  Suppose we are given a time series or a signal $x(t)$ for $0\leq t\leq T$. We
consider the problem of predicting the signal in the interval $T<t\leq T+t_{f}$
from a knowledge of its history and nothing more. We ask the following
question: what is the largest value of $t_{f}$ for which a prediction can be
made? We show that the answer to this question is contained in a fundamental
result of information theory due to Wyner, Ziv, Ornstein, and Weiss. In
particular, for the class of chaotic signals, the upper bound is
$t_{f}\leq\log_{2}T/H$ in the limit $T\rightarrow\infty$, with $H$ being
entropy in a sense that is explained in the text.
  If $\bigl|x(T-s)-x(t^{\ast}-s)\bigr|$ is small for $0\leq s\leq\tau$, where
$\tau$ is of the order of a characteristic time scale, the pattern of events
leading up to $t=T$ is similar to the pattern of events leading up to
$t=t^{\ast}$. It is reasonable to expect $x(t^{\ast}+t_{f})$ to be a good
predictor of $x(T+t_{f}).$ All existing methods for prediction use this idea in
some way or the other. Unfortunately, this intuitively reasonable idea is
fundamentally deficient and all existing methods fall well short of the
Wyner-Ziv entropy bound on $t_{f}$. An optimal predictor should decompose the
distance between the pattern of events leading up to $t=T$ and the pattern
leading up to $t=t^{\ast}$ into stable and unstable components. A good match
should have suitably small unstable components but will in general allow stable
components which are as large as the tolerance for correct prediction. For the
special case of hyperbolic toral automorphisms, we derive an optimal predictor
using Pade approximation.
