Advertisement

Distributions of pattern statistics in sparse Markov models

  • Donald E. K. MartinEmail author
Article
  • 10 Downloads

Abstract

Markov models provide a good approximation to probabilities associated with many categorical time series, and thus they are applied extensively. However, a major drawback associated with them is that the number of model parameters grows exponentially in the order of the model, and thus only very low-order models are considered in applications. Another drawback is lack of flexibility, in that Markov models give relatively few choices for the number of model parameters. Sparse Markov models are Markov models with conditioning histories that are grouped into classes such that the conditional probability distribution for members of each class is constant. The model gives a better handling of the trade-off between bias associated with having too few model parameters and variance from having too many. In this paper, methodology for efficient computation of pattern distributions through Markov chains with minimal state spaces is extended to the sparse Markov framework.

Keywords

Auxiliary Markov chain Pattern distribution Sparse Markov model Variable length Markov chain 

Notes

Acknowledgements

This material is based upon work supported by the National Science Foundation under Grant No. 1811933. The author would like to thank the reviewer for their insightful comments on the original version of the manuscript.

References

  1. Aho, A. V., Corasick, M. J. (1975). Efficient string matching: An aid to bibliographic search. Communications of the ACM, 18, 333–340.MathSciNetzbMATHGoogle Scholar
  2. Aston, J. A. D., Martin, D. E. K. (2007). Waiting time distributions of general runs and patterns in hidden Markov models. Annals of Applied Statistics, 1(2), 585–611.MathSciNetzbMATHGoogle Scholar
  3. Begleiter, R., El-Yaniv, R., Yona, G. (2004). On prediction using variable length Markov models. Journal of Artificial Intelligence, 22, 385–421.zbMATHGoogle Scholar
  4. Belloni, A., Oliveira, R. (2017). Approximate group context tree. The Annals of Statistics, 45(1), 355–385.MathSciNetzbMATHGoogle Scholar
  5. Ben-gal, I., Morag, G., Shmilovici, A. (2003). Context-based statistical process control. Technometrics, 45(4), 293–311.MathSciNetGoogle Scholar
  6. Benson, G., Mak, D. Y. F. (2009). Exact distribution of a spaced seed statistic for DNA homology detection. String processing and information retrieval, Lecture Notes in Computer Science, Vol. 5280, pp. 283–293. Berlin: Springer.Google Scholar
  7. Bercovici, S., Rodriguez, J. M., Elmore, M., Batzoglou, S. (2012). Ancestry inference in complex admixtures via variable-length Markov chain linkage models. Research in computational molecular biology, RECOMB 2012, Lecture Notes in Computer Science, Vol. 7262, pp. 12–28. Berlin: Springer.Google Scholar
  8. Borges, J., Levene, M. (2007). Evaluating variable length Markov chain models for analysis of user web navigation. IEEE Transactions on Knowledge, 19(4), 441–452.Google Scholar
  9. Bratko, A., Cormack, G., Filipic̆, B., Lynam, T., Zupan, B. (2006). Spam filtering using statistical data compression models. Journal of Machine Learning Research, 7, 2673–2698.MathSciNetzbMATHGoogle Scholar
  10. Brookner, E. (1966). Recurrent events in a Markov chain. Information and Control, 9, 215–229.MathSciNetzbMATHGoogle Scholar
  11. Browning, S. R. (2006). Multilocus association mapping using variable-length Markov chains. American Journal of Human Genetics, 78, 903–913.Google Scholar
  12. Buhler, J., Keich, U., Sun, Y. (2005). Designing seeds for similarity search in genomic DNA. Journal of Computer and Systems Science, 70, 342–363.MathSciNetGoogle Scholar
  13. Bühlmann, P., Wyner, A. J. (1999). Variable length Markov chains. Annals of Statistics, 27(2), 480–513.MathSciNetzbMATHGoogle Scholar
  14. Fernández, M., García, J. E., González-López, V. A. (2018). A copula-based partition Markov procedure. Communications in Statistics-Theory and Methods, 47(14), 3408–3417.MathSciNetGoogle Scholar
  15. Fu, J. C., Koutras, M. V. (1994). Distribution theory of runs: A Markov chain approach. Journal of the American Statistical Association, 89, 1050–1058.MathSciNetzbMATHGoogle Scholar
  16. Gabadinho, A., Ritschard, G. (2016). Analyzing state sequences with probabilistic suffix trees. Journal of Statistical Software, 72(3), 1–39.Google Scholar
  17. Gallo, S., Leonardi, F. (2015). Nonparametric statistical inference for the context tree of a stationary ergodic process. Electronic Journal of Statistics, 9, 2076–2098.MathSciNetzbMATHGoogle Scholar
  18. Galves, A., Galves, C., García, J. E., Garcia, N. L., Leonardi, F. (2012). Context tree selection and linguistic rhythm retrieval from written texts. Annals of Applied Statistics, 6, 186–209.MathSciNetzbMATHGoogle Scholar
  19. García, J. E., González-López, V. A. (2010). Minimal Markov models. arXiv:1002.0729.
  20. García, J. E., González-López, V. A. (2017). Consistent estimation of partition Markov models. Entropy, 19, 1050–1058.MathSciNetGoogle Scholar
  21. Hopcroft, J. E. (1971). An \(n\) log \(n\) algorithm for minimizing states in a finite automaton. In Z. Kohavi & A. Paz (Eds.), Theory of Machines and Computation, pp. 189–196. New York: Academic Press.Google Scholar
  22. Jääskinen, V., Xiong, J., Koski, T., Corander, J. (2014). Sparse Markov chains for sequence data. Scandinavian Journal of Statistics, 41, 641–655.MathSciNetzbMATHGoogle Scholar
  23. Keich, U., Li, M., Ma, B., Tromp, J. (2004). On spaced seeds for similarity search. Discrete Applied Mathematics, 138(3), 253–263.MathSciNetzbMATHGoogle Scholar
  24. Koutras, M. V., Alexandrou, V. A. (1995). Runs, scans and urn models: A unified Markov chain approach. Annals of the Institute of Statistical Mathematics, 47, 743–766.MathSciNetzbMATHGoogle Scholar
  25. Lladser, M. E. (2007). Minimal Markov chain embeddings of pattern problems. In Proceedings of the 2007 information theory and applications workshop, University of California, San Diego.Google Scholar
  26. Lladser, M., Betterton, M. D., Knight, R. (2008). Multiple pattern matching: A Markov chain approach. Journal of Mathematical Biology, 56(1–2), 51–92.MathSciNetzbMATHGoogle Scholar
  27. Ma, B., Tromp, J., Li, M. (2002). PatternHunter: Faster and more sensitive homology search. Bioinformatics, 18(3), 440–445.Google Scholar
  28. Mak, D. Y. F., Benson, G. (2009). All hits all the time: Parameter-free calculation of spaced seed sensitivity. Bioinformatics, 25(3), 302–308.Google Scholar
  29. Marshall, T., Rahmann, S. (2008). Probabilistic arithmetic automata and their application to pattern matching statistics. In: Ferragina, P., Landau, G.M. (eds), Proceedings of the 19th annual symposium on combinatorial pattern matching (CPM), Lecture Notes in Computer Science, Vol. 5029, pp. 95–106. Heidelberg: Springer.Google Scholar
  30. Martin, D. E. K. (2018). Minimal auxiliary Markov chains through sequential elimination of states. Communications in Statistics-Simulation and Computation.  https://doi.org/10.1080/03610918.2017.1406505.
  31. Martin, D. E. K., Coleman, D. A. (2011). Distributions of clump statistics for a collection of words. Journal of Applied Probability, 48, 1049–1059.MathSciNetzbMATHGoogle Scholar
  32. Martin, D. E. K., Noé, L. (2017). Faster exact probabilities for statistics of overlapping pattern occurrences. Annals of the Institute of Statistical Mathematics, 69(1), 231–248.MathSciNetGoogle Scholar
  33. Noé, L. (2017). Best hits of 11110110111: Model-free selection and parameter-free sensitivity calculation of spaced seeds. Algorithms for Molecular Biology, 12(1), 1.  https://doi.org/10.1186/s13015-017-0092-1.MathSciNetGoogle Scholar
  34. Noé, L., Martin, D. E. K. (2014). A coverage criterion for spaced seeds and its applications to SVM string-kernels and \(k\)-mer distances. Journal of Computational Biology, 21(12), 947–963.Google Scholar
  35. Nuel, G. (2008). Pattern Markov chains: Optimal Markov chain embedding through deterministic finite automata. Journal of Applied Probability, 45, 226–243.MathSciNetzbMATHGoogle Scholar
  36. Ribeca, P., Raineri, E. (2008). Faster exact Markovian probability functions for motif occurrences: A DFA-only approach. Bioinformatics, 24(24), 2839–2848.Google Scholar
  37. Rissanen, J. (1983). A universal data compression system. IEEE Transactions on Information Theory, 29, 656–664.MathSciNetzbMATHGoogle Scholar
  38. Rissanen, J. (1986). Complexity of strings in the class of Markov sources. IEEE Transactions on Information Theory, 32(4), 526–532.MathSciNetzbMATHGoogle Scholar
  39. Ron, D., Singer, Y., Tishby, N. (1996). The power of amnesia: Learning probabilistic automata with variable memory length. Machine Learning, 25(2–3), 117–149.zbMATHGoogle Scholar
  40. Roos, T., Yu, B. (2009). Sparse Markov source estimation via transformed Lasso. In Proceedings of the IEEE Information Theory Workshop (ITW-2009), pp. 241–245. Taormina, Sicily, Italy.Google Scholar
  41. Shmilovici, A., Ben-gal, I. (2007). Using a VOM model for reconstructing potential coding regions in EST sequences. Computational Statistics, 22, 49–69.MathSciNetzbMATHGoogle Scholar
  42. Weinberger, M., Lempel, A., Ziv, J. (1992). A sequential algorithm for the universal coding of finite memory sources. IEEE Transactions on Information Theory, IT–38, 1002–1024.MathSciNetzbMATHGoogle Scholar
  43. Weinberger, M., Rissanen, J., Feder, M. (1995). A universal finite memory source. IEEE Transactions on Information Theory, 41(3), 643–652.zbMATHGoogle Scholar
  44. Willems, F. M. J., Shtarkov, Y. M., Tjalkens, T. J. (1995). The context-tree weighting method: Basic properties. IEEE Transactions on Information Theory, 41(3), 653–664.zbMATHGoogle Scholar
  45. Xiong, J., Jääskinen, V., Corander, J. (2016). Recursive learning for sparse Markov models. Bayesian Analysis, 11(1), 247–263.MathSciNetzbMATHGoogle Scholar

Copyright information

© The Institute of Statistical Mathematics, Tokyo 2019

Authors and Affiliations

  1. 1.Department of StatisticsNorth Carolina State UniversityRaleighUSA

Personalised recommendations