Gaussian processes are now widely used to perform key machine learning tasks such as nonlinear regression and classification. An attractive feature of Gaussian process models is the behaviour of the error bars, which grow in regions away from observations where there is high uncertainty about the interpolating function. The complexity of these models scales as O(N3) with sample size, which causes difficulties with large data sets. The goals of this work are to develop nonlinear, nonparametric modelling techniques for structure learning and prediction problems in which there are structured dependencies among the observed data, and to equip our models with sparse representations which serve both to handle prior sparse connectivity assumptions and to reduce computational complexity. We present Kernel Dynamical Structure Learning, a Bayesian method for learning the structure of interactions between variables in multivariate time-series. We design a mutual information kernel to handle time-series trajectories, and show that prior knowledge about network sparsity can be incorporated using heavy-tailed priors over parameters. We evaluate the feasibility of our method on synthetic data, and extend the inference methodology to the handling of uncertain input data. Next, we tackle the problem of belief propagation in Bayesian networks with nonlinear node relations. We propose an exact moment-matching approach for nonlinear belief propagation in any tree-structured graph. We call this Gaussian Process Belief Propagation. We extend this approach by the addition of hidden variables which allow nodes sharing common influences to be conditionally independent. This constitutes a novel approach to multi-output regression on bivariate graph structures, and we call this Dependent Gaussian Process Belief Propagation. We describe sparse inference methods for both models, which reduce computational by learning compact parameterisations of the available training data. We then apply our method to the real-world systems biology problem of protein inference in transcriptional networks.

Credit scoring has been described as the most successful application of statistical and operational research methods to financial problems in recent decades. In this thesis methods analogous to those used in spatial modelling and prediction are applied to the problem of application scoring, a branch of credit scoring that involves deciding whether or not to offer credit and how much credit to offer. In particular, Gaussian spatial process (GSP) models, commonly employed in disease mapping, geostatistics and design, are explored in an approach that is novel in the credit scoring field. Credit scoring methods are well established and usually involve computations of scores. By contrast, the focus of this work is to use best linear unbiased predictors in order to predict the probabilities of repayment for credit applications. A spatial structure for the model is provided by reformulating the data. Both theoretical and industry standard methods are used in order to assess the predictive competence of GSP models. In addition, the GSP model approach is compared with standard methods for application scoring and conclusions are made regarding the relevance of such models in this area

No description available.

This thesis concerns the mathematical analysis of random gradient descent evolutions as models for rate-independent dissipative systems under the influence of thermal effects. The basic notions of the theory of gradient descents (especially rate-independent evolutions) are reviewed in chapter 2. Chapters 3 and 4 focus on the scaling regime in which the microstructure dominates the thermal effects and comprise a rigorous justification of rateindependent processes in smooth, convex energies as scaling limits of ratedependent gradient descents in energies that have rapidly-oscillating random microstructure: chapter 3 treats the one-dimensional case with quite a broad class of random microstructures; chapter 4 treats a case in which the microstructure is modeled by a sum of “dent functions” that are scattered in Rn using a suitable point process. Chapters 5 and 6 focus on the opposite scaling regime: a gradient descent system (typically a rate-independent process) is placed in contact with a heat bath. The method used to “thermalize” a gradient descent is an interior-point regularization of the Moreau–Yosida incremental problem for the original gradient descent. Chapter 5 treats the heuristics and generalities; chapter 6 treats the case of 1-homogeneous dissipation (rate independence) and shows that the heat bath destroys the rate independence in a controlled and deterministic way, and that the effective dynamics are a gradient descent in the original energetic potential but with respect to a different and non-trivial effective dissipation potential. The appendices contain some auxiliary definitions and results, most of them standard in the literature, that are used in the main text.

Nous étudions certains théorèmes limite centraux fonctionnels hölderiens pour des processus autorégressifs d’ordre un quasi non stationnaires yn,k = φn yn,k−1 +εk et leurs résidus au sens des moindres carrés avec φn tendant vers 1 et des innovations i.i.d. centrées, de carré intégrable. Dans le cas φn = exp(γ/n) avec γ < 0, la limite en loi est une fonction d’un processus d’Ornstein-Uhlenbeck intégré. Dans le cas φn = 1 − γn /n avec γn tendant vers l'infini plus lentement que n, la convergence vers le mouvement brownien est établie dans l’espace de Hölder en termes de vitesse de divergence γn et d’intégrabilité des innovations εk. Comme application statistique de ces résultats, nous considérons une rupture épidémique dans les innovations de processus autorégressifs d’ordre un quasi non stationnaires AR(1). Deux types de modèles sont considérés. Pour 0 ≤ α < 1 nous construisons une statistique α-hölderienne basée sur les accroissements uniformes des observations ou des résidus pour détecter une courte rupture épidémique dans les processus considérés. Sous certaines hypothèses pour les innovations, nous trouvons la loi limite de la statistique sous l’hypothèse nulle, les conditions de consistance et nous effectuons une analyse de la puissance du test statistique. Nous discutons également l’interaction entre les différents paramètres pour la détectabilité des plus courtes épidémies. / We study some Hölderian functional central limit theorems for the polygonal partial sum processes built on a first order nearly nonstationary autoregressive process yn,k = φn yn,k−1 + εk and its least squares residuals εk with φn converging to 1 and i.i.d. centered square-integrable innovations. In the case where φn = exp( γn /n) with a negative constant γ, we prove that the limiting process depends on Ornstein-Uhlenbeck one. In the case where φn = 1 − γn /n, with γn tending to infinity slower than n, the convergence to Brownian motion is established in Hölder space in terms of the rate of γn and the integrability of the εk’s. As a statistical application of these results, we investigate some epidemic change in the innovations of the first order nearly nonstationary autoregressive process AR(1). Two types of models are considered. For 0 ≤ α < 1, we build the α-Hölderian uniform increments statistics based on the observations and on the least squares residuals to detect the short epidemic change in the process under consideration. Under the assumptions for innovations we find the limit of the statistics under null hypothesis, some conditions of consistency and we perform a test power analysis. We also discuss the interplay between the various parameters to detect the shortest epidemics.

This thesis is about probabilistic simulation techniques. Specifically we consider the exact or perfect sampling of spatial point process models via the dominated CFTP protocol. Fundamental among point process models is the Poisson process, which formalises the notion of complete spatial randomness; synonymous with the Poisson process is the Boolean model. The models treated here are the conditional Boolean model and the area-interaction process. The latter is obtained by weighting a Poisson process according to the area of its associated Boolean model. A fundamental tool employed in the perfect simulation of point processes are spatial birth-death processes. Perfect sampling algorithms for the conditional Boolean and area-interaction models are described. Birth-death processes are also employed in order to develop an exact omnithermal algorithm for the area-interaction process. This enables the simultaneous sampling of the process for a whole range of parameter values using a single realization. A variant of Rejection sampling, namely 2-Stage Rejection, and exact Gibbs samplers for the conditional Boolean and area-interaction processes are also developed here. A quantitative comparison of the methods employing 2-Stage Rejection, spatial birth-death processes and Gibbs samplers is carried, the performance measured by actual run times of the algorithms. Validation of the perfect simulation algorithms is carried out via x2 tests.

Dans cette thèse, nous étudions la dynamique d'une particule dans un milieu inélastique composé de diffuseur et communément appelé gaz de Lorentz inélastique. Dans le cas inerte, le milieu n'est pas affecté par le passage de la particule. L'énergie cinétique de celle-ci croît avec le temps et ce phénomène est appelé « accélération stochastique ». Nous approximons le mouvement de la particule par une chaîne de Markov dont chaque pas correspond à une unique collision entre la particule et un diffuseur. Nous montrons que l'énergie cinétique moyenne de la particule croît avec le temps avec pour exposant 2/5. Ce résultat est montré en utilisant des arguments probabilistes, utilisant des théorèmes de convergence de chaînes de Markov ainsi que la convergence en distribution de la chaîne, correctement changée d'échelle en temps et en espace, vers un processus de Bessel. Nous obtenons également un résultat de convergence pour le vecteur vitesse. Sous un changement d'échelle différent de celui utilisé pour l'énergie cinétique, celui-ci converge en distribution vers un mouvement brownien sphérique. Dans le cas dynamique, l'évolution des degrés de liberté du gaz de Lorentz est affecté par le passage de la particule et le système dynamique considéré est composé de la particule et du milieu. Dans un tel système, le phénomène d'accélération stochastique ne peut pas être observé. En revanche nous montrons que la distribution des vitesses admet un état stationnaire. / In this thesis, we study the dynamics of a particle in an inelastic environment composed of scatterer which is commonly known as inelastic Lorentz gas. In the inert case, the environment is not affected by the particle. The kinetic energy of the latter grows with the time and this phenomenon is called « stochastic acceleration ». We approximate the particle's motion by a Markov chain where each step corresponds to a unique collision of the particle with a scatterer. We show that the particle's averaged kinetic energy grows with the time with the exponent 2/5. The result is proved by using probabilistic arguments, bringing into weak convergence theorems of Markov chain as well as the weak convergence of the chain, correctly rescaled in time and space, to a Bessel process.We thus obtain a convergence result for the velocity vector. Under a different rescaling that the one used for the kinetic energy, the latter converges weakly to a spherical brownian motion. In the dynamical case, the evolution of the degrees of freedom of the Lorentz gas is affected by the particle and the dynamical system considered is constitued of the particle and the environment. In such a system, the stochastic acceleration phenomenon cannot be observed. However, we show that the velocity distribution admits a stationnary state.

An attempt is made in this thesis to study some stochastic models of both reliability and inventory systems with reference to the following aspects: (i) the confidence limits with the introduction of common-cause failures. (ii) the Erlangian repair time distributions. (iii) the product interactions and demand interactions. (iv) the products are perishable. This thesis contains six chapters. Chaper 1 is introductory in nature and gives a review of the literature and the techniques used in the analysis of reliability systems. Chapter 2 is a study of component common-cause failure systems. Such failures may greatly reduce the reliability indices. Two models of such systems (series and parallel) have been studied in this chapter. The expressions such as, reliability, availability and expected number of repairs have been obtained. The confidence limits for the steady state availability of these two systems have also been obtained. A numerical example illustrates the results. A 100 (1 - a) % confidence limit for the steady state availability of a two unit hot and warm standby system has been studied, when the failure of an online unit is constant and the repair time of a failed unit is Erlangian. The general introduction of various inventory systems and the techniques used in the analysis of such systems have been explained in chapter 4. Chapter 5 provides two models of two component continuous review inventory systems. Here we assume that demand occurs according to a poisson process and that a demand can be satisfied only if both the components are available in inventory. Back-orders are not permitted. The two components are bought from outside suppliers and are replenished according to (s, S) policy. In model 1 we assume that the lead-time of the components follow an exponential distribution. By identifying the inventory level as a Markov process, a system of difference-differential equations at any time and the steady-state for the state of inventory level are obtained. Tn model 2 we assume that the lead-time distribution of one product is arbitrary and the other is exponential. Identifying the underlying process as a semi-regenerative process we find the stationary distribution of the inventory level. For both these models, we find out the performance measures such as the mean stationary rate of the number of lost demands, the demands satisfied and the reorders made. Numerical examples for the two models are also considered. Chaper 6 is devoted to the study of a two perishable product inventory model in which the products are substitutable. The perishable rates of product 1 and product 2 are two different constants. Demand for product 1 and product 2 follow two independent Poisson processes. For replenishment of product 1 (s, S) ordering policy is followed and the associated lead-time is arbitrary. Replenishment of product 2 is instantaneous. A demand for product 1 which occurs during its stock-out period can be substituted by product 2 with some probability. Expressions are derived for the stationary distribution of the inventor}' level by identifying the underlying stochastic process as a semi-regenerative process. An expression for the expected profit rate is obtained. A numerical illustration is provided and an optimal reordering level maximising the profit rate is also studied. To sum up, this thesis is an effort to improve the state the of art of (i) complex reliability systems and their estimation study (ii) muitiproduct inventory systems. The salient features of the thesis are: (i) Analysis of a two-component reliability system with common-cause failures. (ii) Estimation study of a complex system in which the repair time for both hot standby and warm standby systems are assumed to be Eriangian. (iii) A multi-product continuous review inventory system with product interaction, with a (s, S) policy. (iv) Introduction of the concept of substitutability for products. (v) Derivation of expressions for various statistical measures. (vi) Effective use of the regeneration point technique in deriving various measures for both reliability and inventory systems. (vii) Illustration of the various results by extensive numerical work. (vii) Consideration of relevant optimization problems. / Mathematical Sciences / PhD (Statistics)

No description available.

An attempt is made in this thesis to study some stochastic models of both reliability and inventory systems with reference to the following aspects: (i) the confidence limits with the introduction of common-cause failures. (ii) the Erlangian repair time distributions. (iii) the product interactions and demand interactions. (iv) the products are perishable. This thesis contains six chapters. Chaper 1 is introductory in nature and gives a review of the literature and the techniques used in the analysis of reliability systems. Chapter 2 is a study of component common-cause failure systems. Such failures may greatly reduce the reliability indices. Two models of such systems (series and parallel) have been studied in this chapter. The expressions such as, reliability, availability and expected number of repairs have been obtained. The confidence limits for the steady state availability of these two systems have also been obtained. A numerical example illustrates the results. A 100 (1 - a) % confidence limit for the steady state availability of a two unit hot and warm standby system has been studied, when the failure of an online unit is constant and the repair time of a failed unit is Erlangian. The general introduction of various inventory systems and the techniques used in the analysis of such systems have been explained in chapter 4. Chapter 5 provides two models of two component continuous review inventory systems. Here we assume that demand occurs according to a poisson process and that a demand can be satisfied only if both the components are available in inventory. Back-orders are not permitted. The two components are bought from outside suppliers and are replenished according to (s, S) policy. In model 1 we assume that the lead-time of the components follow an exponential distribution. By identifying the inventory level as a Markov process, a system of difference-differential equations at any time and the steady-state for the state of inventory level are obtained. Tn model 2 we assume that the lead-time distribution of one product is arbitrary and the other is exponential. Identifying the underlying process as a semi-regenerative process we find the stationary distribution of the inventory level. For both these models, we find out the performance measures such as the mean stationary rate of the number of lost demands, the demands satisfied and the reorders made. Numerical examples for the two models are also considered. Chaper 6 is devoted to the study of a two perishable product inventory model in which the products are substitutable. The perishable rates of product 1 and product 2 are two different constants. Demand for product 1 and product 2 follow two independent Poisson processes. For replenishment of product 1 (s, S) ordering policy is followed and the associated lead-time is arbitrary. Replenishment of product 2 is instantaneous. A demand for product 1 which occurs during its stock-out period can be substituted by product 2 with some probability. Expressions are derived for the stationary distribution of the inventor}' level by identifying the underlying stochastic process as a semi-regenerative process. An expression for the expected profit rate is obtained. A numerical illustration is provided and an optimal reordering level maximising the profit rate is also studied. To sum up, this thesis is an effort to improve the state the of art of (i) complex reliability systems and their estimation study (ii) muitiproduct inventory systems. The salient features of the thesis are: (i) Analysis of a two-component reliability system with common-cause failures. (ii) Estimation study of a complex system in which the repair time for both hot standby and warm standby systems are assumed to be Eriangian. (iii) A multi-product continuous review inventory system with product interaction, with a (s, S) policy. (iv) Introduction of the concept of substitutability for products. (v) Derivation of expressions for various statistical measures. (vi) Effective use of the regeneration point technique in deriving various measures for both reliability and inventory systems. (vii) Illustration of the various results by extensive numerical work. (vii) Consideration of relevant optimization problems. / Mathematical Sciences / PhD (Statistics)