How can I calculate the empircal CDF from an empirical PDF when dimensions are greater than 2?

I would like to calculate the empirical cumulative distribution of a data set with more than 2 dimensions. The only examples I have been able to find are for bivariate data (using hist3 and cumsum). Below is an example 3 dimensional PDF. I would like a robust solution that can accommodate any number of dimensions, if possible. Any help would be very much appreciated. Thank you.

sPDF(1,1,1) = 1;

sPDF(2,1,1) = 2;

sPDF(1,2,1) = 3;

sPDF(2,2,1) = 4;

sPDF(1,1,2) = 5;

sPDF(2,1,2) = 6;

sPDF(1,2,2) = 7;

sPDF(2,2,2) = 8;

Answer 1

[Star Strider]

See if the mvncdf function and its friends do what you want.

[Greg]

Thank you for your response, but this is not what I am asking for. This calculates the CDF for continuous multivariate normal distributions. I am looking for a solution that calculates the empirical CDF based on a multidimensional PDF (this is sometimes referred to as the Probability Mass Function).

[Star Strider]

The only option I can suggest in that situation are the Nonparametric and Empirical Probability Distributions.

[Greg]

For a bivariate example, the following works great to find the joint cumulative distribution function (the discrete form, which is all I need):

CDF = cumsum(cumsum(PDF,1),2);

Unfortunately, cumsum doesn't support > 2 dimensions. Perhaps I should be asking whether anyone can provide me an N-Dimensional cumsum function?

[Star Strider]

I would use the cumtrapz function. You will probably have to apply it in as many dimensions as you have in your matrix. (It will produce a result that is one less dimension than its argument.) I doubt there is anything else in core MATLAB that will do what you want. I’ve used cumtrapz to integrate volumes, but not beyond three dimensions.

[Greg]

Thanks again Star, I'll read through that function. Also, after much googling, I happened to come across the open source program FreeMat, and they have a CumSum function that supports multiple dimensions. This probably warrants a new question, but don`t suppose anyone is able to replicate it in Matlab easily? It produces the result I am expecting.

http://freemat.sourceforge.net/help/elementary_cumsum.html

I used it as follows for a 3 dimensional example:

--> CDF = cumsum(sPDF,1);

--> CDF = cumsum(CDF,2);

--> CDF = cumsum(CDF,3);

--> CDF = CDF./36 ...where 36 is the sum of CDF

[Star Strider]

My pleasure!

The cumsum function can do that. From the cumsum documentation:

• B = cumsum(A,dim) returns the cumulative sum of the elements along dimension dim . For example, if A is a matrix, then cumsum(A,2) returns the cumulative sum of each row.

From the cumtrapz documentation, it can do that as well:

• Z = cumtrapz(_,dim) integrates across the dimension of Y specified by scalar dim , using any of the input arguments in the previous syntaxes. The length of X must be the same as size(Y,dim) .

You simply have to ask them!

[Greg]

Wow. Well that is embarrassing. I thought I tried that earlier... Thank you very much! Recursively calling cumsum is the answer. Thank you for your time and help Star.

[Star Strider]

My pleasure!

I didn’t completely understand what you wanted (my fault) or I’d have suggested it earlier.

[Peter Mills]

What if I want to calculate the empirical cdf from an empirical pdf. Basically the question above but with data containing NaN's. I've tried cumsum but I get NaN's for every value after the first NaN value

[Greg]

You could try interpolating the NaN values first. Something like the following: inpaint This one may also work: inpainting Or you could try fitting a neural network to the data and predict those NaN values.