Reconstructing Semi-Markov Maps with Arbitrary Branch Monotonicity from
Density Sequences
Abstract
This paper presents a novel solution to the inverse Frobenius-Perron
problem of reconstructing an unknown nonlinear and ergodic map from
causal sequences of probability density functions generated by the map.
The original solution to this problem successfully reconstructs members
of the canonical map class (i.e., a subset of the piecewise linear
semi-Markov maps), provided all the map’s branches are monotonically
increasing. The original solution constructs a matrix estimate of the
map’s Frobenius-Perron operator, which governs the evolution of density
functions under iteration of the map, from the density sequences. The
one-dimensional map is reconstructed from this matrix. In contrast, the
proposed solution constructs a higher-order matrix estimate of the
Frobenius-Perron operator. A member of the newly proposed class of
generalized hat maps, a superset of the canonical maps, is constructed
from this matrix estimate. The proposed solution successfully
distinguishes between increasing and decreasing map branches and
enlarges the class of maps that can be successfully reconstructed to
canonical maps with any subset of decreasing branches. When used to
reconstruct any piecewise linear semi-Markov map, the proposed solution
generates a map with consistent invariant density and power spectrum
mode characteristics, regardless of the unknown map’s canonicity or
branch monotonicity. Numerical examples that illustrate the proposed
solution’s characteristics are presented.