Stochastic Models with Applications to Genetics, Cancers, Aids and Other Biomedical Systems.
Title:
Stochastic Models with Applications to Genetics, Cancers, Aids and Other Biomedical Systems.
Author:
Tan, Wai-Yuan.
ISBN:
9789812777966
Personal Author:
Physical Description:
1 online resource (458 pages)
Series:
Series on Concrete and Applicable Mathematics ; v.4
Series on Concrete and Applicable Mathematics
Contents:
Contents -- Preface -- 1 Introduction -- 1.1. Some Basic Concepts of Stochastic Processes and Examples -- 1.2. Markovian and Non-Markovian Processes Markov Chains and Examples -- 1.3. Diffusion Processes and Examples -- 1.4. State Space Models and Hidden Markov Models -- 1.5. The Scope of the Book -- 1.6. Complements and Exercises -- References -- 2 Discrete Time Markov Chain Models in Genetics and Biomedical Systems -- 2.1. Examples from Genetics and AIDS -- 2.2. The Transition Probabilities and Computation -- 2.3. The Structure and Decomposition of Markov Chains -- 2.4. Classification of States and the Dynamic Behavior of Markov Chains -- 2.5. The Absorption Probabilities of Transient States -- 2.5.1. The case when CT is finite -- 2.5.2. The case when CT is infinite -- 2.6. The Moments of First Absorption Times -- 2.6.1. The case when CT is finite -- 2.7. Some Illustrative Examples -- 2.8. Finite Markov Chains -- 2.8.1. The canonical form of transition matrix -- 2.8.2. Absorption probabilities of transient states in finite Markov chains -- 2.9. Stochastic Difference Equation for Markov Chains With Discrete Time -- 2.9.1. Stochastic difference equations for finite Markov chains -- 2.9.2. Markov chains in the HIV epidemic in homosexual or IV drug user populations -- 2.10. Complements and Exercises -- 2.11. Appendix -- 2.11.1. The Hardy-Weinberg law in population genetics -- 2.11.1.1. The Hardy-Weinberg law for a single locus in diploid populations -- 2.11.1.2. The Hardy-Weinberg law for linked loci in diploid populations -- 2.11.2. The inbreeding mating systems -- 2.11.3. Some mathematical methods for computing An the nth power of a square matrix A -- References -- 3 Stationary Distributions and MCMC in Discrete Time Markov Chains -- 3.1. Introduction.
3.2. The Ergodic States and Some Limiting Theorems -- 3.3. Stationary Distributions and Some Examples -- 3.4. Applications of Stationary Distributions and Some MCMC Methods -- 3.4.1. The Gibbs sampling method -- 3.4.2. The weighted bootstrap method for generating random samples -- 3.4.3. The Metropolis-Hastings algorithm -- 3.5. Some Illustrative Examples -- 3.6. Estimation of Linkage Fraction by Gibbs Sampling Method -- 3.7. Complements and Exercises -- 3.8. Appendix: A Lemma for Finite Markov Chains -- References -- 4 Continuous-Time Markov Chain Models in Genetics Cancers and AIDS -- 4.1. Introduction -- 4.2. The Infinitesimal Generators and an Embedded Markov Chain -- 4.3. The Transition Probabilities and Kolmogorov Equations -- 4.4. Kolmogorov Equations for Finite Markov Chains with Continuous Time -- 4.5. Complements and Exercises -- References -- 5 Absorption Probabilities and Stationary Distributions in Continuous-Time Markov Chain Models -- 5.1. Absorption Probabilities and Moments of First Absorption Times of Transient States -- 5.1.1. The case when CT is finite -- 5.2. The Stationary Distributions and Examples -- 5.3. Finite Markov Chains and the HIV Incubation Distribution -- 5.3.1. Some general results in finite Markov chains with continuous Time -- 5.3.2. Non-homogeneous finite chain with continuous time -- 5.4. Stochastic Differential Equations for Markov Chains with Continuons Time -- 5.4.1. The Feller-Arley stochastic birth-death processes -- 5.4.2. The number of initiated cancer tumor cells in the two-stage model of carcinogenesis -- 5.4.3. The number of sensitive and resistant cancer tumor cells under chemotherapy -- 5.4.4. Finite Markov chains with continuous time -- 5.5. Complements and Exercises -- References -- 6 Diffusion Models in Genetics Cancer and AIDS.
6.1. The Transition Probabilities -- 6.2. The Kolmogorov Forward Equation -- 6.3. The Kolmogorov Backward Equation -- 6.4. Diffusion Approximation of Models from Genetics Cancers and AIDS -- 6.5. Diffusion Approximation of Evolutionary Processes -- 6.6. Diffusion Approximation of Finite Birth-Death Processes -- 6.7. Complements and Exercises -- 6.8. Appendix -- 6.8.1. A general proof of Theorem 6.1 -- 6.8.2. Jacobi polynomials and some properties -- 6.8.2.1. Differential equation for Jacobi polynomials -- 6.8.2.2. An explicit form of Jacobi polynomials -- 6.8.2.3. The Rodrigue's formulae and E[J2n(x -- a b)] -- 6.8.3. Some eigenvalue and eigenfunction problems in differential equations -- References -- 7 Asymptotic Distributions Stationary Distributions and Absorption Probabilities in Diffusion Models -- 7.1. Some Approximation Procedures and Asymptotic Distributions in Diffusion Models -- 7.2. Stationary Distributions in Diffusion Processes -- 7.3. The Absorption Probabilities and Moments of First Absorption Times in Diffusion Processes -- 7.3.1. Absorption probabilities -- 7.3.2. The first two moments of first passage times in diffusion processes -- 7.4. Complements and Exercises -- References -- 8 State Space Models and Some Examples from Cancer and AIDS -- 8.1. Some HIV Epidemic Models as Discrete-Time Linear State Space Models -- 8.1.1. A state space model with variable infection for HIV epidemic in homosexual populations -- 8.1.2. A staged state-space model for HIV epidemic in homosexual populations -- 8.2. Some State Space Models with Continuous-Time Stochastic System Model -- 8.2.1. A state space model for drug resistance in cancer chemotherapy -- 8.2.1.1. Stochastic system model -- 8.2.1.2. The observation model -- 8.2.2. A state space model of HIV pathogenesis -- 8.2.2.1 Stochastic system model.
8.2.2.2. The observation model -- 8.3. Some State Space Models in Carcinogenesis -- 8.3.1. The state space model of the extended multi-event model of carcinogenesis -- 8.3.1.1. The stochastic system model -- 8.3.1.2. Stochastic differential equations for Ij cells j = 0l...k-l -- 8.3.1.3. The probability distribution of T(t) -- 8.3.1.4. The probability distribution of intermediate foci in carcinogenesis studies -- 8.3.1.5. The observation model -- 8.3.2. A state space model for extended multiple pathways models of carcinogenesis -- 8.3.2.1. The stochastic system model -- 8.3.2.2. The observation model -- 8.4. Some Classical Theories of Discrete and Linear State Space Models -- 8.4.1. Some general theories -- 8.4.2. Alternative representation of Kalman filters and smoothers -- 8.4.3. Some classical theories for discrete-time linear state space models with missing data -- 8.5. Estimation of HIV Prevalence and AIDS Cases in the San Francisco Homosexual Population -- 8.5.1. Estimation of parameter values in the San Francisco homosexual population -- 8.5.2. The initial distribution -- 8.5.3. The variances and covariances of random noises and measurement error -- 8.5.4. Estimation results -- 8.5.5. Projection results -- 8.6. Complements and Exercises -- References -- 9 Some General Theories of State Space Models and Applications -- 9.1. Some Classical Theories of Linear State Space Models with Continuous-Time Stochastic System Model -- 9.2. The Extended State Space Models with Continuous-Time Stochastic System Model -- 9.3. Estimation of CD4(+) T Cell Counts and Number of HIV in Blood in HIV-infected Individuals.
9.4. A General Bayesian Procedure for Estimating the Unknown Parameters and the State Variables by State Space Models Simultaneously -- 9.4.1. Generating data from P(X
Abstract:
This book presents a systematic treatment of Markov chains, diffusion processes and state space models, as well as alternative approaches to Markov chains through stochastic difference equations and stochastic differential equations. It illustrates how these processes and approaches are applied to many problems in genetics, carcinogenesis, AIDS epidemiology and other biomedical systems. One feature of the book is that it describes the basic MCMC (Markov chain and Monte Carlo) procedures and illustrates how to use the Gibbs sampling method and the multilevel Gibbs sampling method to solve many problems in genetics, carcinogenesis, AIDS and other biomedical systems. As another feature, the book develops many state space models for many genetic problems, carcinogenesis, AIDS epidemiology and HIV pathogenesis. It shows in detail how to use the multilevel Gibbs sampling method to estimate (or predict) simultaneously the state variables and the unknown parameters in cancer chemotherapy, carcinogenesis, AIDS epidemiology and HIV pathogenesis. As a matter of fact, this book is the first to develop many state space models for many genetic problems, carcinogenesis and other biomedical problems. Sample Chapter(s). Chapter 1.1: Some Basic Concepts of Stochastic Processes and Examples (187 KB). Chapter 1.2: Markovian and Non-Markovian Processes, Markov Chains and Examples (317 KB). Chapter 1.3: Diffusion Processes and Examples (183 KB). Chapter 1.4: State Space Models and Hidden Markov Models (191 KB). Chapter 1.5: The Scope of the Book (145 KB). Chapter 1.6: Complements and Exercises (205 KB). Contents: Discrete Time Markov Chain Models in Genetics and Biomedical Systems; Stationary Distributions and MCMC in Discrete Time Markov Chains; Continuous-Time Markov Chain Models in Genetics, Cancers and AIDS; Absorption Probabilities and Stationary Distributions in
Continuous-Time Markov Chain Models; Diffusion Models in Genetics, Cancer and AIDS; Asymptotic Distributions, Stationary Distributions and Absorption Probabilities in Diffusion Models; State Space Models and Some Examples from Cancer and AIDS; Some General Theories of State Space Models and Applications. Readership: Graduate students and researchers in probability & statistics and the life sciences.
Local Note:
Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2017. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.
Genre:
Electronic Access:
Click to View