matlab - pdf for multivariate uniform distribution

Question

The pdf for the multivariate normal distribution in MATLAB is mvnpdf(...). What about the case where multiple variables are uniformly distributed: Is there a function to describe their joint distribution analogous to the multivariate normal distribution? If there is no such function, is there a trick to handle this case?

Answer 1

The simplest way how several variables can be uniformly distributed is if they are mutually independent; in that case you simply have a uniform distribution over the hypercube in the space spanned by the variables. In order to get samples from this distribution, you just separately generate samples for each of the variables.

The point where a "trick" might be necessary is if you have dependencies between the variables even though the marginal distribution for each of them is still uniform. In this case you have to describe the dependency structure, and I'm not aware of any standard way to do this (the way dependencies between normally distributed variables are described by a correlation matrix).

Of course such distributions exist: For two dimensions, one possibility would be to have a joint distribution that looks like a solution to the "eight rooks" problem:

Another one actually derives from the introductory Matlab example, the magic square:

Both of these examples are discrete distributions, but can be produced at arbitrary granularity, or simply interpreted as piecewise constant continuous distributions.

As you can see there are many possibilities for a multivariate distribution each of whose marginal distributions are uniform. The question you have to answer for yourself is what kind of dependencies, if any, you are interested in?

Answer 2

If I'm understanding the question properly, we want to calculate the pdf of a multivariate uniform distribution. By definition, the pdf is constant for all values in the support the distribution. Thus to calculate the pdf all that is required is to calculate the norming constant, which is given by the inverse of the integral of the support. That is to say, the pdf is given by

f(x) = 1 / integral(A)

where A is the support set, and x is an element in A. If an analytic solution to integral(A) is not available, then a numerical integrator can be employed.