Exact goodness-of-fit tests for Markov chains

Biometrics

Volumn 69, Issue 2, 2013, Pages 488-496

  Besag, J ^a   Mondal, D ^b  

^a UNIVERSITY OF WASHINGTON   (United States)

^b University of Chicago   (United States)

Author keywords

Binary time series; Conditional tests; Eulerian walks; Exact p values; Molecular dynamics simulations; Monte Carlo methods; Time reversibility

Indexed keywords

AMINO ACIDS; CHAINS; MARKOV PROCESSES; MOLECULAR DYNAMICS; SAMPLING; STATISTICAL TESTS; STATISTICS;

BINARY TIME SERIES; CONDITIONAL TEST; DYNAMICS SIMULATION; EULERIAN; EULERIAN WALK; EXACT P-VALUE; MOLECULAR DYNAMIC SIMULATION; MONTECARLO METHODS; P-VALUES; TIME REVERSIBILITIES;

MONTE CARLO METHODS;

CLIMATE CONDITIONS; GENETIC ANALYSIS; GRAPHICAL METHOD; MARKOV CHAIN; MONTE CARLO ANALYSIS; PEPTIDE; PROBABILITY; TESTING METHOD; TIME SERIES ANALYSIS;

UNITED STATES; WASHINGTON [UNITED STATES];

EID: 84879602441     PISSN: 0006341X     EISSN: 15410420     Source Type: Journal    
DOI: 10.1111/biom.12009     Document Type: Article

References (45)

1. Abrham, J. and Kotzig, A. (1980). Transformations of Euler tours. Annals of Discrete Mathematics 8, 65-69.
2. Aldous, D. J. (1990). The random walk construction of uniform spanning trees and uniform labelled trees. SIAM Journal of Discrete Mathematics 3, 450-465.
3. Anderson, T.W. (1957). Statistical inference about Markov chains. The Annals of Mathematical Statistics 28, 89-110.
4. Avery, P. and Henderson, D. (1999). Fitting Markov chain models to discrete state series such as DNA sequences. Applied Statistics 48, 53-62.
5. Bacallado, S., Chodera, J. D., and Pande, V. (2009). Bayesian comparison of Markov models of molecular dynamics with detailed balance constraint. Journal of Chemical Physics 131, 045106-01 to 045106-10.
6. Bacallado, S. (2011). Bayesian analysis of variable-order, reversible Markov chains. Annals of Statistics 39, 838-864.
7. Barnard, G. A. (1963). Discussion of paper by M.S. Bartlett. Journal of the Royal Statistical Society B 25, 294.
8. Bartlett, M. S. (1951). The frequency goodness-of-fit test for probability chains. Proceedings of the Cambridge Philosophical Society 47, 86-95.
9. Batty, M. (1981). Symmetry and reversibility in social exchange. The Journal of Mathematical Sociology 8, 1-41.
10. Besag, J. E. and Clifford, P. (1989). Generalized Monte Carlo significance tests. Biometrika 76, 633-642.
11. Besag, J. E. and Clifford, P. (1991). Sequential Monte Carlo p-values. Biometrika 78, 301-304.
12. Billingsley, P. (1961). Statistical methods in Markov chains. The Annals of Mathematical Statistics 32, 12-40.
13. Brightwell, G. and Winkler, P. (2005). Counting Eulerian circuits is #P-complete. Proceedings of the Second Workshop on Analytic Algorithmics and Combinatorics 259-262.
14. Chang L-B. and Geman, S. (2009). Stock prices and the peculiar statistics of large returns. Preliminary paper. Division of Applied Mathematics, University of Brown.
15. Chatfield, C. (1973). Statistical inference regarding Markov chain models. Applied Statistics 22, 7-21.
16. Chodera, J. D., Swope, W. C., Pitera, J. W., and Dill, K. A. (2006) Long-time protein folding dynamics from short-time molecular dynamics simulations. Multiscale Modeling and Simulation 5, 1214-1226.
17. Chodera, J. D., Singhal, N., Pande, V. S., Dill, K. A., Swope, W. C. (2007). Automatic discovery of metastable states for the construction of Markov models of macromolecular conformational dynamics. Journal of Chemical Physics 126, 155101-155117.
18. Dawson, R. and Good, I. J. (1957). Exact Markov probabilities from oriented linear graphs. Annals of the Mathematical Statistics 28, 946-956.
19. de Jong, P. and Greig, M. (1984). First-order Markov chains with a zero diagonal transition matrix. Biometrics 40, 101-107.
20. Diaconis, P. and Rolles, S. W. W. (2006). Bayesian analysis for reversible Markov chains. Annals of Statistics 34, 1270-1292.
21. di Cecco, D. (2011). Conditional exact tests for Markovianity and reversibility in multiple categorical sequences. Test 21, 170-187.
22. Dwass, M. (1957). Modified randomization tests for nonparametric hypotheses. Annals of Mathematical Statistics 28, 181-187.
23. Edwards, D. G. (1980). Stationarity and reversibility in finite Markov chains. Scandinavian Journal of Statistics 7, 203-206.
24. Ehrenfest, T. V. A. and de Bruijn, N. G. (1951). Circuits and trees in oriented linear graphs. Simon Stevin 28, 203-217.
25. Fleischner, H. (1991). Algorithms for Eulerian Trails. In Eulerian Graphs and Related Topics: Part 1, Volume 2, Annals of Discrete Mathematics, 50. Amsterdam, North Holland: Elsevier, pp. X.1-13.
26. Goodman, L. A. (1957). Statistical inference about Markov chains. Annals of Mathematical Statistic 28, 89-110.
27. Gottschau, A. (1992). Exchangeability in Multivariate Markov Chain Models. Biometrics 48, 751-763.
28. Guttorp, P. (1995). Stochastic Modeling of Scientific Data. London: Chapman and Hall.
29. Hall, P. and Titterington, D. M. (1989). The effect of simulation order on level accuracy and power of Monte Carlo tests. Journal of the Royal Statistical Society B 51, 459-467.
30. Hoel, P. G. (1954), A test for Markov chains. Biometrika 41, 430-433
31. Johnson, D. B. (1975). Finding all the elementary circuits of a directed graph. Siam Journal on Computing P4, 77-84.
32. Kandel, D., Matias, Y., Unger, R., and Winkler, P. (1996). Shuffling biological sequences. Discrete Applied Mathematics 71, 171-185.
33. Kelly, F. P. (1979). Reversibility and Stochastic Networks. Chichester: Wiley.
34. Klotz, J. (1973). Statistical inference in Bernoulli trials with dependence. The Annals of Statistics 1, 373-379.
35. Kotzig, A. (1966). Eulerian lines in finite 4-valent graphs and their transformations. In Theory of Graphs, P. Erdos and G. Katona, (eds), Proceedings of the Colloquium held at Tihany, Hungary, (1968). Akademiai Kiado, Publishing House of the Hungarian Academy of Sciences: Budapest. pp.219-230.
36. Krumbein, W. C. and Dacey, M. F. (1969). Markov chains and embedded Markov chains in geology. Mathematical Geology 1, 79-96.
37. McCausland, W. J. (2007). Time reversibility of stationary regular finite-state Markov chains. Journal of Econometrics 136, 303-318.
38. McCullagh, P. (1982). Applications of quasisymmetry. Biometrika 69, 303-308.
39. Prinz, J.-H., Wu, H., Sarich, M., Keller, B., Fischbach, M., Held, M., Chodera, J. D., Schtte, Ch., and No, F. (2011). Markov models of molecular kinetics: Generation and validation. Journal of Chemical Physics 134, 174105-1 to 174105-23.
40. Quintana, F. A. and Newton, M. A. (1998). Assessing the order of dependence for partially exchangeable binary data. Journal of the American Statistical Association 93, 194-202.
41. Richman, D. and Sharp,W. E. (1990). A method for determining the reversibility of a Markov Sequence. Mathematical Geology 22, 749-761.
42. Sharp, W. E. and Severance, K. (1991). Reversible Markov grain sequences in granite. Mathematical Geology 23, 305-324.
43. Smith, W. and Solow, A. R. (1996). An exact McNemar test for paired binary Markov chains. Biometrics 52, 1063-1070.
44. Tutte, W. T. and Smith, C. A. B. (1941). On unicursal paths in a network of degree 4. The American Mathematical Monthly 48, 233-237.
45. Whittle, P. (1955). Some distribution and moment formulae for the Markov chain. Journal of the Royal Statistical Society, Series B 17, 235-242.