Symmetric time warping, Boltzmann pair probabilities and functional genomics.

Given two time series, possibly of different lengths, time warping is a method to construct an optimal alignment obtained by stretching or contracting time intervals. Unlike pairwise alignment of amino acid sequences, classical time warping, originally introduced for speech recognition, is not symmetric in the sense that the time warping distance between two time series is not necessarily equal to the time warping distance of the reversal of the time series. Here we design a new symmetric version of time warping, and present a formal proof of symmetry for our algorithm as well as for one of the variants of Aach and Church [1]. We additionally design quadratic time dynamic programming algorithms to compute both the forward and backward Boltzmann partition functions for symmetric time warping, and hence compute the Boltzmann probability that any two time series points are aligned. In the future, with the availability of increasingly long and accurate time series gene expression data, our algorithm can provide a sense of biological significance for aligned time points - e.g. our algorithm could be used to provide evidence that expression values of two genes have higher Boltzmann probability (say) in the G1 and S phase than in G2 and M phases. Algorithms, source code and web interface, developed by the first author, are made publicly available via the Boltzmann Time Warping web server at bioinformatics.bc.edu/clotelab/.