Test & Analysis Project

MULTIDIMENSIONAL DISTRIBUTIONS

The goodness-of-fit tests can be simply generalized to both discrete and continuous many-dimensional distributions. The only difference is that somehow these many-dimensional distributions must be prepared to the testing phase.

In the discrete case the problem of n variables could be summarized by means of big contingency tables. Of course the discrete corresponding variables that need to be compared with goodness-of-fit tests must have the same grouping. In the easiest case we deal with the joint distribution deriving from 2 discrete variables, respectively with r and c categories. This situation can be represented graphically with a three-dimensional histogram, having the two discrete variables on x and y, while z represents the counts.

The case of n (n>2) discrete variables can always be reduced to the previous one simply dividing the many-dimensional joint distributions into flat layers, i.e. in several r´c separated tables. On each layer the c2 test can be performed, but only if the counts in each cell are higher than 5. This problem can anyhow be solved choosing another kind of grouping for the variables. Otherwise, many-dimensional Anderson-Darling test can be always performed, building the cumulative discrete distributions and comparing the twined marginal distributions.

When we deal with continuous cases, the distributions’ variables and parameters are described by means of n-dimensional real vectors. This implies that for continuous random variables the probability density function f itself will depend on a vector r:=(x1, x2,…,xn). So, the many-dimensional cumulative distribution F(r) will become:

F(r) = ňW f (r) dr.

The goodness-of-fit tests could be still performed to compare these many-dimensional cumulative distributions. Certainly the easiest way to solve the problem would be to try to reduce the study’s dimensionality. This is the aim of all the factor analysis techniques. By means of principal components analysis, in fact, it is always possible to project the n-dimensional cloud of points on a few preferred spatial directions, each one being a linear superposition of the original n variables. This new space must have the property of reducing the deformation deriving from each single projection. The problem is the one of reducing multi-dimensionality without a significant loss of the information coming from the original distributions. Of course this new lower-dimensional space must be the same for both the continuous distributions that we need to compare. First of all we must project the clouds of points of both the distributions on a one-dimensional sub-space. The choice of this vector v is made in terms of least squares and it must assure that the greatest part of the variance of both the clouds will be explained. Then this technique will be iterated and a second, a third, …, spatial directions, independent from the previous ones and explaining the largest variance as possible, must be found. The process stops when the dimensionality desired by the researcher is reached or when the eigenvalues of each spatial direction are higher than 1.

A first approximation would involve an application of the goodness-of-fit tests on the first one-dimensional subspace. This approach is satisfactory only if the global variance explained by this single direction is enough. A better approach would involve at least the first three spatial directions. For this space, a generalization of the distance computed in Kolmogorov-Smirnov test can easily be found.


Barbara Mascialino - Last update: 03/03/2006