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2013 | Buch

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Inference for Diffusion Processes

With Applications in Life Sciences

verfasst von: Christiane Fuchs

Verlag: Springer Berlin Heidelberg

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

Diffusion processes are a promising instrument for realistically modelling the time-continuous evolution of phenomena not only in the natural sciences but also in finance and economics. Their mathematical theory, however, is challenging, and hence diffusion modelling is often carried out incorrectly, and the according statistical inference is considered almost exclusively by theoreticians. This book explains both topics in an illustrative way which also addresses practitioners. It provides a complete overview of the current state of research and presents important, novel insights. The theory is demonstrated using real data applications.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction

Abstract

Diffusion processes are a promising tool to realistically model the time-continuous evolution of phenomena not only in natural sciences but also in finance and economics. Their mathematical theory, however, is challenging. This book explains how diffusion processes can be derived as models for real life phenomena and how model parameters can statistically be estimated. This chapter specifies the objectives and structure of this book.

Christiane Fuchs

Stochastic Modelling

Frontmatter

Chapter 2. Stochastic Modelling in Life Sciences

Abstract

Key mechanisms in life sciences can often be assessed by application of mathematical models. Moreover, real-world phenomena can particularly be captured when such a model allows for random events. This chapter motivates and reviews representative application fields from life sciences and appropriate mathematical models: for the spread of infectious diseases and for processes in molecular biology, biochemistry and genetics. These applications and models recur throughout the entire book. The chapter describes the dynamical evolution of the considered systems in terms of three established types of processes: stochastic jump processes, deterministic state-continuous processes and stochastic diffusion processes. Simulation of such models is explained, and the important role of randomness is discussed.

Christiane Fuchs

Chapter 3. Stochastic Differential Equations and Diffusions in a Nutshell

Abstract

Stochastic differential equations (SDEs) are a powerful and natural tool for the modelling of complex systems that change continuously in time. This chapter provides a short introduction to SDEs and their solutions, which under regularity conditions agree with the class of diffusion processes. In particular, it covers the motivation and introduction of stochastic integrals as opposed to the classical Lebesgue-Stieltjes integral, the definition of diffusion processes, key properties and formulas from stochastic calculus, and finally numerical approximation and exact sampling methods. The chapter serves as a basis for the remaining parts of this book and offers a quick access to stochastic calculus.

Christiane Fuchs

Chapter 4. Approximation of Markov Jump Processes by Diffusions

Abstract

Diffusion processes enable realistic and convenient modelling of dynamic systems. They typically arise as approximations of exact but computationally expensive individual-based stochastic models. However, the correct derivation of an appropriate diffusion approximation is often complicated, and hence their utilisation is not widely spread in the applied sciences. Instead, practitioners often favour rather unrealistic deterministic models and their relatively simple analysis. This chapter motivates the application of diffusion approximations and explains their correct derivation. It reviews and develops different approaches and points out differences and correspondences between them. All methods are formulated for multi-dimensional processes and extended to an even more general framework where systems are characterised by multiple size parameters. The chapter addresses mathematicians who are interested in the theory of diffusion approximations and practitioners who wish to apply diffusion models for their specific problems.

Christiane Fuchs

Chapter 5. Diffusion Models in Life Sciences

Abstract

This chapter investigates models for the spread of infectious diseases as a representative application field which involves large populations. In particular, it covers the standard susceptible–infected–removed (SIR) model and proposes an extension in order to allow for host heterogeneity. The considered dynamics is described in terms of jump processes, deterministic processes and diffusion processes. The latter enables convenient simulation of the random course of an epidemic even for large populations. The purpose of this chapter is on the one hand to illustrate the methods from Chap. 4 for the approximation of Markov jump processes by diffusions. On the other hand, the presented models and their diffusion approximations are the basis for Chap. 8, where Bayesian inference is performed on them.

Christiane Fuchs

Statistical Inference

Frontmatter

Chapter 6. Parametric Inference for Discretely-Observed Diffusions

Abstract

In real applications, diffusion models are often known in parametric form for which one wishes to estimate the model parameters. Statistical inference for diffusions is, however, challenging. The difficulty that underlies most approaches is the general intractability of the transition density for discrete-time observations. This chapter reviews frequentist parametric inference for discretely-observed diffusion processes. In order to get to the heart of the problem, it starts with the formulation of the estimation problem for continuous-time observations and then goes over to discrete time under the assumption that the likelihood function of the parameter is known. Both scenarios are not directly applicable in practice. The remaining techniques covered in this chapter are more advanced. These are approximations of the likelihood function, alternatives to maximum likelihood estimation and a recent approach called the Exact Algorithm.

Christiane Fuchs

Chapter 7. Bayesian Inference for Diffusions with Low-Frequency Observations

Abstract

Most frequentist techniques for parameter estimation in diffusion processes struggle when inter-observation times are large, which is often the case in life sciences. This chapter introduces Bayesian inference methods which estimate missing data such that the union of missing values and observations forms a high-frequency dataset. This facilitates approximation of the likelihood function and hence enables parametric inference even for large inter-observation times. Moreover, the techniques are suitable for irregularly spaced observation intervals, multivariate diffusions with possibly latent components and for observations that are subject to measurement error. This chapter brings together approaches from different authors, explains convergence problems that arise in standard algorithms, and suggests a new sampling scheme which fixes corresponding limitations of existing methods. The universal applicability of this method is proven. The contents of this chapter address both practicioners who wish to implement the estimation schemes and theoreticians who are interested in convergence proofs.

Christiane Fuchs

Applications

Frontmatter

Chapter 8. Application I: Spread of Influenza

Abstract

As a first application of the methods introduced in the first two parts of this book, this chapter investigates the spread of human influenza. More precisely, it analyses a well-known dataset on an influenza outbreak in a British boarding school and the spatial spread of influenza in Germany during the season 2009/10, in which the swine flu virus was prevalent. In the latter example, spatial mixing of individuals is estimated from commuter data. Modelling is based on diffusion approximations derived in Chap. 5. Statistical inference is carried out using a Bayesian approach developed in Chap. 7.

Christiane Fuchs

Chapter 9. Application II: Analysis of Molecular Binding

Abstract

The genetic material of humans and mammals is mainly contained in their cell nuclei, where most genome regulatory processes like DNA replication or transcription take place. These processes are controlled by complex protein networks.

Christiane Fuchs

Chapter 10. Summary and Future Work

Abstract

Stochastic modelling and statistical estimation are important tools for the understanding of complex processes in life sciences. This book motivated the use of diffusion processes for both purposes and contributed to their applicability in practice. This chapter summarises the achievements of this book and points out directions for future work.

Christiane Fuchs

Backmatter

Titel: Inference for Diffusion Processes
verfasst von: Christiane Fuchs
Verlag: Springer Berlin Heidelberg
Electronic ISBN: 978-3-642-25969-2
Print ISBN: 978-3-642-25968-5
DOI: https://doi.org/10.1007/978-3-642-25969-2