Source: J. Appl. Probab.
Volume 45, Number 1
In the framework of patterns in random texts, the Markov chain
embedding techniques consist of turning the occurrences of a
pattern over an order-m Markov sequence into those of a
subset of states into an order-1 Markov chain. In this paper we
use the theory of language and automata to provide space-optimal
Markov chain embedding using the new notion of pattern Markov
chains (PMCs), and we give explicit constructive algorithms to
build the PMC associated to any given pattern problem. The
interest of PMCs is then illustrated through the exact computation
of P-values whose complexity is discussed and compared to other
classical asymptotic approximations. Finally, we consider two
illustrative examples of highly degenerated pattern problems
(structured motifs and PROSITE signatures), which further
illustrate the usefulness of our approach.
Antzoulakos, D. L. (2001). Waiting times for patterns in a sequence of multistate trials. J. Appl. Prob. 38, 508--518.
Biggins, J. D. and Cannings, C. (1987). Markov renewal processes, counters and repeated sequences in Markov chains. Adv. Appl. Prob. 19, 521--545.
Mathematical Reviews (MathSciNet): MR903536
Chadjiconstantinidis, S., Antzoulakos, D. L. and Koutras, M. V. (2000). Joint distribution of successes, failures and patterns in enumeration problems. Adv. Appl. Prob. 32, 866--884.
Chryssaphinou, O. and Papastavridis, S. (1990). The occurrence of a sequence of patterns in repeated dependent experiments. Theory Prob. Appl. 35, 167--173.
Crochemore, M. and Hancart, C. (1997). Automata for matching patterns. In Handbook of Formal Languages, Vol. 2, Linear Modeling: Background and Application, Springer, Berlin, pp. 399--462.
Crochemore, M. and Stefanov, V. T. (2003). Waiting time and complexity for matching patterns with automata. Inf. Proc. Lett. 87, 119--125.
Fu, J. C. (1996). Distribution theory of runs and patterns associated with a sequence of multi-state trials. Statistica Sinica 6, 957--974.
Fu, J. C. and Chang, Y. M. (2002). On probability generating functions for waiting time distributions of compound patterns in a sequence of multistate trials. J. Appl. Prob. 30, 183--208.
Fu, J. C. and Koutras, M. V. (1994). Distribution theory of runs: a Markov chain approach. J. Amer. Statist. Assoc. 89, 1050--1058.
Fu, J. C. and Lou, W. Y. W. (2003). Distribution Theory of Runs and Patterns and Its Applications: A Finite Markov Chain Approach. World Scientific, Singapore.
Gasteiger, E., Jung, E. and Bairoch, A. (2001). SWISS-PROT: Connecting biological knowledge via a protein database. Current Issues Molec. Biol. 3, 47--55.
Glaz, J., Kulldorff, M., Pozdnyakov, V. and Steele, J. M. (2006). Gambling teams and waiting times for patterns in two-state Markov chains. J. Appl. Prob. 43, 127--140.
Guibas, L. J. and Odlyzko, A. M. (1981). String overlaps, pattern matching and transitive games. J. Combinatorial Theory A 30, 183--208.
Mathematical Reviews (MathSciNet): MR611250
Hopcroft, J. E., Motwani, R. and Ullman, J. D. (2001). Introduction to Automata Theory, Languages, and Computation, 2nd edn. ACM Press, New York.
Hulo, N. \et\! (2006). The PROSITE database. Nucleic Acid Res. 34, D227--D230.
Lehoucq, R. B., Sorensen, D. C. and Yang, C. (1998). ARPACK Users' Guide. Solution of Large-Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods. Society for Industrial and Applied Mathematics, Philadelphia, PA.
Li, S.-Y. R. (1980). A martingale approach to the study of occurrence of sequence patterns in repeated experiments. Ann. Prob. 8, 1171--1176.
Mathematical Reviews (MathSciNet): MR602390
Lou, W. Y. W. (1996). On runs and longest run tests: a method of finite Markov chain imbedding. J. Amer. Statist. Assoc. 91, 1595--1601.
Marsan, L. and Sagot, M.-F. (2000). Algorithms for extracting structured motifs using a suffix tree with an application to promoter consensus identification. J. Comput. Biol. 7, 345--362.
Nicodeme, P., Salvy, B. and Flajolet, P. (2002). Motifs statistics. Theoret. Comput. Sci. 28, 593--617.
Nuel, G. (2006a). Effective $p$-value computations using finite Markov chain imbedding (FMCI): application to local score and to pattern statistics. Algo. Molec. Biol. 1.
Nuel, G. (2006b). Numerical solutions for patterns statistics on Markov chains. Statist. Appl. Gen. Molec. Biol. 5.
Nuel, G. (2007). Cumulative distribution function of a geometric Poisson distribution. J. Statist. Comput. Simul. 77.
Reinert, G., Schbath, S. and Waterman, M. (2000). Probabilistic and statistical properties of words, an overview. J. Comput. Biology 7, 1--46.
Robin, S. and Daudin, J.-J. (1999). Exact distribution of word occurrences in a random sequence of letters. J. Appl. Prob. 36, 179--193.
Robin, S. and Daudin, J.-J. (2001). Exact distribution of the distances between any occurrences of a set of words. Ann. Inst. Statist. Math. 36, 895--905.
Robin, S. \et\! (2002). Occurrence probability of structured motifs in random sequences. J. Comput. Biol. 9, 761--773.
Stefanov, V. T. (2000). On some waiting time problems. J. Appl. Prob. 37, 756--764.
Stefanov, V. T. (2003). The intersite distances between pattern occurrences in strings generated by general discrete- and continuous-time models: an algorithmic approach. J. Appl. Prob. 40, 881--892.
Stefanov, V. T. and Pakes, A. G. (1997). Explicit distributional results in pattern formation. Ann. Appl. Prob. 7, 666--678.
Stefanov, V. T. and Pakes, A. G. (1999). Explicit distributional results in pattern formation. II. Austral. N. Z. J. Statist. 41, 79--90, 254.
Stefanov, V. T., Robin, S. and Schbath, S. (2007). Waiting times for clumps of patterns and for structured motifs in random sequences. Discrete Appl. Math. 155, 868--880.