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Fitting a Weibull distribution to scores over time in MATLAB

Worked examples using the Weibull distribution are here.

The Weibull has three parameters, one which represents time and in particular equals the 63rd percentile for time, a shape parameter taking values between 0.5 and 8 with a shape parameter of 1 equal to the exponential distribution, one of 2.5 equal to the Log Normal distribution and one of 3.6 equal to the Normal distribution. The third parameter is the scalar multiplying up the average probability associated with each time to the average observed raw score. Note that the Weibull distribution is constrained to go through the origin.

If the above link is broken details there are reproduced (minus the plots) below:

This example shows the difference between fitting a curve to a set of points, and fitting a probability distribution to a sample of data.

A common question is, "I have some data and I want to fit a Weibull distribution. What MATLAB® functions can I use to do Weibull curve fitting?" Before answering that question, we need to figure out what kind of data analysis is really appropriate.

Curve Fitting

Consider an experiment where we measure the concentration of a compound in blood samples taken from several subjects at various times after taking an experimental medication. 

time = [ 0.1   0.1   0.3   0.3   1.3   1.7   2.1   2.6   3.9   3.9 ...
         5.1   5.6   6.2   6.4   7.7   8.1   8.2   8.9   9.0   9.5 ...
         9.6  10.2  10.3  10.8  11.2  11.2  11.2  11.7  12.1  12.3 ...
        12.3  13.1  13.2  13.4  13.7  14.0  14.3  15.4  16.1  16.1 ...
        16.4  16.4  16.7  16.7  17.5  17.6  18.1  18.5  19.3  19.7];
conc = [0.01  0.08  0.13  0.16  0.55  0.90  1.11  1.62  1.79  1.59 ...
        1.83  1.68  2.09  2.17  2.66  2.08  2.26  1.65  1.70  2.39 ...
        2.08  2.02  1.65  1.96  1.91  1.30  1.62  1.57  1.32  1.56 ...
        1.36  1.05  1.29  1.32  1.20  1.10  0.88  0.63  0.69  0.69 ...
        0.49  0.53  0.42  0.48  0.41  0.27  0.36  0.33  0.17  0.20];
plot(time,conc,'o');
xlabel('Time');
ylabel('Blood Concentration');

Notice that we have one response variable, blood concentration, and one predictor variable, time after ingestion. The predictor data are assumed to be measured with little or no error, while the response data are assumed to be affected by experimental error. The main objective in analyzing data like these is often to define a model that predicts the response variable. That is, we are trying to describe the trend line, or the mean response of y (blood concentration), as a function of x (time). This is curve fitting with bivariate data.

Based on theoretical models of absorption into and breakdown in the bloodstream, we might, for example, decide that the concentrations ought to follow a Weibull curve as a function of time. The Weibull curve, which takes the form

$$y = c * (x/a)(b-1) * exp(-(x/a)b),$$

is defined with three parameters: the first scales the curve along the horizontal axis, the second defines the shape of the curve, and the third scales the curve along the vertical axis. Notice that while this curve has almost the same form as the Weibull probability density function, it is not a density because it includes the parameter c, which is necessary to allow the curve's height to adjust to data.

We can fit the Weibull model using nonlinear least squares. 

modelFun =  @(p,x) p(3) .* (x ./ p(1)).^(p(2)-1) .* exp(-(x ./ p(1)).^p(2));
startingVals = [10 2 5];
coefEsts = nlinfit(time, conc, modelFun, startingVals);
xgrid = linspace(0,20,100);
line(xgrid, modelFun(coefEsts, xgrid), 'Color','r');

This scatterplot suggests that the measurement errors do not have equal variance, but rather, that their variance is proportional to the height of the mean curve. One way to address this problem would be to use weighted least squares. However, there is another potential problem with this fit.

The Weibull curve we're using, as with other similar models such as Gaussian, gamma, or exponential curves, is often used as a model when the response variable is nonnegative. Ordinary least squares curve fitting is appropriate when the experimental errors are additive and can be considered as independent draws from a symmetric distribution, with constant variance. However, if that were true, then in this example it would be possible to measure negative blood concentrations, which is clearly not reasonable.

It might be more realistic to assume multiplicative errors, symmetric on the log scale. We can fit a Weibull curve to the data under that assumption by logging both the concentrations and the original Weibull curve itself. That is, we can use nonlinear least squares to fit the curve

$$log(y) = log(c) + (b-1)*log(x/a) - (x/a)^b.$$ 

coefEsts2 = nlinfit(time, log(conc), @(p,x)log(modelFun(p,x)), startingVals);
line(xgrid, modelFun(coefEsts2, xgrid), 'Color',[0 .5 0]);
legend({'Raw Data' 'Additive Errors Model' 'Multiplicative Errors Model'});

This model fit should be accompanied by estimates of precision and followed by checks on the model's goodness of fit. For example, we should make residual plots on the log scale to check the assumption of constant variance for the multiplicative errors. For simplicity we'll leave that out here.

In this example, using the multiplicative errors model made little difference in the model's predictions, but that's not always the case. An example where it does matter is described in the Pitfalls in Fitting Nonlinear Models by Transforming to Linearity example.

Functions for Curve Fitting

MATLAB and several toolboxes contain functions that can used to perform curve fitting. The Statistics and Machine Learning Toolbox™ includes the functions nlinfit, for nonlinear least squares curve fitting, and glmfit, for fitting Generalized Linear Models. The Curve Fitting Toolbox™ provides command line and graphical tools that simplify many of the tasks in curve fitting, including automatic choice of starting coefficient values for many models, as well as providing robust and nonparametric fitting methods. Many complicated types of curve fitting analyses, including models with constraints on the coefficients, can be done using functions in the Optimization Toolbox™. The MATLAB function polyfit fits polynomial models, and fminsearch can be used in many other kinds of curve fitting.

Distribution Fitting

Now consider an experiment where we've measured the time to failure for 50 identical electrical components. 

life = [ 6.2 16.1 16.3 19.0 12.2  8.1  8.8  5.9  7.3  8.2 ...
        16.1 12.8  9.8 11.3  5.1 10.8  6.7  1.2  8.3  2.3 ...
         4.3  2.9 14.8  4.6  3.1 13.6 14.5  5.2  5.7  6.5 ...
         5.3  6.4  3.5 11.4  9.3 12.4 18.3 15.9  4.0 10.4 ...
         8.7  3.0 12.1  3.9  6.5  3.4  8.5  0.9  9.9  7.9];

Notice that only one variable has been measured -- the components' lifetimes. There is no notion of response and predictor variables; rather, each observation consists of just a single measurement. The objective of an analysis for data like these is not to predict the lifetime of a new component given a value of some other variable, but rather to describe the full distribution of possible lifetimes. This is distribution fitting with univariate data.

One simple way to visualize these data is to make a histogram. 

binWidth = 2;
binCtrs = 1:binWidth:19;
hist(life,binCtrs);
xlabel('Time to Failure');
ylabel('Frequency');
ylim([0 10]);

It might be tempting to think of this histogram as a set of (x,y) values, where each x is a bin center and each y is a bin height. 

h_gca = gca;
h = h_gca.Children;
h.FaceColor = [.98 .98 .98];
h.EdgeColor = [.94 .94 .94];
counts = hist(life,binCtrs);
hold on
plot(binCtrs,counts,'o');
hold off

We might then try to fit a curve through those points to model the data. Since lifetime data often follow a Weibull distribution, and we might even think to use the Weibull curve from the curve fitting example above.

coefEsts = nlinfit(binCtrs,counts,modelFun,startingVals);

However, there are some potential pitfalls in fitting a curve to a histogram, and this simple fit is not appropriate. First, the bin counts are nonnegative, implying that measurement errors cannot be symmetric. Furthermore, the bin counts have different variability in the tails than in the center of the distribution. They also have a fixed sum, implying that they are not independent measurements. These all violate basic assumptions of least squares fitting.

It's also important to recognize that the histogram really represents a scaled version of an empirical probability density function (PDF). If we fit a Weibull curve to the bar heights, we would have to constrain the curve to be properly normalized.

These problems might be addressed by using a more appropriate least squares fit. But another concern is that for continuous data, fitting a model based on the histogram bin counts rather than the raw data throws away information. In addition, the bar heights in the histogram are very dependent on the choice of bin edges and bin widths. While it is possible to fit distributions in this way, it's usually not the best way.

For many parametric distributions, maximum likelihood is a much better way to estimate parameters. Maximum likelihood does not suffer from any of the problems mentioned above, and it is often the most efficient method in the sense that results are as precise as is possible for a given amount of data.

For example, the function wblfit uses maximum likelihood to fit a Weibull distribution to data. The Weibull PDF takes the form

$$y = (a/b) * (x/a){(b-1)} * exp(-(x/a)b).$$

Notice that this is almost the same functional form as the Weibull curve used in the curve fitting example above. However, there is no separate parameter to independently scale the vertical height. Being a PDF, the function always integrates to 1.

We'll fit the data with a Weibull distribution, then plot a histogram of the data, scaled to integrate to 1, and superimpose the fitted PDF. 

paramEsts = wblfit(life);
n = length(life);
prob = counts / (n * binWidth);
bar(binCtrs,prob,'hist');
h_gca = gca;
h = h_gca.Children;
h.FaceColor = [.9 .9 .9];
xlabel('Time to Failure');
ylabel('Probability Density');
ylim([0 0.1]);
xgrid = linspace(0,20,100);
pdfEst = wblpdf(xgrid,paramEsts(1),paramEsts(2));
line(xgrid,pdfEst)

Maximum likelihood can, in a sense, be thought of as finding a Weibull PDF to best match the histogram. But it is not done by minimizing the sum of squared differences between the PDF and the bar heights.

As with the curve fitting example above, this model fit should be accompanied by estimates of precision and followed by checks on the model's goodness of fit; for simplicity we'll leave that out here.

Functions for Distribution Fitting

The Statistics and Machine Learning Toolbox includes functions, such as wblfit, for fitting many different parametric distributions using maximum likelihood, and the function mle can be used to fit custom distributions for which dedicated fitting functions are not explicitly provided. The function ksdensity fits a nonparametric distribution model to data. The Statistics and Machine Learning Toolbox also provides the GUI dfittool, which simplifies many of the tasks in distribution fitting, including generation of various visualizations and diagnostic plots. Many types of complicated distributions, including distributions with constraints on the parameters, can be fit using functions in the Optimization Toolbox. Finally, the MATLAB function fminsearch can be used in many kinds of maximum likelihood distribution fitting.

Although fitting a curve to a histogram is usually not optimal, there are sensible ways to apply special cases of curve fitting in certain distribution fitting contexts. One method, applied on the cumulative probability (CDF) scale instead of the PDF scale, is described in the Fitting Custom Univariate Distributions example.

Summary

It's not uncommon to do curve fitting with a model that is a scaled version of a common probability density function, such as the Weibull, Gaussian, gamma, or exponential. Curve fitting and distribution fitting can be easy to confuse in these cases, but the two are very different kinds of data analysis.

• Curve fitting involves modelling the trend or mean of a response variable as a function of a second predictor variable. The model usually must include a parameter to scale the height of the curve, and may also include an intercept term. The appropriate plot for the data is an x-y scatterplot.

• Distribution fitting involves modelling the probability distribution of a single variable. The model is a normalized probability density function. The appropriate plot for the data is a histogram.