Abstract
The connection between a disease and some genetic disorders can be studied by looking for recurrent alterations in genomic profiles of humans affected by the disease. Modeling each patient profile as a 2-state jump Markov process, we define a recurrent alteration in terms of a sojourn time of a cumulated process. The cumulated process happen to be a birth-and-death process. We provide the exact Laplace transforms of relevant hitting times associated with the latter. Based on this exact results, we derive an upper bound of a sojourn time probability of interest. Our upper bound is exactly equal to the p-value nominated in all earlier literature where an attempt is made for an assessment of the significance of genomic alterations. Moreover, that p-value has never been evaluated exactly in previous works. Finally, we give abaques which are useful for the assessment of recurrent genomic alterations for a variety of realistic alteration frequencies and lengths.
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References
Abate J, Whitt W (2006) A unified framework for numerically inverting Laplace transforms. INFORMS J Comput 18:408–421
Ball F, Stefanov VT (2001) Further approaches to computing fundamental characteristics of birth-death processes. J. Appl Probab 38:995–1005
Chin K, DeVries S, Fridlyand J, Spellman PT, Roydasgupta R, Kuo WL, Lapuk A, Neve RM, Qian Z, Ryder T, Chen F, Feiler H, Tokuyasu T, Kingsley C, Dairkee S, Meng Z, Chew K, Pinkel D, Jain A, Ljung BM, Esserman L, Albertson DG, Waldman FM, Gray JW (2006) Genomic and transcriptional aberrations linked to breast cancer pathophysiologies. Cancer Cell 10(6):529–541
Kijima M (1997) Markov processes for stochastic modeling. Chapman and Hall, London
Rapaport F, Leslie C (2010) Determining frequent patterns of copy number alterations in cancer. PLoS ONE 5(8):e12, 028
Robin S, Stefanov VT (2009) Simultaneous occurrences of runs in independent Markov chains. Method Comput Appl Probab 11(2):267–275. doi:10.1007/s11009-008-9093-3
Rouveirol C, Stransky N, Hupé P, La Rosa P, Viara E, Barillot E, Radvanyi F (2006) Computation of recurrent minimal genomic alterations from array-CGH data. Bioinformatics 22(7):849–856
Rueda OM, Diaz-Uriarte R (2010) Finding recurrent copy number alteration regions: a review of methods. Curr Bioinforma 5(1):1–17 . doi:10.2174/157489310790596402
Vert JP, Bleakley K (2010) Fast detection of multiple change-points shared by many signals using group lars. In: Lafferty JD, Williams CKI, Shawe-Taylor J, Zemel RS, Culotta A (eds) NIPS. Curran Associates, Inc., pp 2343–2351. http://dblp.uni-trier.de/db/conf/nips/nips2010.html#VertB10
Yau C, Mouradov D, Jorissen R, Colella S, Mirza G, Steers G, Harris A, Ragoussis J, Sieber O, Holmes C (2010) A statistical approach for detecting genomic aberrations in heterogeneous tumor samples from single nucleotide polymorphism genotyping data. Genome Biol 11(9):R92. doi:10.1186/gb-2010-11-9-r92. http://genomebiology.com/content/11/9/R92
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Robin, S., Stefanov, V.T. Detection of Significant Genomic Alterations via Simultaneous Minimal Sojourns at a State by Independent Continuous-time Markov Chains. Methodol Comput Appl Probab 17, 479–487 (2015). https://doi.org/10.1007/s11009-013-9374-3
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DOI: https://doi.org/10.1007/s11009-013-9374-3