This correspondence analyzes the distribution of loop lengths in graphical models for turbo decoding. The properties of such loops are of significant interest in the context of iterative decoding algorithms based on belief propagation. We estimate the probability that there exist no loops of length less than or equal to c at a randomly chosen node in the acyclic directed graphical (ADC) model for turbo decoding, using a combination of counting arguments and approximations. When K, the number of information bits, is large, this probability is approximately e -2/sup c-1/-4/K, for c/spl ges/4, where nodes for input information bits are ignored for convenience. The analytical results are validated by simulations. For example, for turbo codes with K=64,000, a randomly chosen node has a less than 1% chance of being on a loop of length less than or equal to 10, but has a greater than 99.9% chance of being on a loop of length less than or equal to 20.

This paper addresses the problem of automatically detecting specific patterns or shapes in time-series data. A novel and exible approach is proposed based on segmental semi-Markov models. Unlike dynamic time-warping or template-matching, the proposed framework provides a systematic and coherent framework for leveraging both prior knowledge and training data. The pattern of interest is modeled as a K-state segmental hidden Markov model where each state is responsible for the generation of a component of the overall shape using a state-based regression function. The distance (in time) between segments is modeled as a semi-Markov process, allowing exible deformation of time. The model can be constructed from a single training example. Recognition of a pattern in a new time series is achieved by a recursive Viterbi-like algorithm which scales linearly in the length of the sequence. The method is successfully demonstrated on real data sets, including an application to end-point detection in s...

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