Random Articles
Reseach Article
An Analysis of Continuous Time Markov Chains using Generator Matrices
| International Journal of Computer Applications |
| Foundation of Computer Science (FCS), NY, USA |
| Volume 35 - Number 10 |
| Year of Publication: 2011 |
| Authors: G. Padma, Dr. C. Vijayalakshmi |
10.5120/4437-6185
|
G. Padma, Dr. C. Vijayalakshmi . An Analysis of Continuous Time Markov Chains using Generator Matrices. International Journal of Computer Applications. 35, 10 ( December 2011), 20-24. DOI=10.5120/4437-6185
Abstract
This paper mainly analyzes the applications of the Generator matrices in a Continuous Time Markov Chain (CTMC). Hidden Markov models [HMMs] together with related probabilistic models such as Stochastic Context-Free Grammars [SCFGs] are the basis of many algorithms for the analysis of biological sequences. Combined with the continuous-time Markov chain theory of likelihood based phylogeny, stochastic grammar approaches are finding broad application in comparative sequence analysis, in particular the annotation of multiple alignments, simultaneous alignment. It was originally used to annotate individual sequences, then in later stages stochastic grammars were soon also combined with phylogenetic models to annotate the alignments. Thus, trees have been combined with HMMs to predict genes and conserved regions in DNA sequences, secondary structures and transmembrane topologies in protein sequences and base pairing structures in RNA sequences. The importance of Generator matrix is analyzed in deriving the various properties of continuous time Markov chins with examples from the phylogenetic tree.
References
- Albert, A ,1962, Estimating the infinitesimal generator of a continuous time, finite state markov process, Ann. Mathemat. Stat., 33, pp (727–753).
- Bruno WJ, 1996, “Modelling residue usage in aligned protein sequences via maximum likelihood”. Molecular Biology and Evolution, 13(10), pp(1368-1374).
- Dayhoff .MO, Eck. RV, Park .CM ,1972, A model of evolutionary changein proteins”. In Atlas of Protein Sequence and Structure, Volume 5. Edited by Dayhoff MO, National Biomedical Research Foundation, Washington, DC, pp. 89?99.
- Darren.J,2006, Wilkinson. Stochastic Modeling for Systems Biology, Chapman & Hall/CRC.
- Holmes I, Rubin GM,2002, “An Expectation Maximization algorithm for training hidden substitution models, Journal of Molecular Biology, 317(5), pp. 757?768.
- Jukes TH,Cantor C,1969, Evolution of protein molecules, In Mammalian Protein Metabolism Academic Press, New York, pp 121-132.
- Kall .L, Krogh .A, Sonnhammer. EL, 2004, A combined transmembrane topology and signal peptide prediction method,Journal ofMolecular Biology, 338(5),pp1027-1036.
- Koshi JM, Goldstein RA ,1995, Context-dependent optimal substitution matrices, Protein Engineering, 8, pp. 641?645.
- Meyn, S.P. and Tweedie, R.L, 1993, Markov Chains and Stochastic Stability. London: Springer-Verlag. ISBN 0-387-19832-6, Cambridge University Press.
- Michalek S, Timmer J, 1999, “Estimating rate constants in hidden Markov models by the EM algorithm”, IEEE Transactions in Signal Processing, 47,p.p(226-228).
- Norris, J.R., 1999, Markov chains, volume 2 of Cambridge Series in Statistical and Probabilistic Mathematics, Cambridge University Press, Cambridge, ISBN0-521-3966, pages (60-70).
- Patricia Buendia, Brice Cadwallader and Victor DeGruttola, 2009, “A phylogenetic and Markov model approach for the reconstruction of mutational pathways of drug resistance Bioinformatics, Vol. 25, No. 19, pp. 2522–2529.
- Padma.G, Dr.C.Vijayalakshmi,2010, “Analysis of Stochastic Differential equations using Propagators”, Publications of Manonmaniam Sundaranar university,Tirunelveli,India, ISBN- 978-81-908352-8-2, pp(43?47).
- Padma.G, Dr.C.Vijayalakshmi,2010, “Propagators of Markov Processes” ,Proceedings of the National Conference at B.S. Abdur Rahman University, Chennai, Vol. No1,pp(171-175).
- Siepel A, Haussler D, 2004, “Phylogenetic estimation of context dependent substitution rates by maximum likelihood”, Molecular Biology and Evolution, 21(3),pp( 468-488).
- Simon Tavaré, C. Adams, Olivier Fedrigo, Gavin j. P. Naylor ,2001, A model for phylogenetic inference using Structural and chemical covariates, Pacific Symposium on Bio Computing, 6, pp(215?225).
- Yang, Z, 1993, “Maximum-likelihood estimation of phylogeny from DNA sequences when substitution rates differ over sites”, Mol. Biol. Evol., 10, pp(1396–1401).
- Yang Z,1994, “Estimating the pattern of nucleotide substitution”,Journal of Molecular Evolution, 39,pp(105-111).
- Alexander Kreinin and Marina Sidelnikova,2001, “Regularization Algorithms forTransition Matrices” Journal of Algo research quarterly, pp (23-40).
- Yasunari Inamura, 2006, “Estimating Continuous Time TransitionMatrices From Discretely Observed Data” Bank of Japan Working Paper Series,pp (1-40).
Index Terms
Keywords