擴散過程推斷理論及其在生命科學中的套用

《擴散過程推斷理論及其在生命科學中的套用》是2016年7月1日世界圖書出版公司出版的著作，作者是[德] Christiane，Fuchs（C.福克斯）。

基本介紹

• 書名：《擴散過程推斷理論及其在生命科學中的套用》
• 作者：[德] Christiane，Fuchs（C.福克斯）
• 出版社：世界圖書出版公司
• 出版時間：2016年07月01日
• ISBN：9787519211684

內容簡介

《擴散過程推斷理論及其在生命科學中的套用》是一部理論和實踐相結合的參考圖書，是數學、統計學以及一些前沿學科生物信息學和生物學的不可或缺的學習參考用書。和已有的同類型圖書相比，本書是在一個框架下處理擴散用的模型和推斷，因此囊括了實際套用中所需的所有步驟，有基本的微分方程、機率論和統計知識即可學習該書。

作者簡介

《擴散過程推斷理論及其在生命科學中的套用》是一部理論和實踐相結合的參考圖書，作者為數學、統計學以及一些前沿學科生物信息學和生物學的不可或缺的學習參考用書。

目錄

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