Sequential Weighting Algorithms for Multi-Alphabet Sources∗

The sequential Context Tree Weighting procedure [6] achieves the asymptotically optimal redundancy behavior of k/2 · (log n)/n, where k is the number of free parameters of the source and n is the sequence length. For FSMX sources with an alphabet A, this number, k, is (|A| − 1) · |Ka|. However, especially the FSMX sources often use only a few possible letters in every state. It would be nice if the linear term in the redundancy was determined by the number of letters actually used in the states in stead of the alphabet size. We shall discuss sequential methods achieving this redundancy behavior. First we describe a ‘natural’ use of the weighting principle on an appropriate class of sources and next we introduce a binary derived model where we use an appropriate estimator.