regression with ARIMA(0,0,4) errors : Inf\n# Regression with ARIMA(0,0,4)(0,0,1)[12] errors : Inf\n# Regression with ARIMA(0,0,4)(0,0,1)[12] errors : 1167.795\n# ARIMA(0,0,4)(1,0,0)[12] with zero mean : Inf\n# Regression with ARIMA(0,0,4)(1,0,0)[12] errors : Inf\n# Regression with ARIMA(0,0,5) errors : Inf\n# Regression with ARIMA(0,0,5) errors : Inf\n# Regression with ARIMA(1,0,0) errors : 1269.039\n# Regression with ARIMA(1,0,0) errors : 1236.551\n# Regression with ARIMA(1,0,0)(0,0,1)[12] errors : 1191.463\n# Regression with ARIMA(1,0,0)(0,0,1)[12] errors : 1163.925\n# Regression with ARIMA(1,0,0)(0,0,2)[12] errors : 1144.206\n# Regression with ARIMA(1,0,0)(0,0,2)[12] errors : 1122.139\n# ARIMA(1,0,0)(1,0,0)[12] with zero mean : Inf\n# ARIMA(1,0,0)(1,0,0)[12] with non-zero mean : Inf\n# ARIMA(1,0,0)(1,0,1)[12] with zero mean : Inf\n# ARIMA(1,0,0)(1,0,1)[12] with non-zero mean : Inf\n# ARIMA(1,0,0)(1,0,2)[12] with zero mean : Inf\n# ARIMA(1,0,0)(1,0,2)[12] with non-zero mean : Inf\n# ARIMA(1,0,0)(2,0,0)[12] with zero mean : Inf\n# Regression with ARIMA(1,0,0)(2,0,0)[12] errors : Inf\n# ARIMA(1,0,0)(2,0,1)[12] with zero mean : Inf\n# ARIMA(1,0,0)(2,0,1)[12] with non-zero mean : Inf\n# ARIMA(1,0,0)(2,0,2)[12] with zero mean : Inf\n# ARIMA(1,0,0)(2,0,2)[12] with non-zero mean : Inf\n# Regression with ARIMA(1,0,1) errors : 1269.741\n# Regression with ARIMA(1,0,1) errors : 1237.127\n# Regression with ARIMA(1,0,1)(0,0,1)[12] errors : 1191.287\n# Regression with ARIMA(1,0,1)(0,0,1)[12] errors : 1165.649\n# Regression with ARIMA(1,0,1)(0,0,2)[12] errors : 1145.611\n# Regression with ARIMA(1,0,1)(0,0,2)[12] errors : 1123.56\n# ARIMA(1,0,1)(1,0,0)[12] with zero mean : Inf\n# ARIMA(1,0,1)(1,0,0)[12] with non-zero mean : Inf\n# ARIMA(1,0,1)(1,0,1)[12] with zero mean : Inf\n# ARIMA(1,0,1)(1,0,1)[12] with non-zero mean : Inf\n# ARIMA(1,0,1)(1,0,2)[12] with zero mean : Inf\n# ARIMA(1,0,1)(1,0,2)[12] with non-zero mean : Inf\n# ARIMA(1,0,1)(2,0,0)[12] with zero mean : Inf\n# Regression with ARIMA(1,0,1)(2,0,0)[12] errors : Inf\n# ARIMA(1,0,1)(2,0,1)[12] with zero mean : Inf\n# ARIMA(1,0,1)(2,0,1)[12] with non-zero mean : Inf\n# Regression with ARIMA(1,0,2) errors : Inf\n# ARIMA(1,0,2) with non-zero mean : Inf\n# Regression with ARIMA(1,0,2)(0,0,1)[12] errors : Inf\n# ARIMA(1,0,1)(2,0,1)[12] with non-zero mean : Inf\n# Regression with ARIMA(1,0,2) errors : Inf\n# ARIMA(1,0,2) with non-zero mean : Inf\n# Regression with ARIMA(1,0,2)(0,0,1)[12] errors : Inf\n# Regression with ARIMA(1,0,2)(0,0,1)[12] errors : Inf\n# Regression with ARIMA(1,0,2)(0,0,2)[12] errors : Inf\n# Regression with ARIMA(1,0,2)(0,0,2)[12] errors : Inf\n# ARIMA(1,0,2)(1,0,0)[12] with zero mean : Inf\n# ARIMA(1,0,2)(1,0,0)[12] with non-zero mean : Inf\n# ARIMA(1,0,2)(1,0,1)[12] with zero mean : Inf\n# ARIMA(1,0,2)(1,0,1)[12] with non-zero mean : Inf\n# ARIMA(1,0,2)(2,0,0)[12] with zero mean : Inf\n# Regression with ARIMA(1,0,2)(2,0,0)[12] errors : Inf\n# Regression with ARIMA(1,0,3) errors : Inf\n# Regression with ARIMA(1,0,3) errors : Inf\n# Regression with ARIMA(1,0,3)(0,0,1)[12] errors : 1196.785\n# Regression with ARIMA(1,0,3)(0,0,1)[12] errors : Inf\n# ARIMA(1,0,3)(1,0,0)[12] with zero mean : Inf\n# ARIMA(1,0,3)(1,0,0)[12] with non-zero mean : Inf\n# Regression with ARIMA(1,0,4) errors : Inf\n# Regression with ARIMA(1,0,4) errors : Inf\n# Regression with ARIMA(2,0,0) errors : 1270.227\n# Regression with ARIMA(2,0,0) errors : 1235.453\n# Regression with ARIMA(2,0,0)(0,0,1)[12] errors : 1191.855\n# Regression with ARIMA(2,0,0)(0,0,1)[12] errors : 1165.096\n# Regression with ARIMA(2,0,0)(0,0,2)[12] errors : 1145.887\n# Regression with ARIMA(2,0,0)(0,0,2)[12] errors : 1123.031\n# ARIMA(2,0,0)(1,0,0)[12] with zero mean : Inf\n# ARIMA(2,0,0)(1,0,0)[12] with non-zero mean : Inf\n# ARIMA(2,0,0)(1,0,1)[12] with zero mean : Inf\n# ARIMA(2,0,0)(1,0,1)[12] with non-zero mean : Inf\n# ARIMA(2,0,0)(1,0,2)[12] with zero mean : Inf\n# ARIMA(2,0,0)(1,0,2)[12] with non-zero mean : Inf\n# ARIMA(2,0,0)(2,0,0)[12] with zero mean : Inf\n# ARIMA(2,0,0)(2,0,0)[12] with non-zero mean : Inf\n# ARIMA(2,0,0)(2,0,1)[12] with zero mean : Inf\n# Regression with ARIMA(2,0,0)(2,0,1)[12] errors : Inf\n# Regression with ARIMA(2,0,1) errors : Inf\n# Regression with ARIMA(2,0,1) errors : Inf\n# Regression with ARIMA(2,0,1)(0,0,1)[12] errors : Inf\n# Regression with ARIMA(2,0,1)(0,0,1)[12] errors : Inf\n# Regression with ARIMA(2,0,1)(0,0,2)[12] errors : Inf\n# Regression with ARIMA(2,0,1)(0,0,2)[12] errors : Inf\n# ARIMA(2,0,1)(1,0,0)[12] with zero mean : Inf\n# ARIMA(2,0,1)(1,0,0)[12] with non-zero mean : Inf\n# Regression with ARIMA(2,0,1)(1,0,1)[12] errors : Inf\n# ARIMA(2,0,1)(1,0,1)[12] with non-zero mean : Inf\n# Regression with ARIMA(2,0,1)(2,0,0)[12] errors : Inf\n# Regression with ARIMA(2,0,1)(2,0,0)[12] errors : Inf\n# Regression with ARIMA(2,0,2) errors : Inf\n# Regression with ARIMA(2,0,2) errors : Inf\n# ARIMA(2,0,2)(0,0,1)[12] with zero mean : Inf\n# Regression with ARIMA(2,0,2)(0,0,1)[12] errors : Inf\n# ARIMA(2,0,2)(1,0,0)[12] with zero mean : Inf\n# ARIMA(2,0,2)(1,0,0)[12] with non-zero mean : Inf\n# ARIMA(2,0,3) with zero mean : Inf\n# Regression with ARIMA(2,0,3) errors : Inf\n# Regression with ARIMA(3,0,0) errors : 1271.552\n# Regression with ARIMA(3,0,0) errors : 1234.442\n# Regression with ARIMA(3,0,0)(0,0,1)[12] errors : 1193.944\n# Regression with ARIMA(3,0,0)(0,0,1)[12] errors : 1165.004\n# Regression with ARIMA(3,0,0)(0,0,2)[12] errors : 1147.337\n# Regression with ARIMA(3,0,0)(0,0,2)[12] errors : 1124.69\n# ARIMA(3,0,0)(1,0,0)[12] with zero mean : Inf\n# ARIMA(3,0,0)(1,0,0)[12] with non-zero mean : Inf\n# ARIMA(3,0,0)(1,0,1)[12] with zero mean : Inf\n# ARIMA(3,0,0)(1,0,1)[12] with non-zero mean : Inf\n# ARIMA(3,0,0)(2,0,0)[12] with zero mean : Inf\n# ARIMA(3,0,0)(2,0,0)[12] with non-zero mean : Inf\n# Regression with ARIMA(3,0,1) errors : Inf\n# Regression with ARIMA(3,0,1) errors : Inf\n# Regression with ARIMA(3,0,1)(0,0,1)[12] errors : Inf\n# Regression with ARIMA(3,0,1)(0,0,1)[12] errors : Inf\n# ARIMA(3,0,1)(1,0,0)[12] with zero mean : Inf\n# ARIMA(3,0,1)(1,0,0)[12] with non-zero mean : Inf\n# Regression with ARIMA(3,0,2) errors : Inf\n# Regression with ARIMA(3,0,2) errors : Inf\n# Regression with ARIMA(4,0,0) errors : 1268.176\n# Regression with ARIMA(4,0,0) errors : 1236.501\n# Regression with ARIMA(4,0,0)(0,0,1)[12] errors : 1191.278\n# Regression with ARIMA(4,0,0)(0,0,1)[12] errors : 1167.39\n# ARIMA(4,0,0)(1,0,0)[12] with zero mean : Inf\n# ARIMA(4,0,0)(1,0,0)[12] with non-zero mean : Inf\n# Regression with ARIMA(4,0,1) errors : Inf\n# Regression with ARIMA(4,0,1) errors : Inf\n# Regression with ARIMA(5,0,0) errors : 1270.388\n# Regression with ARIMA(5,0,0) errors : 1226.449\n# \n# \n# \n# Best model: Regression with ARIMA(1,0,0)(0,0,2)[12] errors \n# \n\nsummary(fit)\n# Series: d2$prevalence \n# Regression with ARIMA(1,0,0)(0,0,2)[12] errors \n# \n# Coefficients:\n# ar1 sma1 sma2 intercept time step1 ramp1 step2 ramp2 step3 ramp3\n# 0.5256 1.0413 0.6987 188.8179 1.5717 3.3284 2.1966 -5.6181 0.4640 0.4378 4.6447\n# s.e. 0.0800 0.1536 0.1312 20.6895 0.4724 13.0492 3.5219 23.9955 3.8197 19.1790 2.5724\n# \n# sigma^2 = 494.9: log likelihood = -547.61\n# AIC=1119.22 AICc=1122.14 BIC=1152.67\n# \n# Training set error measures:\n# ME RMSE MAE MPE MAPE MASE ACF1\n# Training set -0.2760281 21.20232 16.21419 -0.9807653 5.578825 0.3604042 0.05861382\n</code></pre>\n<h2>Question</h2>\n<p>Should I just apply the same strategy to adults? Or can such models handle correlated/stratified time series?</p>\n<p>I guess that there is a better strategy than fitting separate models within each subgroup. However, I suspect that I cannot simply stack the time series and include <code>age</code> in <code>xreg</code>. Additionally, I probably need an interaction terms (e.g. <code>time * age</code>).</p>\n<p>I have also read about <a href=\"https://stat.ethz.ch/R-manual/R-devel/library/nlme/html/gls.html\" rel=\"nofollow noreferrer\"><code>nlme::gls()</code></a> but I would like guidance/confirmation whether it is more appropriate.</p>\n<hr />\n<p>BTW, I am not satisfied with the fit, but I will explore that once I have decided the modelling strategy.</p>\n<pre class=\"lang-r prettyprint-override\"><code>res <- residuals(fit)\nl <- 36\n\npar(mfrow = c(3, 1))\nplot(res)\nacf(res, lag.max = l)\npacf(res, lag.max = l)\ndev.off()\n\nBox.test(res, lag = 24, type = "Ljung-Box", fitdf = 3)\n# \n# Box-Ljung test\n# \n# data: res\n# X-squared = 49.814, df = 21, p-value = 0.0003872\n# \n</code></pre>\n<p><a href=\"https://i.sstatic.net/pzs11NTf.png\" rel=\"nofollow noreferrer\"><img src=\"https://i.sstatic.net/pzs11NTf.png\" alt=\"residuals\" /></a></p>\n", "question_id": 676979, "question_license": "CC BY-SA 4.0", "question_score": 1, "question_text": "Background\n\n\n\n\n\nData: time series of the monthly prevalence of patients receiving a given medication by age group (children or adults) from January 2016 to December 2026\n\n\n\n\nInterventions: three successive regulatory interventions implemented on September 2021, June 2022, and June 2023 to facilitate prescription of the medication (dotted lines on the figure below)\n\n\n\n\nAim: to estimate pre-intervention slope, post-intervention 1 slope, post-intervention 2 slope, and post-intervention 3 slope depending on age group to assess the impact of the interventions (i.e., interrupted time series analysis, ITS)\n\n\n\n\n\nmonth <- seq(\n from = as.Date(\"2016-01-01\"),\n to = as.Date(\"2025-12-31\"),\n by = \"month\"\n)\nmonth <- rep(month, each = 2)\nage <- rep(c(\"Children\", \"Adults\"), 120)\nprevalence <- c(\n 214.1, 11.3, 194.9, 11.1, 220.3, 11.8, 200.7, 11.7, 213.5, \n 11.7, 197.6, 11.5, 128.2, 10.3, 138.6, 9.9, 215, 12.1, 201.9, \n 11.9, 223.8, 12.2, 204.9, 12.4, 234.1, 12.7, 202, 11.9, 247, \n 13.6, 209.8, 13, 236.7, 13.4, 216.9, 13.1, 145.1, 12, 144.3, \n 11, 235.1, 13.7, 227.5, 13.9, 242.5, 13.9, 229.5, 13.8, 255.3, \n 14.5, 226.3, 13.6, 260.8, 15.2, 229.7, 14.2, 245.5, 14.7, 232.6, \n 14.4, 158.8, 13.4, 165, 12.5, 244.3, 14.8, 249.4, 15.7, 259.4, \n 15.5, 244, 15.3, 275.7, 16.2, 240.7, 14.9, 274.4, 16.2, 256.8, \n 16.3, 274.3, 16.4, 240.3, 15.5, 178.2, 15.5, 180.3, 13.6, 264, \n 16.3, 271.3, 17.4, 277.8, 16.9, 261.5, 16.8, 296.9, 17.8, 265.8, \n 17.2, 270.4, 17.9, 218.8, 16, 214.7, 16, 221.8, 16.7, 183.6, \n 16.8, 185.4, 14.9, 273.6, 18.7, 276.7, 19.3, 285, 19.4, 282.8, \n 19.9, 308.3, 20.2, 279.2, 19.3, 331.1, 22, 289.3, 21.2, 303, \n 20.9, 307.5, 21.8, 221.2, 20.9, 215.4, 18.9, 324, 23.4, 323.3, \n 24.3, 328.3, 24.7, 326.5, 25.8, 349.2, 26.1, 319.1, 25.5, 379.3, \n 29, 341.7, 28.3, 365.1, 28.6, 351.7, 30.1, 253.7, 28.6, 266.9, \n 27, 375, 33.4, 373.5, 34.4, 385.8, 35.5, 368.1, 36, 413.7, 38.8, \n 376, 37.4, 440.1, 42.5, 390.3, 40.8, 409.5, 42.6, 402, 44.9, \n 297.9, 43.4, 297.1, 40.2, 430.9, 49.3, 433.4, 51.2, 454.1, 53.1, \n 441.4, 53.4, 472.9, 57, 455.8, 57, 496.3, 62.6, 475, 63.1, 483.4, \n 63.3, 466, 65.1, 367.3, 66.2, 368.4, 58.6, 499.4, 70.8, 528.3, \n 76.1, 533.8, 76, 525.6, 77.2, 588.3, 84.8, 532.4, 81.1, 599.6, \n 90.9, 580.3, 90.8, 601.5, 93.8, 560.6, 95.1, 455.2, 96.9, 440.6, \n 83.7, 629.2, 107.7, 655.7, 112.8, 653.8, 113.2, 674.9, 118.7\n)\nd <- data.frame(month, age, prevalence)\n\npar(mfrow = c(1, 2))\nplot(prevalence ~ month, d, type = \"l\", main = \"Children\", subset = age == \"Children\")\nplot(prevalence ~ month, d, type = \"l\", main = \"Adults\", subset = age == \"Adults\")\ndev.off()\n\n\n\n\n\n[image: time series; source: https://i.sstatic.net/4a0pdydL.png] (https://i.sstatic.net/4a0pdydL.png)\n\n\n\n\nWhat I did\n\n\n\n\nI started with children.\n\n\n\n\nI constructed the ITS variables for the 3 interventions (step1, ramp1, step2, ramp2, step3, ramp3) following Schaffer et al. (https://doi.org/10.1186/s12874-021-01235-8) and Xiao et al. (https://doi.org/10.1093/ije/dyaa148)\n\n\n\n\nd2 <- subset(d, age == \"Children\")\nd2$prevalence <- ts(d2$prevalence, start = 2016, frequency = 12)\n\ninterventions <- as.Date(c(\"2021-09-13\", \"2022-06-02\", \"2023-06-29\"))\n\nstep1 <- as.numeric(d2$month >= interventions[1])\nramp1 <- ifelse(step1 == 1, cumsum(step1) - 1, 0)\n\nstep2 <- as.numeric(d2$month >= interventions[2])\nramp2 <- ifelse(step2 == 1, cumsum(step2) - 1, 0)\n\nstep3 <- as.numeric(d2$month >= interventions[3])\nramp3 <- ifelse(step3 == 1, cumsum(step3) - 1, 0)\n\ntime <- seq(from = 0, along.with = d2$month)\n\nxreg <- cbind(time, step1, ramp1, step2, ramp2, step3, ramp3)\n\ncbind(d2, xreg)\n# month age prevalence time step1 ramp1 step2 ramp2 step3 ramp3\n# 1 2016-01-01 Children 214.1 0 0 0 0 0 0 0\n# 3 2016-02-01 Children 194.9 1 0 0 0 0 0 0\n# 5 2016-03-01 Children 220.3 2 0 0 0 0 0 0\n# 7 2016-04-01 Children 200.7 3 0 0 0 0 0 0\n# 9 2016-05-01 Children 213.5 4 0 0 0 0 0 0\n# 11 2016-06-01 Children 197.6 5 0 0 0 0 0 0\n# 13 2016-07-01 Children 128.2 6 0 0 0 0 0 0\n# 15 2016-08-01 Children 138.6 7 0 0 0 0 0 0\n# 17 2016-09-01 Children 215.0 8 0 0 0 0 0 0\n# 19 2016-10-01 Children 201.9 9 0 0 0 0 0 0\n# 21 2016-11-01 Children 223.8 10 0 0 0 0 0 0\n# 23 2016-12-01 Children 204.9 11 0 0 0 0 0 0\n# 25 2017-01-01 Children 234.1 12 0 0 0 0 0 0\n# 27 2017-02-01 Children 202.0 13 0 0 0 0 0 0\n# 29 2017-03-01 Children 247.0 14 0 0 0 0 0 0\n# 31 2017-04-01 Children 209.8 15 0 0 0 0 0 0\n# 33 2017-05-01 Children 236.7 16 0 0 0 0 0 0\n# 35 2017-06-01 Children 216.9 17 0 0 0 0 0 0\n# 37 2017-07-01 Children 145.1 18 0 0 0 0 0 0\n# 39 2017-08-01 Children 144.3 19 0 0 0 0 0 0\n# 41 2017-09-01 Children 235.1 20 0 0 0 0 0 0\n# 43 2017-10-01 Children 227.5 21 0 0 0 0 0 0\n# 45 2017-11-01 Children 242.5 22 0 0 0 0 0 0\n# 47 2017-12-01 Children 229.5 23 0 0 0 0 0 0\n# 49 2018-01-01 Children 255.3 24 0 0 0 0 0 0\n# 51 2018-02-01 Children 226.3 25 0 0 0 0 0 0\n# 53 2018-03-01 Children 260.8 26 0 0 0 0 0 0\n# 55 2018-04-01 Children 229.7 27 0 0 0 0 0 0\n# 57 2018-05-01 Children 245.5 28 0 0 0 0 0 0\n# 59 2018-06-01 Children 232.6 29 0 0 0 0 0 0\n# 61 2018-07-01 Children 158.8 30 0 0 0 0 0 0\n# 63 2018-08-01 Children 165.0 31 0 0 0 0 0 0\n# 65 2018-09-01 Children 244.3 32 0 0 0 0 0 0\n# 67 2018-10-01 Children 249.4 33 0 0 0 0 0 0\n# 69 2018-11-01 Children 259.4 34 0 0 0 0 0 0\n# 71 2018-12-01 Children 244.0 35 0 0 0 0 0 0\n# 73 2019-01-01 Children 275.7 36 0 0 0 0 0 0\n# 75 2019-02-01 Children 240.7 37 0 0 0 0 0 0\n# 77 2019-03-01 Children 274.4 38 0 0 0 0 0 0\n# 79 2019-04-01 Children 256.8 39 0 0 0 0 0 0\n# 81 2019-05-01 Children 274.3 40 0 0 0 0 0 0\n# 83 2019-06-01 Children 240.3 41 0 0 0 0 0 0\n# 85 2019-07-01 Children 178.2 42 0 0 0 0 0 0\n# 87 2019-08-01 Children 180.3 43 0 0 0 0 0 0\n# 89 2019-09-01 Children 264.0 44 0 0 0 0 0 0\n# 91 2019-10-01 Children 271.3 45 0 0 0 0 0 0\n# 93 2019-11-01 Children 277.8 46 0 0 0 0 0 0\n# 95 2019-12-01 Children 261.5 47 0 0 0 0 0 0\n# 97 2020-01-01 Children 296.9 48 0 0 0 0 0 0\n# 99 2020-02-01 Children 265.8 49 0 0 0 0 0 0\n# 101 2020-03-01 Children 270.4 50 0 0 0 0 0 0\n# 103 2020-04-01 Children 218.8 51 0 0 0 0 0 0\n# 105 2020-05-01 Children 214.7 52 0 0 0 0 0 0\n# 107 2020-06-01 Children 221.8 53 0 0 0 0 0 0\n# 109 2020-07-01 Children 183.6 54 0 0 0 0 0 0\n# 111 2020-08-01 Children 185.4 55 0 0 0 0 0 0\n# 113 2020-09-01 Children 273.6 56 0 0 0 0 0 0\n# 115 2020-10-01 Children 276.7 57 0 0 0 0 0 0\n# 117 2020-11-01 Children 285.0 58 0 0 0 0 0 0\n# 119 2020-12-01 Children 282.8 59 0 0 0 0 0 0\n# 121 2021-01-01 Children 308.3 60 0 0 0 0 0 0\n# 123 2021-02-01 Children 279.2 61 0 0 0 0 0 0\n# 125 2021-03-01 Children 331.1 62 0 0 0 0 0 0\n# 127 2021-04-01 Children 289.3 63 0 0 0 0 0 0\n# 129 2021-05-01 Children 303.0 64 0 0 0 0 0 0\n# 131 2021-06-01 Children 307.5 65 0 0 0 0 0 0\n# 133 2021-07-01 Children 221.2 66 0 0 0 0 0 0\n# 135 2021-08-01 Children 215.4 67 0 0 0 0 0 0\n# 137 2021-09-01 Children 324.0 68 0 0 0 0 0 0\n# 139 2021-10-01 Children 323.3 69 1 0 0 0 0 0\n# 141 2021-11-01 Children 328.3 70 1 1 0 0 0 0\n# 143 2021-12-01 Children 326.5 71 1 2 0 0 0 0\n# 145 2022-01-01 Children 349.2 72 1 3 0 0 0 0\n# 147 2022-02-01 Children 319.1 73 1 4 0 0 0 0\n# 149 2022-03-01 Children 379.3 74 1 5 0 0 0 0\n# 151 2022-04-01 Children 341.7 75 1 6 0 0 0 0\n# 153 2022-05-01 Children 365.1 76 1 7 0 0 0 0\n# 155 2022-06-01 Children 351.7 77 1 8 0 0 0 0\n# 157 2022-07-01 Children 253.7 78 1 9 1 0 0 0\n# 159 2022-08-01 Children 266.9 79 1 10 1 1 0 0\n# 161 2022-09-01 Children 375.0 80 1 11 1 2 0 0\n# 163 2022-10-01 Children 373.5 81 1 12 1 3 0 0\n# 165 2022-11-01 Children 385.8 82 1 13 1 4 0 0\n# 167 2022-12-01 Children 368.1 83 1 14 1 5 0 0\n# 169 2023-01-01 Children 413.7 84 1 15 1 6 0 0\n# 171 2023-02-01 Children 376.0 85 1 16 1 7 0 0\n# 173 2023-03-01 Children 440.1 86 1 17 1 8 0 0\n# 175 2023-04-01 Children 390.3 87 1 18 1 9 0 0\n# 177 2023-05-01 Children 409.5 88 1 19 1 10 0 0\n# 179 2023-06-01 Children 402.0 89 1 20 1 11 0 0\n# 181 2023-07-01 Children 297.9 90 1 21 1 12 1 0\n# 183 2023-08-01 Children 297.1 91 1 22 1 13 1 1\n# 185 2023-09-01 Children 430.9 92 1 23 1 14 1 2\n# 187 2023-10-01 Children 433.4 93 1 24 1 15 1 3\n# 189 2023-11-01 Children 454.1 94 1 25 1 16 1 4\n# 191 2023-12-01 Children 441.4 95 1 26 1 17 1 5\n# 193 2024-01-01 Children 472.9 96 1 27 1 18 1 6\n# 195 2024-02-01 Children 455.8 97 1 28 1 19 1 7\n# 197 2024-03-01 Children 496.3 98 1 29 1 20 1 8\n# 199 2024-04-01 Children 475.0 99 1 30 1 21 1 9\n# 201 2024-05-01 Children 483.4 100 1 31 1 22 1 10\n# 203 2024-06-01 Children 466.0 101 1 32 1 23 1 11\n# 205 2024-07-01 Children 367.3 102 1 33 1 24 1 12\n# 207 2024-08-01 Children 368.4 103 1 34 1 25 1 13\n# 209 2024-09-01 Children 499.4 104 1 35 1 26 1 14\n# 211 2024-10-01 Children 528.3 105 1 36 1 27 1 15\n# 213 2024-11-01 Children 533.8 106 1 37 1 28 1 16\n# 215 2024-12-01 Children 525.6 107 1 38 1 29 1 17\n# 217 2025-01-01 Children 588.3 108 1 39 1 30 1 18\n# 219 2025-02-01 Children 532.4 109 1 40 1 31 1 19\n# 221 2025-03-01 Children 599.6 110 1 41 1 32 1 20\n# 223 2025-04-01 Children 580.3 111 1 42 1 33 1 21\n# 225 2025-05-01 Children 601.5 112 1 43 1 34 1 22\n# 227 2025-06-01 Children 560.6 113 1 44 1 35 1 23\n# 229 2025-07-01 Children 455.2 114 1 45 1 36 1 24\n# 231 2025-08-01 Children 440.6 115 1 46 1 37 1 25\n# 233 2025-09-01 Children 629.2 116 1 47 1 38 1 26\n# 235 2025-10-01 Children 655.7 117 1 48 1 39 1 27\n# 237 2025-11-01 Children 653.8 118 1 49 1 40 1 28\n# 239 2025-12-01 Children 674.9 119 1 50 1 41 1 29\n\n\n\n\n\nAccording to this thread (https://stats.stackexchange.com/q/665216/), I included the ITS variables for the 3 interventions in a single model. I used forecast::auto.arima() to select (p, q) and (P, Q) parameters automatically.\n\n\n\n\nAccording to Schaffer et al. (https://doi.org/10.1186/s12874-021-01235-8), I included a time variable and set $d = 0$ and $D = 0$ (but I am not sure that I need both $d = 0$ and $D = 0$): (emphasis mine)\n\n\n\n\n\n\n\nIn ITS analysis, ARIMA forecasts $Y_t$ in the absence of the intervention (the “counterfactual”) and determines how the observed diverges from this forecast. Unlike segmented regression, including time or seasonal dummy variables in the ARIMA model is not necessary, as ARIMA can eliminate trends and seasonality through differencing. If the trend is eliminated via differencing then the pre- and post-intervention trends cannot be estimated from the model. However, if estimation of the pre- and/or post-intervention slope is desired, this can be accommodated by including time as a covariate and incorporating AR and MA terms to address autocorrelation (e.g. ARMA models) [21 (https://doi.org/10.1186/s12889-017-4998-9), 22 (https://doi.org/10.3111/13696998.2011.626097)].\n\n\n\n\n\n\n\nfit <- forecast::auto.arima(\n d2$prevalence,\n d = 0,\n D = 0,\n stepwise = FALSE,\n trace = TRUE,\n approximation = FALSE,\n xreg = xreg\n)\n# \n# Regression with ARIMA(0,0,0) errors : 1422.701\n# Regression with ARIMA(0,0,0) errors : 1256.821\n# Regression with ARIMA(0,0,0)(0,0,1)[12] errors : Inf\n# Regression with ARIMA(0,0,0)(0,0,1)[12] errors : 1186.586\n# Regression with ARIMA(0,0,0)(0,0,2)[12] errors : Inf\n# Regression with ARIMA(0,0,0)(0,0,2)[12] errors : Inf\n# ARIMA(0,0,0)(1,0,0)[12] with zero mean : Inf\n# ARIMA(0,0,0)(1,0,0)[12] with non-zero mean : Inf\n# ARIMA(0,0,0)(1,0,1)[12] with zero mean : Inf\n# ARIMA(0,0,0)(1,0,1)[12] with non-zero mean : Inf\n# ARIMA(0,0,0)(1,0,2)[12] with zero mean : Inf\n# ARIMA(0,0,0)(1,0,2)[12] with non-zero mean : Inf\n# ARIMA(0,0,0)(2,0,0)[12] with zero mean : Inf\n# ARIMA(0,0,0)(2,0,0)[12] with non-zero mean : Inf\n# ARIMA(0,0,0)(2,0,1)[12] with zero mean : Inf\n# ARIMA(0,0,0)(2,0,1)[12] with non-zero mean : Inf\n# ARIMA(0,0,0)(2,0,2)[12] with zero mean : Inf\n# ARIMA(0,0,0)(2,0,2)[12] with non-zero mean : Inf\n# Regression with ARIMA(0,0,1) errors : 1348.712\n# Regression with ARIMA(0,0,1) errors : 1237.018\n# Regression with ARIMA(0,0,1)(0,0,1)[12] errors : Inf\n# Regression with ARIMA(0,0,1)(0,0,1)[12] errors : 1167.011\n# Regression with ARIMA(0,0,1)(0,0,2)[12] errors : Inf\n# Regression with ARIMA(0,0,1)(0,0,2)[12] errors : Inf\n# ARIMA(0,0,1)(1,0,0)[12] with zero mean : Inf\n# ARIMA(0,0,1)(1,0,0)[12] with non-zero mean : Inf\n# ARIMA(0,0,1)(1,0,1)[12] with zero mean : Inf\n# ARIMA(0,0,1)(1,0,1)[12] with non-zero mean : Inf\n# ARIMA(0,0,1)(1,0,2)[12] with zero mean : Inf\n# ARIMA(0,0,1)(1,0,2)[12] with non-zero mean : Inf\n# ARIMA(0,0,1)(2,0,0)[12] with zero mean : Inf\n# ARIMA(0,0,1)(2,0,0)[12] with non-zero mean : Inf\n# ARIMA(0,0,1)(2,0,1)[12] with zero mean : Inf\n# ARIMA(0,0,1)(2,0,1)[12] with non-zero mean : Inf\n# ARIMA(0,0,1)(2,0,2)[12] with zero mean : Inf\n# Regression with ARIMA(0,0,1)(2,0,2)[12] errors : Inf\n# Regression with ARIMA(0,0,2) errors : 1304.633\n# Regression with ARIMA(0,0,2) errors : 1235.236\n# Regression with ARIMA(0,0,2)(0,0,1)[12] errors : 1226.35\n# Regression with ARIMA(0,0,2)(0,0,1)[12] errors : 1164.06\n# Regression with ARIMA(0,0,2)(0,0,2)[12] errors : Inf\n# Regression with ARIMA(0,0,2)(0,0,2)[12] errors : 1122.153\n# ARIMA(0,0,2)(1,0,0)[12] with zero mean : Inf\n# Regression with ARIMA(0,0,2)(1,0,0)[12] errors : Inf\n# ARIMA(0,0,2)(1,0,1)[12] with zero mean : Inf\n# ARIMA(0,0,2)(1,0,1)[12] with non-zero mean : Inf\n# ARIMA(0,0,2)(1,0,2)[12] with zero mean : Inf\n# ARIMA(0,0,2)(1,0,2)[12] with non-zero mean : Inf\n# ARIMA(0,0,2)(2,0,0)[12] with zero mean : Inf\n# ARIMA(0,0,2)(2,0,0)[12] with non-zero mean : Inf\n# ARIMA(0,0,2)(2,0,1)[12] with zero mean : Inf\n# ARIMA(0,0,2)(2,0,1)[12] with non-zero mean : Inf\n# Regression with ARIMA(0,0,3) errors : 1301.979\n# Regression with ARIMA(0,0,3) errors : Inf\n# Regression with ARIMA(0,0,3)(0,0,1)[12] errors : Inf\n# Regression with ARIMA(0,0,3)(0,0,1)[12] errors : 1166.444\n# Regression with ARIMA(0,0,3)(0,0,2)[12] errors : Inf\n# Regression with ARIMA(0,0,3)(0,0,2)[12] errors : Inf\n# ARIMA(0,0,3)(1,0,0)[12] with zero mean : Inf\n# Regression with ARIMA(0,0,3)(1,0,0)[12] errors : Inf\n# ARIMA(0,0,3)(1,0,1)[12] with zero mean : Inf\n# ARIMA(0,0,3)(1,0,1)[12] with non-zero mean : Inf\n# ARIMA(0,0,3)(2,0,0)[12] with zero mean : Inf\n# ARIMA(0,0,3)(2,0,0)[12] with non-zero mean : Inf\n# Regression with ARIMA(0,0,4) errors : Inf\n# Regression with ARIMA(0,0,4) errors : Inf\n# Regression with ARIMA(0,0,4)(0,0,1)[12] errors : Inf\n# Regression with ARIMA(0,0,4)(0,0,1)[12] errors : 1167.795\n# ARIMA(0,0,4)(1,0,0)[12] with zero mean : Inf\n# Regression with ARIMA(0,0,4)(1,0,0)[12] errors : Inf\n# Regression with ARIMA(0,0,5) errors : Inf\n# Regression with ARIMA(0,0,5) errors : Inf\n# Regression with ARIMA(1,0,0) errors : 1269.039\n# Regression with ARIMA(1,0,0) errors : 1236.551\n# Regression with ARIMA(1,0,0)(0,0,1)[12] errors : 1191.463\n# Regression with ARIMA(1,0,0)(0,0,1)[12] errors : 1163.925\n# Regression with ARIMA(1,0,0)(0,0,2)[12] errors : 1144.206\n# Regression with ARIMA(1,0,0)(0,0,2)[12] errors : 1122.139\n# ARIMA(1,0,0)(1,0,0)[12] with zero mean : Inf\n# ARIMA(1,0,0)(1,0,0)[12] with non-zero mean : Inf\n# ARIMA(1,0,0)(1,0,1)[12] with zero mean : Inf\n# ARIMA(1,0,0)(1,0,1)[12] with non-zero mean : Inf\n# ARIMA(1,0,0)(1,0,2)[12] with zero mean : Inf\n# ARIMA(1,0,0)(1,0,2)[12] with non-zero mean : Inf\n# ARIMA(1,0,0)(2,0,0)[12] with zero mean : Inf\n# Regression with ARIMA(1,0,0)(2,0,0)[12] errors : Inf\n# ARIMA(1,0,0)(2,0,1)[12] with zero mean : Inf\n# ARIMA(1,0,0)(2,0,1)[12] with non-zero mean : Inf\n# ARIMA(1,0,0)(2,0,2)[12] with zero mean : Inf\n# ARIMA(1,0,0)(2,0,2)[12] with non-zero mean : Inf\n# Regression with ARIMA(1,0,1) errors : 1269.741\n# Regression with ARIMA(1,0,1) errors : 1237.127\n# Regression with ARIMA(1,0,1)(0,0,1)[12] errors : 1191.287\n# Regression with ARIMA(1,0,1)(0,0,1)[12] errors : 1165.649\n# Regression with ARIMA(1,0,1)(0,0,2)[12] errors : 1145.611\n# Regression with ARIMA(1,0,1)(0,0,2)[12] errors : 1123.56\n# ARIMA(1,0,1)(1,0,0)[12] with zero mean : Inf\n# ARIMA(1,0,1)(1,0,0)[12] with non-zero mean : Inf\n# ARIMA(1,0,1)(1,0,1)[12] with zero mean : Inf\n# ARIMA(1,0,1)(1,0,1)[12] with non-zero mean : Inf\n# ARIMA(1,0,1)(1,0,2)[12] with zero mean : Inf\n# ARIMA(1,0,1)(1,0,2)[12] with non-zero mean : Inf\n# ARIMA(1,0,1)(2,0,0)[12] with zero mean : Inf\n# Regression with ARIMA(1,0,1)(2,0,0)[12] errors : Inf\n# ARIMA(1,0,1)(2,0,1)[12] with zero mean : Inf\n# ARIMA(1,0,1)(2,0,1)[12] with non-zero mean : Inf\n# Regression with ARIMA(1,0,2) errors : Inf\n# ARIMA(1,0,2) with non-zero mean : Inf\n# Regression with ARIMA(1,0,2)(0,0,1)[12] errors : Inf\n# ARIMA(1,0,1)(2,0,1)[12] with non-zero mean : Inf\n# Regression with ARIMA(1,0,2) errors : Inf\n# ARIMA(1,0,2) with non-zero mean : Inf\n# Regression with ARIMA(1,0,2)(0,0,1)[12] errors : Inf\n# Regression with ARIMA(1,0,2)(0,0,1)[12] errors : Inf\n# Regression with ARIMA(1,0,2)(0,0,2)[12] errors : Inf\n# Regression with ARIMA(1,0,2)(0,0,2)[12] errors : Inf\n# ARIMA(1,0,2)(1,0,0)[12] with zero mean : Inf\n# ARIMA(1,0,2)(1,0,0)[12] with non-zero mean : Inf\n# ARIMA(1,0,2)(1,0,1)[12] with zero mean : Inf\n# ARIMA(1,0,2)(1,0,1)[12] with non-zero mean : Inf\n# ARIMA(1,0,2)(2,0,0)[12] with zero mean : Inf\n# Regression with ARIMA(1,0,2)(2,0,0)[12] errors : Inf\n# Regression with ARIMA(1,0,3) errors : Inf\n# Regression with ARIMA(1,0,3) errors : Inf\n# Regression with ARIMA(1,0,3)(0,0,1)[12] errors : 1196.785\n# Regression with ARIMA(1,0,3)(0,0,1)[12] errors : Inf\n# ARIMA(1,0,3)(1,0,0)[12] with zero mean : Inf\n# ARIMA(1,0,3)(1,0,0)[12] with non-zero mean : Inf\n# Regression with ARIMA(1,0,4) errors : Inf\n# Regression with ARIMA(1,0,4) errors : Inf\n# Regression with ARIMA(2,0,0) errors : 1270.227\n# Regression with ARIMA(2,0,0) errors : 1235.453\n# Regression with ARIMA(2,0,0)(0,0,1)[12] errors : 1191.855\n# Regression with ARIMA(2,0,0)(0,0,1)[12] errors : 1165.096\n# Regression with ARIMA(2,0,0)(0,0,2)[12] errors : 1145.887\n# Regression with ARIMA(2,0,0)(0,0,2)[12] errors : 1123.031\n# ARIMA(2,0,0)(1,0,0)[12] with zero mean : Inf\n# ARIMA(2,0,0)(1,0,0)[12] with non-zero mean : Inf\n# ARIMA(2,0,0)(1,0,1)[12] with zero mean : Inf\n# ARIMA(2,0,0)(1,0,1)[12] with non-zero mean : Inf\n# ARIMA(2,0,0)(1,0,2)[12] with zero mean : Inf\n# ARIMA(2,0,0)(1,0,2)[12] with non-zero mean : Inf\n# ARIMA(2,0,0)(2,0,0)[12] with zero mean : Inf\n# ARIMA(2,0,0)(2,0,0)[12] with non-zero mean : Inf\n# ARIMA(2,0,0)(2,0,1)[12] with zero mean : Inf\n# Regression with ARIMA(2,0,0)(2,0,1)[12] errors : Inf\n# Regression with ARIMA(2,0,1) errors : Inf\n# Regression with ARIMA(2,0,1) errors : Inf\n# Regression with ARIMA(2,0,1)(0,0,1)[12] errors : Inf\n# Regression with ARIMA(2,0,1)(0,0,1)[12] errors : Inf\n# Regression with ARIMA(2,0,1)(0,0,2)[12] errors : Inf\n# Regression with ARIMA(2,0,1)(0,0,2)[12] errors : Inf\n# ARIMA(2,0,1)(1,0,0)[12] with zero mean : Inf\n# ARIMA(2,0,1)(1,0,0)[12] with non-zero mean : Inf\n# Regression with ARIMA(2,0,1)(1,0,1)[12] errors : Inf\n# ARIMA(2,0,1)(1,0,1)[12] with non-zero mean : Inf\n# Regression with ARIMA(2,0,1)(2,0,0)[12] errors : Inf\n# Regression with ARIMA(2,0,1)(2,0,0)[12] errors : Inf\n# Regression with ARIMA(2,0,2) errors : Inf\n# Regression with ARIMA(2,0,2) errors : Inf\n# ARIMA(2,0,2)(0,0,1)[12] with zero mean : Inf\n# Regression with ARIMA(2,0,2)(0,0,1)[12] errors : Inf\n# ARIMA(2,0,2)(1,0,0)[12] with zero mean : Inf\n# ARIMA(2,0,2)(1,0,0)[12] with non-zero mean : Inf\n# ARIMA(2,0,3) with zero mean : Inf\n# Regression with ARIMA(2,0,3) errors : Inf\n# Regression with ARIMA(3,0,0) errors : 1271.552\n# Regression with ARIMA(3,0,0) errors : 1234.442\n# Regression with ARIMA(3,0,0)(0,0,1)[12] errors : 1193.944\n# Regression with ARIMA(3,0,0)(0,0,1)[12] errors : 1165.004\n# Regression with ARIMA(3,0,0)(0,0,2)[12] errors : 1147.337\n# Regression with ARIMA(3,0,0)(0,0,2)[12] errors : 1124.69\n# ARIMA(3,0,0)(1,0,0)[12] with zero mean : Inf\n# ARIMA(3,0,0)(1,0,0)[12] with non-zero mean : Inf\n# ARIMA(3,0,0)(1,0,1)[12] with zero mean : Inf\n# ARIMA(3,0,0)(1,0,1)[12] with non-zero mean : Inf\n# ARIMA(3,0,0)(2,0,0)[12] with zero mean : Inf\n# ARIMA(3,0,0)(2,0,0)[12] with non-zero mean : Inf\n# Regression with ARIMA(3,0,1) errors : Inf\n# Regression with ARIMA(3,0,1) errors : Inf\n# Regression with ARIMA(3,0,1)(0,0,1)[12] errors : Inf\n# Regression with ARIMA(3,0,1)(0,0,1)[12] errors : Inf\n# ARIMA(3,0,1)(1,0,0)[12] with zero mean : Inf\n# ARIMA(3,0,1)(1,0,0)[12] with non-zero mean : Inf\n# Regression with ARIMA(3,0,2) errors : Inf\n# Regression with ARIMA(3,0,2) errors : Inf\n# Regression with ARIMA(4,0,0) errors : 1268.176\n# Regression with ARIMA(4,0,0) errors : 1236.501\n# Regression with ARIMA(4,0,0)(0,0,1)[12] errors : 1191.278\n# Regression with ARIMA(4,0,0)(0,0,1)[12] errors : 1167.39\n# ARIMA(4,0,0)(1,0,0)[12] with zero mean : Inf\n# ARIMA(4,0,0)(1,0,0)[12] with non-zero mean : Inf\n# Regression with ARIMA(4,0,1) errors : Inf\n# Regression with ARIMA(4,0,1) errors : Inf\n# Regression with ARIMA(5,0,0) errors : 1270.388\n# Regression with ARIMA(5,0,0) errors : 1226.449\n# \n# \n# \n# Best model: Regression with ARIMA(1,0,0)(0,0,2)[12] errors \n# \n\nsummary(fit)\n# Series: d2$prevalence \n# Regression with ARIMA(1,0,0)(0,0,2)[12] errors \n# \n# Coefficients:\n# ar1 sma1 sma2 intercept time step1 ramp1 step2 ramp2 step3 ramp3\n# 0.5256 1.0413 0.6987 188.8179 1.5717 3.3284 2.1966 -5.6181 0.4640 0.4378 4.6447\n# s.e. 0.0800 0.1536 0.1312 20.6895 0.4724 13.0492 3.5219 23.9955 3.8197 19.1790 2.5724\n# \n# sigma^2 = 494.9: log likelihood = -547.61\n# AIC=1119.22 AICc=1122.14 BIC=1152.67\n# \n# Training set error measures:\n# ME RMSE MAE MPE MAPE MASE ACF1\n# Training set -0.2760281 21.20232 16.21419 -0.9807653 5.578825 0.3604042 0.05861382\n\n\n\n\n\nQuestion\n\n\n\n\nShould I just apply the same strategy to adults? Or can such models handle correlated/stratified time series?\n\n\n\n\nI guess that there is a better strategy than fitting separate models within each subgroup. However, I suspect that I cannot simply stack the time series and include age in xreg. Additionally, I probably need an interaction terms (e.g. time * age).\n\n\n\n\nI have also read about nlme::gls() (https://stat.ethz.ch/R-manual/R-devel/library/nlme/html/gls.html) but I would like guidance/confirmation whether it is more appropriate.\n\n\n\n\n\n\n\nBTW, I am not satisfied with the fit, but I will explore that once I have decided the modelling strategy.\n\n\n\n\nres <- residuals(fit)\nl <- 36\n\npar(mfrow = c(3, 1))\nplot(res)\nacf(res, lag.max = l)\npacf(res, lag.max = l)\ndev.off()\n\nBox.test(res, lag = 24, type = \"Ljung-Box\", fitdf = 3)\n# \n# Box-Ljung test\n# \n# data: res\n# X-squared = 49.814, df = 21, p-value = 0.0003872\n# \n\n\n\n\n\n[image: residuals; source: https://i.sstatic.net/pzs11NTf.png] (https://i.sstatic.net/pzs11NTf.png)", "revision_attribution": [{"content_license": "CC BY-SA 4.0", "contributor": {"display_name": "Thomas", "profile_url": "https://stats.stackexchange.com/users/263046/thomas", "user_type": "registered"}, "created_at": "2026-08-27T20:48:18+00:00", "raw_file": "raw/codex_api_v1/84bb317e707465a80f008019404ef1f770e152d74ef692d3b81c48903ec855cc_1790825232001984700_0.json", "raw_sha256": "75778d86d5325e15f4b491a748dc70ae6ac6ee98840d584e01ed28416cb71fbe", "revision_guid": "4F68B416-BBB1-4FBC-8F34-014B8106BA46", "revision_number": 1, "revision_type": "single_user", "revision_url": "https://stats.stackexchange.com/revisions/4F68B416-BBB1-4FBC-8F34-014B8106BA46/view-source"}, {"content_license": "CC BY-SA 4.0", "contributor": {"display_name": "Thomas", "profile_url": "https://stats.stackexchange.com/users/263046/thomas", "user_type": "registered"}, "created_at": "2026-08-28T00:03:27+00:00", "raw_file": "raw/codex_api_v1/84bb317e707465a80f008019404ef1f770e152d74ef692d3b81c48903ec855cc_1790825232001984700_0.json", "raw_sha256": "75778d86d5325e15f4b491a748dc70ae6ac6ee98840d584e01ed28416cb71fbe", "revision_guid": "FB0908B9-A046-40C1-A08B-E6284215F76C", "revision_number": 2, "revision_type": "single_user", "revision_url": "https://stats.stackexchange.com/revisions/FB0908B9-A046-40C1-A08B-E6284215F76C/view-source"}, {"content_license": "CC BY-SA 4.0", "contributor": {"display_name": "Thomas", "profile_url": "https://stats.stackexchange.com/users/263046/thomas", "user_type": "registered"}, "created_at": "2026-08-28T00:11:08+00:00", "raw_file": "raw/codex_api_v1/84bb317e707465a80f008019404ef1f770e152d74ef692d3b81c48903ec855cc_1790825232001984700_0.json", "raw_sha256": "75778d86d5325e15f4b491a748dc70ae6ac6ee98840d584e01ed28416cb71fbe", "revision_guid": "9E9FBC78-84A9-484D-B830-4D1751368045", "revision_number": 3, "revision_type": "single_user", "revision_url": "https://stats.stackexchange.com/revisions/9E9FBC78-84A9-484D-B830-4D1751368045/view-source"}, {"content_license": "CC BY-SA 4.0", "contributor": {"display_name": "Thomas", "profile_url": "https://stats.stackexchange.com/users/263046/thomas", "user_type": "registered"}, "created_at": "2026-08-28T09:19:31+00:00", "raw_file": "raw/codex_api_v1/84bb317e707465a80f008019404ef1f770e152d74ef692d3b81c48903ec855cc_1790825232001984700_0.json", "raw_sha256": "75778d86d5325e15f4b491a748dc70ae6ac6ee98840d584e01ed28416cb71fbe", "revision_guid": "F52E35E2-A4F3-4D9C-BBE7-6F9246725A84", "revision_number": 4, "revision_type": "single_user", "revision_url": "https://stats.stackexchange.com/revisions/F52E35E2-A4F3-4D9C-BBE7-6F9246725A84/view-source"}], "source_site": "stats", "source_url": "https://stats.stackexchange.com/questions/676979/strategy-for-interrupted-time-series-of-correlated-stratified-time-series", "split": "validation", "split_group": "4776dab9c10254abf857639b7a8deff481b81182a68f5330c41a896a7463144f", "tags": ["r", "time-series", "arima", "generalized-least-squares", "intervention-analysis"], "thread_id": "stats:676979", "title": "Strategy for interrupted time series of correlated/stratified time series"}}
|
| {"accepted_answer": null, "code_blocks": [{"block_index": 0, "code_text": "import sys\nimport numpy as np\nimport matplotlib.pyplot as plt\nfrom scipy.integrate import solve_ivp\nfrom math import pi\nimport random\n\n# constants\nl_1 = 0.1\nl_2 = 0.5\nm_1 = 0.8\nm_2 = 0.3\ng = 9.81\n\n# Cauchy problem\n\n# y = [theta_1, theta_2, theta_1_dot, theta_2_dot]\n# y0 = [theta_1_0, theta_2_0, theta_1_dot_0, theta_2_dot_0]\n\n# generating initial conditions for some energy E\n\nnumber_of_initial_conditions = 50\n\ntheta_1_0 = []\n# theta_2_0 = []\ntheta_2_0 = np.linspace(-pi/2, pi/2, number_of_initial_conditions)\ntheta_1_dot_0 = []\ntheta_2_dot_0 = []\n\n# Energy = np.linspace(5,5.6,8) # [0,100] 1.\n# Energy = np.linspace(5.6,6.2,8) # [0,300] 2.\n# Energy = np.linspace(6.2,6.4,8) # [0,500] 3.\n# Energy = np.linspace(6.4,6.8,8) # [0,1000] 4.\n# Energy = np.linspace(5.9,6.3,8) # [0,700] 2-3.\n# Energy = np.linspace(6.3,6.8,12) # [0,1200] 3-4.\n# Energy = np.linspace(5.4,6.8,20) # [0,2000] 0.\n# Energy = np.linspace(5.6, 6.4, 16) # [0,700] 2_3\n# Energy = np.linspace(6.2,6.8,18) # [0,1200] 3_4.\n\n# Energy = [5.865]\nEnergy = [6.09]\n# Energy = [6.075]\n# Energy = [6.095]\n# Energy = [6.079]\n# Energy = [6.12]\n\n# number_of_initial_conditions = 1\n\ntotal_numbers_of_initial_conditions = number_of_initial_conditions*len(Energy)\n\ndef initial_conditions(E, number_of_initial_conditions):\n\n for i in range(number_of_initial_conditions):\n theta_1_0.append(0)\n # theta_2_0.append(np.random.uniform(-pi/4, pi/4))\n theta_2_dot_0.append(0)\n bucket = []\n bucket.append(np.sqrt(2*(E+l_1*g*(m_1+m_2)+l_2*m_2*g*np.cos(theta_2_0[i])-(l_1+l_2)*(m_1+m_2)*g)/(l_1**2*(m_1+m_2))))\n bucket.append(-np.sqrt(2*(E+l_1*g*(m_1+m_2)+l_2*m_2*g*np.cos(theta_2_0[i])-(l_1+l_2)*(m_1+m_2)*g)/(l_1**2*(m_1+m_2))))\n theta_1_dot_0.append(bucket[random.randint(0,1)])\n\nfor i in range(len(Energy)):\n initial_conditions(Energy[i], number_of_initial_conditions)\n\ndef f(t,y):\n return [y[2], \n y[3], \n (-g*np.sin(y[0])*(2*m_1+m_2)-g*m_2*np.sin(y[0]-2*y[1])-2*m_2*np.sin(y[0]-y[1])*(y[2]**2*l_1*np.cos(y[0]-y[1])+l_2*y[3]**2))/(l_1*(2*m_1+m_2-m_2*np.cos(2*y[0]-2*y[1]))), \n (2*np.sin(y[0]-y[1])*(np.cos(y[0])*g*(m_1+m_2)+l_1*(m_1+m_2)*y[2]**2+m_2*l_2*y[3]**2*np.cos(y[0]-y[1])))/(l_2*(2*m_1+m_2-m_2*np.cos(2*y[0]-2*y[1])))]\n\n# energy of the system\ndef E(y_1, y_2, y_3, y_4):\n total_energy = 0.5*l_1**2*y_3**2*(m_1+m_2)+0.5*m_2*l_2**2*y_4**2+m_2*l_1*l_2*y_3*y_4*np.cos(y_1-y_2)-l_1*g*np.cos(y_1)*(m_1+m_2)-l_2*m_2*g*np.cos(y_2)+(l_1+l_2)*g*(m_1+m_2)\n return total_energy\n\n# Poincaré section\ndef Poincare_section(total_numbers_of_initial_conditions):\n theta_2 = []\n theta_2_dot = []\n\n colors = []\n\n color = [\n \"#2C3E50\", \"#3A3F64\", \"#484078\", \"#56428D\", \"#6B469E\",\n \"#804AAF\", \"#954EBF\", \"#A753C4\", \"#BA58C8\", \"#CE5DCD\",\n \"#E062C9\", \"#E971B4\", \"#F1809F\", \"#F98F8A\", \"#FFA07A\",\n \"#FF9C65\", \"#FF9850\", \"#FF943B\", \"#FF9026\", \"#FF8C11\",\n \"#F97F0D\", \"#F3730A\", \"#ED6606\", \"#E75A03\", \"#E04E00\",\n \"#D4431E\", \"#C8383C\", \"#BC2D5A\", \"#B02178\", \"#A41596\",\n \"#9710A3\", \"#880EA7\", \"#790CAB\", \"#6A0AAF\", \"#5C08B2\",\n \"#4D06B6\", \"#3E04BA\", \"#2F02BD\", \"#2000C1\", \"#1800B8\",\n \"#1000AF\", \"#0800A6\", \"#00009D\", \"#00008F\", \"#000081\",\n \"#0B006C\", \"#160057\", \"#210043\", \"#2C002E\", \"#37001A\"\n ]\n\n for i in range(total_numbers_of_initial_conditions):\n # color = [\"#\"+''.join([random.choice('0123456789ABCDEF') for r in range(6)])\n # for s in range(1)]\n\n number_of_points = len(solutions[i].t)\n\n for j in range(number_of_points-1):\n if solutions[i].y[0][j] <= 0 and solutions[i].y[0][j+1] >= 0:\n colors.append(color[i])\n theta_2.append(solutions[i].y[1][j+1])\n theta_2_dot.append(solutions[i].y[3][j+1])\n\n for m in range(len(theta_2)):\n plt.scatter((theta_2[m]+np.pi)%(2 * np.pi) - np.pi, theta_2_dot[m], c=colors[m], s=0.1)\n\n plt.xlabel(r\"$\\theta_2\\ [rad]$\")\n plt.ylabel(r\"$\\dot{\\theta_2}\\ [rad \\cdot s^{-1}]$\")\n plt.show()\n\n# solving the equations of motion\nsolutions = []\n\nerror = 0.001\n\n# defining time interval\nt_span = [0,750]\n\nfor i in range(total_numbers_of_initial_conditions):\n y0 = [theta_1_0[i], theta_2_0[i], theta_1_dot_0[i], theta_2_dot_0[i]]\n system_energy = E(y0[0], y0[1], y0[2], y0[3])\n print(y0)\n print(system_energy)\n solutions.append(solve_ivp(f, t_span, y0, dense_output=True, rtol = 1e-12, atol = 1e-14))\n number_of_points = len(solutions[i].t)\n \n for j in range(number_of_points):\n solution_energy = E(solutions[i].y[0][j], solutions[i].y[1][j], solutions[i].y[2][j], solutions[i].y[3][j])\n if np.abs(solution_energy - system_energy) > error:\n sys.exit('Maximum energy drift of {} exceeded.'.format(error))\n\nPoincare_section(total_numbers_of_initial_conditions)\n\nx_1 = []\nx_2 = []\ny_1 = []\ny_2 = []\n\nfor i in range(len(solutions[0].t)):\n x_1.append(l_1*np.sin(solutions[0].y[0][i]))\n x_2.append(x_1[i]+l_2*np.sin(solutions[0].y[1][i]))\n y_1.append(-l_1*np.cos(solutions[0].y[0][i]))\n y_2.append(y_1[i]-l_2*np.cos(solutions[0].y[1][i]))\n\nplt.plot(solutions[0].y[0], solutions[0].y[1])\nplt.xlabel(r\"$\\theta_1\\ [rad]$\")\nplt.ylabel(r\"$\\theta_2\\ [rad]$\")\nplt.show()\n\nplt.plot(x_1, y_1)\nplt.show()\nplt.plot(x_2, y_2)\nplt.show()\n\nplt.plot(x_1, y_1)\nplt.plot(x_2, y_2)\nplt.show()\n", "language": "unspecified", "language_note": "Syntax-screened scientific code; language not asserted; not executed", "post_id": 45171, "sha256": "1d67820d96c570166fb54e56067d4cef4f8c2e54fa592f7764a9d6f5c940d00b", "source_url": "https://scicomp.stackexchange.com/questions/45171/poincar%c3%a9-section-for-double-pendulum-code-improvement"}, {"block_index": 0, "code_text": "import time\n\nimport numpy as np\nimport matplotlib.pyplot as plt\nfrom scipy.integrate import solve_ivp, OdeSolution\n\nfrom scipy.constants import g\nfrom scipy.integrate._ivp.ivp import OdeResult\n\nl_1 = 0.1\nl_2 = 0.5\nm_1 = 0.8\nm_2 = 0.3\n\n\ndef initial_conditions(\n E: float, n_initial_conditions: int, rand: np.random.Generator,\n) -> tuple[\n np.ndarray, np.ndarray, np.ndarray, np.ndarray,\n]:\n theta_1_0 = np.zeros(n_initial_conditions)\n theta_2_0 = np.linspace(-0.5*np.pi, 0.5*np.pi, n_initial_conditions)\n theta_2_dot_0 = np.zeros(n_initial_conditions)\n\n abs_t10 = np.sqrt(\n 2*(\n (\n E + l_2*m_2*g*np.cos(theta_2_0)\n )/(m_1 + m_2)\n - l_2*g\n )\n )/l_1\n\n theta_1_dot_0 = abs_t10*rand.choice((-1, 1), size=n_initial_conditions)\n\n return theta_1_0, theta_2_0, theta_1_dot_0, theta_2_dot_0\n\n\ndef f(t: float, y: np.ndarray) -> tuple[\n float, float, float, float,\n]:\n y0, y1, y2, y3 = y\n sin_y01 = 2*np.sin(y0 - y1)\n cos_y01 = np.cos(y0 - y1)\n m12m2 = 2*m_1 + m_2\n den = m12m2 - m_2*np.cos(2*y0 - 2*y1)\n y22l1 = y2**2*l_1\n y32l2 = l_2*y3**2\n\n res = (\n y2,\n y3,\n (\n -g*np.sin(y0)*m12m2 - m_2*(\n g*np.sin(y0 - 2*y1)\n + sin_y01*(\n y22l1*cos_y01 + y32l2\n )\n )\n ) / (l_1*den),\n (\n sin_y01*(\n (m_1 + m_2)*(\n np.cos(y0)*g + y22l1\n )\n + m_2*y32l2*cos_y01\n )\n ) / (l_2*den),\n )\n return res\n\n\ndef E(y_1: float, y_2: float, y_3: float, y_4: float) -> float:\n \"\"\"energy of the system\"\"\"\n total_energy = (\n 0.5*l_1**2*y_3**2*(m_1 + m_2)\n + 0.5*m_2*l_2**2*y_4**2\n + m_2*l_1*l_2*y_3*y_4*np.cos(y_1 - y_2)\n - l_1*g*np.cos(y_1)*(m_1 + m_2)\n - l_2*m_2*g*np.cos(y_2)\n + (l_1 + l_2)*g*(m_1 + m_2)\n )\n return total_energy\n\n\ndef plot_poincaré_section(solutions: list[OdeSolution]) -> plt.Figure:\n theta_2 = []\n theta_2_dot = []\n colors = []\n\n color = (\n \"#2C3E50\", \"#3A3F64\", \"#484078\", \"#56428D\", \"#6B469E\",\n \"#804AAF\", \"#954EBF\", \"#A753C4\", \"#BA58C8\", \"#CE5DCD\",\n \"#E062C9\", \"#E971B4\", \"#F1809F\", \"#F98F8A\", \"#FFA07A\",\n \"#FF9C65\", \"#FF9850\", \"#FF943B\", \"#FF9026\", \"#FF8C11\",\n \"#F97F0D\", \"#F3730A\", \"#ED6606\", \"#E75A03\", \"#E04E00\",\n \"#D4431E\", \"#C8383C\", \"#BC2D5A\", \"#B02178\", \"#A41596\",\n \"#9710A3\", \"#880EA7\", \"#790CAB\", \"#6A0AAF\", \"#5C08B2\",\n \"#4D06B6\", \"#3E04BA\", \"#2F02BD\", \"#2000C1\", \"#1800B8\",\n \"#1000AF\", \"#0800A6\", \"#00009D\", \"#00008F\", \"#000081\",\n \"#0B006C\", \"#160057\", \"#210043\", \"#2C002E\", \"#37001A\"\n )\n\n for solution, colori in zip(solutions, color):\n n_points = len(solution.t)\n y0, y1, y2, y3 = solution.y\n\n for j in range(n_points - 1):\n if y0[j] <= 0 <= y0[j + 1]:\n colors.append(colori)\n theta_2.append(y1[j + 1])\n theta_2_dot.append(y3[j + 1])\n\n fig, ax = plt.subplots()\n\n for m in range(len(theta_2)):\n ax.scatter((theta_2[m] + np.pi) % (2 * np.pi) - np.pi, theta_2_dot[m], c=colors[m], s=0.1)\n\n ax.set_xlabel(r\"$\\theta_2\\ [rad]$\")\n ax.set_ylabel(r\"$\\dot{\\theta_2}\\ [rad \\cdot s^{-1}]$\")\n return fig\n\n\ndef solve_motion_equations(\n theta_1_0: np.ndarray,\n theta_2_0: np.ndarray,\n theta_1_dot_0: np.ndarray,\n theta_2_dot_0: np.ndarray,\n strict: bool = True,\n error_max: float = 1e-3,\n rtol: float = 1e-4, atol: float = 1e-6,\n) -> list[OdeResult]:\n \"\"\"solving the equations of motion\"\"\"\n solutions = []\n\n # defining time interval\n t_span = (0, 750)\n jac_sparsity = np.array(( # All methods but LSODA\n (0, 0, 1, 0),\n (0, 0, 0, 1),\n (1, 1, 1, 1),\n (1, 1, 1, 1),\n ))\n # LSODA band parameters only reduce the Jacobian by one element; probably not worth it\n\n for i in range(len(theta_1_0)):\n y0 = (theta_1_0[i], theta_2_0[i], theta_1_dot_0[i], theta_2_dot_0[i])\n t0 = time.perf_counter()\n solutions.append(solve_ivp(\n fun=f, t_span=t_span, y0=y0, dense_output=True, method='LSODA',\n # jac_sparsity=jac_sparsity,\n rtol=rtol, atol=atol,\n ))\n t1 = time.perf_counter()\n print(t1 - t0)\n\n system_energy = E(*y0)\n for y in solutions[i].y.T:\n solution_energy = E(*y)\n error = np.abs(solution_energy - system_energy)\n if strict and error > error_max:\n message = f'Energy drift of {error_max} exceeds maximum {error_max}'\n raise ValueError(message)\n\n return solutions\n\n\ndef fill_xy(solutions: list[OdeSolution]) -> tuple[\n np.ndarray, np.ndarray, np.ndarray, np.ndarray,\n]:\n x_1 = l_1*np.sin(solutions[0].y[0])\n x_2 = l_2*np.sin(solutions[0].y[1]) + x_1\n y_1 = -l_1*np.cos(solutions[0].y[0])\n y_2 = -l_2*np.cos(solutions[0].y[1]) + y_1\n return x_1, x_2, y_1, y_2\n\n\ndef plot(\n solutions: list[OdeSolution],\n x_1: np.ndarray, x_2: np.ndarray,\n y_1: np.ndarray, y_2: np.ndarray,\n) -> None:\n fig, ax = plt.subplots()\n ax.plot(solutions[0].y[0], solutions[0].y[1])\n ax.set_xlabel(r\"$\\theta_1\\ [rad]$\")\n ax.set_ylabel(r\"$\\theta_2\\ [rad]$\")\n\n fig, ax = plt.subplots()\n ax.plot(x_1, y_1)\n ax.set_xlabel('x_1')\n ax.set_ylabel('y_1')\n\n fig, ax = plt.subplots()\n ax.plot(x_2, y_2)\n ax.set_xlabel('x_2')\n ax.set_ylabel('y_2')\n\n fig, ax = plt.subplots()\n ax.plot(x_1, y_1, label='y_1')\n ax.plot(x_2, y_2, label='y_2')\n ax.legend()\n\n\ndef main() -> None:\n n_initial_conditions = 3 # 50\n rand = np.random.default_rng(seed=0)\n\n theta_1_0, theta_2_0, theta_1_dot_0, theta_2_dot_0 = initial_conditions(\n E=6.09, n_initial_conditions=n_initial_conditions, rand=rand,\n )\n solutions = solve_motion_equations(\n theta_1_0=theta_1_0, theta_1_dot_0=theta_1_dot_0,\n theta_2_0=theta_2_0, theta_2_dot_0=theta_2_dot_0,\n error_max=0.1,\n )\n plot_poincaré_section(solutions)\n x_1, x_2, y_1, y_2 = fill_xy(solutions)\n plot(solutions, x_1, x_2, y_1, y_2)\n plt.show()\n\n\nif __name__ == '__main__':\n main()\n", "language": "unspecified", "language_note": "Syntax-screened scientific code; language not asserted; not executed", "post_id": 45267, "sha256": "ae3ad9e84c6f094fe7cfe99f0f34dd3421a235cc15f4e45eab8309949d15b321", "source_url": "https://scicomp.stackexchange.com/a/45267"}], "other_answers": [], "product": "code_qa", "question": "I wanted to try numerical analysis of a chaotic system. So I decided to write my own code for the Poincaré section of a double pendulum in Python. The code works and the Poincaré section should be correct, but I can't get a nice picture of the section, like you see in textbooks or on the internet. I don't know if the problem is in the choice of energies, the (number of) initial conditions, or the integration \"time interval\" and steps. Can someone more experienced advise me on a good approach to obtaining nice Poincaré sections? The code can also be improved, so if anyone has any suggestions for improvements, I would be very happy to hear them!\n\n\n\n\nHere is the code: https://github.com/DJopek/chaos/blob/main/double_pendulum.py (https://github.com/DJopek/chaos/blob/main/double_pendulum.py)\n\n\n\n\nimport sys\nimport numpy as np\nimport matplotlib.pyplot as plt\nfrom scipy.integrate import solve_ivp\nfrom math import pi\nimport random\n\n# constants\nl_1 = 0.1\nl_2 = 0.5\nm_1 = 0.8\nm_2 = 0.3\ng = 9.81\n\n# Cauchy problem\n\n# y = [theta_1, theta_2, theta_1_dot, theta_2_dot]\n# y0 = [theta_1_0, theta_2_0, theta_1_dot_0, theta_2_dot_0]\n\n# generating initial conditions for some energy E\n\nnumber_of_initial_conditions = 50\n\ntheta_1_0 = []\n# theta_2_0 = []\ntheta_2_0 = np.linspace(-pi/2, pi/2, number_of_initial_conditions)\ntheta_1_dot_0 = []\ntheta_2_dot_0 = []\n\n# Energy = np.linspace(5,5.6,8) # [0,100] 1.\n# Energy = np.linspace(5.6,6.2,8) # [0,300] 2.\n# Energy = np.linspace(6.2,6.4,8) # [0,500] 3.\n# Energy = np.linspace(6.4,6.8,8) # [0,1000] 4.\n# Energy = np.linspace(5.9,6.3,8) # [0,700] 2-3.\n# Energy = np.linspace(6.3,6.8,12) # [0,1200] 3-4.\n# Energy = np.linspace(5.4,6.8,20) # [0,2000] 0.\n# Energy = np.linspace(5.6, 6.4, 16) # [0,700] 2_3\n# Energy = np.linspace(6.2,6.8,18) # [0,1200] 3_4.\n\n# Energy = [5.865]\nEnergy = [6.09]\n# Energy = [6.075]\n# Energy = [6.095]\n# Energy = [6.079]\n# Energy = [6.12]\n\n# number_of_initial_conditions = 1\n\ntotal_numbers_of_initial_conditions = number_of_initial_conditions*len(Energy)\n\ndef initial_conditions(E, number_of_initial_conditions):\n\n for i in range(number_of_initial_conditions):\n theta_1_0.append(0)\n # theta_2_0.append(np.random.uniform(-pi/4, pi/4))\n theta_2_dot_0.append(0)\n bucket = []\n bucket.append(np.sqrt(2*(E+l_1*g*(m_1+m_2)+l_2*m_2*g*np.cos(theta_2_0[i])-(l_1+l_2)*(m_1+m_2)*g)/(l_1**2*(m_1+m_2))))\n bucket.append(-np.sqrt(2*(E+l_1*g*(m_1+m_2)+l_2*m_2*g*np.cos(theta_2_0[i])-(l_1+l_2)*(m_1+m_2)*g)/(l_1**2*(m_1+m_2))))\n theta_1_dot_0.append(bucket[random.randint(0,1)])\n\nfor i in range(len(Energy)):\n initial_conditions(Energy[i], number_of_initial_conditions)\n\ndef f(t,y):\n return [y[2], \n y[3], \n (-g*np.sin(y[0])*(2*m_1+m_2)-g*m_2*np.sin(y[0]-2*y[1])-2*m_2*np.sin(y[0]-y[1])*(y[2]**2*l_1*np.cos(y[0]-y[1])+l_2*y[3]**2))/(l_1*(2*m_1+m_2-m_2*np.cos(2*y[0]-2*y[1]))), \n (2*np.sin(y[0]-y[1])*(np.cos(y[0])*g*(m_1+m_2)+l_1*(m_1+m_2)*y[2]**2+m_2*l_2*y[3]**2*np.cos(y[0]-y[1])))/(l_2*(2*m_1+m_2-m_2*np.cos(2*y[0]-2*y[1])))]\n\n# energy of the system\ndef E(y_1, y_2, y_3, y_4):\n total_energy = 0.5*l_1**2*y_3**2*(m_1+m_2)+0.5*m_2*l_2**2*y_4**2+m_2*l_1*l_2*y_3*y_4*np.cos(y_1-y_2)-l_1*g*np.cos(y_1)*(m_1+m_2)-l_2*m_2*g*np.cos(y_2)+(l_1+l_2)*g*(m_1+m_2)\n return total_energy\n\n# Poincaré section\ndef Poincare_section(total_numbers_of_initial_conditions):\n theta_2 = []\n theta_2_dot = []\n\n colors = []\n\n color = [\n \"#2C3E50\", \"#3A3F64\", \"#484078\", \"#56428D\", \"#6B469E\",\n \"#804AAF\", \"#954EBF\", \"#A753C4\", \"#BA58C8\", \"#CE5DCD\",\n \"#E062C9\", \"#E971B4\", \"#F1809F\", \"#F98F8A\", \"#FFA07A\",\n \"#FF9C65\", \"#FF9850\", \"#FF943B\", \"#FF9026\", \"#FF8C11\",\n \"#F97F0D\", \"#F3730A\", \"#ED6606\", \"#E75A03\", \"#E04E00\",\n \"#D4431E\", \"#C8383C\", \"#BC2D5A\", \"#B02178\", \"#A41596\",\n \"#9710A3\", \"#880EA7\", \"#790CAB\", \"#6A0AAF\", \"#5C08B2\",\n \"#4D06B6\", \"#3E04BA\", \"#2F02BD\", \"#2000C1\", \"#1800B8\",\n \"#1000AF\", \"#0800A6\", \"#00009D\", \"#00008F\", \"#000081\",\n \"#0B006C\", \"#160057\", \"#210043\", \"#2C002E\", \"#37001A\"\n ]\n\n for i in range(total_numbers_of_initial_conditions):\n # color = [\"#\"+''.join([random.choice('0123456789ABCDEF') for r in range(6)])\n # for s in range(1)]\n\n number_of_points = len(solutions[i].t)\n\n for j in range(number_of_points-1):\n if solutions[i].y[0][j] <= 0 and solutions[i].y[0][j+1] >= 0:\n colors.append(color[i])\n theta_2.append(solutions[i].y[1][j+1])\n theta_2_dot.append(solutions[i].y[3][j+1])\n\n for m in range(len(theta_2)):\n plt.scatter((theta_2[m]+np.pi)%(2 * np.pi) - np.pi, theta_2_dot[m], c=colors[m], s=0.1)\n\n plt.xlabel(r\"$\\theta_2\\ [rad]$\")\n plt.ylabel(r\"$\\dot{\\theta_2}\\ [rad \\cdot s^{-1}]$\")\n plt.show()\n\n# solving the equations of motion\nsolutions = []\n\nerror = 0.001\n\n# defining time interval\nt_span = [0,750]\n\nfor i in range(total_numbers_of_initial_conditions):\n y0 = [theta_1_0[i], theta_2_0[i], theta_1_dot_0[i], theta_2_dot_0[i]]\n system_energy = E(y0[0], y0[1], y0[2], y0[3])\n print(y0)\n print(system_energy)\n solutions.append(solve_ivp(f, t_span, y0, dense_output=True, rtol = 1e-12, atol = 1e-14))\n number_of_points = len(solutions[i].t)\n \n for j in range(number_of_points):\n solution_energy = E(solutions[i].y[0][j], solutions[i].y[1][j], solutions[i].y[2][j], solutions[i].y[3][j])\n if np.abs(solution_energy - system_energy) > error:\n sys.exit('Maximum energy drift of {} exceeded.'.format(error))\n\nPoincare_section(total_numbers_of_initial_conditions)\n\nx_1 = []\nx_2 = []\ny_1 = []\ny_2 = []\n\nfor i in range(len(solutions[0].t)):\n x_1.append(l_1*np.sin(solutions[0].y[0][i]))\n x_2.append(x_1[i]+l_2*np.sin(solutions[0].y[1][i]))\n y_1.append(-l_1*np.cos(solutions[0].y[0][i]))\n y_2.append(y_1[i]-l_2*np.cos(solutions[0].y[1][i]))\n\nplt.plot(solutions[0].y[0], solutions[0].y[1])\nplt.xlabel(r\"$\\theta_1\\ [rad]$\")\nplt.ylabel(r\"$\\theta_2\\ [rad]$\")\nplt.show()\n\nplt.plot(x_1, y_1)\nplt.show()\nplt.plot(x_2, y_2)\nplt.show()\n\nplt.plot(x_1, y_1)\nplt.plot(x_2, y_2)\nplt.show()\n\n\n\n\n\nPoincaré section obtained by conditions in the code:\n[image: Poincaré section obtained by conditions in the code; source: https://i.sstatic.net/U0nOulED.png] (https://i.sstatic.net/U0nOulED.png)\n\n\n\n\nEDIT:\nPoincaré section obtained by conditions in the code with labels[image: Poincaré section obtained by conditions in the code with labels; source: https://i.sstatic.net/oJaBUfA4.png] (https://i.sstatic.net/oJaBUfA4.png)", "record_id": "Scientific-Code-and-Analysis-QA:scicomp:45171", "selected_answer": {"answer_html": "<blockquote>\n<p>Poincaré section obtained by conditions in the code</p>\n</blockquote>\n<p>I very much cannot reproduce what you've generated, after refactoring - I suspect due to moving to the more sophisticated LSODA and its accompanying parameters. I get results more similar to the resonances seen in e.g. <a href=\"https://duetosymmetry.com/tool/poincare-section-clicker-toy/\" rel=\"nofollow noreferrer\">Leo Stein's Poincaré sections</a>.</p>\n<blockquote>\n<p>The code can also be improved, so if anyone has any suggestions</p>\n</blockquote>\n<p>Well... There's a lot. Since this is the Computational Science site and not the Code Review site, I will elide much of my feedback by saying "improve your Python". For the numerics:</p>\n<ul>\n<li>Your <code>g</code> is both wrong and unnecessary; get the correct value from Scipy instead</li>\n<li>Don't use the <code>math</code> module; stick to Numpy</li>\n<li><code>f()</code> is both pretty slow and wholly illegible. It can be somewhat sped up by identifying common expressions, and should be broken up into many more lines. The same applies to <code>E()</code>.</li>\n<li>The current parameters to <code>solve_ivp</code> produce a very slow solution. Switch to LSODA and go easy on those tolerances.</li>\n<li>For plotting, only call <code>plt.show()</code> once.</li>\n</ul>\n<p><a href=\"https://i.sstatic.net/INKhjIWk.png\" rel=\"nofollow noreferrer\"><img src=\"https://i.sstatic.net/INKhjIWk.png\" alt=\"poincare\" /></a></p>\n<pre class=\"lang-py prettyprint-override\"><code>import time\n\nimport numpy as np\nimport matplotlib.pyplot as plt\nfrom scipy.integrate import solve_ivp, OdeSolution\n\nfrom scipy.constants import g\nfrom scipy.integrate._ivp.ivp import OdeResult\n\nl_1 = 0.1\nl_2 = 0.5\nm_1 = 0.8\nm_2 = 0.3\n\n\ndef initial_conditions(\n E: float, n_initial_conditions: int, rand: np.random.Generator,\n) -> tuple[\n np.ndarray, np.ndarray, np.ndarray, np.ndarray,\n]:\n theta_1_0 = np.zeros(n_initial_conditions)\n theta_2_0 = np.linspace(-0.5*np.pi, 0.5*np.pi, n_initial_conditions)\n theta_2_dot_0 = np.zeros(n_initial_conditions)\n\n abs_t10 = np.sqrt(\n 2*(\n (\n E + l_2*m_2*g*np.cos(theta_2_0)\n )/(m_1 + m_2)\n - l_2*g\n )\n )/l_1\n\n theta_1_dot_0 = abs_t10*rand.choice((-1, 1), size=n_initial_conditions)\n\n return theta_1_0, theta_2_0, theta_1_dot_0, theta_2_dot_0\n\n\ndef f(t: float, y: np.ndarray) -> tuple[\n float, float, float, float,\n]:\n y0, y1, y2, y3 = y\n sin_y01 = 2*np.sin(y0 - y1)\n cos_y01 = np.cos(y0 - y1)\n m12m2 = 2*m_1 + m_2\n den = m12m2 - m_2*np.cos(2*y0 - 2*y1)\n y22l1 = y2**2*l_1\n y32l2 = l_2*y3**2\n\n res = (\n y2,\n y3,\n (\n -g*np.sin(y0)*m12m2 - m_2*(\n g*np.sin(y0 - 2*y1)\n + sin_y01*(\n y22l1*cos_y01 + y32l2\n )\n )\n ) / (l_1*den),\n (\n sin_y01*(\n (m_1 + m_2)*(\n np.cos(y0)*g + y22l1\n )\n + m_2*y32l2*cos_y01\n )\n ) / (l_2*den),\n )\n return res\n\n\ndef E(y_1: float, y_2: float, y_3: float, y_4: float) -> float:\n """energy of the system"""\n total_energy = (\n 0.5*l_1**2*y_3**2*(m_1 + m_2)\n + 0.5*m_2*l_2**2*y_4**2\n + m_2*l_1*l_2*y_3*y_4*np.cos(y_1 - y_2)\n - l_1*g*np.cos(y_1)*(m_1 + m_2)\n - l_2*m_2*g*np.cos(y_2)\n + (l_1 + l_2)*g*(m_1 + m_2)\n )\n return total_energy\n\n\ndef plot_poincaré_section(solutions: list[OdeSolution]) -> plt.Figure:\n theta_2 = []\n theta_2_dot = []\n colors = []\n\n color = (\n "#2C3E50", "#3A3F64", "#484078", "#56428D", "#6B469E",\n "#804AAF", "#954EBF", "#A753C4", "#BA58C8", "#CE5DCD",\n "#E062C9", "#E971B4", "#F1809F", "#F98F8A", "#FFA07A",\n "#FF9C65", "#FF9850", "#FF943B", "#FF9026", "#FF8C11",\n "#F97F0D", "#F3730A", "#ED6606", "#E75A03", "#E04E00",\n "#D4431E", "#C8383C", "#BC2D5A", "#B02178", "#A41596",\n "#9710A3", "#880EA7", "#790CAB", "#6A0AAF", "#5C08B2",\n "#4D06B6", "#3E04BA", "#2F02BD", "#2000C1", "#1800B8",\n "#1000AF", "#0800A6", "#00009D", "#00008F", "#000081",\n "#0B006C", "#160057", "#210043", "#2C002E", "#37001A"\n )\n\n for solution, colori in zip(solutions, color):\n n_points = len(solution.t)\n y0, y1, y2, y3 = solution.y\n\n for j in range(n_points - 1):\n if y0[j] <= 0 <= y0[j + 1]:\n colors.append(colori)\n theta_2.append(y1[j + 1])\n theta_2_dot.append(y3[j + 1])\n\n fig, ax = plt.subplots()\n\n for m in range(len(theta_2)):\n ax.scatter((theta_2[m] + np.pi) % (2 * np.pi) - np.pi, theta_2_dot[m], c=colors[m], s=0.1)\n\n ax.set_xlabel(r"<span class=\"math-container\">$\\theta_2\\ [rad]$</span>")\n ax.set_ylabel(r"<span class=\"math-container\">$\\dot{\\theta_2}\\ [rad \\cdot s^{-1}]$</span>")\n return fig\n\n\ndef solve_motion_equations(\n theta_1_0: np.ndarray,\n theta_2_0: np.ndarray,\n theta_1_dot_0: np.ndarray,\n theta_2_dot_0: np.ndarray,\n strict: bool = True,\n error_max: float = 1e-3,\n rtol: float = 1e-4, atol: float = 1e-6,\n) -> list[OdeResult]:\n """solving the equations of motion"""\n solutions = []\n\n # defining time interval\n t_span = (0, 750)\n jac_sparsity = np.array(( # All methods but LSODA\n (0, 0, 1, 0),\n (0, 0, 0, 1),\n (1, 1, 1, 1),\n (1, 1, 1, 1),\n ))\n # LSODA band parameters only reduce the Jacobian by one element; probably not worth it\n\n for i in range(len(theta_1_0)):\n y0 = (theta_1_0[i], theta_2_0[i], theta_1_dot_0[i], theta_2_dot_0[i])\n t0 = time.perf_counter()\n solutions.append(solve_ivp(\n fun=f, t_span=t_span, y0=y0, dense_output=True, method='LSODA',\n # jac_sparsity=jac_sparsity,\n rtol=rtol, atol=atol,\n ))\n t1 = time.perf_counter()\n print(t1 - t0)\n\n system_energy = E(*y0)\n for y in solutions[i].y.T:\n solution_energy = E(*y)\n error = np.abs(solution_energy - system_energy)\n if strict and error > error_max:\n message = f'Energy drift of {error_max} exceeds maximum {error_max}'\n raise ValueError(message)\n\n return solutions\n\n\ndef fill_xy(solutions: list[OdeSolution]) -> tuple[\n np.ndarray, np.ndarray, np.ndarray, np.ndarray,\n]:\n x_1 = l_1*np.sin(solutions[0].y[0])\n x_2 = l_2*np.sin(solutions[0].y[1]) + x_1\n y_1 = -l_1*np.cos(solutions[0].y[0])\n y_2 = -l_2*np.cos(solutions[0].y[1]) + y_1\n return x_1, x_2, y_1, y_2\n\n\ndef plot(\n solutions: list[OdeSolution],\n x_1: np.ndarray, x_2: np.ndarray,\n y_1: np.ndarray, y_2: np.ndarray,\n) -> None:\n fig, ax = plt.subplots()\n ax.plot(solutions[0].y[0], solutions[0].y[1])\n ax.set_xlabel(r"<span class=\"math-container\">$\\theta_1\\ [rad]$</span>")\n ax.set_ylabel(r"<span class=\"math-container\">$\\theta_2\\ [rad]$</span>")\n\n fig, ax = plt.subplots()\n ax.plot(x_1, y_1)\n ax.set_xlabel('x_1')\n ax.set_ylabel('y_1')\n\n fig, ax = plt.subplots()\n ax.plot(x_2, y_2)\n ax.set_xlabel('x_2')\n ax.set_ylabel('y_2')\n\n fig, ax = plt.subplots()\n ax.plot(x_1, y_1, label='y_1')\n ax.plot(x_2, y_2, label='y_2')\n ax.legend()\n\n\ndef main() -> None:\n n_initial_conditions = 3 # 50\n rand = np.random.default_rng(seed=0)\n\n theta_1_0, theta_2_0, theta_1_dot_0, theta_2_dot_0 = initial_conditions(\n E=6.09, n_initial_conditions=n_initial_conditions, rand=rand,\n )\n solutions = solve_motion_equations(\n theta_1_0=theta_1_0, theta_1_dot_0=theta_1_dot_0,\n theta_2_0=theta_2_0, theta_2_dot_0=theta_2_dot_0,\n error_max=0.1,\n )\n plot_poincaré_section(solutions)\n x_1, x_2, y_1, y_2 = fill_xy(solutions)\n plot(solutions, x_1, x_2, y_1, y_2)\n plt.show()\n\n\nif __name__ == '__main__':\n main()\n</code></pre>\n", "answer_id": 45267, "answer_text": "Poincaré section obtained by conditions in the code\n\n\n\n\n\n\n\nI very much cannot reproduce what you've generated, after refactoring - I suspect due to moving to the more sophisticated LSODA and its accompanying parameters. I get results more similar to the resonances seen in e.g. Leo Stein's Poincaré sections (https://duetosymmetry.com/tool/poincare-section-clicker-toy/).\n\n\n\n\n\n\n\nThe code can also be improved, so if anyone has any suggestions\n\n\n\n\n\n\n\nWell... There's a lot. Since this is the Computational Science site and not the Code Review site, I will elide much of my feedback by saying \"improve your Python\". For the numerics:\n\n\n\n\n\nYour g is both wrong and unnecessary; get the correct value from Scipy instead\n\n\n\n\nDon't use the math module; stick to Numpy\n\n\n\n\nf() is both pretty slow and wholly illegible. It can be somewhat sped up by identifying common expressions, and should be broken up into many more lines. The same applies to E().\n\n\n\n\nThe current parameters to solve_ivp produce a very slow solution. Switch to LSODA and go easy on those tolerances.\n\n\n\n\nFor plotting, only call plt.show() once.\n\n\n\n\n\n[image: poincare; source: https://i.sstatic.net/INKhjIWk.png] (https://i.sstatic.net/INKhjIWk.png)\n\n\n\n\nimport time\n\nimport numpy as np\nimport matplotlib.pyplot as plt\nfrom scipy.integrate import solve_ivp, OdeSolution\n\nfrom scipy.constants import g\nfrom scipy.integrate._ivp.ivp import OdeResult\n\nl_1 = 0.1\nl_2 = 0.5\nm_1 = 0.8\nm_2 = 0.3\n\n\ndef initial_conditions(\n E: float, n_initial_conditions: int, rand: np.random.Generator,\n) -> tuple[\n np.ndarray, np.ndarray, np.ndarray, np.ndarray,\n]:\n theta_1_0 = np.zeros(n_initial_conditions)\n theta_2_0 = np.linspace(-0.5*np.pi, 0.5*np.pi, n_initial_conditions)\n theta_2_dot_0 = np.zeros(n_initial_conditions)\n\n abs_t10 = np.sqrt(\n 2*(\n (\n E + l_2*m_2*g*np.cos(theta_2_0)\n )/(m_1 + m_2)\n - l_2*g\n )\n )/l_1\n\n theta_1_dot_0 = abs_t10*rand.choice((-1, 1), size=n_initial_conditions)\n\n return theta_1_0, theta_2_0, theta_1_dot_0, theta_2_dot_0\n\n\ndef f(t: float, y: np.ndarray) -> tuple[\n float, float, float, float,\n]:\n y0, y1, y2, y3 = y\n sin_y01 = 2*np.sin(y0 - y1)\n cos_y01 = np.cos(y0 - y1)\n m12m2 = 2*m_1 + m_2\n den = m12m2 - m_2*np.cos(2*y0 - 2*y1)\n y22l1 = y2**2*l_1\n y32l2 = l_2*y3**2\n\n res = (\n y2,\n y3,\n (\n -g*np.sin(y0)*m12m2 - m_2*(\n g*np.sin(y0 - 2*y1)\n + sin_y01*(\n y22l1*cos_y01 + y32l2\n )\n )\n ) / (l_1*den),\n (\n sin_y01*(\n (m_1 + m_2)*(\n np.cos(y0)*g + y22l1\n )\n + m_2*y32l2*cos_y01\n )\n ) / (l_2*den),\n )\n return res\n\n\ndef E(y_1: float, y_2: float, y_3: float, y_4: float) -> float:\n \"\"\"energy of the system\"\"\"\n total_energy = (\n 0.5*l_1**2*y_3**2*(m_1 + m_2)\n + 0.5*m_2*l_2**2*y_4**2\n + m_2*l_1*l_2*y_3*y_4*np.cos(y_1 - y_2)\n - l_1*g*np.cos(y_1)*(m_1 + m_2)\n - l_2*m_2*g*np.cos(y_2)\n + (l_1 + l_2)*g*(m_1 + m_2)\n )\n return total_energy\n\n\ndef plot_poincaré_section(solutions: list[OdeSolution]) -> plt.Figure:\n theta_2 = []\n theta_2_dot = []\n colors = []\n\n color = (\n \"#2C3E50\", \"#3A3F64\", \"#484078\", \"#56428D\", \"#6B469E\",\n \"#804AAF\", \"#954EBF\", \"#A753C4\", \"#BA58C8\", \"#CE5DCD\",\n \"#E062C9\", \"#E971B4\", \"#F1809F\", \"#F98F8A\", \"#FFA07A\",\n \"#FF9C65\", \"#FF9850\", \"#FF943B\", \"#FF9026\", \"#FF8C11\",\n \"#F97F0D\", \"#F3730A\", \"#ED6606\", \"#E75A03\", \"#E04E00\",\n \"#D4431E\", \"#C8383C\", \"#BC2D5A\", \"#B02178\", \"#A41596\",\n \"#9710A3\", \"#880EA7\", \"#790CAB\", \"#6A0AAF\", \"#5C08B2\",\n \"#4D06B6\", \"#3E04BA\", \"#2F02BD\", \"#2000C1\", \"#1800B8\",\n \"#1000AF\", \"#0800A6\", \"#00009D\", \"#00008F\", \"#000081\",\n \"#0B006C\", \"#160057\", \"#210043\", \"#2C002E\", \"#37001A\"\n )\n\n for solution, colori in zip(solutions, color):\n n_points = len(solution.t)\n y0, y1, y2, y3 = solution.y\n\n for j in range(n_points - 1):\n if y0[j] <= 0 <= y0[j + 1]:\n colors.append(colori)\n theta_2.append(y1[j + 1])\n theta_2_dot.append(y3[j + 1])\n\n fig, ax = plt.subplots()\n\n for m in range(len(theta_2)):\n ax.scatter((theta_2[m] + np.pi) % (2 * np.pi) - np.pi, theta_2_dot[m], c=colors[m], s=0.1)\n\n ax.set_xlabel(r\"$\\theta_2\\ [rad]$\")\n ax.set_ylabel(r\"$\\dot{\\theta_2}\\ [rad \\cdot s^{-1}]$\")\n return fig\n\n\ndef solve_motion_equations(\n theta_1_0: np.ndarray,\n theta_2_0: np.ndarray,\n theta_1_dot_0: np.ndarray,\n theta_2_dot_0: np.ndarray,\n strict: bool = True,\n error_max: float = 1e-3,\n rtol: float = 1e-4, atol: float = 1e-6,\n) -> list[OdeResult]:\n \"\"\"solving the equations of motion\"\"\"\n solutions = []\n\n # defining time interval\n t_span = (0, 750)\n jac_sparsity = np.array(( # All methods but LSODA\n (0, 0, 1, 0),\n (0, 0, 0, 1),\n (1, 1, 1, 1),\n (1, 1, 1, 1),\n ))\n # LSODA band parameters only reduce the Jacobian by one element; probably not worth it\n\n for i in range(len(theta_1_0)):\n y0 = (theta_1_0[i], theta_2_0[i], theta_1_dot_0[i], theta_2_dot_0[i])\n t0 = time.perf_counter()\n solutions.append(solve_ivp(\n fun=f, t_span=t_span, y0=y0, dense_output=True, method='LSODA',\n # jac_sparsity=jac_sparsity,\n rtol=rtol, atol=atol,\n ))\n t1 = time.perf_counter()\n print(t1 - t0)\n\n system_energy = E(*y0)\n for y in solutions[i].y.T:\n solution_energy = E(*y)\n error = np.abs(solution_energy - system_energy)\n if strict and error > error_max:\n message = f'Energy drift of {error_max} exceeds maximum {error_max}'\n raise ValueError(message)\n\n return solutions\n\n\ndef fill_xy(solutions: list[OdeSolution]) -> tuple[\n np.ndarray, np.ndarray, np.ndarray, np.ndarray,\n]:\n x_1 = l_1*np.sin(solutions[0].y[0])\n x_2 = l_2*np.sin(solutions[0].y[1]) + x_1\n y_1 = -l_1*np.cos(solutions[0].y[0])\n y_2 = -l_2*np.cos(solutions[0].y[1]) + y_1\n return x_1, x_2, y_1, y_2\n\n\ndef plot(\n solutions: list[OdeSolution],\n x_1: np.ndarray, x_2: np.ndarray,\n y_1: np.ndarray, y_2: np.ndarray,\n) -> None:\n fig, ax = plt.subplots()\n ax.plot(solutions[0].y[0], solutions[0].y[1])\n ax.set_xlabel(r\"$\\theta_1\\ [rad]$\")\n ax.set_ylabel(r\"$\\theta_2\\ [rad]$\")\n\n fig, ax = plt.subplots()\n ax.plot(x_1, y_1)\n ax.set_xlabel('x_1')\n ax.set_ylabel('y_1')\n\n fig, ax = plt.subplots()\n ax.plot(x_2, y_2)\n ax.set_xlabel('x_2')\n ax.set_ylabel('y_2')\n\n fig, ax = plt.subplots()\n ax.plot(x_1, y_1, label='y_1')\n ax.plot(x_2, y_2, label='y_2')\n ax.legend()\n\n\ndef main() -> None:\n n_initial_conditions = 3 # 50\n rand = np.random.default_rng(seed=0)\n\n theta_1_0, theta_2_0, theta_1_dot_0, theta_2_dot_0 = initial_conditions(\n E=6.09, n_initial_conditions=n_initial_conditions, rand=rand,\n )\n solutions = solve_motion_equations(\n theta_1_0=theta_1_0, theta_1_dot_0=theta_1_dot_0,\n theta_2_0=theta_2_0, theta_2_dot_0=theta_2_dot_0,\n error_max=0.1,\n )\n plot_poincaré_section(solutions)\n x_1, x_2, y_1, y_2 = fill_xy(solutions)\n plot(solutions, x_1, x_2, y_1, y_2)\n plt.show()\n\n\nif __name__ == '__main__':\n main()", "answer_url": "https://scicomp.stackexchange.com/a/45267", "author": "Reinderien", "author_url": "https://scicomp.stackexchange.com/users/41212/reinderien", "author_user_type": "registered", "content_license": "CC BY-SA 4.0", "created_at": "2025-10-26T03:59:29+00:00", "is_accepted": false, "provenance": {"attribution_required": true, "collected_at": "2026-10-01T03:28:47.033952+00:00", "license": "CC BY-SA 4.0", "license_url": "https://creativecommons.org/licenses/by-sa/4.0/", "raw_file": "raw/codex_api_v1/cb6e5f564c84a92c4e121c0637c99292018f8935652e423630f2001099a2410f_1790825327386675300_0.json", "raw_sha256": "1a957289f56621bebd21cb45dd99baaedb2c3bc11cb28d700558e4bbcaa19695", "source_api": "Stack Exchange API 2.3", "source_url": "https://api.stackexchange.com/2.3/questions/45545;45541;45538;45533;45532;45530;45523;45513;45510;45507;45499;45493;45488;45487;45479;45474;45472;45463;45461;45447;45444;45436;45428;45425;45424;45423;45422;45416;45414;45410;45404;45401;45396;45391;45389;45387;45380;45377;45376;45375;45369;45366;45365;45363;45362;45359;45350;45347;45344;45336;45334;45331;45326;45322;45316;45313;45311;45309;45305;45302;45300;45291;45289;45285;45276;45269;45263;45262;45261;45253;45247;45246;45238;45236;45230;45229;45208;45201;45200;45185;45183;45171;45167;45165;45158;45154;45146;45141;45139;45134;45129;45127;45122;45114;45112;45108;45105;45100;45098;45096/answers?filter=withbody&order=asc&page=1&pagesize=100&site=scicomp&sort=creation", "transformation": "API HTML retained; mechanical HTML-to-text; no LLM rewriting"}, "question_id": 45171, "revision_attribution": [{"content_license": "CC BY-SA 4.0", "contributor": {"display_name": "Reinderien", "profile_url": "https://scicomp.stackexchange.com/users/41212/reinderien", "user_type": "registered"}, "created_at": "2025-10-26T03:59:29+00:00", "raw_file": "raw/codex_api_v1/932c57dde4e92dfe2af05c19643e98b6494a44cb31534e6703e92349c4e07eb4_1790825333093747000_0.json", "raw_sha256": "646e25c86dbb0c4df1ebe365442e733a253ac993340d2ed18168bbf783f75cac", "revision_guid": "1952C7E4-BD5B-442D-86E8-1FF1EBB10DF9", "revision_number": 1, "revision_type": "single_user", "revision_url": "https://scicomp.stackexchange.com/revisions/1952C7E4-BD5B-442D-86E8-1FF1EBB10DF9/view-source"}], "score": 3, "updated_at": "2025-10-26T03:59:29+00:00"}, "selection_rule": "accepted; otherwise maximum score >= 1, tie lowest ID; no correctness label", "split": "validation", "thread": {"accepted_answer_id": null, "answers": [{"answer_html": "<blockquote>\n<p>Poincaré section obtained by conditions in the code</p>\n</blockquote>\n<p>I very much cannot reproduce what you've generated, after refactoring - I suspect due to moving to the more sophisticated LSODA and its accompanying parameters. I get results more similar to the resonances seen in e.g. <a href=\"https://duetosymmetry.com/tool/poincare-section-clicker-toy/\" rel=\"nofollow noreferrer\">Leo Stein's Poincaré sections</a>.</p>\n<blockquote>\n<p>The code can also be improved, so if anyone has any suggestions</p>\n</blockquote>\n<p>Well... There's a lot. Since this is the Computational Science site and not the Code Review site, I will elide much of my feedback by saying "improve your Python". For the numerics:</p>\n<ul>\n<li>Your <code>g</code> is both wrong and unnecessary; get the correct value from Scipy instead</li>\n<li>Don't use the <code>math</code> module; stick to Numpy</li>\n<li><code>f()</code> is both pretty slow and wholly illegible. It can be somewhat sped up by identifying common expressions, and should be broken up into many more lines. The same applies to <code>E()</code>.</li>\n<li>The current parameters to <code>solve_ivp</code> produce a very slow solution. Switch to LSODA and go easy on those tolerances.</li>\n<li>For plotting, only call <code>plt.show()</code> once.</li>\n</ul>\n<p><a href=\"https://i.sstatic.net/INKhjIWk.png\" rel=\"nofollow noreferrer\"><img src=\"https://i.sstatic.net/INKhjIWk.png\" alt=\"poincare\" /></a></p>\n<pre class=\"lang-py prettyprint-override\"><code>import time\n\nimport numpy as np\nimport matplotlib.pyplot as plt\nfrom scipy.integrate import solve_ivp, OdeSolution\n\nfrom scipy.constants import g\nfrom scipy.integrate._ivp.ivp import OdeResult\n\nl_1 = 0.1\nl_2 = 0.5\nm_1 = 0.8\nm_2 = 0.3\n\n\ndef initial_conditions(\n E: float, n_initial_conditions: int, rand: np.random.Generator,\n) -> tuple[\n np.ndarray, np.ndarray, np.ndarray, np.ndarray,\n]:\n theta_1_0 = np.zeros(n_initial_conditions)\n theta_2_0 = np.linspace(-0.5*np.pi, 0.5*np.pi, n_initial_conditions)\n theta_2_dot_0 = np.zeros(n_initial_conditions)\n\n abs_t10 = np.sqrt(\n 2*(\n (\n E + l_2*m_2*g*np.cos(theta_2_0)\n )/(m_1 + m_2)\n - l_2*g\n )\n )/l_1\n\n theta_1_dot_0 = abs_t10*rand.choice((-1, 1), size=n_initial_conditions)\n\n return theta_1_0, theta_2_0, theta_1_dot_0, theta_2_dot_0\n\n\ndef f(t: float, y: np.ndarray) -> tuple[\n float, float, float, float,\n]:\n y0, y1, y2, y3 = y\n sin_y01 = 2*np.sin(y0 - y1)\n cos_y01 = np.cos(y0 - y1)\n m12m2 = 2*m_1 + m_2\n den = m12m2 - m_2*np.cos(2*y0 - 2*y1)\n y22l1 = y2**2*l_1\n y32l2 = l_2*y3**2\n\n res = (\n y2,\n y3,\n (\n -g*np.sin(y0)*m12m2 - m_2*(\n g*np.sin(y0 - 2*y1)\n + sin_y01*(\n y22l1*cos_y01 + y32l2\n )\n )\n ) / (l_1*den),\n (\n sin_y01*(\n (m_1 + m_2)*(\n np.cos(y0)*g + y22l1\n )\n + m_2*y32l2*cos_y01\n )\n ) / (l_2*den),\n )\n return res\n\n\ndef E(y_1: float, y_2: float, y_3: float, y_4: float) -> float:\n """energy of the system"""\n total_energy = (\n 0.5*l_1**2*y_3**2*(m_1 + m_2)\n + 0.5*m_2*l_2**2*y_4**2\n + m_2*l_1*l_2*y_3*y_4*np.cos(y_1 - y_2)\n - l_1*g*np.cos(y_1)*(m_1 + m_2)\n - l_2*m_2*g*np.cos(y_2)\n + (l_1 + l_2)*g*(m_1 + m_2)\n )\n return total_energy\n\n\ndef plot_poincaré_section(solutions: list[OdeSolution]) -> plt.Figure:\n theta_2 = []\n theta_2_dot = []\n colors = []\n\n color = (\n "#2C3E50", "#3A3F64", "#484078", "#56428D", "#6B469E",\n "#804AAF", "#954EBF", "#A753C4", "#BA58C8", "#CE5DCD",\n "#E062C9", "#E971B4", "#F1809F", "#F98F8A", "#FFA07A",\n "#FF9C65", "#FF9850", "#FF943B", "#FF9026", "#FF8C11",\n "#F97F0D", "#F3730A", "#ED6606", "#E75A03", "#E04E00",\n "#D4431E", "#C8383C", "#BC2D5A", "#B02178", "#A41596",\n "#9710A3", "#880EA7", "#790CAB", "#6A0AAF", "#5C08B2",\n "#4D06B6", "#3E04BA", "#2F02BD", "#2000C1", "#1800B8",\n "#1000AF", "#0800A6", "#00009D", "#00008F", "#000081",\n "#0B006C", "#160057", "#210043", "#2C002E", "#37001A"\n )\n\n for solution, colori in zip(solutions, color):\n n_points = len(solution.t)\n y0, y1, y2, y3 = solution.y\n\n for j in range(n_points - 1):\n if y0[j] <= 0 <= y0[j + 1]:\n colors.append(colori)\n theta_2.append(y1[j + 1])\n theta_2_dot.append(y3[j + 1])\n\n fig, ax = plt.subplots()\n\n for m in range(len(theta_2)):\n ax.scatter((theta_2[m] + np.pi) % (2 * np.pi) - np.pi, theta_2_dot[m], c=colors[m], s=0.1)\n\n ax.set_xlabel(r"<span class=\"math-container\">$\\theta_2\\ [rad]$</span>")\n ax.set_ylabel(r"<span class=\"math-container\">$\\dot{\\theta_2}\\ [rad \\cdot s^{-1}]$</span>")\n return fig\n\n\ndef solve_motion_equations(\n theta_1_0: np.ndarray,\n theta_2_0: np.ndarray,\n theta_1_dot_0: np.ndarray,\n theta_2_dot_0: np.ndarray,\n strict: bool = True,\n error_max: float = 1e-3,\n rtol: float = 1e-4, atol: float = 1e-6,\n) -> list[OdeResult]:\n """solving the equations of motion"""\n solutions = []\n\n # defining time interval\n t_span = (0, 750)\n jac_sparsity = np.array(( # All methods but LSODA\n (0, 0, 1, 0),\n (0, 0, 0, 1),\n (1, 1, 1, 1),\n (1, 1, 1, 1),\n ))\n # LSODA band parameters only reduce the Jacobian by one element; probably not worth it\n\n for i in range(len(theta_1_0)):\n y0 = (theta_1_0[i], theta_2_0[i], theta_1_dot_0[i], theta_2_dot_0[i])\n t0 = time.perf_counter()\n solutions.append(solve_ivp(\n fun=f, t_span=t_span, y0=y0, dense_output=True, method='LSODA',\n # jac_sparsity=jac_sparsity,\n rtol=rtol, atol=atol,\n ))\n t1 = time.perf_counter()\n print(t1 - t0)\n\n system_energy = E(*y0)\n for y in solutions[i].y.T:\n solution_energy = E(*y)\n error = np.abs(solution_energy - system_energy)\n if strict and error > error_max:\n message = f'Energy drift of {error_max} exceeds maximum {error_max}'\n raise ValueError(message)\n\n return solutions\n\n\ndef fill_xy(solutions: list[OdeSolution]) -> tuple[\n np.ndarray, np.ndarray, np.ndarray, np.ndarray,\n]:\n x_1 = l_1*np.sin(solutions[0].y[0])\n x_2 = l_2*np.sin(solutions[0].y[1]) + x_1\n y_1 = -l_1*np.cos(solutions[0].y[0])\n y_2 = -l_2*np.cos(solutions[0].y[1]) + y_1\n return x_1, x_2, y_1, y_2\n\n\ndef plot(\n solutions: list[OdeSolution],\n x_1: np.ndarray, x_2: np.ndarray,\n y_1: np.ndarray, y_2: np.ndarray,\n) -> None:\n fig, ax = plt.subplots()\n ax.plot(solutions[0].y[0], solutions[0].y[1])\n ax.set_xlabel(r"<span class=\"math-container\">$\\theta_1\\ [rad]$</span>")\n ax.set_ylabel(r"<span class=\"math-container\">$\\theta_2\\ [rad]$</span>")\n\n fig, ax = plt.subplots()\n ax.plot(x_1, y_1)\n ax.set_xlabel('x_1')\n ax.set_ylabel('y_1')\n\n fig, ax = plt.subplots()\n ax.plot(x_2, y_2)\n ax.set_xlabel('x_2')\n ax.set_ylabel('y_2')\n\n fig, ax = plt.subplots()\n ax.plot(x_1, y_1, label='y_1')\n ax.plot(x_2, y_2, label='y_2')\n ax.legend()\n\n\ndef main() -> None:\n n_initial_conditions = 3 # 50\n rand = np.random.default_rng(seed=0)\n\n theta_1_0, theta_2_0, theta_1_dot_0, theta_2_dot_0 = initial_conditions(\n E=6.09, n_initial_conditions=n_initial_conditions, rand=rand,\n )\n solutions = solve_motion_equations(\n theta_1_0=theta_1_0, theta_1_dot_0=theta_1_dot_0,\n theta_2_0=theta_2_0, theta_2_dot_0=theta_2_dot_0,\n error_max=0.1,\n )\n plot_poincaré_section(solutions)\n x_1, x_2, y_1, y_2 = fill_xy(solutions)\n plot(solutions, x_1, x_2, y_1, y_2)\n plt.show()\n\n\nif __name__ == '__main__':\n main()\n</code></pre>\n", "answer_id": 45267, "answer_text": "Poincaré section obtained by conditions in the code\n\n\n\n\n\n\n\nI very much cannot reproduce what you've generated, after refactoring - I suspect due to moving to the more sophisticated LSODA and its accompanying parameters. I get results more similar to the resonances seen in e.g. Leo Stein's Poincaré sections (https://duetosymmetry.com/tool/poincare-section-clicker-toy/).\n\n\n\n\n\n\n\nThe code can also be improved, so if anyone has any suggestions\n\n\n\n\n\n\n\nWell... There's a lot. Since this is the Computational Science site and not the Code Review site, I will elide much of my feedback by saying \"improve your Python\". For the numerics:\n\n\n\n\n\nYour g is both wrong and unnecessary; get the correct value from Scipy instead\n\n\n\n\nDon't use the math module; stick to Numpy\n\n\n\n\nf() is both pretty slow and wholly illegible. It can be somewhat sped up by identifying common expressions, and should be broken up into many more lines. The same applies to E().\n\n\n\n\nThe current parameters to solve_ivp produce a very slow solution. Switch to LSODA and go easy on those tolerances.\n\n\n\n\nFor plotting, only call plt.show() once.\n\n\n\n\n\n[image: poincare; source: https://i.sstatic.net/INKhjIWk.png] (https://i.sstatic.net/INKhjIWk.png)\n\n\n\n\nimport time\n\nimport numpy as np\nimport matplotlib.pyplot as plt\nfrom scipy.integrate import solve_ivp, OdeSolution\n\nfrom scipy.constants import g\nfrom scipy.integrate._ivp.ivp import OdeResult\n\nl_1 = 0.1\nl_2 = 0.5\nm_1 = 0.8\nm_2 = 0.3\n\n\ndef initial_conditions(\n E: float, n_initial_conditions: int, rand: np.random.Generator,\n) -> tuple[\n np.ndarray, np.ndarray, np.ndarray, np.ndarray,\n]:\n theta_1_0 = np.zeros(n_initial_conditions)\n theta_2_0 = np.linspace(-0.5*np.pi, 0.5*np.pi, n_initial_conditions)\n theta_2_dot_0 = np.zeros(n_initial_conditions)\n\n abs_t10 = np.sqrt(\n 2*(\n (\n E + l_2*m_2*g*np.cos(theta_2_0)\n )/(m_1 + m_2)\n - l_2*g\n )\n )/l_1\n\n theta_1_dot_0 = abs_t10*rand.choice((-1, 1), size=n_initial_conditions)\n\n return theta_1_0, theta_2_0, theta_1_dot_0, theta_2_dot_0\n\n\ndef f(t: float, y: np.ndarray) -> tuple[\n float, float, float, float,\n]:\n y0, y1, y2, y3 = y\n sin_y01 = 2*np.sin(y0 - y1)\n cos_y01 = np.cos(y0 - y1)\n m12m2 = 2*m_1 + m_2\n den = m12m2 - m_2*np.cos(2*y0 - 2*y1)\n y22l1 = y2**2*l_1\n y32l2 = l_2*y3**2\n\n res = (\n y2,\n y3,\n (\n -g*np.sin(y0)*m12m2 - m_2*(\n g*np.sin(y0 - 2*y1)\n + sin_y01*(\n y22l1*cos_y01 + y32l2\n )\n )\n ) / (l_1*den),\n (\n sin_y01*(\n (m_1 + m_2)*(\n np.cos(y0)*g + y22l1\n )\n + m_2*y32l2*cos_y01\n )\n ) / (l_2*den),\n )\n return res\n\n\ndef E(y_1: float, y_2: float, y_3: float, y_4: float) -> float:\n \"\"\"energy of the system\"\"\"\n total_energy = (\n 0.5*l_1**2*y_3**2*(m_1 + m_2)\n + 0.5*m_2*l_2**2*y_4**2\n + m_2*l_1*l_2*y_3*y_4*np.cos(y_1 - y_2)\n - l_1*g*np.cos(y_1)*(m_1 + m_2)\n - l_2*m_2*g*np.cos(y_2)\n + (l_1 + l_2)*g*(m_1 + m_2)\n )\n return total_energy\n\n\ndef plot_poincaré_section(solutions: list[OdeSolution]) -> plt.Figure:\n theta_2 = []\n theta_2_dot = []\n colors = []\n\n color = (\n \"#2C3E50\", \"#3A3F64\", \"#484078\", \"#56428D\", \"#6B469E\",\n \"#804AAF\", \"#954EBF\", \"#A753C4\", \"#BA58C8\", \"#CE5DCD\",\n \"#E062C9\", \"#E971B4\", \"#F1809F\", \"#F98F8A\", \"#FFA07A\",\n \"#FF9C65\", \"#FF9850\", \"#FF943B\", \"#FF9026\", \"#FF8C11\",\n \"#F97F0D\", \"#F3730A\", \"#ED6606\", \"#E75A03\", \"#E04E00\",\n \"#D4431E\", \"#C8383C\", \"#BC2D5A\", \"#B02178\", \"#A41596\",\n \"#9710A3\", \"#880EA7\", \"#790CAB\", \"#6A0AAF\", \"#5C08B2\",\n \"#4D06B6\", \"#3E04BA\", \"#2F02BD\", \"#2000C1\", \"#1800B8\",\n \"#1000AF\", \"#0800A6\", \"#00009D\", \"#00008F\", \"#000081\",\n \"#0B006C\", \"#160057\", \"#210043\", \"#2C002E\", \"#37001A\"\n )\n\n for solution, colori in zip(solutions, color):\n n_points = len(solution.t)\n y0, y1, y2, y3 = solution.y\n\n for j in range(n_points - 1):\n if y0[j] <= 0 <= y0[j + 1]:\n colors.append(colori)\n theta_2.append(y1[j + 1])\n theta_2_dot.append(y3[j + 1])\n\n fig, ax = plt.subplots()\n\n for m in range(len(theta_2)):\n ax.scatter((theta_2[m] + np.pi) % (2 * np.pi) - np.pi, theta_2_dot[m], c=colors[m], s=0.1)\n\n ax.set_xlabel(r\"$\\theta_2\\ [rad]$\")\n ax.set_ylabel(r\"$\\dot{\\theta_2}\\ [rad \\cdot s^{-1}]$\")\n return fig\n\n\ndef solve_motion_equations(\n theta_1_0: np.ndarray,\n theta_2_0: np.ndarray,\n theta_1_dot_0: np.ndarray,\n theta_2_dot_0: np.ndarray,\n strict: bool = True,\n error_max: float = 1e-3,\n rtol: float = 1e-4, atol: float = 1e-6,\n) -> list[OdeResult]:\n \"\"\"solving the equations of motion\"\"\"\n solutions = []\n\n # defining time interval\n t_span = (0, 750)\n jac_sparsity = np.array(( # All methods but LSODA\n (0, 0, 1, 0),\n (0, 0, 0, 1),\n (1, 1, 1, 1),\n (1, 1, 1, 1),\n ))\n # LSODA band parameters only reduce the Jacobian by one element; probably not worth it\n\n for i in range(len(theta_1_0)):\n y0 = (theta_1_0[i], theta_2_0[i], theta_1_dot_0[i], theta_2_dot_0[i])\n t0 = time.perf_counter()\n solutions.append(solve_ivp(\n fun=f, t_span=t_span, y0=y0, dense_output=True, method='LSODA',\n # jac_sparsity=jac_sparsity,\n rtol=rtol, atol=atol,\n ))\n t1 = time.perf_counter()\n print(t1 - t0)\n\n system_energy = E(*y0)\n for y in solutions[i].y.T:\n solution_energy = E(*y)\n error = np.abs(solution_energy - system_energy)\n if strict and error > error_max:\n message = f'Energy drift of {error_max} exceeds maximum {error_max}'\n raise ValueError(message)\n\n return solutions\n\n\ndef fill_xy(solutions: list[OdeSolution]) -> tuple[\n np.ndarray, np.ndarray, np.ndarray, np.ndarray,\n]:\n x_1 = l_1*np.sin(solutions[0].y[0])\n x_2 = l_2*np.sin(solutions[0].y[1]) + x_1\n y_1 = -l_1*np.cos(solutions[0].y[0])\n y_2 = -l_2*np.cos(solutions[0].y[1]) + y_1\n return x_1, x_2, y_1, y_2\n\n\ndef plot(\n solutions: list[OdeSolution],\n x_1: np.ndarray, x_2: np.ndarray,\n y_1: np.ndarray, y_2: np.ndarray,\n) -> None:\n fig, ax = plt.subplots()\n ax.plot(solutions[0].y[0], solutions[0].y[1])\n ax.set_xlabel(r\"$\\theta_1\\ [rad]$\")\n ax.set_ylabel(r\"$\\theta_2\\ [rad]$\")\n\n fig, ax = plt.subplots()\n ax.plot(x_1, y_1)\n ax.set_xlabel('x_1')\n ax.set_ylabel('y_1')\n\n fig, ax = plt.subplots()\n ax.plot(x_2, y_2)\n ax.set_xlabel('x_2')\n ax.set_ylabel('y_2')\n\n fig, ax = plt.subplots()\n ax.plot(x_1, y_1, label='y_1')\n ax.plot(x_2, y_2, label='y_2')\n ax.legend()\n\n\ndef main() -> None:\n n_initial_conditions = 3 # 50\n rand = np.random.default_rng(seed=0)\n\n theta_1_0, theta_2_0, theta_1_dot_0, theta_2_dot_0 = initial_conditions(\n E=6.09, n_initial_conditions=n_initial_conditions, rand=rand,\n )\n solutions = solve_motion_equations(\n theta_1_0=theta_1_0, theta_1_dot_0=theta_1_dot_0,\n theta_2_0=theta_2_0, theta_2_dot_0=theta_2_dot_0,\n error_max=0.1,\n )\n plot_poincaré_section(solutions)\n x_1, x_2, y_1, y_2 = fill_xy(solutions)\n plot(solutions, x_1, x_2, y_1, y_2)\n plt.show()\n\n\nif __name__ == '__main__':\n main()", "answer_url": "https://scicomp.stackexchange.com/a/45267", "author": "Reinderien", "author_url": "https://scicomp.stackexchange.com/users/41212/reinderien", "author_user_type": "registered", "content_license": "CC BY-SA 4.0", "created_at": "2025-10-26T03:59:29+00:00", "is_accepted": false, "provenance": {"attribution_required": true, "collected_at": "2026-10-01T03:28:47.033952+00:00", "license": "CC BY-SA 4.0", "license_url": "https://creativecommons.org/licenses/by-sa/4.0/", "raw_file": "raw/codex_api_v1/cb6e5f564c84a92c4e121c0637c99292018f8935652e423630f2001099a2410f_1790825327386675300_0.json", "raw_sha256": "1a957289f56621bebd21cb45dd99baaedb2c3bc11cb28d700558e4bbcaa19695", "source_api": "Stack Exchange API 2.3", "source_url": "https://api.stackexchange.com/2.3/questions/45545;45541;45538;45533;45532;45530;45523;45513;45510;45507;45499;45493;45488;45487;45479;45474;45472;45463;45461;45447;45444;45436;45428;45425;45424;45423;45422;45416;45414;45410;45404;45401;45396;45391;45389;45387;45380;45377;45376;45375;45369;45366;45365;45363;45362;45359;45350;45347;45344;45336;45334;45331;45326;45322;45316;45313;45311;45309;45305;45302;45300;45291;45289;45285;45276;45269;45263;45262;45261;45253;45247;45246;45238;45236;45230;45229;45208;45201;45200;45185;45183;45171;45167;45165;45158;45154;45146;45141;45139;45134;45129;45127;45122;45114;45112;45108;45105;45100;45098;45096/answers?filter=withbody&order=asc&page=1&pagesize=100&site=scicomp&sort=creation", "transformation": "API HTML retained; mechanical HTML-to-text; no LLM rewriting"}, "question_id": 45171, "revision_attribution": [{"content_license": "CC BY-SA 4.0", "contributor": {"display_name": "Reinderien", "profile_url": "https://scicomp.stackexchange.com/users/41212/reinderien", "user_type": "registered"}, "created_at": "2025-10-26T03:59:29+00:00", "raw_file": "raw/codex_api_v1/932c57dde4e92dfe2af05c19643e98b6494a44cb31534e6703e92349c4e07eb4_1790825333093747000_0.json", "raw_sha256": "646e25c86dbb0c4df1ebe365442e733a253ac993340d2ed18168bbf783f75cac", "revision_guid": "1952C7E4-BD5B-442D-86E8-1FF1EBB10DF9", "revision_number": 1, "revision_type": "single_user", "revision_url": "https://scicomp.stackexchange.com/revisions/1952C7E4-BD5B-442D-86E8-1FF1EBB10DF9/view-source"}], "score": 3, "updated_at": "2025-10-26T03:59:29+00:00"}], "domain": "computational_science", "external_links": ["https://duetosymmetry.com/tool/poincare-section-clicker-toy/", "https://github.com/DJopek/chaos/blob/main/double_pendulum.py", "https://i.sstatic.net/INKhjIWk.png", "https://i.sstatic.net/U0nOulED.png", "https://i.sstatic.net/oJaBUfA4.png"], "medical_sensitive": false, "patient_specific": false, "provenance": {"attribution_required": true, "collected_at": "2026-10-01T03:28:44.584971+00:00", "license": "CC BY-SA 4.0", "license_url": "https://creativecommons.org/licenses/by-sa/4.0/", "raw_file": "raw/codex_api_v1/272744bcc95b58699f7df6e6979b288bb8b00a80ab9c7982dbe170a6d0b63584_1790825325025901700_0.json", "raw_sha256": "27a81850a40920682c29ada76a2b55e80340a5c82764a3e9dde8a9c2ba23db4e", "source_api": "Stack Exchange API 2.3", "source_url": "https://api.stackexchange.com/2.3/search/advanced?answers=1&filter=withbody&order=desc&page=1&pagesize=100&site=scicomp&sort=creation", "transformation": "API HTML retained; mechanical HTML-to-text; no LLM rewriting"}, "question_author": "Dávid Jopek", "question_author_url": "https://scicomp.stackexchange.com/users/54327/d%c3%a1vid-jopek", "question_author_user_type": "registered", "question_created_at": "2025-07-15T18:16:26+00:00", "question_html": "<p>I wanted to try numerical analysis of a chaotic system. So I decided to write my own code for the Poincaré section of a double pendulum in Python. The code works and the Poincaré section should be correct, but I can't get a nice picture of the section, like you see in textbooks or on the internet. I don't know if the problem is in the choice of energies, the (number of) initial conditions, or the integration "time interval" and steps. Can someone more experienced advise me on a good approach to obtaining nice Poincaré sections? The code can also be improved, so if anyone has any suggestions for improvements, I would be very happy to hear them!</p>\n<p>Here is the code: <a href=\"https://github.com/DJopek/chaos/blob/main/double_pendulum.py\" rel=\"nofollow noreferrer\">https://github.com/DJopek/chaos/blob/main/double_pendulum.py</a></p>\n<pre class=\"lang-py prettyprint-override\"><code>import sys\nimport numpy as np\nimport matplotlib.pyplot as plt\nfrom scipy.integrate import solve_ivp\nfrom math import pi\nimport random\n\n# constants\nl_1 = 0.1\nl_2 = 0.5\nm_1 = 0.8\nm_2 = 0.3\ng = 9.81\n\n# Cauchy problem\n\n# y = [theta_1, theta_2, theta_1_dot, theta_2_dot]\n# y0 = [theta_1_0, theta_2_0, theta_1_dot_0, theta_2_dot_0]\n\n# generating initial conditions for some energy E\n\nnumber_of_initial_conditions = 50\n\ntheta_1_0 = []\n# theta_2_0 = []\ntheta_2_0 = np.linspace(-pi/2, pi/2, number_of_initial_conditions)\ntheta_1_dot_0 = []\ntheta_2_dot_0 = []\n\n# Energy = np.linspace(5,5.6,8) # [0,100] 1.\n# Energy = np.linspace(5.6,6.2,8) # [0,300] 2.\n# Energy = np.linspace(6.2,6.4,8) # [0,500] 3.\n# Energy = np.linspace(6.4,6.8,8) # [0,1000] 4.\n# Energy = np.linspace(5.9,6.3,8) # [0,700] 2-3.\n# Energy = np.linspace(6.3,6.8,12) # [0,1200] 3-4.\n# Energy = np.linspace(5.4,6.8,20) # [0,2000] 0.\n# Energy = np.linspace(5.6, 6.4, 16) # [0,700] 2_3\n# Energy = np.linspace(6.2,6.8,18) # [0,1200] 3_4.\n\n# Energy = [5.865]\nEnergy = [6.09]\n# Energy = [6.075]\n# Energy = [6.095]\n# Energy = [6.079]\n# Energy = [6.12]\n\n# number_of_initial_conditions = 1\n\ntotal_numbers_of_initial_conditions = number_of_initial_conditions*len(Energy)\n\ndef initial_conditions(E, number_of_initial_conditions):\n\n for i in range(number_of_initial_conditions):\n theta_1_0.append(0)\n # theta_2_0.append(np.random.uniform(-pi/4, pi/4))\n theta_2_dot_0.append(0)\n bucket = []\n bucket.append(np.sqrt(2*(E+l_1*g*(m_1+m_2)+l_2*m_2*g*np.cos(theta_2_0[i])-(l_1+l_2)*(m_1+m_2)*g)/(l_1**2*(m_1+m_2))))\n bucket.append(-np.sqrt(2*(E+l_1*g*(m_1+m_2)+l_2*m_2*g*np.cos(theta_2_0[i])-(l_1+l_2)*(m_1+m_2)*g)/(l_1**2*(m_1+m_2))))\n theta_1_dot_0.append(bucket[random.randint(0,1)])\n\nfor i in range(len(Energy)):\n initial_conditions(Energy[i], number_of_initial_conditions)\n\ndef f(t,y):\n return [y[2], \n y[3], \n (-g*np.sin(y[0])*(2*m_1+m_2)-g*m_2*np.sin(y[0]-2*y[1])-2*m_2*np.sin(y[0]-y[1])*(y[2]**2*l_1*np.cos(y[0]-y[1])+l_2*y[3]**2))/(l_1*(2*m_1+m_2-m_2*np.cos(2*y[0]-2*y[1]))), \n (2*np.sin(y[0]-y[1])*(np.cos(y[0])*g*(m_1+m_2)+l_1*(m_1+m_2)*y[2]**2+m_2*l_2*y[3]**2*np.cos(y[0]-y[1])))/(l_2*(2*m_1+m_2-m_2*np.cos(2*y[0]-2*y[1])))]\n\n# energy of the system\ndef E(y_1, y_2, y_3, y_4):\n total_energy = 0.5*l_1**2*y_3**2*(m_1+m_2)+0.5*m_2*l_2**2*y_4**2+m_2*l_1*l_2*y_3*y_4*np.cos(y_1-y_2)-l_1*g*np.cos(y_1)*(m_1+m_2)-l_2*m_2*g*np.cos(y_2)+(l_1+l_2)*g*(m_1+m_2)\n return total_energy\n\n# Poincaré section\ndef Poincare_section(total_numbers_of_initial_conditions):\n theta_2 = []\n theta_2_dot = []\n\n colors = []\n\n color = [\n "#2C3E50", "#3A3F64", "#484078", "#56428D", "#6B469E",\n "#804AAF", "#954EBF", "#A753C4", "#BA58C8", "#CE5DCD",\n "#E062C9", "#E971B4", "#F1809F", "#F98F8A", "#FFA07A",\n "#FF9C65", "#FF9850", "#FF943B", "#FF9026", "#FF8C11",\n "#F97F0D", "#F3730A", "#ED6606", "#E75A03", "#E04E00",\n "#D4431E", "#C8383C", "#BC2D5A", "#B02178", "#A41596",\n "#9710A3", "#880EA7", "#790CAB", "#6A0AAF", "#5C08B2",\n "#4D06B6", "#3E04BA", "#2F02BD", "#2000C1", "#1800B8",\n "#1000AF", "#0800A6", "#00009D", "#00008F", "#000081",\n "#0B006C", "#160057", "#210043", "#2C002E", "#37001A"\n ]\n\n for i in range(total_numbers_of_initial_conditions):\n # color = ["#"+''.join([random.choice('0123456789ABCDEF') for r in range(6)])\n # for s in range(1)]\n\n number_of_points = len(solutions[i].t)\n\n for j in range(number_of_points-1):\n if solutions[i].y[0][j] <= 0 and solutions[i].y[0][j+1] >= 0:\n colors.append(color[i])\n theta_2.append(solutions[i].y[1][j+1])\n theta_2_dot.append(solutions[i].y[3][j+1])\n\n for m in range(len(theta_2)):\n plt.scatter((theta_2[m]+np.pi)%(2 * np.pi) - np.pi, theta_2_dot[m], c=colors[m], s=0.1)\n\n plt.xlabel(r"<span class=\"math-container\">$\\theta_2\\ [rad]$</span>")\n plt.ylabel(r"<span class=\"math-container\">$\\dot{\\theta_2}\\ [rad \\cdot s^{-1}]$</span>")\n plt.show()\n\n# solving the equations of motion\nsolutions = []\n\nerror = 0.001\n\n# defining time interval\nt_span = [0,750]\n\nfor i in range(total_numbers_of_initial_conditions):\n y0 = [theta_1_0[i], theta_2_0[i], theta_1_dot_0[i], theta_2_dot_0[i]]\n system_energy = E(y0[0], y0[1], y0[2], y0[3])\n print(y0)\n print(system_energy)\n solutions.append(solve_ivp(f, t_span, y0, dense_output=True, rtol = 1e-12, atol = 1e-14))\n number_of_points = len(solutions[i].t)\n \n for j in range(number_of_points):\n solution_energy = E(solutions[i].y[0][j], solutions[i].y[1][j], solutions[i].y[2][j], solutions[i].y[3][j])\n if np.abs(solution_energy - system_energy) > error:\n sys.exit('Maximum energy drift of {} exceeded.'.format(error))\n\nPoincare_section(total_numbers_of_initial_conditions)\n\nx_1 = []\nx_2 = []\ny_1 = []\ny_2 = []\n\nfor i in range(len(solutions[0].t)):\n x_1.append(l_1*np.sin(solutions[0].y[0][i]))\n x_2.append(x_1[i]+l_2*np.sin(solutions[0].y[1][i]))\n y_1.append(-l_1*np.cos(solutions[0].y[0][i]))\n y_2.append(y_1[i]-l_2*np.cos(solutions[0].y[1][i]))\n\nplt.plot(solutions[0].y[0], solutions[0].y[1])\nplt.xlabel(r"<span class=\"math-container\">$\\theta_1\\ [rad]$</span>")\nplt.ylabel(r"<span class=\"math-container\">$\\theta_2\\ [rad]$</span>")\nplt.show()\n\nplt.plot(x_1, y_1)\nplt.show()\nplt.plot(x_2, y_2)\nplt.show()\n\nplt.plot(x_1, y_1)\nplt.plot(x_2, y_2)\nplt.show()\n</code></pre>\n<p>Poincaré section obtained by conditions in the code:\n<a href=\"https://i.sstatic.net/U0nOulED.png\" rel=\"nofollow noreferrer\"><img src=\"https://i.sstatic.net/U0nOulED.png\" alt=\"Poincaré section obtained by conditions in the code\" /></a></p>\n<p>EDIT:\nPoincaré section obtained by conditions in the code with labels<a href=\"https://i.sstatic.net/oJaBUfA4.png\" rel=\"nofollow noreferrer\"><img src=\"https://i.sstatic.net/oJaBUfA4.png\" alt=\"Poincaré section obtained by conditions in the code with labels\" /></a></p>\n", "question_id": 45171, "question_license": "CC BY-SA 4.0", "question_score": 3, "question_text": "I wanted to try numerical analysis of a chaotic system. So I decided to write my own code for the Poincaré section of a double pendulum in Python. The code works and the Poincaré section should be correct, but I can't get a nice picture of the section, like you see in textbooks or on the internet. I don't know if the problem is in the choice of energies, the (number of) initial conditions, or the integration \"time interval\" and steps. Can someone more experienced advise me on a good approach to obtaining nice Poincaré sections? The code can also be improved, so if anyone has any suggestions for improvements, I would be very happy to hear them!\n\n\n\n\nHere is the code: https://github.com/DJopek/chaos/blob/main/double_pendulum.py (https://github.com/DJopek/chaos/blob/main/double_pendulum.py)\n\n\n\n\nimport sys\nimport numpy as np\nimport matplotlib.pyplot as plt\nfrom scipy.integrate import solve_ivp\nfrom math import pi\nimport random\n\n# constants\nl_1 = 0.1\nl_2 = 0.5\nm_1 = 0.8\nm_2 = 0.3\ng = 9.81\n\n# Cauchy problem\n\n# y = [theta_1, theta_2, theta_1_dot, theta_2_dot]\n# y0 = [theta_1_0, theta_2_0, theta_1_dot_0, theta_2_dot_0]\n\n# generating initial conditions for some energy E\n\nnumber_of_initial_conditions = 50\n\ntheta_1_0 = []\n# theta_2_0 = []\ntheta_2_0 = np.linspace(-pi/2, pi/2, number_of_initial_conditions)\ntheta_1_dot_0 = []\ntheta_2_dot_0 = []\n\n# Energy = np.linspace(5,5.6,8) # [0,100] 1.\n# Energy = np.linspace(5.6,6.2,8) # [0,300] 2.\n# Energy = np.linspace(6.2,6.4,8) # [0,500] 3.\n# Energy = np.linspace(6.4,6.8,8) # [0,1000] 4.\n# Energy = np.linspace(5.9,6.3,8) # [0,700] 2-3.\n# Energy = np.linspace(6.3,6.8,12) # [0,1200] 3-4.\n# Energy = np.linspace(5.4,6.8,20) # [0,2000] 0.\n# Energy = np.linspace(5.6, 6.4, 16) # [0,700] 2_3\n# Energy = np.linspace(6.2,6.8,18) # [0,1200] 3_4.\n\n# Energy = [5.865]\nEnergy = [6.09]\n# Energy = [6.075]\n# Energy = [6.095]\n# Energy = [6.079]\n# Energy = [6.12]\n\n# number_of_initial_conditions = 1\n\ntotal_numbers_of_initial_conditions = number_of_initial_conditions*len(Energy)\n\ndef initial_conditions(E, number_of_initial_conditions):\n\n for i in range(number_of_initial_conditions):\n theta_1_0.append(0)\n # theta_2_0.append(np.random.uniform(-pi/4, pi/4))\n theta_2_dot_0.append(0)\n bucket = []\n bucket.append(np.sqrt(2*(E+l_1*g*(m_1+m_2)+l_2*m_2*g*np.cos(theta_2_0[i])-(l_1+l_2)*(m_1+m_2)*g)/(l_1**2*(m_1+m_2))))\n bucket.append(-np.sqrt(2*(E+l_1*g*(m_1+m_2)+l_2*m_2*g*np.cos(theta_2_0[i])-(l_1+l_2)*(m_1+m_2)*g)/(l_1**2*(m_1+m_2))))\n theta_1_dot_0.append(bucket[random.randint(0,1)])\n\nfor i in range(len(Energy)):\n initial_conditions(Energy[i], number_of_initial_conditions)\n\ndef f(t,y):\n return [y[2], \n y[3], \n (-g*np.sin(y[0])*(2*m_1+m_2)-g*m_2*np.sin(y[0]-2*y[1])-2*m_2*np.sin(y[0]-y[1])*(y[2]**2*l_1*np.cos(y[0]-y[1])+l_2*y[3]**2))/(l_1*(2*m_1+m_2-m_2*np.cos(2*y[0]-2*y[1]))), \n (2*np.sin(y[0]-y[1])*(np.cos(y[0])*g*(m_1+m_2)+l_1*(m_1+m_2)*y[2]**2+m_2*l_2*y[3]**2*np.cos(y[0]-y[1])))/(l_2*(2*m_1+m_2-m_2*np.cos(2*y[0]-2*y[1])))]\n\n# energy of the system\ndef E(y_1, y_2, y_3, y_4):\n total_energy = 0.5*l_1**2*y_3**2*(m_1+m_2)+0.5*m_2*l_2**2*y_4**2+m_2*l_1*l_2*y_3*y_4*np.cos(y_1-y_2)-l_1*g*np.cos(y_1)*(m_1+m_2)-l_2*m_2*g*np.cos(y_2)+(l_1+l_2)*g*(m_1+m_2)\n return total_energy\n\n# Poincaré section\ndef Poincare_section(total_numbers_of_initial_conditions):\n theta_2 = []\n theta_2_dot = []\n\n colors = []\n\n color = [\n \"#2C3E50\", \"#3A3F64\", \"#484078\", \"#56428D\", \"#6B469E\",\n \"#804AAF\", \"#954EBF\", \"#A753C4\", \"#BA58C8\", \"#CE5DCD\",\n \"#E062C9\", \"#E971B4\", \"#F1809F\", \"#F98F8A\", \"#FFA07A\",\n \"#FF9C65\", \"#FF9850\", \"#FF943B\", \"#FF9026\", \"#FF8C11\",\n \"#F97F0D\", \"#F3730A\", \"#ED6606\", \"#E75A03\", \"#E04E00\",\n \"#D4431E\", \"#C8383C\", \"#BC2D5A\", \"#B02178\", \"#A41596\",\n \"#9710A3\", \"#880EA7\", \"#790CAB\", \"#6A0AAF\", \"#5C08B2\",\n \"#4D06B6\", \"#3E04BA\", \"#2F02BD\", \"#2000C1\", \"#1800B8\",\n \"#1000AF\", \"#0800A6\", \"#00009D\", \"#00008F\", \"#000081\",\n \"#0B006C\", \"#160057\", \"#210043\", \"#2C002E\", \"#37001A\"\n ]\n\n for i in range(total_numbers_of_initial_conditions):\n # color = [\"#\"+''.join([random.choice('0123456789ABCDEF') for r in range(6)])\n # for s in range(1)]\n\n number_of_points = len(solutions[i].t)\n\n for j in range(number_of_points-1):\n if solutions[i].y[0][j] <= 0 and solutions[i].y[0][j+1] >= 0:\n colors.append(color[i])\n theta_2.append(solutions[i].y[1][j+1])\n theta_2_dot.append(solutions[i].y[3][j+1])\n\n for m in range(len(theta_2)):\n plt.scatter((theta_2[m]+np.pi)%(2 * np.pi) - np.pi, theta_2_dot[m], c=colors[m], s=0.1)\n\n plt.xlabel(r\"$\\theta_2\\ [rad]$\")\n plt.ylabel(r\"$\\dot{\\theta_2}\\ [rad \\cdot s^{-1}]$\")\n plt.show()\n\n# solving the equations of motion\nsolutions = []\n\nerror = 0.001\n\n# defining time interval\nt_span = [0,750]\n\nfor i in range(total_numbers_of_initial_conditions):\n y0 = [theta_1_0[i], theta_2_0[i], theta_1_dot_0[i], theta_2_dot_0[i]]\n system_energy = E(y0[0], y0[1], y0[2], y0[3])\n print(y0)\n print(system_energy)\n solutions.append(solve_ivp(f, t_span, y0, dense_output=True, rtol = 1e-12, atol = 1e-14))\n number_of_points = len(solutions[i].t)\n \n for j in range(number_of_points):\n solution_energy = E(solutions[i].y[0][j], solutions[i].y[1][j], solutions[i].y[2][j], solutions[i].y[3][j])\n if np.abs(solution_energy - system_energy) > error:\n sys.exit('Maximum energy drift of {} exceeded.'.format(error))\n\nPoincare_section(total_numbers_of_initial_conditions)\n\nx_1 = []\nx_2 = []\ny_1 = []\ny_2 = []\n\nfor i in range(len(solutions[0].t)):\n x_1.append(l_1*np.sin(solutions[0].y[0][i]))\n x_2.append(x_1[i]+l_2*np.sin(solutions[0].y[1][i]))\n y_1.append(-l_1*np.cos(solutions[0].y[0][i]))\n y_2.append(y_1[i]-l_2*np.cos(solutions[0].y[1][i]))\n\nplt.plot(solutions[0].y[0], solutions[0].y[1])\nplt.xlabel(r\"$\\theta_1\\ [rad]$\")\nplt.ylabel(r\"$\\theta_2\\ [rad]$\")\nplt.show()\n\nplt.plot(x_1, y_1)\nplt.show()\nplt.plot(x_2, y_2)\nplt.show()\n\nplt.plot(x_1, y_1)\nplt.plot(x_2, y_2)\nplt.show()\n\n\n\n\n\nPoincaré section obtained by conditions in the code:\n[image: Poincaré section obtained by conditions in the code; source: https://i.sstatic.net/U0nOulED.png] (https://i.sstatic.net/U0nOulED.png)\n\n\n\n\nEDIT:\nPoincaré section obtained by conditions in the code with labels[image: Poincaré section obtained by conditions in the code with labels; source: https://i.sstatic.net/oJaBUfA4.png] (https://i.sstatic.net/oJaBUfA4.png)", "revision_attribution": [{"content_license": "CC BY-SA 4.0", "contributor": {"display_name": "Dávid Jopek", "profile_url": "https://scicomp.stackexchange.com/users/54327/d%c3%a1vid-jopek", "user_type": "registered"}, "created_at": "2025-07-15T18:16:26+00:00", "raw_file": "raw/codex_api_v1/8b02bbba3a1a1a633af4866e7fea0bd6789190b52c4b8380b0538b5c670e7131_1790825346551636200_0.json", "raw_sha256": "33050439b68e1f78cef06bf4c2ce07c0ec45a5fdf656fa1a9efd04691108f5e0", "revision_guid": "650653BB-F748-411A-AA5A-3EB629C84696", "revision_number": 1, "revision_type": "single_user", "revision_url": "https://scicomp.stackexchange.com/revisions/650653BB-F748-411A-AA5A-3EB629C84696/view-source"}, {"content_license": "CC BY-SA 4.0", "contributor": {"display_name": "Dávid Jopek", "profile_url": "https://scicomp.stackexchange.com/users/54327/d%c3%a1vid-jopek", "user_type": "registered"}, "created_at": "2025-07-15T20:05:50+00:00", "raw_file": "raw/codex_api_v1/8b02bbba3a1a1a633af4866e7fea0bd6789190b52c4b8380b0538b5c670e7131_1790825346551636200_0.json", "raw_sha256": "33050439b68e1f78cef06bf4c2ce07c0ec45a5fdf656fa1a9efd04691108f5e0", "revision_guid": "95B2D6D6-09DE-443D-9C99-ED7070A24120", "revision_number": 2, "revision_type": "single_user", "revision_url": "https://scicomp.stackexchange.com/revisions/95B2D6D6-09DE-443D-9C99-ED7070A24120/view-source"}, {"content_license": "CC BY-SA 4.0", "contributor": {"display_name": "Reinderien", "profile_url": "https://scicomp.stackexchange.com/users/41212/reinderien", "user_type": "registered"}, "created_at": "2025-10-26T02:04:23+00:00", "raw_file": "raw/codex_api_v1/932c57dde4e92dfe2af05c19643e98b6494a44cb31534e6703e92349c4e07eb4_1790825333093747000_0.json", "raw_sha256": "646e25c86dbb0c4df1ebe365442e733a253ac993340d2ed18168bbf783f75cac", "revision_guid": "9709635F-9F4A-425B-AB66-BE60E6D8B46F", "revision_number": 3, "revision_type": "single_user", "revision_url": "https://scicomp.stackexchange.com/revisions/9709635F-9F4A-425B-AB66-BE60E6D8B46F/view-source"}], "source_site": "scicomp", "source_url": "https://scicomp.stackexchange.com/questions/45171/poincar%c3%a9-section-for-double-pendulum-code-improvement", "split": "validation", "split_group": "381f73da7d816e1e3458beb0f1f8068c97e333253953f864c60d0e95d85a8757", "tags": ["python", "numerics", "ode", "computational-physics", "chaotic-systems"], "thread_id": "scicomp:45171", "title": "Poincaré section for double pendulum code improvement"}}
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