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Literature on change point analysis mostly requires a sudden change in the data distribution, either in a few parameters or the distribution as a whole. We are interested in the scenario that the variance of data may make a significant jump while the mean of data changes in a smooth fashion. It is motivated by a liver procurement experiment with organ surface temperature monitoring. Blindly applying the existing change point analysis methods to the example can yield erratic change point estimates since the smoothly-changing mean violates the sudden-change assumption. In my dissertation, we propose a penalized weighted least squares approach with an iterative estimation procedure that naturally integrates variance change point detection and smooth mean function estimation. Given the variance components, the mean function is estimated by smoothing splines as the minimizer of the penalized weighted least squares. Given the mean function, we propose a likelihood ratio test statistic for identifying the variance change point. The null distribution of the test statistic is derived together with the rates of convergence of all the parameter estimates. Simulations show excellent performance of the proposed method. Application analysis offers numerical support to the non-invasive organ viability assessment by surface temperature monitoring. The method above can only yield the variance change point of temperature at a single point on the surface of the organ at a time. In practice, an organ is often transplanted as a whole or in part. Therefore, it is generally of more interest to study the variance change point for a chunk of organ. With this motivation, we extend our method to study variance change point for a chunk of the organ surface. Now the variances become functions on a 2D space of locations (longitude and latitude) and the mean is a function on a 3D space of location and time. We model the variance functions by thin-plate splines and the mean function by the tensor product of thin-plate splines and cubic splines. However, the additional dimensions in these functions incur serious computational problems since the sample size, as a product of the number of locations and the number of sampling time points, becomes too large to run the standard multi-dimensional spline models. To overcome the computational hurdle, we introduce a multi-stages subsampling strategy into our modified iterative algorithm. The strategy involves several down-sampling or subsampling steps educated by preliminary statistical measures. We carry out extensive simulations to show that the new method can efficiently cut down the computational cost and make a practically unsolvable problem solvable with reasonable time and satisfactory parameter estimates. Application of the new method to the liver surface temperature monitoring data shows its effectiveness in providing accurate status change information for a portion of or the whole organ. / Ph. D. / The viability evaluation is the key issue in the organ transplant operation. The donated organ must be viable at the time of being transplanted to the recipient. Nowadays, viability evaluation can be assessed by analyzing the temperature data monitored on the organ surface. In my dissertation, I have developed two new statistical methods to evaluate the viability status of a prepared organ by studying the organ surface temperature. The first method I have developed can be used to detect the change of viability status at a spot on the organ surface. The second method I have developed can be used to detect the change of viability condition for the selected organ chunks. In practice, combining these two methods together can provide accurate viability status change information for a portion of or the whole organ effectively.

In this study we are interested on the following 4-dimensional system of stochastic differential equations.dS=(-βSI+μ(K-S)+αI+ζR)dt-σ_1 SIF_1 (S,E,I,R)dW_1+σ_4 RF_4 (S,E,I,R)dW_4 dE=(βSI-(μ+η)E)dt+σ_1 SIF_1 (S,E,I,R)dW_1-σ_2 EF_2 (S,E,I,R)dW_2 dI=(ηE-(α+γ+μ)I)dt+σ_2 EF_2 (S,E,I,R)dW_2-σ_3 IF_3 (S,E,I,R)dW_3 dR=(γI-(μ+ζ)R)dt+σ_3 IF_3 (S,E,I,R)dW_3-σ_4 RF_4 (S,E,I,R)dW_4 with variance parameters σ_i≥0 and constants α,β,η,γ,μ ζ≥0. This system may be used to model the dynamics of susceptible, exposed, infected and recovering individuals subject to a present virus with state-dependent random transitions. Our main goal is to prove the existence of a bounded, unique, strong (pathwise), global solution to this system, and to discuss asymptotic stochastic and moment stability of the two equilibrium points, namely the disease free and the endemic equilibria. In this model, as suggested by our advisor, diffusion coefficients can be any local Lipschitz continuous functions on bounded domain D={(S,E,I,R)∈R_+^4:00 of maximum carrying capacity and W_i are independent and identical Wiener processes defined on a complete probability space (Ω,F,{F_t }_(t≥0),P). At the end we carry out some simulations to illustrate our results.

The stochastic SEIR(S) model with random total population is given by the system of stochastic differential equations:dS=(-βSI+μ(K-S)+αI+ζR)dt-σ_1 SIF_1 (S,E,I,R)dW_1+σ_4 RF_4 (S,E,I,R)dW_4+σ_5 S(K-N)dW_5\\ dE=(βSI-(μ+η)E)dt+σ_1 SIF_1 (S,E,I,R)dW_1-σ_2 EF_2 (S,E,I,R)dW_2+σ_5 E(K-N)dW_5 \\ dI=(ηE-(α+γ+μ)I)dt+σ_2 EF_2 (S,E,I,R)dW_2-σ_3 IF_3 (S,E,I,R)dW_3+σ_5 I(K-N)dW_5 \\ dR=(γI-(μ+ζ)R)dt+σ_3 IF_3 (S,E,I,R)dW_3-σ_4 RF_4 (S,E,I,R)dW_4+σ_5 R(K-N)dW_5, where σ_i>0 and constants α, β, η, γ, ζ, μ≥0. K represents the maximum carrying capacity for the total population and W_k=(W_k (t))_(t≥0) are independent, standard Wiener processes on a complete probability space (Ω,F,(F_t )_(t≥0),P). The SDE for the total population N=S+E+I+R has the form dN(t)=μ(K-N)dt+σ_5 N(K-N)dW_5 on D_0=(0,K). The goal of our study is to prove the existence of unique, Markovian, continuous time solutions on the 4D prism D={(S,E,I,R)∈R_+^4:0≤S, E,I,R≤K, S+E+I+R≤K}. Then using the method of Lyapunov functions we prove the asymptotic stochastic and moment stability of disease-free and endemic equilibria. Finally, we use numerical simulations to illustrate our results.

Dans ce travail, nous proposons des procédures de test paramétriques et nonparamétrique localement et asymptotiquement optimales au sens de Hajek et Le Cam, pour trois modèles d'asymétrie. <p>La construction de modèles d'asymétrie est un sujet de recherche qui a connu un grand développement ces dernières années, et l'obtention des tests optimaux (pour trois modèles différents) est une étape essentielle en vue de leur mise en application. <p>Notre approche est fondée sur la théorie de Le Cam d'une part, pour obtenir les propriétés de normalité asymptotique, bases de la construction des tests paramétriques optimaux, et la théorie de Hajek d'autre part, qui, via un principe d'invariance permet d'obtenir les procédures non-paramétriques.<p><p>Nous considérons dans ce travail deux classes de distributions univariées asymétriques, l'une fondée sur un développement d'Edgeworth (décrit dans le Chapitre 1), et l'autre construite en utilisant un paramètre d'échelle différent pour les valeurs positives et négatives (le modèle de Fechner, décrit dans le Chapitre 2).<p>Le modèle d'asymétrie elliptique étudié dans le dernier chapitre est une généralisation multivariée du modèle du Chapitre 2.<p>Pour chacun de ces modèles, nous proposons de tester l'hypothèse de symétrie par rapport à un centre fixé, puis par rapport à un centre non spécifié.<p><p>Après avoir décrit le modèle pour lequel nous construisons les procédures optimales, nous obtenons la propriété de normalité locale asymptotique. A partir de ce résultat, nous sommes capable de construire les tests paramétriques localement et asymptotiquement optimaux. Ces tests ne sont toutefois valides que si la densité sous-jacente f est correctement spécifiée. Ils ont donc le mérite de déterminer les bornes d'efficacité paramétrique, mais sont difficilement applicables. <p>Nous adaptons donc ces tests afin de pouvoir tester les hypothèses de symétrie par rapport à un centre fixé ou non, lorsque la densité sous-jacente est considérée comme un paramètre de nuisance. <p>Les tests que nous obtenons restent localement et asymptotiquement optimaux sous f, mais restent valides sous une large classe de densités. <p><p>A partir des propriétés d'invariance du sous-modèle identifié par l'hypothèse nulle, nous obtenons les tests de rangs signés localement et asymptotiquement optimaux sous f, et valide sous une vaste classe de densité. Nous présentons en particulier, les tests fondés sur les scores normaux (ou tests de van der Waerden), qui sont optimaux sous des hypothèses Gaussiennes, tout en étant valides si cette hypothèse n'est pas vérifiée.<p>Afin de comparer les performances des tests paramétriques et non paramétriques présentés, nous calculons les efficacités asymptotiques relatives des tests non paramétriques par rapport aux tests pseudo-Gaussiens, sous une vaste classe de densités non-Gaussiennes, et nous proposons quelques simulations. / Doctorat en sciences, Orientation statistique / info:eu-repo/semantics/nonPublished

Thesis (MSc)--Stellenbosch University, 2013. / ENGLISH ABSTRACT: There are many examples of cooperation in action in society and in nature. In some cases cooperation leads to the increase of the overall welfare of those involved, and in other cases cooperation may be to the detriment of the larger society. The presence of cooperation seems natural if there is a direct bene t to individuals who choose to cooperate. However, in examples of cooperation this bene t is not always immediately obvious. The so called prisoner's dilemma is often used as an analogy to study cooperation and tease out the factors that lead to cooperation. In classical game theory, each player is assumed to be rational and hence typically seeks to select his strategy in such a way as to maximise his own expected pay-o . In the case of the classical prisoner's dilemma, this causes both players to defect. In evolutionary game theory, on the other hand, it is assumed that players have limited knowledge of the game and only bounded rationality. Games in evolutionary game theory are repeated in rounds and players are a orded the opportunity to adapt and learn as this repetition occurs. Past studies have revealed that cooperation may be a viable strategy if the prisoner's dilemma is placed in an evolutionary context, where the evolutionary tness of a strategy is directly related to the pay-o achieved by the player adopting the strategy. One of the mechanisms that promote the persistence of cooperation in the evolutionary prisoner's dilemma is structured interaction between players. A mathematical framework for representing the evolutionary prisoner's dilemma (ESPD) is developed in this thesis. The mathematical framework is used to undertake an analytical approach (i.e. avoiding the use of simulation) towards investigating the dynamics of the ESPD with a path, cycle, plane grid or toroidal grid as underlying graph. The objective of this investigation is to determine the likelihood of the emergence of persistent cooperation between players. The ESPD on a path or a cycle admits two fundamentally di erent parameter regions; large values of the temptation-to-defect parameter are not capable of inducing persistent cooperation, while small values of this parameter allow for the possibility of persistent cooperation. It is found that the likelihood of cooperation increases towards certainty as the order of the underlying graph increases if the underlying graph is a path or cycle. The state space of the ESPD with a plane or toroidal grid graph as underlying graph grows very quickly as a function of the graph order. The automorphism classes of game states are enumerated to determine exactly how fast the size of the state space of the game grows as a function of the order of the underlying graph. Finally, the dynamics of the ESPD is investigated for a grid graph as underlying graph (in cases where the state space is small enough) by means of constructing the corresponding state graphs of the ESPD. / AFRIKAANSE OPSOMMING: Daar is baie voorbeelde van samewerking in the gemeenskap en in die natuur. In sommige gevalle lei samewerking tot 'n toename in die algehele welvaart van die betrokkenes, terwyl samewerking in ander gevalle tot nadeel van die bre er gemeenskap mag wees. Die voorkoms van samewerking blyk natuurlik te wees indien daar 'n direkte voordeel vir die individue is wat kies om saam te werk. In voorbeelde van samewerking is s o 'n voordeel egter nie altyd voor-diehand- liggend nie. Die sogenaamde prisoniersdilemma word dikwels as voorbeeld in die studie van samewerking gebruik om die faktore wat na samewerking lei, te ontbloot. In klassieke speleteorie word daar aangeneem dat elke speler rasioneel is en dus poog om sy spelstrategie op s o 'n manier te kies dat sy eie verwagte uitbetaling gemaksimeer word. In die geval van die klassieke prisoniersdilemma veroorsaak dit dat beide spelers mekaar verraai. In evolusion^ere speleteorie, daarenteen, word daar slegs aangeneem dat elke speler oor beperkte kennis van die spel en begrensde rasionaliteit beskik. Spele in evolusion^ere speleteorie word in rondtes herhaal en spelers word die geleentheid gebied om gedurende hierdie herhalingsproses aan te pas en te leer. Vorige studies het getoon dat samewerking 'n lewensvatbare strategie is indien die prisoniersdilemma in 'n evolusion^ere konteks gespeel word, waar die evolusion^ere ksheid van 'n strategie direk afhang van die uitbetaling van 'n speler wat die strategie volg. Een van die meganismes wat volhoubare samewerking in die evolusion^ere prisoniersdilemma voortbring, is gestruktureerde interaksie tussen spelers. 'n Wiskundige raamwerk word vir die voorstelling van die evolusion^ere prisoniersdilemma in hierdie tesis ontwikkel. Hierdie wiskundige raamwerk word gebruik om 'n analitiese studie (met ander woorde sonder die gebruik van simulasie) van die dinamika van die prisoniersdilemma op 'n pad, siklus, rooster in die vlak, of rooster op die torus as onderliggende gra ek van stapel te stuur. Die doel van hierdie studie is om die waarskynlikheid vir die ontstaan van volhoubare samewerking tussen spelers te bepaal. Die prisoniersdilemma op 'n pad of siklus as onderliggende gra ek het twee fundamenteel verskillende parametergebiede tot gevolg; groot waardes van die versoeking-om-te-verraai parameter lei nie tot volhoubare samewerking nie, terwyl volhoubare samewerking wel vir klein waardes van hierdie parameter moontlik is. Daar word gevind dat die kans vir volhoubare samewerking toeneem tot sekerheid namate die orde van die onderliggende gra ek groei. Die toestandsruimte van die prisoniersdilemma met 'n rooster in die vlak of 'n rooster op die torus as onderliggende gra ek groei baie vinnig as 'n funksie van die orde van die gra ek. Die outomor smeklasse van die speltoestande word getel met die doel om te bepaal presies hoe vinnig die toestandsruimte van die spel as 'n funksie van die orde van die onderliggende gra ek groei. Die dinamika van die prisoniersdilemma met 'n rooster in die vlak of 'n rooster op die torus as onderliggende gra ek word laastens deur middel van konstruksies van die ooreenstemmende toestandsgra eke ondersoek (in gevalle waar die toestandsruimte klein genoeg is).

Observations of Jupiter show a super-rotating (prograde) equatorial jet that has persisted for decades. Shallow water simulations run in the Jovian parameter regime reproduce the mixture of robust vortices and alternating zonal jets observed on Jupiter, but the equatorial jet is invariably sub-rotating (retrograde). Recent work has obtained super-rotating equatorial jets by extending the standard shallow water equations to relax the height field towards its mean value. This Newtonian cooling-like term is intended to model radiative cooling to space, but its addition breaks key conservation properties for mass and momentum. In this thesis the radiatively damped thermal shallow water equations are proposed as an alternative model for Jovian atmospheres. They extend standard shallow water theory by permitting horizontal variations of the thermodynamic properties of the fluid. The additional temperature equation allows a Newtonian cooling term to be included while conserving mass and momentum. Simulations reproduce equatorial jets in the correct directions for both Jupiter and Neptune (which sub-rotates). Quasi-geostrophic theory filters out rapidly moving inertia-gravity waves. A local quasi-geostrophic theory of the radiatively damped thermal shallow water equations is derived, and then extended to cover whole planets. Simulations of this global thermal quasi-geostrophic theory show the same transition, from sub- to super-rotating equatorial jets, seen in simulations of the original thermal shallow water model as the radiative time scale is decreased. Thus the mechanism responsible for setting the direction of the equatorial jet must exist within quasi-geostrophic theory. Such a mechanism is developed by calculating the competing effects of Newtonian cooling and Rayleigh friction upon the zonal mean zonal acceleration induced by equatorially trapped Rossby waves. These waves transport no momentum in the absence of dissipation. Dissipation by Newtonian cooling creates an eastward zonal mean zonal acceleration, consistent with the formation of super-rotating equatorial jets in simulations, while the corresponding acceleration is westward for dissipation by Rayleigh friction.

L'objectif de cette thèse est d'étudier le comportement local d'une mesure de probabilité, notamment au travers d'un indice de régularité locale. Dans la première partie, nous établissons la normalité asymptotique de l'estimateur des kn plus proches voisins de la densité et de l'histogramme. Dans la deuxième, nous définissons un estimateur du mode sous des hypothèses affaiblies. Nous montrons que l'indice de régularité intervient dans ces deux problèmes. Enfin, nous construisons dans une troisième partie différents estimateurs pour l'indice de régularité à partir d'estimateurs de la fonction de répartition, dont nous réalisons une revue bibliographique.

When traditional linearised theory is used to study free-surface flows past a surface-piercing object or over an obstruction in a stream, the geometry of the object is usually lost, having been assumed small in one or several of its dimensions. In order to preserve the nonlinear nature of the geometry, asymptotic expansions in the low-Froude or low-Bond limits can be derived, but here, the solution invariably predicts a waveless free-surface at every order. This is because the waves are in fact, exponentially small, and thus beyond-all-orders of regular asymptotics; their formation is a consequence of the divergence of the asymptotic series and the associated Stokes Phenomenon. In this thesis, we will apply exponential asymptotics to the study of two new problems involving nonlinear geometries. In the first, we examine the case of free-surface flow over a step including the effects of both gravity and surface tension. Here, we shall see that the availability of multiple singularities in the geometry, coupled with the interplay of gravitational and cohesive effects, leads to the discovery of a remarkable new set of solutions. In the second problem, we study the waves produced by bluff-bodied ships in low-Froude flows. We will derive the analytical form of the exponentially small waves for a wide range of hull geometries, including single-cornered and multi-cornered ships, and then provide comparisons with numerical computations. A particularly significant result is our confirmation of the thirty-year old conjecture by Vanden-Broeck & Tuck (1977) regarding the impossibility of waveless single-cornered ships.