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Following grafts are taken from this article . I'm newbie to bootstrap and trying to implement the parametric, semiparametric and nonparametric bootstrapping bootstrapping for linear mixed model with R boot package. R Code Here is my R code: library(SASmixed) library(lme4) library(boot) fm1Cult <- lmer(drywt ~ In...
I am trying to model the effect that proximity to transportation infrastructure has on home prices. It is common (and reasonable) to use some type of spatial dependence model, such as the Spatial Durbin Model, $$y = \rho W y + X\beta + XW\lambda + \varepsilon $$ In this model, the outcome is a result of autocorrelat...
I have a database where experiments are stored. Each experiment has a flag, which says that it was properly registered. I am trying to show with a statistical test, that there is a trend in time that experients are increasingly properly registered (ratio column; rising from 56% in 2006 to 72% in 2011) > print(ascii(c,i...
(Asked this on MathOverflow, got redirected here) Say I want to create a histogram of $N$ samples from some simple compactly supported distribution on $\mathbb{R}$, where $N$ is very large, say $N = 10^{30}$. The histogram has $K$ disjoint bins, where $K$ is a more reasonable number like $10$ or $1000$. Obviously it's ...
My question is the following. Let's say I have two probability distributions: $f(x|b), g(x|c)$ $b$ and $c$ are discrete events while $x$ is a continuous variable, i.e., when the button b is pressed there is some distribution for the amount of rain fall the next day, $x$. When the button $c$ is pressed there is a diffe...
I've fit a model with lm(), and I now want to analyze it to see where it's overfit, etc... I'm imagining a plot that has the index of each observation on the x-axis, and the corresponding responses plotted as points on the y-axis. In (vertical) line with each point is a set of colored lines stacked on top of each other...
I am trying to get predictions for observations from a lme object. This is supposed to be quite straightforward. Yet, since I am get different types of errors for different trials, it seems to me I am missing something. My model is the following: model <- lme(log(child_mortality) ~ as.factor(cluster)*time + ...
I'm not sure how to ask this so I've written two versions: the short version is a pure stats question and the longer version explains my application. The short version: I have two sets of probabilities. The first set has a range from $5.8\times 10^{-3}$ to $2.52\times 10^{-9}$ and the second ranges from $3.51\times 10^...
I'm new to neural networks and machine learning and I was wondering how you use time series data to set the weights of a regular FNN, and how you use the ending weights to forecast the time series. In essence how can you transform the time series data into weights and back again for the output.
The regular linear regression model is easy to present; we are minimizing the sum of the least square errors: $$\sum_{i=1}^{n} ( y_i - \hat{y}_i)^2$$ How do we present the mixed model? More specifically, we are wondering about how to present the following simple model: lmefit = lmer(MathAch ~ SES + (1 |School) , MathS...
I have data with a starting (y) value that sequentially increments/decrements as (x) time measured in days passes. I found this link for creating a linear regression of the data http://www.easycalculation.com/statistics/regression.php I would like to automate the slope calculation in excel. Does anyone have an idea ...
$R = S + xx^t$ where $x \in \mathbb{R}^n$ and $R$ and $S$ are $n \times n$ covariance matrices. Is there anything I can say about the eigenvectors of $R$ and $S$? Or at least the largest eigenvalue and eigenvector pair of R? Is there a way to express the eigenvectors of $R$ in terms of $S$ and $x$?
If you roll a die 8 times, what is the probabiliy of getting both 1 and 2. $P(1) = .3$ $ P(2) = .1$ $P(3) = .15$ $ P(4) = .15$ $P(5) = .15$ $P(6) = .15$
What is background risk or baseline risk? and also what does APE refer to in EBM?
The question I am working with is: Setup a 90% confidence interval estimate for the average processing time. I gathered the information below from the spreadsheet $n = 27$ $\bar{X} =48.888$ Sample standard deviation $= 25.283$ $\sigma/\sqrt{n} = 4.871$ I am confused because I thought that to setup the confidence lev...
Question that seems complex but probably isn't. I'm comparing 10 metagenomes (http://en.wikipedia.org/wiki/Metagenomics), that are supposedly unrelated (they are 10 geographically separate locations), for a specific group of operons (http://en.wikipedia.org/wiki/Operon). I'm using Hidden Markov Model protein profiles ...
Imagine some paired data ${(x_i,y_i)}_{i=1}^n$ representing the results of two different measurement methods and the question is about the quantification of the bias between the two methods. Let us assume that the normality assumption is reasonable for $x_i-y_i$ hence we simply compute an estimate and a confidence inte...
I ran a simple randomized experiment with 1 control and 1 experimental condition (final N = 80). The dependent variable (frequency of a shown behavior) is clearly not normally distributed so I thought about bootstrapping my analysis (an independent t-test). The t-test without bootstrapping resulted in a significant eff...
Practical situation: I’ve got 120 days of data collected during rainy season. On an average it rained on 52.5 of those days. What is the probability of it raining at least once in 30 days? If it rains at least once in 30 days, then what is the probability that it continues to rain for 4 days in a stretch in those 30 d...
I have pairs of timeseries that are estimating the same quantity over the years. It is some survival data: number of "dead" subjects during the year over the number of total subjects at the beginning of the year. The underlying data is regularly revised, so I need to assess the effect of the changes. Below an example i...
I have 15 supervisors who have ranked 21 employees from 1 (best) to 21 (worst), but the rankings are strictly ordinal-there is no interval data available. Can I compare any two (or more) supervisors' rankings and determine how closely their evaluations are aligned? I'll be using Excel 2010 (or pen/paper) for my analys...
I am using rlm in the R MASS package to regress a multivariate linear model. It works well for a number of samples but I am getting quasi-null coefficients for a particular model: Call: rlm(formula = Y ~ X1 + X2 + X3 + X4, data = mymodel, maxit = 50, na.action = na.omit) Residuals: Min 1Q Median ...
I am self-studying this book and need assistance on one of the exercises, specifically 3.9.2: Suppose that, for a certain population, we can predict log earnings from log height as follows: A person who is 66 inches tall is predicted to have earnings of $30K. Every increase of 1% in height corresponds to a predicted i...
I am an occasional data-mining methods user and I simply wonder what are the main challenges and research directions for method designers. Thanks for your expertise, Peuhp
I once heard that, generative learning, discriminative learning and max-margin learning can be separated in terms of their respective definition of loss function. I am not sure how to achieve that?
Can somebody tell me or give a reference about the "Market Basket Problem" in data mining. What is this problem about? Any scientific reference with application will be greatly appreciated.
What are the key differences between the following two models? lmefit = lmer(MathAch ~ SES + (1 |School) , MathScores) lmfit = lm(MathAch ~ SES + factor(School) -1 , MathScores) To me, they seem to be the same, except that lmefit takes less parameters (because it used Normal distribution to model the levels at the ...
I would like to calculate the jensen-shannon divergence for he following 3 distributions. Is the calculation below correct? (I followed the JSD formula from wikipedia): P1 a:1/2 b:1/2 c:0 P2 a:0 b:1/10 c:9/10 P3 a:1/3 b:1/3 c:1/3 All distributions have equal weights, ie 1/3. JSD(P1, P2, P3) = H[(1/6, 1...
Intro: I have a dataset with a classical "large p, small n problem". The number available samples n=150 while the number of possible predictors p=400. The outcome is a continuous variable. I want to find the most "important" descriptors, i.e., those that are best candidates for explaining the outcome and helping to bu...
I'm evaluating the performance of a statistical estimator under a number of parameter settings. The estimator is unbiased for all of the parameters, so I'm reporting the sample variance as a measure of quality, that is, If I am interested in approximating: $$ \int h(x)\pi(x)dx $$ Where $h(x) : \mathbb{R}^d \rightarrow ...
I've been working with WEKA to build class predictors using this (rather old..) breast cancer dataset. The dataset is divided into a training and a test set. I've been testing different learning schemes (mostly focused on feature selection) using 10-Fold cross-validation experiments on the training set. Unfortunatel...
Suppose I have samples drawn from categories A, B, C. Within those categories, I have subcategories d,e,f which are found in all 3 categories. I want to visualize how many samples I have form categories A, B, C and the proportional composition of subcategories d,e,f within each category. One way to do this is a bar plo...
I am using R to examine the relationship between two variables in a small data set ($n=16$). My problem is that I'm not really sure how to handle the analysis (read: I'm in deeper waters than I've traveled before). Do I use Spearman's Rho to calculate the rank coefficient? Or do I assume normality and use standard par...
Stacked area charts work great when charting raw values, averages, counts, and percentage of total. The sum of the values across the different groups will add up to that of the entire set. But what about in the case when the value being charted is a Median, Geo Mean, or any Percentile? In these cases the sum of the val...
Given: Crude incidence rate per 100,000: 164 population: 211,101 95% CI: 147-182 What is the standard deviation of the incidence rate : My calc: Standard error:(182-147) / (2 x 1.96) = 8.93 Standard deviation = 8.93 x sqrt(211,101) = 4103 This can't be right, and I think it must be to do with the 100,000 Is this corre...
Sorry, this is a homework help question: In a bacterial growth experiment, ten μ (exponential growth rate constant) data points have been collected: one each of nine different concentrations of an antibiotic, plus one control (no antibiotic). What is the most appropriate statistical treatment for such a data set? The...
I have data coming in the following format: UserId, GroupId, Frequency (how many times the user chose the group), Max timestamp (the last time the user chose the group). Based on this dataset we need to figure out which groups look alike. I was thinking of using collaborative filtering but I have 3 concerns: We don't ...
I wonder what is wrong with my own answer for the problem which I have shared below, could you please help me some about that: " Suppose that we have 12 observation units $ ( n=12 ) $ in a statistical investigation. What is the "t-table" value $ ( {\downarrow}v^{ \backslash ^{\large{p} \, \rightarrow} } \; ) $ for $ ...
I'm examining the effect of 3 different interventions on a DV with age and gender included in my model. Now, I'm curious as to whether I could apply the exact same analyses to test for non-inferiority? Specifically, as I examine the ANOVA table, I am provided with the adjusted mean difference between Intervention2 an...
I am studying Statistics for business at introductory level, and having difficulty in handling the amount of information, partly because I am coming back to study after 5 years and I didn't do much stats at school. I don't understand binomial distribution, inference and all the types of "errors" on my textbook. I did p...
I am beginner of survival analysis, so I am confused about some basic ideas in this area. I think the basic assumption in survival analysis is that time of event for a subject can be estimated given all relevant time-invariable attributes of subjects. Therefore, ideally, if we can regress over the failure time correct...
I am doing multiple linear regression. I have 21 observations and 5 variables. My aim is just finding the relation between variables Is my data set enough to do multiple regression? The t-test result revealed 3 of my variables are not significant. Do I need to do my regression again with the significant variables (or ...
The other day I had a consultation with an epidemiologist. She is an MD with a public health degree in epidemiology and has a lot of statistical savvy. She mentors her research fellows and residents and helps them with statistical issues. She understands hypothesis testing pretty well. She had a typical problem of c...
I am looking for a C++ library for statistics to play with outliers detection in time series (amongst other). What I need: Robust estimators, correlations, hypothesis tests, etc; No dependencies with external libraries; No GPL; Would be a plus: Lightweight; Free; portable; Active and supported;
In a setting where one observes $X_1,\ldots,X_n$ distributed from a distribution with density $f$, I wonder if there is an unbiased estimator (based on the $X_i$'s) of the Hellinger distance to another distribution with density $f_0$, namely $$ \mathfrak{H}(f,f_0) = \left\{ 1 - \int_\mathcal{X} \sqrt{f(x)f_0(x)} \text{...
Note : this question is a repost, as my previous question had to be deleted for legal reasons. While comparing PROC MIXED from SAS with the function lme from the nlme package in R, I stumbled upon some rather confusing differences. More specifically, the degrees of freedom in the different tests differ between PROC MI...
I have to find 10 clusters of 100 samples with dimension 100. I have access to two k-means implementations. Both of them initialize the means with 10 randomly picked samples. When I run these algorithms on my samples, I can see that they immediately converge: After one iteration, the solution is stable with the followi...
I have seen somewhere that classical distances (like Euclidean distance) become weakly discriminant when we have multidimensional and sparse data. Why? Do you have an example of two sparse data vectors where the Euclidean distance does not perform well? In this case which similarity should we use?
I am trying to understand ordinary kriging. Say I have 3 elevation measurements: Z1, Z2, and Z3 taken at X positions: X1, X2 and X3. I am also assuming some semivariogram: g(h) and that the process is stationary. From what I understand the kriging weights in this case should be ?: $ L = M^{-1}*Gs $ Where: $L=\ \left( \...
This thread discusses an issue found in robust regression estimation: when the model fit is particularly low, although significant, the Huber estimator with a standard $k$ value of, say, 2.5, downweights most of the observations. The result is very low coefficients and, loosely speaking, under-estimation of the predict...
I'have fitted a mixed model in SPSS with fixed and random effects, and I would like to know what do the Fixed effects contrasts (type III) mean... In my model they are significant (e.g F=23.9; p=.012), but in the parameters estimates table the fixed variables are not significant... How should I interpret it? Thank you ...
I was recently looking at the National Survey of Family Growth (US Census Bureau) and noticed that the numbers on the median column were not whole numbers, but rather numbers like 5.1 and 4.6. (Not 5.1 thousand or 5.1 million, just 5.1.) This seemed unusual to me, so I looked to see if there was a note. There was: For...
Suppose I'm running an experiment that can have 2 outcomes, and I'm assuming that the underlying "true" distribution of the 2 outcomes is a binomial distribution with parameters $n$ and $p$: ${\rm Binomial}(n, p)$. I can compute the standard error, $SE_X = \frac{\sigma_X}{\sqrt{n}}$, from the form of the variance of ...
In my project, one variable named Organizational Culture has 5 dimensions namely Employee Development, Harmony, Customer Orientation, Social Responsibility and Innovation. In SPSS, I need to configure these 5 dimensions into 4 types so that: The highly integrative culture has high scores on all 5 dimensions. The marke...
I realize the methodology pursued by the Frequentist and Bayesian camps generally differ. However, one method of estimation that they do share is optimization of a certain function: Frequentists maximize the likelihood function, giving the Max. Likelihood (ML) estimator. Bayesians maximize the posterior function, givi...
Suppose I am entered into a contest, with the following rules: Every person may get up to 6 entries All the entries will be pooled, and 25% of the entries will be selected to be winners, with a maximum of 25. Each person can only win once, regardless of the number of their entries. If someone's name gets drawn again, ...
There was a concern raised by reviewers of a manuscript of mine over the proper execution of a Pearson's correlation. In brief, this was undertaken in order to determine the relationship between the extent of wheel running (y axis) and ethanol intake (x axis) across three, separate 10 day periods in 7 animals. In the ...
The likelihood ratio (a.k.a. deviance) $G^2$ statistic and lack-of-fit (or goodness-of-fit) test is fairly straightforward to obtain for a logistic regression model (fit using the glm(..., family = binomial) function) in R. However, it can be easy to have some cell counts end up low enough that the test is unreliable. ...
We're designing a trial to test three behavioral interventions (call them A, B, and C) for a health condition. Treatment A is an established intervention and B and C are considered add-ons to A. We want to test the following contrasts: Outcomes of packages involving Treatment A vs. No treatment. Outcomes of adding B,...
In Steven Pinker's book Better Angels of Our Nature, he notes that Probability is a matter of perspective. Viewed at sufficiently close range, individual events have determinate causes. Even a coin flip can be predicted from the starting conditions and the laws of physics, and a skilled magician can exploit thos...
In R, I have an $N \times K$ matrix $P$ where the $i$'th row of $P$ corresponds to a distribution on $\{1, ..., K\}$. Essentially, I need to sample from each row efficiently. A naive implementation is: X = rep(0, N); for(i in 1:N){ X[i] = sample(1:K, 1, prob = P[i, ]); } This is much too slow. In principle I could...
Which statistical method can be used to find the relationships between independent and dependent variables instead of regression? And their advantages over regression. It should be mentioned I am not able to do regression due to less number of observations that I have 5 independent variables 3 categorical and two conti...
I have a measurement for two groups of subjects where each subject contains 3 replicates. For example, the data is like below: Group Subject Replicate Value 1 A 1 0.2 1 A 2 0.3 1 A 3 0.25 1 B 1 0.4 1 B 2 0.3 1 ...
I get from a specific experiment 3 outputs. Each is the information of some physical quantity in the direction x,y or z. The way we extract the information from those signals is by fitting. In the data extracted from fitting, some parameters depend only on the direction of the physical quantity we want to measure, and ...
The likelihood could be defined by several ways, for instance : the function $L$ from $\Theta\times{\cal X}$ which maps $(\theta,x)$ to $L(\theta \mid x)$ i.e. $L:\Theta\times{\cal X} \rightarrow \mathbb{R} $. the random function $L(\cdot \mid X)$ we could also consider that the likelihood is only the "observed" likel...
Does the pdf of an mvn variable even exist when there is high correlation? I want to use an algorithm (actually it is the cross-entropy method for estimating a rare-event probability) that needs the pdf value of the mvn distribution, that is mvnpdf(x,mu,Sigma). However, my Sigma is close to singular, meaning there is ...
I would like to construct predictions for a mixed model (logistic via glmer) on a new data set using only the fixed effects, holding the random effects to 0. But I am having trouble setting up the model matrix to be able to calculate them. Since the mer class doesn't have a predict method, and since I want to omit t...
I conducted a pre and post survey that consisted of five questions with answers being either agree or disagree. The survey was to assess if a persons perception of Pit Bulls was improved after they interacted with it. The first or pre survey was before the individual interacted with the dog. After they did, the respo...
I have the following data for 200 cases/subjects: time to death over a 15 year period, $t$. A datum with a value of 180 months means that the subject did not die over the 15 year period. the frequency of three particular types of event associated with each subject $a_{i}$, $b_{i}$ and $c_{i}$, Apart from the obvious ...
We're trying to develop a simple tool that will help our teams optimise their clients' Facebook posting strategies. In our experience, time of post can have a big impact on audience response; but those times will differ from audience to audience based on their behaviours (can they access Facebook from work? What times ...
We always say that statistics is just dealing with data. But we also know that informatics is also getting knowledge from data analysis. For example, bioinformatics people can totally go without biostatistics. I want to know what is the essential difference between statistics and informatics.
I have many vectors $a_i\in\mathbb{R}^p:||a||=1,1\leq i \leq n$. I would like to test whether the $a_i$'s are randomly spread out on $S^{p-1}$ (the $p$ variate unit circle). Can anyone point to a test for that? Thanks in advance,
What happens if you reject normality of residuals when estimating with least squares? Is it too important to have normality on the residuals?
I'm conducting a project where I determine which factors (3 in total) contribute to the accuracy in which one would hit a dartboard. Each factor has two levels, low and high. For example, one of the factors is distance and the lowest distance (closest to the dart board) is 2 meters while the highest distance is 4 meter...
What is covariance in plain language and how is it linked to the terms dependence, correlation and variance-covariance structure with respect to repeated-measures designs?
I have a raw discrete data(curves). I need to find methods for detecting features of each curve. Some example features: 1) Stable growing 2) Fast growing 3) Stable falling 4) Fast falling 5) And so on Here is good example of such curves type(russian): How can i do it?
I have several sets of 10 raters which I want to compare. Each rater can cast only Yes or No vote, however this decision is skewed and the Yes votes make only about 10% of all votes (and this is expected, i.e. the such proportion is objectively true). Which of the inter-rater agreement statistics would be suitable in ...
I have the data of a test that could be used to distinguish normal and tumor cells. According to ROC curve it looks good for this purpose (area under curve is 0.9): My questions are: How to determine cutoff point for this test and its confidence interval where readings should be judged as ambiguous? What is the best...
I have a large $(10000 \times 5001)$ table representing $10000$ samples and $5001$ different features of these samples. One of these features represents an output variable of each sample. In other words, I have $5000$ input variables and one output variable for each sample. I know that most of these inputs are irreleva...
I have 4 models in the hierarchical regression output. The models are as follows: Model 1: Control variable Model 2: Control variable and 11 independent variables. Model 3: Control variable, 11 independent variables and moderator. Model 4: Control variable, 11 independent variables, moderator and interaction terms. My ...
There are several threads on this site discussing how to determine if the OLS residuals are asymptotically normally distributed. Another way to evaluate the normality of the residuals with R code is provided in this excellent answer. This is another discussion on the practical difference between standardized and observ...
I have a population with age range 0-6 years old. A survey asked various "Choose all that apply" questions and I'd like to compare the age densities for each subset (Did choose option X, Did Not choose option X), to look for age ranges where significant differences appear to have occurred. The reported ages are actuall...
I got these values from the measurement: 20.1, 20.2, 19.9, 20, 20.5, 20.5, 20, 19.8, 19.9, 20 I know that the actual distance is 20 km and the error of the measurement is not affected by the systematic error. What is the accuracy of a telescope? [Answer is: $\sigma^2 = 0.061$] How to calculate it?
I have three variables some of which are occasionally 0 which is a na and not a sample. I would like to calculate the standard deviation of the remaining two items but there isn't a deviation if like there is an averageif does anyone know how to do an STDEVIF? Thanks
I'm trying to calculate a single breakpoint on $x$ from several thousand $(x,y)$ points. Due to the density distribution as per the 2D histogram below, the least squares method is proving unsuitable. An orthogonal regression is a much better fit for the lower $x$-axis, but is it possible to conduct a piecewise orthogon...
Possible Duplicate: Machine learning cookbook / reference card / cheatsheet? Each classifier has it's own advantages and disadvantages. E.g. train/test speed, classification/regression (and how many classes can be handled), how many degrees of freedom, suitable/not suitable for abstract kinds of datasets (of possib...
My original question was going to be : How do you estimate the proportion of species in a population from count data of their individual body parts, provided that you can identify each part as belonging to a specific species. Is it just enough to take the average number of body parts found per species? (keeping in min...
I recently happened to read about empirical Bayes (Casella, 1985, An introduction to empirical Bayes data analysis) and it looked a lot like random effects model; in that both have estimates shrunken to global mean. But I have not read it throughly... Does anyone have any insight about the similarity and differences be...
I have recently started working on my master's thesis, which is a collaboration between my university and an IT company. The problem from the company's point of view is to identify correlations in their web marketing campaigns for various properties, with the purpose of applying this information to future ads in order ...
I would like to fit a Gaussian mixture model to some data. The data is 1D and I want to constrain all the Gaussians to have equal variance. I would also like to have a uniform background noise cluster to pick up points that are not in a real cluster. I don't know how many clusters there should be, and this can vary ov...
I got these values from the measurement by a telescope: 20.1, 20.2, 19.9, 20, 20.5, 20.5, 20, 19.8, 19.9, 20. I know that the actual distance is 20 km and the error of the measurement is not affected by the systematic error. What is the accuracy of the telescope? [Answer is: ${σ}^2$=0.061] Now I am supposed to calculat...
I am working on a problem which looks like this: Input Variables Categorical a b c d Continuous e Output Variables Discrete(Integers) v x y Continuous z The major issue that I am facing is that Output Variables are not totally independent of each other and there is no relation that can be established betwee...
I am making a task for a class, where I need to code simple moving average algorithm. In my copybook, I have this written: $$\bar{y_{i}}=\frac{y_{i-1}+y_{i}+y_{i+1}}{3}$$ for $n=3$ MA case, where near the start and end the values of smoothed time series are: $$\bar{y_{1}}=\frac{5y_{1}+2y_{2}-y_{3}}{6}$$ $$\bar{y_{n}}=\...
In some literature, I have read that a regression with multiple explanatory variables, if in different units, needed to be standardized. (Standardizing consists in subtracting the mean and dividing by the standard deviation.) In which other cases do I need to standardize my data? Are there cases in which I should on...
I'm just starting to look into exponential smoothing models. Is there a way to fit a linear regression with exponentially-smoothed errors, similar to the standard technique of fitting a regression with ARMA errors? Or is this generally considered a bad idea? Of course, I can come up with heuristics, like fit with AR(1)...
Netflix and Google among others use customer usage (number of videos streamed or unique searches conducted) as a proxy for value, or for determining which version of a test is best. I am curious how they would measure this but I'm not great at statistics. If they split their customer base into even groups, the usage wi...
These are the values: 18.2 17.6 ------------------------------------------------------------------------------------------------------------ 18.9 18.6 -------------------------------------------------------------------...
I have a question regarding the impact of regression to the mean when using forecasting with exponential smoothing. Unfortunately, I do not have access to the book on which this analysis was based [@Rob Hyndman, Forecasting with Exponential Smoothing: The State Space Approach (New York: Springer, 2008)], nor am I fami...
I'm looking for an article, program, algorithm that can clearly explain whats going on inside a Multinomial Naive Bayes classifier compared to a Gaussian Naive Bayes Classifier.
I have a gene expression profile with negative values. Is there any way to do an Nonnegative matrix factorization (NMF) on a matrix which has a few negative values? (I couldn't approximate the negatives by 1 because the range of negative values are really high.) Thanks.