An overview on the distribution of word counts in Markov chains

Journal of Computational Biology

Volumn 7, Issue 1-2, 2000, Pages 193-201

  Schbath, Sophie ^a  

^a CEMAGREF   (France)

Author keywords

Clumps; Markovian random sequence; Overlapping occurrences; Renewals; Word count distribution

Indexed keywords

COVARIANCE; DNA SEQUENCE; PRIORITY JOURNAL; PROBABILITY; REVIEW; STATISTICAL ANALYSIS; STATISTICAL MODEL; WORD RECOGNITION;

BIOMETRY; MARKOV CHAINS; MODELS, STATISTICAL; PATTERN RECOGNITION, AUTOMATED; PROBABILITY; SEQUENCE ANALYSIS, DNA; SEQUENCE ANALYSIS, PROTEIN;

EID: 0034048881     PISSN: 10665277     EISSN: None     Source Type: Journal    
DOI: 10.1089/10665270050081469     Document Type: Review

References (62)

1. Arratia, R., Goldstein, L., and Gordon, L., 1989. Two moments suffice for Poisson approximations: the Chen-Stein method. Ann. Prob. 17, 9-25.
2. Arratia, R., Goldstein, L., and Gordon, L., 1990. Poisson approximation and the Chen-Stein method. Statistical Science. 5, 403-434.
3. Arratia, R., and Waterman, M.S. 1985. Critical phenomena in sequence matching. Ann. Prob. 13, 1236-1249.
4. Arratia, R., and Waterman, M.S. 1989. The Erd̈s-Rényi strong law for pattern matching with a given porportion of mismatches. Ann. Prob. 17, 1152-1169.
5. Avery, P.J. 1987. The analysis of intron data and their use in the detection of short signals. J. Mol. Evol. 26, 335-340.
6. Banjevic, D. 1988. On some statistics connected with runs in Markov chains. J. Appl. Prob. 25, 815-821.
7. Benevento, R. 1984. The occurrence of sequence patterns in ergodic Markov chains. Stoch. Proc. Appl. 17, 369-373.
8. Biggins, J.D., and Cannings, C. 1987. Markov renewal processes, counters and repeated sequences in Markov chains. Adv. Appl. Prob. 19, 521-545.
9. Blom, G. 1982. On the mean number of random digits until a given sequence occurs. J. Appl. Prob. 19, 136-143.
10. Blom, G., and Thorburn, D. 1982. How many random digits are required until given sequences are obtained? J. Appl. Prob. 19, 518-531.
11. Bouvier, A., Gélis, F., and Schbath, S. 1999. R'MES : Recherche de Mots Exceptionnels dans les Séquences d'ADN-Version 2. Guide de l'utilisateur. INTRA, Biométrie, F78352 Jouy-en-Josas. (http://www-bia.inra.fr/J/AB/ genome/RMES/welcome.html).
12. Breen, S., Waterman, M.S., and Zhang, N. 1985. Renewal theory for several patterns. J. Appl. Prob. 22, 228-234.
13. Brendel, V., Beckmann, J.S., and Trifonov, E.N. 1986. Linguistics of nucleotide sequences: Morphology and comparison of vocabularies. J. Biomol. Struct. Dynamics. 4, 11-21.
14. Chryssaphinou, O., and Papastavridis, S. 1988. A limit theorem on the number of overlapping appearances of a pattern in a sequence of independent trials. Prob. Theory Rel. Fields. 79, 129-143.
15. Deheuvels, P., Devroye, L., and Lynch, J. 1986. Exact convergence rate in the limit theorems of Erdös-Rényi and Shepp. Ann. Prob. 14, 209-223.
16. Dembo, A., and Karlin, S. 1992. Poisson approximations for r-scan processes. Ann. Appl. Prob. 2, 329-357.
17. Erdös, P., and Rényi, A. 1970. On a new law of large numbers. J. Analyse Math. 23, 103-111.
18. Erhardsson, T. 1997. Compound Poisson approximation for Markov chains. PhD thesis, Royal Institute of Technology, Stockholm.
19. Erhardsson, T. 1999a. Compound Poisson approximation for counts of rare patterns in Markov chains and extreme sojourns in birth-death chains. Preprint.
20. Erhardsson, T. 1999b. Compound Poisson approximation for Markov chains using Stein's method. Ann. Prob. 27, 565-596.
21. Feller, W. 1968. An introduction to Probability Theory and its Applications. Wiley, New York.
22. Fu, J.C. 1993. Poisson convergence in reliability of a large linearly connected system as related to coin tossing. Statistica Sinica. 3, 261-275.
23. Gerber, H.U., and Li, S.-Y. 1981. The occurrence of sequence patterns in repeated experiments and hitting times in a Markov chain. Stock. Proc. Appl. 11, 101-108.
24. Geske, M.X., Godbole, A.P., Schaffner, A.A., Skolnick, A.M., and Wallstrom, G. L. 1995. Compound Poisson approximations for word patterns under Markovian hypotheses. J. Appl. Prob. 32, 877-892.
25. Gélis, F., and Schbath, S. 1996. R'MES : Recherche de Mots Exceptionnels dans les Séquences d'ADN - Version 1. Guide de l'utilisateur. INRA, Biométrie, F78352 Jouy-en-Josas. http://www-bia.inra.fr/J/AB/genome/RMES/ welcome.html.
26. Godbole, A.P. 1991. Poisson approximations for runs and patterns of rare events. Adv. Appl. Prob. 23, 851-865.
27. Godbole, A.P., and Schaffner, A.A. 1993. Improved Poisson approximations for word patterns. Adv. Appl. Prob. 25, 334-347.
28. Godbole, A. 1994. Applications of the Chen-Stein method to the theory of patterns and runs: an annotated bibliography, 253-261. In Runs and Patterns in Probability, A. Godbole and S. Papastavridis, ed., 253-261. Kluwer.
29. Guibas, L.J., and Odlyzko, A.M. 1980. Long repetitive patterns in random sequences. Z. Wahrscheinlichkeitsth. 53, 241-262.
30. Guibas, L.J., and Odlyzko, A.M. 1981a. Periods in strings. J. Combinatorial Theory A. 30, 19-42.
31. Guibas, L.J., and Odlyzko, A.M. 1981b. String overlaps, pattern matching, and nontransitive games. J. Combinatorial Theory A. 30, 183-208.
32. Hirano, K., and Aki, S. 1993. On number of occurrences of success runs of specified length in a two-state Markov chain. Statistica Sinica. 3, 313-320.
33. Karlin, S., and Ost, F. 1987. Counts of long aligned word matches among random letter sequences. Ann. Prob. 19, 293-351.
34. Kleffe, J., and Borodovsky, M. 1992. First and second moment of counts of words in random texts generated by Markov chains. Comp. Applic. Biosci. 8, 433-441.
35. Kusolitsch, N. 1982. Longest runs in Markov chains. Prob. Statist. Inference. 223-230.
36. Li, S. 1980. A martingale approach to the study of occurrence of sequences patterns in repeated experiments. Ann. Prob. 8, 1171-1176.
37. Lundstrom, R. 1990. Stochastic models and statistical methods for DNA sequence data. PhD thesis, University of Utah.
38. Nemetz, T., and Kusolitsch, N. 1982. On the longest run of coincidences. Z. Wahrscheinlichkeitsth. 61, 59-73.
39. Novak, S.Y. 1994. Poisson approximation for the number of long match patterns in a random sequences. Theory Prob. Appl. 39.
40. Novak, S.Y. 1995. Long match patterns in random sequences. Siberian Adv. Math. 5, 128-140.
41. Pevzner, P.A., Borodovsky, M.Y., and Mironov, A.A. 1989. Linguistics of nucleotides sequences I: The significance of deviations from mean statistical characteristics and prediction of the frequencies of occurrence of words. J. Biomol. Struct. Dynamics. 6, 1013-1026.
42. Phillips, G.J., Arnold, J., and Ivarie, R. 1987. The effect of codon usage on the oligonucleotide composition of the e. coli genome and identification of over- and underrepresented sequences by Markov chain analysis. Nucl. Acids Res. 15, 2627-2638.
43. Prum, B., Rodolphe, F., and Turckheim, É. de 1995. Finding words with unexpected frequencies in DNA sequences. J. R. Statist. Soc. B. 57, 205-220.
44. Rajarshi, M.B. 1974. Success runs in a two-state Markov chain. J. Appl. Prob. 11, 190-192.
45. Reinert, G., Schbath, S., and Waterman, M. 1999. Probabilistic and statistical properties of words. In Applied Combinatorics on Words, J. Berstel and D. Perrin, ed. to appear.
46. Reinert, G., and Schbath, S. 1998. Compound Poisson and Poisson process approximations for occurrences of multiple words in Markov chains. J. Comp. Biol. 5, 223-253.
47. Régnier, M. 1999. A unified approach to word occurrence probabilities. To appear in Discrete Applied Mathematics, Special Issue on Computational Biology. Régnier, M., and Szpankowski, W. 1997. On pattern frequency occurences in a Markovian sequence. Purdue University, CSD-TR-96-043; preliminary draft at ISIT'97; submitted.
48. Robin, S., and Daudin, J.-J. 1999. Exact distribution of word occurrences in a random sequence of letters. J. Appl. Prob. 36, 179-193.
49. Roos, M. 1994. Stein's method for compound Poisson approximation: the local approach. Ann. Appl. Prob. 4, 1177-1187.
50. Rudander, J. 1996. On the first occurrence of a given pattern in a semi-Markov process. PhD thesis, University of Uppsala.
51. Samarova, S. 1981. On the length of the longest head run for a Markov chain with two states. Theory Prob. Appl. 26, 498-509.
52. Schbath, S. 1995a. Compound Poisson approximation of word counts in DNA sequences. ESAIM: Probability and Statistics. 1, 1-16. http://www.emath.fr/ps/.
53. Schbath, S. 1995b. Étude asymptotique du nombre d'occurrences d'un mot dans une chaîne de Markov et application à la recherche de mots de fréquence exceptionnelle dans les séquences d'ADN. PhD thesis, Université René Descartes, Paris V.
54. Schbath, S. 1997. An efficient statistic to detect over- and under-represented words in DNA sequences. J. Comp. Biol. 4, 61-82.
55. Schbath, S., Prum, B., and Turckheim, É. de 1995. Exceptional motifs in different Markov chain models for a statistical analysis of DNA sequences. J. Comp. Biol. 2, 417-437.
56. Schwager. 1983. Run probabilities in sequences of Markov-dependent trials. J. Amer. Statist. Assoc. 78, 168-175.
57. Stefanov, V., and Pakes, A.G. 1997. Explicit distributional results in pattern formation. Ann. Appl. Prob. 7, 666-678.
58. Stuckle, E.E., Emmrich, C., Grob, U., and Nielsen, P.J. 1990. Statistical analysis of nucleotide sequences. Nucl. Acids Res. 18, 6641-6647.
59. Tanushev, M.S. 1996. Central limit theorem for several patterns in a Markov chain sequence of letters. Preprint.
60. Tanushev, M.S., and Arratia, R. 1997. Central limit theorem for renewal theory for several patterns. J. Comp. Biol. 4, 35-44.
61. Thorburn, D. 1983. On the mean number of trials until the last trials satisfy a given condition. Stoch. Proc. Appl. 16, 211-217.
62. Waterman, M.S. 1995. Introduction to computational biology. Chapman and Hall.