Numerical integration and NaN

Question

James 2011-5-11

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I want to calculate the following integral:

integral( pdf(data_one) / pdf(data_two) )

I managed to do this, but I had to resort to a hack to avoid divisions by zero. This hack works but creates deviations that annoy me - what is the proper way to do this?

data_axis = 1:5;
data_one = [1,2,3,4,5];
data_two = [1,1,1,1,5];
% Calculate probability densities
pdf_one = histc(data_one,data_axis); % [0.2,0.2,0.2,0.2,0.2]
pdf_two = histc(data_two,data_axis); % [0.8,0.0,0.0,0.0,0.2]
% HACK: Histc may have generated 0-buckets
%pdf_one = (pdf_one + 0.1);
%pdf_two = (pdf_two + 0.1);
% Normalize the result
pdf_one = pdf_one./sum(pdf_one);
pdf_two = pdf_two./sum(pdf_two);
complicated_integral = trapz(pdf_one./pdf_two);

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Answer 1

Walter Roberson 2011-5-11

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在 MATLAB Online 中打开

Probability Density is not defined for discrete distributions.

http://en.wikipedia.org/wiki/Probability_density_function#Further_details

Not every probability distribution has a density function: the distributions of discrete random variables do not; nor does the Cantor distribution, even though it has no discrete component, i.e., does not assign positive probability to any individual point.

and

If a probability distribution admits a density, then the probability of every one-point set {a} is zero; the same holds for finite and countable sets.

This implies that you are asking to calculate something which is not defined, or else that your calculation is incorrect, perhaps attempting to sample the counts at a point rather than over a range.

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Richard Crozier 2011-5-18

If you know what the underlying form of the pdf of the data from some scientific knowledge of the system from which you have sampled the data, could you not fit the distribution to the datasets and integrate the resulting continuous functions? Of course the accuracy of this is doubtful for some distributions, e.g. an exponential distribution.

James 2011-5-18

That is a nice idea which I will have to implement anyways for other parts of the project.

Though I fear that in this case, it would ultimately not solve my problem. My main problem currently comes from the fact that I am doing this:

pdf_one./pdf_two

= [0.2,0.2,0.2,0.2,0.2] ./ [0.8,0.0,0.0,0.0,0.2]

= [0.2500, Inf, Inf, Inf, 1.0000]

And then try to trapezoid-integrate over it, which is pretty stupid. Whenever there is no probability of a value appearing in said pdf_two (=0), I am diving by zero.

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