Measure that takes samples that is minimized in expectation for a uniformly-distributed random variable?












2














I am having trouble thinking of a function that operates on a set of samples, that is, single-valued random variables between zero and one, $x_i in (0,1), iin{1,2,...I}$, and provides a measure of "non uniformity" of the samples:



$$
F({x_i}) : [0,1]^I to R
$$
This measure would be uniquely minimized (in expectation) when the samples are from a uniform distribution.



My first reaction was to bin these samples by cutting $[0,1]$ into equal-sized peices. Then the standard deviation of the counts across the bins would be my measure, minimized when the counts in each bin are equal.



But the binning allows for "cheating", that is, the samples may not actually be uniformly distributed, but only appear that way due to the binning procedure; a different choice of bins would show the problem.



Any better ideas?



The motivation is to evaluate the quality of a CDF fitting procedure. The fitting procedure is optimal when the CDFs (different CDF for each sample) map all the "real" samples $y_i in R$ uniformly to the [0,1] interval.






sampling descriptive-statistics uniform minimum uniformity







share|cite|improve this question





























    2














    I am having trouble thinking of a function that operates on a set of samples, that is, single-valued random variables between zero and one, $x_i in (0,1), iin{1,2,...I}$, and provides a measure of "non uniformity" of the samples:



    $$
    F({x_i}) : [0,1]^I to R
    $$
    This measure would be uniquely minimized (in expectation) when the samples are from a uniform distribution.



    My first reaction was to bin these samples by cutting $[0,1]$ into equal-sized peices. Then the standard deviation of the counts across the bins would be my measure, minimized when the counts in each bin are equal.



    But the binning allows for "cheating", that is, the samples may not actually be uniformly distributed, but only appear that way due to the binning procedure; a different choice of bins would show the problem.



    Any better ideas?



    The motivation is to evaluate the quality of a CDF fitting procedure. The fitting procedure is optimal when the CDFs (different CDF for each sample) map all the "real" samples $y_i in R$ uniformly to the [0,1] interval.






    sampling descriptive-statistics uniform minimum uniformity







    share|cite|improve this question



























      2












      2








      2







      I am having trouble thinking of a function that operates on a set of samples, that is, single-valued random variables between zero and one, $x_i in (0,1), iin{1,2,...I}$, and provides a measure of "non uniformity" of the samples:



      $$
      F({x_i}) : [0,1]^I to R
      $$
      This measure would be uniquely minimized (in expectation) when the samples are from a uniform distribution.



      My first reaction was to bin these samples by cutting $[0,1]$ into equal-sized peices. Then the standard deviation of the counts across the bins would be my measure, minimized when the counts in each bin are equal.



      But the binning allows for "cheating", that is, the samples may not actually be uniformly distributed, but only appear that way due to the binning procedure; a different choice of bins would show the problem.



      Any better ideas?



      The motivation is to evaluate the quality of a CDF fitting procedure. The fitting procedure is optimal when the CDFs (different CDF for each sample) map all the "real" samples $y_i in R$ uniformly to the [0,1] interval.






      sampling descriptive-statistics uniform minimum uniformity







      share|cite|improve this question















      I am having trouble thinking of a function that operates on a set of samples, that is, single-valued random variables between zero and one, $x_i in (0,1), iin{1,2,...I}$, and provides a measure of "non uniformity" of the samples:



      $$
      F({x_i}) : [0,1]^I to R
      $$
      This measure would be uniquely minimized (in expectation) when the samples are from a uniform distribution.



      My first reaction was to bin these samples by cutting $[0,1]$ into equal-sized peices. Then the standard deviation of the counts across the bins would be my measure, minimized when the counts in each bin are equal.



      But the binning allows for "cheating", that is, the samples may not actually be uniformly distributed, but only appear that way due to the binning procedure; a different choice of bins would show the problem.



      Any better ideas?



      The motivation is to evaluate the quality of a CDF fitting procedure. The fitting procedure is optimal when the CDFs (different CDF for each sample) map all the "real" samples $y_i in R$ uniformly to the [0,1] interval.






      sampling descriptive-statistics uniform minimum uniformity




      sampling descriptive-statistics uniform minimum uniformity






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




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      edited Dec 9 at 6:18

























      asked Dec 8 at 5:00









      OrangeSherbet

      1155




      1155






















          2 Answers
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          active

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          5














          Like @Chris said, you can use some kind of divergence or information theory ideas. KL divergence like he said is good, or more generally f-divergence, etc., but more simply just use entropy. You said minimized when uniform, so entropy would be maximized, so do negative entropy. You said maps to real numbers, maybe you want for easier comparison, [0,1] interval, so use (negative) normalized entropy (divide by logN):



          https://math.stackexchange.com/questions/395121/how-entropy-scales-with-sample-size



          Or since you said the end goal is to compare some empirical CDF's, then depending on your problem maybe there is a more direct way you can do this like:



          KS test:
          https://en.wikipedia.org/wiki/Kolmogorov%E2%80%93Smirnov_test



          Anderson-Darling:
          https://en.wikipedia.org/wiki/Anderson%E2%80%93Darling_test



          Cramer von Mises
          https://en.wikipedia.org/wiki/Cram%C3%A9r%E2%80%93von_Mises_criterion



          etc.






          share|cite|improve this answer

















          • 2




            The biggest revelation for me was realizing I could calculate the "empirical CDF" using the samples, and then use distance measures between the empirical CDF and the cdf for the uniform distribution (a straight line from [0,0] to [1,1]), all without binning anything.
            – OrangeSherbet
            Dec 8 at 10:52





















          2














          Take your samples and form a random variable $X$. To measure the non-uniformity of the samples measure the KL divergence between $X$ and $X'$ as $KL(X||X')$.






          share|cite|improve this answer





















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            2 Answers
            2






            active

            oldest

            votes








            2 Answers
            2






            active

            oldest

            votes









            active

            oldest

            votes






            active

            oldest

            votes









            5














            Like @Chris said, you can use some kind of divergence or information theory ideas. KL divergence like he said is good, or more generally f-divergence, etc., but more simply just use entropy. You said minimized when uniform, so entropy would be maximized, so do negative entropy. You said maps to real numbers, maybe you want for easier comparison, [0,1] interval, so use (negative) normalized entropy (divide by logN):



            https://math.stackexchange.com/questions/395121/how-entropy-scales-with-sample-size



            Or since you said the end goal is to compare some empirical CDF's, then depending on your problem maybe there is a more direct way you can do this like:



            KS test:
            https://en.wikipedia.org/wiki/Kolmogorov%E2%80%93Smirnov_test



            Anderson-Darling:
            https://en.wikipedia.org/wiki/Anderson%E2%80%93Darling_test



            Cramer von Mises
            https://en.wikipedia.org/wiki/Cram%C3%A9r%E2%80%93von_Mises_criterion



            etc.






            share|cite|improve this answer

















            • 2




              The biggest revelation for me was realizing I could calculate the "empirical CDF" using the samples, and then use distance measures between the empirical CDF and the cdf for the uniform distribution (a straight line from [0,0] to [1,1]), all without binning anything.
              – OrangeSherbet
              Dec 8 at 10:52


















            5














            Like @Chris said, you can use some kind of divergence or information theory ideas. KL divergence like he said is good, or more generally f-divergence, etc., but more simply just use entropy. You said minimized when uniform, so entropy would be maximized, so do negative entropy. You said maps to real numbers, maybe you want for easier comparison, [0,1] interval, so use (negative) normalized entropy (divide by logN):



            https://math.stackexchange.com/questions/395121/how-entropy-scales-with-sample-size



            Or since you said the end goal is to compare some empirical CDF's, then depending on your problem maybe there is a more direct way you can do this like:



            KS test:
            https://en.wikipedia.org/wiki/Kolmogorov%E2%80%93Smirnov_test



            Anderson-Darling:
            https://en.wikipedia.org/wiki/Anderson%E2%80%93Darling_test



            Cramer von Mises
            https://en.wikipedia.org/wiki/Cram%C3%A9r%E2%80%93von_Mises_criterion



            etc.






            share|cite|improve this answer

















            • 2




              The biggest revelation for me was realizing I could calculate the "empirical CDF" using the samples, and then use distance measures between the empirical CDF and the cdf for the uniform distribution (a straight line from [0,0] to [1,1]), all without binning anything.
              – OrangeSherbet
              Dec 8 at 10:52
















            5












            5








            5






            Like @Chris said, you can use some kind of divergence or information theory ideas. KL divergence like he said is good, or more generally f-divergence, etc., but more simply just use entropy. You said minimized when uniform, so entropy would be maximized, so do negative entropy. You said maps to real numbers, maybe you want for easier comparison, [0,1] interval, so use (negative) normalized entropy (divide by logN):



            https://math.stackexchange.com/questions/395121/how-entropy-scales-with-sample-size



            Or since you said the end goal is to compare some empirical CDF's, then depending on your problem maybe there is a more direct way you can do this like:



            KS test:
            https://en.wikipedia.org/wiki/Kolmogorov%E2%80%93Smirnov_test



            Anderson-Darling:
            https://en.wikipedia.org/wiki/Anderson%E2%80%93Darling_test



            Cramer von Mises
            https://en.wikipedia.org/wiki/Cram%C3%A9r%E2%80%93von_Mises_criterion



            etc.






            share|cite|improve this answer












            Like @Chris said, you can use some kind of divergence or information theory ideas. KL divergence like he said is good, or more generally f-divergence, etc., but more simply just use entropy. You said minimized when uniform, so entropy would be maximized, so do negative entropy. You said maps to real numbers, maybe you want for easier comparison, [0,1] interval, so use (negative) normalized entropy (divide by logN):



            https://math.stackexchange.com/questions/395121/how-entropy-scales-with-sample-size



            Or since you said the end goal is to compare some empirical CDF's, then depending on your problem maybe there is a more direct way you can do this like:



            KS test:
            https://en.wikipedia.org/wiki/Kolmogorov%E2%80%93Smirnov_test



            Anderson-Darling:
            https://en.wikipedia.org/wiki/Anderson%E2%80%93Darling_test



            Cramer von Mises
            https://en.wikipedia.org/wiki/Cram%C3%A9r%E2%80%93von_Mises_criterion



            etc.







            share|cite|improve this answer












            share|cite|improve this answer



            share|cite|improve this answer










            answered Dec 8 at 6:35









            123learn

            761




            761








            • 2




              The biggest revelation for me was realizing I could calculate the "empirical CDF" using the samples, and then use distance measures between the empirical CDF and the cdf for the uniform distribution (a straight line from [0,0] to [1,1]), all without binning anything.
              – OrangeSherbet
              Dec 8 at 10:52
















            • 2




              The biggest revelation for me was realizing I could calculate the "empirical CDF" using the samples, and then use distance measures between the empirical CDF and the cdf for the uniform distribution (a straight line from [0,0] to [1,1]), all without binning anything.
              – OrangeSherbet
              Dec 8 at 10:52










            2




            2




            The biggest revelation for me was realizing I could calculate the "empirical CDF" using the samples, and then use distance measures between the empirical CDF and the cdf for the uniform distribution (a straight line from [0,0] to [1,1]), all without binning anything.
            – OrangeSherbet
            Dec 8 at 10:52






            The biggest revelation for me was realizing I could calculate the "empirical CDF" using the samples, and then use distance measures between the empirical CDF and the cdf for the uniform distribution (a straight line from [0,0] to [1,1]), all without binning anything.
            – OrangeSherbet
            Dec 8 at 10:52















            2














            Take your samples and form a random variable $X$. To measure the non-uniformity of the samples measure the KL divergence between $X$ and $X'$ as $KL(X||X')$.






            share|cite|improve this answer


























              2














              Take your samples and form a random variable $X$. To measure the non-uniformity of the samples measure the KL divergence between $X$ and $X'$ as $KL(X||X')$.






              share|cite|improve this answer
























                2












                2








                2






                Take your samples and form a random variable $X$. To measure the non-uniformity of the samples measure the KL divergence between $X$ and $X'$ as $KL(X||X')$.






                share|cite|improve this answer












                Take your samples and form a random variable $X$. To measure the non-uniformity of the samples measure the KL divergence between $X$ and $X'$ as $KL(X||X')$.







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered Dec 8 at 6:00









                Chris

                514211




                514211






























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