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Bayesian inference for diffusion processes with applications in life sciences [Elektronische Ressource] / Christiane Dargatz

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Bayesian Inference for Diffusion Processeswith Applications in Life SciencesChristiane DargatzMünchen 2010Erstgutachter: Prof. Dr. Ludwig FahrmeirZweitgutachter: Prof. Gareth O. Roberts, Ph.D.Rigorosum: 22. September 2010Bayesian Inference for Diffusion Processeswith Applications in Life SciencesChristiane DargatzDissertationan der Fakultät für Mathematik, Informatik und Statistikder Ludwig–Maximilians–Universität Münchenvorgelegt vonChristiane Dargatzaus HannoverMünchen, den 9. August 2010AcknowledgementsI would like to thank a number of people who have accompanied me during the writing ofthis thesis.First and foremost, my sincere gratitude goes to my supervisors Ludwig Fahrmeir andGareth Roberts, who enriched my work through their advice, ideas and encouragement.I also thank Leonhard Held for his directions during the first stage of my thesis.My research has financially been supported by the German Research Foundation (DFG),the German Academic Exchange Service (DAAD) and the LMU Mentoring programme, inwhich Francesca Biagini has been a dedicated mentor to me.I deeply appreciate the careful proof-reading and helpful comments by Michael Höhle.Furthermore, I am grateful to my former and present colleagues for their interest in myresearch and their friendship, in particular to the members of the Semwiso group, myFRAP collaborators, the advocates of good teaching, my fellow women’s representatives,the Cozi Family and my office mates.

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Published 01 January 2010
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Bayesian Inference for Diffusion Processes
with Applications in Life Sciences
Christiane Dargatz
München 2010Erstgutachter: Prof. Dr. Ludwig Fahrmeir
Zweitgutachter: Prof. Gareth O. Roberts, Ph.D.
Rigorosum: 22. September 2010Bayesian Inference for Diffusion Processes
with Applications in Life Sciences
Christiane Dargatz
Dissertation
an der Fakultät für Mathematik, Informatik und Statistik
der Ludwig–Maximilians–Universität München
vorgelegt von
Christiane Dargatz
aus Hannover
München, den 9. August 2010Acknowledgements
I would like to thank a number of people who have accompanied me during the writing of
this thesis.
First and foremost, my sincere gratitude goes to my supervisors Ludwig Fahrmeir and
Gareth Roberts, who enriched my work through their advice, ideas and encouragement.
I also thank Leonhard Held for his directions during the first stage of my thesis.
My research has financially been supported by the German Research Foundation (DFG),
the German Academic Exchange Service (DAAD) and the LMU Mentoring programme, in
which Francesca Biagini has been a dedicated mentor to me.
I deeply appreciate the careful proof-reading and helpful comments by Michael Höhle.
Furthermore, I am grateful to my former and present colleagues for their interest in my
research and their friendship, in particular to the members of the Semwiso group, my
FRAP collaborators, the advocates of good teaching, my fellow women’s representatives,
the Cozi Family and my office mates.
My family has been a constant source of support, and I greatly acknowledge their personal
way of understanding my work. I owe my heartful gratitude to Florian Fuchs, who has
been a strong and close partner during all stages of my thesis and who stayed awake until
the last sentence was written.
Christiane Dargatz
München, October 2010Abstract
Diffusion processes are a promising instrument to realistically model the time-continuous
evolution of natural phenomena in life sciences. However, approximation of a given system
is often carried out heuristically, leading to diffusions that do not correctly reflect the true
dynamics of the original process. Moreover, statistical inference for diffusions proves to be
challenging in practice as the likelihood function is typically intractable.
This thesis contributes to stochastic modelling and statistical estimation of real problems
in life sciences by means of diffusion processes. In particular, it creates a framework
from existing and novel techniques for the correct approximation of pure Markov jump
processes by diffusions. Concerning statistical inference, the thesis reviews existing practices
and analyses and further develops a well-known Bayesian approach which introduces
auxiliary observations by means of Markov chain Monte Carlo (MCMC) techniques. This
procedure originally suffers from convergence problems which stem from a deterministic
link between the model parameters and the quadratic variation of a continuously observed
diffusion path. This thesis formulates a neat modification of the above approach for general
multi-dimensional diffusions and provides the mathematical and empirical proof that the
so-constructed MCMC scheme converges.
The potential of the newly developed modelling and estimation methods is demonstrated
in two real-data application studies: the spatial spread of human influenza in Germany and
the in vivo binding behaviour of proteins in cell nuclei.Zusammenfassung
Diffusionsprozesse eignen sich besonders für die realistische Modellierung des zeitstetigen
Verlaufs von natürlichen Vorgängen in den Lebenswissenschaften. Bei der Approximation
eines gegebenen Systems wird jedoch häufig heuristisch vorgegangen, was zu Diffusions-
modellen führt, welche die Dynamik des ursprünglichen Prozesses nicht wirklichkeitsgetreu
widerspiegeln. Auch die statistische Inferenz stellt sich in der Praxis im Allgemeinen als
anspruchsvoll heraus, da die Likelihoodfunktion meist nicht in analytisch expliziter Form
bekannt ist.
Diese Arbeit untersucht und konzipiert Methoden zur stochastischen Modellierung und
statistischen Inferenz auf Basis von Diffusionsprozessen für Anwendungen in den Lebenswis-
senschaften. Dazu werden existierende Verfahren zur korrekten Approximation von Markov-
Sprungprozessen durch Diffusionsprozesse zusammengestellt, erweitert und durch neue
Ansätze ergänzt. Zur statistischen Inferenz wird ein Überblick über vorhandene Konzepte
gegeben und insbesondere eine etablierte bayesianische Methodik anschaulich erklärt und
weiterentwickelt. Dieser Ansatz fügt zusätzliche Datenpunkte zu bereits vorhandenen
Beobachtungen mittels Markov Chain Monte Carlo (MCMC) Verfahren hinzu. In ihrer
ursprünglichen Form ist diese Technik nur begrenzt einsetzbar, da sie einer Konvergenz-
problematik unterliegt, welche durch einen deterministischen Zusammenhang zwischen den
Modellparametern und der quadratischen Variation eines in stetiger Zeit beobachteten
Diffusionspfades verursacht wird. Diese Arbeit modifiziert das Verfahren für allgemeine
mehrdimensionale Diffusionsprozesse so, dass dieses Problem gelöst wird. Dies wird sowohl
analytisch als auch durch Simulationsstudien empirisch bewiesen.
Die Einsatzmöglichkeiten der neu entwickelten Modellierungs- und Schätzverfahren werden
anhand von zwei Anwendungen gezeigt: bei der räumlichen Ausbreitung von Influenza in
Deutschland und am Bindungsverhalten von Proteinen in Kernen von lebenden Zellen.