# Evolvable Warps for Data Normalization

### Jeremy Gilbert and Daniel Ashlock

Submitted to CEC 2016

The traditional method of fitting an approximate cumulative
probability distribution to a data set is to bin the data in narrow
bins and obtain a step function approximation. This technique suffices
for many applications, but the resulting object is not a
differentiable function making recovery of the underlying probability
distribution function impossible. In this study, a unique group
theoretic representation is used to define evolvable data warps that
can be used to recover continuous, infinitely differentiable versions
of the inverse cumulative distribution function. The use of a group
theoretic representation permits a simple calculation to transform the
evolved object into a cumulative distribution function and, via
differentiation, into a probability distribution function. The group
used to define the evolvable data warps is the group of bijection of
the unit interval. The generators used by evolution are chosen to be
differentiable in order to enable the computation of probability
distribution functions.