Time-series information and learning

Given a time series X-1,...,X-n,... taking values in a large (high-dimensional) space X, we would like to find a function f from X to a small (low-dimensional or finite) space Y such that the time series f(X-1),..., f(X-n),... retains all the information about the time-series dependence in the original sequence, or as much as possible thereof. This goal is formalized in this work, and it is shown that the target function f can be found as the one that maximizes a certain quantity that can be expressed in terms of entropies of the series (f(X-i)) i is an element of N. This quantity can be estimated empirically, and does not involve estimating the distribution on the original time series (X-i) i is an element of N.