Stationary Stochastic Processes and Fractal Data Compression

It is shown that the invariant measure of a stationary nonatomic stochastic process yields an iterated function system with probabilities and an associated dynamical system that provide the basis for optimal lossless data compression algorithms. The theory is illustrated for the case of finite-order Markov processes: For a zero-order process, it produces the arithmetic compression method; while for higher order processes it yields dynamical systems, constructed from piecewise affine mappings from the interval [0, 1] into itself, that may be used to store information efficiently. The theory leads to a new geometrical approach to the development of compression algorithms.