--- license: mit language: - en tags: - fractal - fractal-dimension - box-counting - hurst-exponent - self-affine - time-series - signal-processing - benchmark - ground-truth - synthetic - roughness - estimator-validation pretty_name: Fractal Dimension Benchmark — synthetic signals with exactly known dimension size_categories: - n<1K configs: - config_name: default data_files: data/signals.parquet - config_name: index data_files: data/signals_index.csv --- # Fractal Dimension Benchmark **Ground truth for fractal dimension estimators.** 76 synthetic time series whose box-counting dimension is known exactly — in closed form, per file, not in expectation. If you are building or testing a **fractal dimension estimator**, a **Hurst exponent estimator**, a **roughness measure**, a box-counting implementation, a variogram or detrended-fluctuation method, this is a test set where you already know every answer. Dimensions span **1.05 to 1.95** in steps of 0.05, at 19,684 samples each. Deterministic and reproducible bit for bit. Companion tool and derivation: [10.5281/zenodo.22052384](https://doi.org/10.5281/zenodo.22052384) --- ## Why fBm is not enough The usual test signal for a dimension estimator is **fractional Brownian motion**. Its box dimension is `2 − H`, which is correct **on average**. But any single realization you hand someone deviates from that by an unknown amount, so when your estimator reads 1.47 on a signal labelled 1.50, you cannot tell whether your estimator is off, or the realization is. You are checking a ruler against a stick that is roughly a metre, usually. These signals are deterministic. Given the file, the true value is a closed-form expression. Given the seed, the file regenerates bit for bit on any machine. --- ## The construction Each signal is a *closed-leg recursion*: replace a segment with three legs whose signed heights sum to one, then replace each leg with a scaled copy of the whole figure, repeatedly. With equal durations the box dimension is exactly ``` D = 1 + log₃( |d₁| + |d₂| + |d₃| ) ``` so to hit a target `D` you solve `Σ|dᵢ| = 3^(D−1)` subject to `Σdᵢ = 1`. The balanced solution ``` b = (3^(D−1) − 1) / 2 , s = 1 + b , legs = ( s/2 , −b , s/2 ) ``` keeps every leg under 1 in magnitude across the whole range, which is the condition for the limit curve to exist. This is Barnsley's affine fractal interpolation function (1986). The dimension formula is classical. Nothing here is claimed as new mathematics — what is offered is the test set. --- ## Two kinds of signal, and the difference matters **`shuffled_*` — 57 signals, 3 seeds at each dimension.** The leg *order* is shuffled at every subdivision. Because addition does not care about order, `Σ|dᵢ|` is untouched and the box dimension is **still exactly** the stated value. What changes is that the curve is no longer exactly self-similar, only statistically so — which averages away multifractality and makes these the right target for a plain single-number estimator. **`fixed_*` — 19 signals, one at each dimension.** The same legs in fixed order. Identical exact box dimension, but genuinely **multifractal**: roughness varies from point to point, so an increment-based estimator will legitimately disagree with the box dimension. These test whether your estimator *notices* that one number is not enough. An estimator that scores well on `fixed_*` without flagging the disagreement is not measuring what it claims to measure. --- ## Contents ``` data/signals.parquet all 76 signals, 1,495,984 rows, long format: signal_id, kind, target_dimension, true_dimension, seed, t, value data/signals_index.csv 76 rows, one per signal, with the leg coefficients ``` Dimensions run from **1.05 to 1.95** in steps of 0.05, at 19,684 samples per signal. The `t` column is the sample index; samples are evenly spaced by construction, so there is no clock time. Sort by `t` within a `signal_id` before measuring. --- ## Loading it ```python from datasets import load_dataset import numpy as np, pandas as pd ds = load_dataset("Trackertracker2/closed-leg-benchmark", split="train") df = ds.to_pandas() # one signal g = df[df.signal_id == "shuffled_D1.50_s1"] y = g.value.to_numpy() print(g.true_dimension.iloc[0]) # 1.500000000000 ``` Scoring your estimator across the whole shuffled set: ```python errs = [] for sid, g in df[df.kind == "shuffled"].groupby("signal_id"): y = g.sort_values("t").value.to_numpy().astype(float) errs.append(your_estimator(y) - g.true_dimension.iloc[0]) errs = np.abs(errs) print(f"mean |error| {errs.mean():.4f} worst {errs.max():.4f}") ``` Straight from parquet, without the `datasets` library: ```python df = pd.read_parquet("https://huggingface.co/datasets/Trackertracker2/" "closed-leg-benchmark/resolve/main/data/signals.parquet") ``` --- ## A reference score The estimator in the companion deposit — three variogram-of-order-*p* estimators at *p* = ½, 1, 2, taking the median — scores: | set | mean abs. error | worst | |---|---|---| | `shuffled_*` | **0.021** | 0.093 | | `fixed_*` | 0.088 | 0.164 | Its accuracy is **not uniform**. Errors are smallest around *D* = 1.6 and grow toward the top of the range, reaching about 0.09 near *D* = 1.95. That degradation is a property of the estimator, not of the benchmark, and surfacing it is the point. An estimator that does well at 1.5 and poorly at 1.9 is worth knowing about before you trust it on data whose answer you do not have. --- ## Caveats, stated plainly - These are **self-affine graphs**, not general rough signals. An estimator tuned to them may still behave differently on physical data. This benchmark tests correctness against known ground truth; it does not certify performance in the field. - **19,684 points** is a moderate record length. Estimators needing longer series will be penalised. That is fair, but it should be said. - Large errors on `fixed_*` are **expected**, not failures — those signals are multifractal by construction. Read them alongside whatever multifractality diagnostic your tool provides. - The signals occupy a **narrow structural class**. Good scores here are necessary for trusting an estimator, not sufficient. --- ## Citation The full tool, specification and derivation: - Roughness — a validated measure of texture for ordered signals. [10.5281/zenodo.22052384](https://doi.org/10.5281/zenodo.22052384) - The Closed-Leg Atlas — a browser for self-affine and self-similar curves. [10.5281/zenodo.22040679](https://doi.org/10.5281/zenodo.22040679) ```bibtex @software{koch_roughness_2026, author = {Koch, W. A.}, title = {Roughness: a validated measure of texture for ordered signals}, year = {2026}, doi = {10.5281/zenodo.22052384}, url = {https://doi.org/10.5281/zenodo.22052384} } ``` W. A. Koch · [ORCID 0009-0001-1341-7871](https://orcid.org/0009-0001-1341-7871) Prepared with AI assistance for numerical verification and code. Every value in the answer key is a closed-form expression, independently recomputed.