出版日期:2016-5-1
ISBN:9787519211681
作者:Christiane Fuchs
内容概要
《扩散过程推断理论及其在生命科学中的应用》是一部理论和实践相结合的参考图书，作者为数学、统计学以及一些前沿学科生物信息学和生物学的不可或缺的学习参考用书。
书籍目录
Introduction
1.1 Aims of This Book
1.2 Outline of This Book
Part Ⅰ Stochastic Modelling
2 Stochastic Modelling in Life Sciences
2.1 Compartment Models
2.2 Modelling the Spread of Infectious Diseases
2.2.1 History of Epidemic Modelling
2.2.2 SIR Model
2.2.3 Model Extensions
2.3 Modelling Processes in Molecular Biology，Biochemistry and Genetics
2.3.1 History of Chemical Reaction Modelling
2.3.2 Chemical Reaction Kinetics
2.3.3 Reaction Kinetics in the Biological Sciences
2.4 Algorithms for Simulation
2.4.1 Simulation of Continuous—Time Markov Jump Processes
2.4.2 Simulation of Solutions of ODEs and SDEs
2.5 Conclusion
References
3 Stochastic Differential Equations and Diffusions in a Nutshell
3.1 Brownian Motion and Gaussian White Noise
3.1.1 Brownian Motion
3.1.2 Brownian Bridge
3.1.3 Gaussian White Noise
3.1.4 Excursus： Lovy Processes
3.2 Ito Calculus
3.2.1 Stochastic Integral and Stochastic Differential Equations
3.2.2 Different Stochastic Integrals
3.2.3 Existence and Uniqueness of Solutions
3.2.4 Transition Density and Likelihood Function
3.2.5 Ito Diffusion Processes
3.2.6 Sample Path Properties
3.2.7 Ergodicity
3.2.8 Kolmogorov Forward and Backward Equations
3.2.9 Infinitesimal Generator
3.2.10 Ito Formula
3.2.11 Lamperti Transformation
3.2.12 Girsanov Formula
3.3 Approximation and Simulation
3.3.1 Convergence and Consistency
3.3.2 Numerical Approximation
3.3.3 Simulation of Brownian Bridge
3.4 Concluding Remarks
References
4 Approximation of Markov Jump Processes by Diffusions
4.1 Characterisation of Processes
4.2 Motivation and Purpose
4.3 Approximation Methods
4.3.1 Convergence of the Master Equation
4.3.2 Convergence of the Infinitesimal Generator
4.3.3 Langevin Approach
4.3.4 Kramers—Moyal Expansion
4.3.5 Van Kampen Expansion
4.3.6 Other Approaches
4.4 Extensions to Systems with Multiple Size Parameters
4.4.1 Convergence of the Master Equation
4.4.2 Convergence of the Infinitesimal Generator
4.4.3 Langevin Approach
4.4.4 Kramers—Moyal Expansion
4.4.5 Van Kampen Expansion
4.5 Choice of Stochastic Integral
4.6 Discussion and Conclusion
References
5 Diffusion Models in Life Sciences
5.1 Standard SIR Model
5.1.1 Model
5.1.2 Jump Process
5.1.3 Diffusion Approximation
5.1.4 Summary
5.1.5 Illustration
5.2 Multitype SIR Model
5.2.1 Model
5.2.2 Jump Process
5.2.3 Diffusion Approximation
5.2.4 Summary
5.2.5 Illustration and Further Remarks
5.3 Existence and Uniqueness of Solutions
5.4 Conclusion
References
Part Ⅱ Statistical Inference
6 Parametric Inference for Discretely—Observed Diffusions
6.1 Preliminaries
6.1.1 Time—Continuous Observation
6.1.2 Time—Discrete Observation
6.1.3 Time Scheme
6.2 Naive Maximum Likelihood Approach
6.3 Approximation of the Likelihood Function
6.3.1 Analytical Approximation of the Likelihood Function
6.3.2 Numerical Solutions of the Kolmogorov Forward Equation
6.3.3 Simulated Maximum Likelihood Estimation
6.3.4 Local Linearisation
6.4 Alternatives to Maximum Likelihood Estimation
6.4.1 Estimating Functions
6.4.2 Generalised Method of Moments
6.4.3 Simulated Moments Estimation
6.4.4 Indirect Inference
6.4.5 Efficient Method of Moments
6.5 Exact Algorithm
6.6 Discussion and Conclusion
References
7 Bayesian Inference for Diffusions with Low—Frequency Observations
7.1 Concepts of Bayesian Data Augmentation for Diffusions
7.1.1 Preliminaries and Notation
7.1.2 Path Update
7.1.3 Parameter Update
7.1.4 Generalisation to Several Observation Times
7.1.5 Generalisation to Several Observed Diffusion Paths
7.1.6 Practical Concerns
7.1.7 Example： Ornstein—Uhlenbeck Process
7.1.8 Discussion
7.2 Extension to Latent Data and Observation with Error
7.2.1 Latent Data
7.2.2 Observation with Error
7.3 Convergence Problems
7.4 Improvements of Convergence
7.4.1 Changing the Factorisation of the Dominating Measure
7.4.2 Time Change Transformations
7.4.3 Particle Filters
7.4.4 Innovation Scheme on Infinite—Dimensional State Spaces
7.5 Discussion and Conclusion
References
Part Ⅲ Applications
Application Ⅰ：Spread of Influenza
8.1 Simulation Study
8.1.1 Data
8.1.2 Parameter Estimation
8.2 Example： Influenza in a Boarding School
8.2.1 Data
8.2.2 Parameter Estimation
8.3 Example： Influenza in Germany
8.3.1 Data
8.3.2 Parameter Estimation
8.4 Conclusion and Outlook
References
Application Ⅱ： Analysis of Molecular Binding
9.1 Problem Statement
9.1.1 Data Acquisition by Fluorescence Recovery After Photobleaching
9.1.2 Research Questions
9.2 Preliminary Analysis
9.2.1 Impact of Binding
9.2.2 Impact of Diffusion
9.3 General Model
9.3.1 Compartmental Description
9.3.2 Diffusion Approximation
9.3.3 Deterministic Approximation
9.3.4 Simulation Study
9.4 Refinement of the General Model
9.4.1 Compartmental Description
9.4.2 Diffusion Approximation
9.4.3 Deterministic Approximation
9.4.4 Simulation Study
9.5 Extension of the General Model to Multiple Mobility Classes
9.5.1 Compartmental Description
9.5.2 Diffusion Approximation
9.5.3 Deterministic Approximation
9.5.4 Simulation Study
9.6 Data Preparation
9.6.1 Triple Normalisation
9.6.2 Double Normalisation
9.6.3 Single Normalisation
9.7 Application
9.7.1 Data
9.7.2 Bayesian Estimation
9.7.3 Least Squares Estimation
9.7.4 Conclusion
9.8 Diffusion—Coupled FRAP
9.9 Conclusion and Outlook
References
……
Conclusion and Appendix
Index
作者简介
《扩散过程推断理论及其在生命科学中的应用》是一部理论和实践相结合的参考图书，是数学、统计学以及一些前沿学科生物信息学和生物学的不可或缺的学习参考用书。和已有的同类型图书相比，本书是在一个框架下处理扩散用的模型和推断，因此囊括了实际应用中所需的所有步骤，有基本的微分方程、概率论和统计知识即可学习该书。