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We studied invariant measures and invariant densities for dynamical systems with random switching (switching systems, in short). These switching systems can be described by a two-component Markov process whose first component is a stochastic process on a finite-dimensional smooth manifold and whose second component is a stochastic process on a finite collection of smooth vector fields that are defined on the manifold. We identified sufficient conditions for uniqueness and absolute continuity of the invariant measure associated to this Markov process. These conditions consist of a Hoermander-type hypoellipticity condition and a recurrence condition. In the case where the manifold is the real line or a subset of the real line, we studied regularity properties of the invariant densities of absolutely continuous invariant measures. We showed that invariant densities are smooth away from critical points of the vector fields. Assuming in addition that the vector fields are analytic, we derived the asymptotically dominant term for invariant densities at critical points.

The widespread use of ordinary differential equation (ODE) models has long been underrepresented in the statistical literature. The most common methods for estimating parameters from ODE models are nonlinear least squares and an MCMC based method. Both of these methods depend on a likelihood involving the numerical solution to the ODE. The challenge faced by these methods is parameter spaces that are difficult to navigate, exacerbated by the wide variety of behaviours that a single ODE model can produce with respect to small changes in parameter values. / In this work, two competing methods, generalized profile estimation and Bayesian collocation tempering are described. Both of these methods use a basis expansion to approximate the ODE solution in the likelihood, where the shape of the basis expansion, or data smooth, is guided by the ODE model. This approximation to the ODE, smooths out the likelihood surface, reducing restrictions on parameter movement. / Generalized Profile Estimation maximizes the profile likelihood for the ODE parameters while profiling out the basis coefficients of the data smooth. The smoothing parameter determines the balance between fitting the data and the ODE model, and consequently is used to build a parameter cascade, reducing the dimension of the estimation problem. Generalized profile estimation is described with under a constraint to ensure the smooth follows known behaviour such as monotonicity or non-negativity. / Bayesian collocation tempering, uses a sequence posterior densities with smooth approximations to the ODE solution. The level of the approximation is determined by the value of the smoothing parameter, which also determines the level of smoothness in the likelihood surface. In an algorithm similar to parallel tempering, parallel MCMC chains are run to sample from the sequence of posterior densities, while allowing ODE parameters to swap between chains. This method is introduced and tested against a variety of alternative Bayesian models, in terms of posterior variance and rate of convergence. / The performance of generalized profile estimation and Bayesian collocation tempering are tested and compared using simulated data sets from the FitzHugh-Nagumo ODE system and real data from nylon production dynamics.

I present a method that regroups cis-regulatory modules by shared sequences motifs. The goal of this approach is to search for clusters of modules that may share some function, using only sequence similarity. The proposed similarity measure is based on a variable-order Markov model likelihood scoring of sequences. I also introduce an extension of the variable-order Markov model which could better perform the required task. Results. I show that my method may recover subsets of sequences sharing a pattern in a set of generated sequences. I found that the proposed approach is successful in finding groups of modules that shared a type of transcription factor binding site.

Diss. Västerås : Mälardalens högskola, 2007. / Härtill 5 uppsatser. S. 20-23: Bibliografi.

Thesis (Ph. D.)--Albert-Ludwigs-Universität Freiburg im Breisgau, 2006. / Description based on print version record. Includes bibliographical references (p. 201-221) and index.

Diss. (sammanfattning) Västerås : Mälardalens högskola, 2007. / Härtill 6 uppsatser.

Thesis (Ph. D.)--Carleton University, 2003. / Includes bibliographical references (p. 151-158). Also available in electronic format on the Internet.

Thesis (M.C.S.) - Carleton University, 2006. / Includes bibliographical references (p. 78-81). Also available in electronic format on the Internet.