# Spherical-Homoscedastic Distributions: The Equivalency of Spherical and Normal Distributions in Classification

Onur C. Hamsici, Aleix M. Martinez.

Year: 2007, Volume: 8, Issue: 56, Pages: 1583−1623

#### Abstract

Many feature representations, as in genomics, describe directional
data where all feature vectors share a common norm. In other
cases, as in computer vision, a norm or variance normalization
step, where all feature vectors are normalized to a common length,
is generally used. These representations and pre-processing step
map the original data from ℜ^{p} to the surface of a
hypersphere *S*^{p-1}. Such representations should then be
modeled using spherical distributions. However, the difficulty
associated with such spherical representations has prompted
researchers to model their spherical data using Gaussian
distributions instead---as if the data were represented in
ℜ^{p} rather than *S*^{p-1}.
This opens the question to
whether the classification results calculated with the Gaussian
approximation are the same as those obtained when using the
original spherical distributions. In this paper, we show that in
some particular cases (which we named spherical-homoscedastic) the
answer to this question is positive. In the more general case
however, the answer is negative. For this reason, we further
investigate the additional error added by the Gaussian modeling.
We conclude that the more the data deviates from
spherical-homoscedastic, the less advisable it is to employ the
Gaussian approximation. We then show how our derivations can be
used to define optimal classifiers for spherical-homoscedastic
distributions. By using a kernel which maps the original space
into one where the data adapts to the spherical-homoscedastic
model, we can derive non-linear classifiers with potential
applications in a large number of problems. We conclude this paper
by demonstrating the uses of spherical-homoscedasticity in the
classification of images of objects, gene expression sequences,
and text data.