On State Occupancies, First Passage Times and Duration in Non-Homogeneous Semi-Markov Chains
24 Jul 2021-Vol. 9, Iss: 15, pp 1745
TL;DR: In this paper, the authors provide closed analytic forms for three types of probabilities which describe attributes of considerable research interest in semi-Markov modeling: (a) the number of transitions to a state through time (Occupancy), (b) the amount of transitions or the time required to observe the first passage to a states (First passage time), and (c) the period of time required after a state is entered before the first real transition is made to another state (Duration).
Abstract: Semi-Markov processes generalize the Markov chains framework by utilizing abstract sojourn time distributions. They are widely known for offering enhanced accuracy in modeling stochastic phenomena. The aim of this paper is to provide closed analytic forms for three types of probabilities which describe attributes of considerable research interest in semi-Markov modeling: (a) the number of transitions to a state through time (Occupancy), (b) the number of transitions or the amount of time required to observe the first passage to a state (First passage time) and (c) the number of transitions or the amount of time required after a state is entered before the first real transition is made to another state (Duration). The non-homogeneous in time recursive relations of the above probabilities are developed and a description of the corresponding geometric transforms is produced. By applying appropriate properties, the closed analytic forms of the above probabilities are provided. Finally, data from human DNA sequences are used to illustrate the theoretical results of the paper.
Citations
More filters
04 Oct 2021
TL;DR: Probability resembles the ancient Roman God Janus since, like Janus, probability also has a face with two different sides, which correspond to the metaphorical gateways and transitions between the past and the future as mentioned in this paper.
Abstract: Probability resembles the ancient Roman God Janus since, like Janus, probability also has a face with two different sides, which correspond to the metaphorical gateways and transitions between the past and the future [...]
1 citations
TL;DR: In this article , a non-homogeneous semi-Markov-decision-process (NHSMDP) based naive algorithm that relies on prior knowledge about the layout of a building and uses recurring updates of the shooter's location (based on automatic processing of images from a camera network) to provide an optimized egress plan for evacuees.
References
More filters
TL;DR: It is shown that CpG islands in methylated genomes are maintained, despite a tendency for 5mCpG to mutate by deamination to TpG+CpA, by the structural stability of a high G+C content alone, and that C pG islands associated with exons result from some selective importance of the arginine codon CGX.
3,321 citations
Book•
01 Jan 2001TL;DR: In this article, first passage in an interval is illustrated in simple geometries, and the first passage is in a semi-infinite system and a non-fractal system.
Abstract: Preface Errata 1. First-passage fundamentals 2. First passage in an interval 3. Semi-infinite system 4. Illustrations of first passage in simple geometries 5. Fractal and nonfractal networks 6. Systems with spherical symmetry 7. Wedge domains 8. Applications to simple reactions References Index.
1,891 citations
01 Oct 2001
TL;DR: In this paper, first passage in an interval is illustrated in simple geometries, and the first passage is in a semi-infinite system and a non-fractal system.
Abstract: Preface Errata 1. First-passage fundamentals 2. First passage in an interval 3. Semi-infinite system 4. Illustrations of first passage in simple geometries 5. Fractal and nonfractal networks 6. Systems with spherical symmetry 7. Wedge domains 8. Applications to simple reactions References Index.
1,282 citations
Book•
01 Jan 1995TL;DR: A Tree Representation of a Sequence Hashing a Sequence Repeats in a Sequence Sequence Sequence Comparison by Hashing Sequence Comparison with at most l mismatches Sequence comparison by Statistical Content Dynamic Programming Alignment of Two Sequences The number of Alignments Shortest and Longest Paths in a Network Global Distance Alignment Global Similarity Alignment Fitting One Sequence into Another Local Alignment and Clumps Linear Space Algorithms Tracebacks Inversions Map Alignment Parametric Sequence Comparisons Multiple Sequence Alignment The Cystic Fibrosis Gene Dynamic Programming in r-Dimensions Weight
Abstract: Preface Introduction Molecular Biology Mathematics, Statistics, and Computer Science Some Molecular Biology DNA and Proteins The Central Dogma The Genetic Code Transfer RNA and Protein Sequences Genes Are Not Simple Biological Chemistry Restriction Maps Introduction Graphs Interval Graphs Measuring Fragment Sizes Multiple Maps Double Digest Problem Classifying Multiple Solutions Algorithms for DDP Algorithms and Complexity DDP is N P-Complete Approaches to DDP Simulated Annealing: TSP and DDP Mapping with Real Data Cloning and Clone Libraries A Finite Number of Random Clones Libraries by Complete Digestion Libraries by Partial Digestion Genomes per Microgram Physical Genome Maps: Oceans, Islands, and Anchors Mapping by Fingerprinting Mapping by Anchoring An Overview of Clone Overlap Putting It Together Sequence Assembly Shotgun Sequencing Sequencing by Hybridization Shotgun Sequencing Revisited Databases and Rapid Sequence Analysis DNA and Protein Sequence Databases A Tree Representation of a Sequence Hashing a Sequence Repeats in a Sequence Sequence Comparison by Hashing Sequence Comparison with at most l Mismatches Sequence Comparison by Statistical Content Dynamic Programming Alignment of Two Sequences The Number of Alignments Shortest and Longest Paths in a Network Global Distance Alignment Global Similarity Alignment Fitting One Sequence into Another Local Alignment and Clumps Linear Space Algorithms Tracebacks Inversions Map Alignment Parametric Sequence Comparisons Multiple Sequence Alignment The Cystic Fibrosis Gene Dynamic Programming in r-Dimensions Weighted-Average Sequences Profile Analysis Alignment by Hidden Markov Models Consensus Word Analysis Probability and Statistics for Sequence Alignment Global Alignment Local Alignment Extreme Value Distributions The Chein-Stein Method Poisson Approximation and Long Matches Sequence Alignment with Scores Probability and Statistics for Sequence Patterns A Central Limit Theorem Nonoverlapping Pattern Counts Poisson Approximation Site Distributions RNA Secondary Structure Combinatorics Minimum Free-energy Structures Consensus folding Trees and Sequences Trees Distance Parsimony Maximum Likelihood Trees Sources and Perspectives Molecular Biology Physical Maps and Clone Libraries Sequence Assembly Sequence Comparisons Probability and Statistics RNA Secondary Structure Trees and Sequences References Problem Solutions and Hints Mathematical Notation Algorithm Index Author Index Subject Index
523 citations