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Thread: How well a few values fit a given distribution?

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    How well a few values fit a given distribution?




    Dear all,

    I have approximated two one-dimensional random variables A and B using Gaussian Mixture Models (3 gaussians). I used gmdistribution.fit Matlab function with 10000 values. The resulting distributions are called f_{A}(x) and f_{B}(x) (shown in attached Figure 1, being f_{A}(x) in red and f_{B}(x) in blue).

    Now I have a few values (e.g. v=[-1 0 0.5 1 5 6 6.5 7 14], as in Figure 1). This vector produces a very sparse histogram, since there are not many values.

    How probable is that these values were generated by f_{A}(x) distribution? or How probable is that these values were generated by f_{B}(x) distribution? I would like to obtain a probability value in order to classify the set of values to category A (f_{A}(x) distribution) or category B (f_{B}(x) distribution).

    I had several ideas:

    -> Joint distribution (product of probabilities)... but, what happens with outliers? Since f_{A}(x) and f_{B}(x) are approximations of the real distribution, a outlier might produce zero probability for some value of x (see attached Figure 2). So I am not sure about this way.

    -> Average probability: This is a trivial solution I though, and probably it's wrong.

    -> Hypothesis tests (Squared Chi, or Kolmogorov–Smirnov): In the case of squared Chi, data should be binned... what is the optimal size of these bins? In addition, these hypothesis tests produce a p-value, which is not the probability value I am looking for (as far as I understood).

    Thanks in advance for your advice.
    Best regards,
    Emliio.
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  2. #2
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    Re: How well a few values fit a given distribution?


    Any idea?

    My questions are:

    How to measure the goodness of fit between a few samples and a non-gaussian distribution?

    Given two possible distributions: What is the probability that some few samples are generated by each of them?

    Thanks,
    Emilio.

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