MATLAB: Log likelihood for each distributions.

fitdistlikelihoodStatistics and Machine Learning Toolbox

Hi! I was wondering how to compute in Matlab the log likelihood.

I think which the "fitdist" function includes "likelihood" function(e.g. normlike, gamlike and wbllike, etc).

However, the result of likelihood value is not same result which I was using "Weilbull" distribution.

The "Weibull" and "fitdist(data,'weibull') -> PD.NLogL" were not same value.

Plz understatnd me.

Thanks all.

Best regards.

Jin.

Best Answer

It's not clear to me what fails to match with what. These match:

>> x = -5*log(rand(100,1));
>> pd = fitdist(x,'weibull');
>> format compact
>> negloglik(pd)
ans =
  267.1545
>> sum(-log(pdf(pd,x)))
ans =
  267.1545
>> wbllike(pd.ParameterValues,x)
ans =
  267.1545

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I find the same thing. I got 52/1000 rejected the first way. That's close to the 5% value that you would expect. I got fewer the second way. That's because by estimating the parameter from the data used in the test, you are using a fit that is "too good." Imagine if you estimated the p parameter of a binomial distribution -- this would always give an exact match between the observed and expected counts of 1's and 0's. This case is not so extreme, but is similar.

The chi2gof function does provide a way to do an approximate adjustment for that, if you tell it that you have estimated one parameter:

[H(sim) P(sim) STATS] = chi2gof(X,'cdf',@(z)poisscdf(z,mean(X)),'nparams',1);

The concept is described further in lecture 12 of the MIT course notes.

As for your other question, the chi2gof function defines a set of bin edges and calls your function to compute the cdf values at these edges. The "@(z)" indicates that your function accepts one input to be represented by the symbol z, and the rest of the expression defines how the cdf is to be computed at the provided z values.

MATLAB: How to compare appropriateness of fitted Weibull and lognormal models via the likelihood ratio test

Unfortunately the likelihood ratio test cannot be used this way. It is only for comparing nested models (i.e., one model has to be a logical subset of the other--for example, with some parameter constrained to a particular value in the subset model but that same parameter free to take on any value in the larger model). The Weibull and lognormal distributions do not have this subset relationship, so they can't be compared with an lrt. For non-nested models, you probably have to make comparisons using AIC or some such model goodness measure.

I don't really understand what you are saying about rejecting the null. It seems to me that if you generate the data from a Weibull then you wouldn't expect to reject the Weibull in favor of some other distribution like the lognormal. (Well, maybe if the other distribution were much more flexible than the Weibull, but that doesn't apply to the lognormal AFAIK).

By the way, these statements are redundant:

wblPD = fitdist(x, 'weibull');
lnPD = fitdist(x, 'lognormal');
wblPms = mle(x, 'distribution', 'weibull');
lnPms = mle(x, 'distribution', 'lognormal');

The outputs of fitdist are structures containing the very same parameter estimates that you get from mle.

I hope some of this helps you to move forward.

Best Answer

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