Functional Magnetic Resonance Imaging (fMRI) is a noninvasive imaging method that reflects local changes in brain activity. FMRI group studies involves the analysis of the functional images acquired for each of a group of subjects under the same experimental conditions. We propose a spatial marked point-process model for the activation patterns of the subjects in a group study. Each pattern is described as the sum of individual centres of activation. The marked point-process that we propose allows the researcher to enforce repulsion between all pairs of centres of an individual subject that are within a specified minimum distance of each other. It also allows the researcher to enforce attraction between similarly-located centres from different subjects. This attraction helps to compensate for the misalignment of corresponding functional areas across subjects and is a novel method of addressing the problem of imperfect inter-subject registration of functional images. We use a Bayesian framework and choose prior distributions according to current understanding of brain activity. Simulation studies and exploratory studies of our reference dataset are used to fine-tune the prior distributions. We perform inference via Markov chain Monte Carlo. The fitted model gives a summary of the activation in terms of its location, height and size. We use this summary both to identify brain regions that were activated in response to the stimuli under study and to quantify the discrepancies between the activation maps of subjects. Applied to our reference dataset, our measure is successful in separating out those subjects with activation patterns that do not agree with the overall group pattern. In addition, our measure is sensitive to subjects with a large number of activation centres relative to the other subjects in the group. The activation summary given by our model makes it possible to pursue a range of inferential questions that cannot be addressed with ease by current model-based approaches.

Dans cette thèse, nous nous intéressons à des matrices aléatoires en lien avec des permutations. Nous abordons l'étude de leurs spectres de plusieurs manières, et à différentes échelles d'observation. Dans un premier temps, nous prolongeons l'étude de Wieand à propos des nombres de valeurs propres appartenant à certains arcs fixés du cercle unité. Pour cela nous tirons parti des travaux réalisés par Ben Arous et Dang sur les statistiques linéaires du spectre de matrices de permutation pour une famille de lois à un paramètre englobant le cas de la loi uniforme sur le groupe symétrique, appelée famille des lois d'Ewens. Une partie innovante de notre travail réside dans la généralisation à des arcs non nécessairement fixés. Nous obtenons en effet des résultats similaires en autorisant les longueurs des arcs à décroître lentement vers zéro avec la taille des matrices. Dans un deuxième temps, nous regardons le spectre à échelle microscopique. En nous inspirant des travaux de Najnudel et Nikeghbali en rapport avec la convergence de mesures empiriques des angles propres normalisés, nous commençons par donner un sens à la convergence en terme de comptages de points sur des intervalles fixés. A partir du processus ponctuel limite, nous montrons que le nombre de points dans un intervalle a des fluctuations asymptotiquement gaussiennes lorsque la longueur de l'intervalle tend vers l'infini. Enfin, nous adaptons certains résultats de Chhaibi, Najnudel et Nikeghbali sur le polynôme caractéristique de matrices du CUE à échelle microscopique, et les développons dans notre cadre. De manière analogue mais avec d'autres techniques de preuves, nous obtenons des convergences des polynômes caractéristiques vers des fonctions entières, et cela pour une grande famille de lois pour le tirage des permutations, incluant les lois d'Ewens. / In this thesis, our goal is to study random matrices related to permutations. We tackle the study of their spectra in various ways, and at different scales. First, we extend the work of Wieand about the numbers of eigenvalues lying in some fixed arcs of the unit circle. We take advantage of the results of Ben Arous and Dang on the linear statistics of the spectrum of permutation matrices for a one-parameter family of deformations of the uniform law on the symmetric group, called Ewens' measures. One of the most innovative parts of our work is the generalization to non-fixed arcs. Indeed we get similar results when we let the lengths of the arcs decrease to zero slower than 1/n. Then, we look at the spectrum at microscopic scale. Inspired by the work of Najnudel and Nikeghbali about the convergence of empirical measures of rescaled eigenangles, we give a meaning to the convergence in terms of indicator functions of intervals. From the limiting point process, we show that the number of points in any interval is asymptotically normal as the length of the interval goes to infinity. Finally, we adapt some results of Chhaibi, Najnudel and Nikeghbali on the characteristic polynomial of the CUE at microscopic scale, and develop them in our framework. Analogously but with different techniques of proof, we get that the characteristic polynomials converge to entire functions, and this for a large family of laws including the Ewens' measures.

Dans cette thèse, nous nous sommes intéressés à la détection de galaxies lointaines dans les données hyperspectrales MUSE. Ces galaxies, en particulier, sont difficiles à observer, elles sont spatialement peu étendues du fait de leur distance, leur spectre est composé d'une seule raie d'émission dont la position est inconnue et dépend de la distance de la galaxie, et elles présentent un rapport signal-à-bruit très faible. Ces galaxies lointaines peuvent être considérées comme des sources quasi-ponctuelles dans les trois dimensions du cube. Il existe peu de méthodes dans la littérature qui permettent de détecter des sources dans des données en trois dimensions. L'approche proposée dans cette thèse repose sur la modélisation de la configuration de galaxies par un processus ponctuel marqué. Ceci consiste à représenter la position des galaxies comme une configuration de points auxquels nous ajoutons des caractéristiques géométriques, spectrales, etc, qui transforment un point en objet. Cette approche présente l'avantage d'avoir une représentation mathématique proche du phénomène physique et permet de s'affranchir des approches pixelliques qui sont pénalisées par les dimensions conséquentes des données (300 x 300 x 3600 pixels). La détection des galaxies et l'estimation de leurs caractéristiques spatiales, spectrales ou d'intensité sont réalisées dans un cadre entièrement bayésien, ce qui conduit à un algorithme générique et robuste, où tous les paramètres sont estimés sur la base des seules données observées, la détection des objets d'intérêt étant effectuée conjointement.La dimension des données et la difficulté du problème de détection nous ont conduit à envisager une phase de prétraitement des données visant à définir des zones de recherche dans le cube. Des approches de type tests multiples permettent de construire des cartes de proposition des objets. La détection bayésienne est guidée par ces cartes de pré-détection (définition de la fonction d'intensité du processus ponctuel marqué), la proposition des objets est réalisée sur les pixels sélectionnés sur ces cartes. La qualité de la détection peut être caractérisée par un critère de contrôle des erreurs.L'ensemble des traitements développés au cours de cette thèse a été validé sur des données synthétiques, et appliqué ensuite à un jeu de données réelles acquises par MUSE suite à sa mise en service en 2014. L'analyse de la détection obtenue est présentée dans le manuscrit. / Detecting the faintest galaxies in the hyperspectral MUSE data is particularly challenging because they have a small spatial extension, a very sparse spectrum that contains only one narrow emission line, which position in the spectral range is unknown. Moreover, their signal-to-noise ratio are very low. These galaxies are modeled as quasi point sources in the three dimensions of the data cube. We propose a method for the detection of a galaxy configuration based on a marked point process in a nonparametric Bayesian framework. A galaxy is modeled by a point (its position in the spatial domain), and marks (geometrical, spectral features) are added to transform a point into an object. These processes yield a natural sparse representation of massive data (300 x 300 x 3600 pixels). The fully Bayesian framework leads to a general and robust algorithm where the parameters of the objects are estimated in a fully data-driven way. Preprocessing strategies are drawn to tackle the massive dimensions of the data and the complexity of the detection problem, they allow to reduce the exploration of the data to areas that probably contain sources. Multiple testing approaches have been proposed to build proposition map. This map is also used to define the intensity of the point process, textit{i.e.} it describes the probability density function of the point process. It also gives a global error control criterion for the detection. The performance of the proposed algorithm is illustrated on synthetic data and real hyperspectral data acquired by the MUSE instrument for young galaxy detection.

Parmi les modèles permettant d'introduire de l'interaction entre les points, nous trouvons très large famille des modèles gibbsiens de processus ponctuels spatiaux issus de la physique statistique, permettant de modéliser à la fois des motifs répulsifs ou attractifs. Dans cette thèse, nous nous intéressons à l'inférence semi-paramétrique de ces modèles caractérisés par l'intensité conditionnelle de Papangelou. Deux contextes sont étudiés. Dans le premier thème, nous décrivons une procédure d'estimation du terme d'interaction du premier ordre (qui peut être aussi appelé l'intensité de Poisson) de l'intensité conditionnelle de Papangelou. L'idée sur laquelle l'estimation est basée permet, sous l'hypothèse d'une portée finie, de négliger les termes d'interaction d'ordre supérieur quelle que soit leur nature. La consistance forte et la normalité asymptotique de l'estimateur sont prouvées. Une étude par simulations illustre la performance de l'estimateur sur une fenêtre d'observation finie. Dans le second thème, nous nous focalisons sur la classe la plus connue et utilisée; le processus ponctuel à interaction par paires. Nous construisons une nouvelle méthode d'estimation de la fonction d'interaction de paires dans l'esprit des estimations non paramétriques par lissage à partir d'une réalisation du processus ponctuel spatial à interaction par paires. Deux cas sont étudiées: le cas stationnaire et le cas isotrope. Ces estimateurs exploitent à nouveau la propriété de portée finie des processus ponctuels et intégrent l'estimation du paramètre de l'intensité de Poisson vue dans le premier thème. Nous présentons les propriétés asymptotiques telles que la consistance forte ponctuelle, la consistance forte globale avec différentes vitesses de consistance, le comportement de l'erreur quadratique moyenne et la normalité asymptotique de ces estimateurs. / Among models allowing to introduce interaction between points, we find the large class of Gibbs models coming from statistical physics. Such models can produce repulsive as well as attractive point pattern. In this thesis, we are interested in the semi-parametric inference of such models characterized by the Papangelou conditional intensity. Two frameworks are considered. First, we describe a procédure which intends to estimate the first-order interaction term (also called Poisson intensity) of the Papangelou conditional intensity. Under the assumption of finite range of the process, the idea upon which the procedure is based allows us to neglect higher-order interaction terms. We study the stong consistency and the asymptotic normality and conduct a simulation study which highlights the efficiency of the method for finite observation window. Second, we focus on the main class of Gibbs models which is the class of pairwise interaction point processes. We construct a kernel-based estimator of the pairwise interaction function. Two cases are studied: the stationary case and the isotropic case.The estimators, we propose, exploit the finite range property and the estimator of the Poisson intensity defined in the first part. We present asymptotic properties, namely the strong consistency, the behavior of the mean squared error and the asymptotic normality.

La thèse s'inscrit dans l'étude de la structuration des forêts à partir des propriétés de la canopée telles que décrites par la distribution spatiale ou la taille des houppiers des arbres dominants. L'approche suivie est fondée sur la théorie des Processus Ponctuels Marqués (PPM) qui permet de modéliser ces houppiers comme des disques sur images considérées comme un espace 2D. Le travail a consisté à évaluer le potentiel des PPM pour détecter automatiquement les houppiers d'arbres dans des images optiques de très résolution spatiale acquises sur des forêts de mangroves et des plantations d'Eucalyptus. Pour les mangroves, nous avons également travaillé sur des images simulées de réflectance et des données Lidar. Différentes adaptations (paramétrage, modèles d'énergie) de la méthode de PPM ont été testées et comparées grâce à des indices quantitatifs de comparaison entre résultats de la détection et références de positionnement issues du terrain, de photo-interprétation ou de maquettes forestières.Dans le cas des mangroves, les tailles de houppier estimées par détection restent cohérentes avec les sorties des modèles allométriques disponibles. Les résultats thématiques indiquent que la détection par PPM permet de cartographier dans une jeune plantation d'Eucalyptus la densité locale d'arbres dont la taille des houppiers est proche de la résolution spatiale de l'image (0.5m). Cependant, la qualité de la détection diminue quand le couvert se complexifie. Ce travail dresse plusieurs pistes de recherche tant mathématique, comme la prise en compte des objets de forme complexe, que thématiques, comme l'apport des informations forestières à des échelles pertinentes pour la mise au point de méthodes de télédétection. / This PhD work aims at providing information on the forest structure through the analysis of canopy properties as described by the spatial distribution and the crown size of dominant trees. Our approach is based on the Marked Point Processes (MPP) theory, which allows modeling tree crowns observed in remote sensing images by discs belonging a two dimensional space. The potential of MPP to detect the trees crowns automatically is evaluated by using very high spatial resolution optical satellite images of both Eucalyptus plantations and mangrove forest. Lidar and simulated reflectance images are also analyzed for the mangrove application. Different adaptations (parameter settings, energy models) of the MPP method are tested and compared through the development of quantitative indices that allow comparison between detection results and tree references derived from the field, photo-interpretation or the forest mockups.In the case of mangroves, the estimated crown sizes from detections are consistent with the outputs from the available allometric models. Other results indicate that tree detection by MPP allows mapping, the local density of trees of young Eucalyptus plantations even if crown size is close to the image spatial resolution (0.5m). However, the quality of detection by MPP decreases with canopy closeness. To improve the results, further work may involve MPP detection using objects with finer shapes and forest data measurements collected at the tree plant scale.

Dans ce travail de thèse, nous étudions les fluctuations macroscopiques dans un modèle de boules aléatoires. Un modèle de boules aléatoires est une agrégation de boules dans Rd dont les centres et les rayons sont aléatoires. On marque également chaque boule par un poids aléatoire. On considère la masse M induite par le système de boules pondérées sur une configuration μ de Rd. Pour réaliser l’étude macroscopique des fluctuations de M, on réalise un "dézoom" sur la configuration de boules. Mathématiquement cela revient à diminuer le rayon moyen tout en augmentant le nombre moyen de centres par unité de volume. La question a déjà été étudiée lorsque les composantes des triplets (centre, rayon, poids) sont indépen- dantes et que ces triplets sont engendrés selon un processus ponctuel de Poisson sur Rd × R+ × R. On observe alors trois comportements distincts selon le rapport de force entre la vitesse de diminution des rayons et la vitesse d’augmentation de la densité des boules. Nous proposons de généraliser ces résultats dans trois directions distinctes. La première partie de ce travail de thèse consiste à introduire de la dépendance entre les centres et les rayons et de l’inhomogénéité dans la répartition des centres. Dans le modèle que nous proposons, le comportement stochastique des rayons dépend de l’emplacement de la boule. Dans les travaux précédents, les convergences obtenues pour les fluctuations de M sont au mieux des convergences fonctionnelles en dimension finie. Nous obtenons, dans la deuxième partie de ce travail, de la convergence fonctionnelle sur un ensemble de configurations μ de dimension infinie. Dans une troisième et dernière partie, nous étudions un modèle de boules aléatoires (non pondérées) sur C dont les couples (centre, rayon) sont engendrés par un processus ponctuel déterminantal. Contrairement au processus ponctuel de Poisson, le processus ponctuel déterminantal présente des phénomènes de répulsion entre ses points ce qui permet de modéliser davantage de problèmes physiques. / In this thesis, we study the macroscopic fluctuations in random balls models. A random balls model is an aggregation of balls in Rd whose centers and radii are random. We also mark each balls with a random weight. We consider the mass M induced by the system of weighted balls on a configuration μ of Rd. In order to investigate the macroscopic fluctuations of M, we realize a zoom-out on the configuration of balls. Mathematically, we reduce the mean radius while increasing the mean number of centers by volume unit. The question has already been studied when the centers, the radii and the weights are independent and the triplets (center, radius, weight) are generated according to a Poisson point process on Rd ×R+ ×R. Then, we observe three different behaviors depending on the comparison between the speed of the decreasing of the radii and the speed of the increasing of the density of centers. We propose to generalize these results in three different directions. The first part of this thesis consists in introducing dependence between the radii and the centers and inhomogeneity in the distribution of the centers. In the model we propose, the stochastic behavior of the radii depends on the location of the ball. In the previous works, the convergences obtained for the fluctuations of M are at best functional convergences in finite dimension. In the second part of this work, we obtain functional convergence on an infinite dimensional set of configurations μ. In the third and last part, we study a random balls model (non-weighted) on C where the couples (center, radius) are generated according to determinantal point process. Unlike to the Poisson point process, the determinantal point process exhibits repulsion phenomena between its points which allows us to model more physical problems.

Les neurones interagissent à travers leur potentiel de membrane qui a en général une évolution temporelle complexe due aux nombreuses entrées synaptiques irrégulières reçues. Cette évolution est mieux décrite en termes probabilistes, en raison de ces entrées irrégulières ou «bruit synaptique». L'évolution temporelle du potentiel de membrane est stochastique mais aussi déterministe: stochastique, car conduite par des entrées synaptiques qui arrivent de façon aléatoire dans le temps, et déterministe, car un neurone biologique a une évolution temporelle très similaire quand soumis à une même séquence d'entrées synaptiques. Nous étudions les propriétés statistiques d'un modèle simplifié de neurone soumis à des entrées à taux variable d'où en résulte l'évolution non-stationnaire du potentiel de membrane. Nous considérons un modèle passif de membrane neuronale, sans mécanisme de décharge neuronale, soumis à des entrées à courant ou à conductance sous la forme d'un processus de «shot noise». Les fluctuations du potentiel de membrane sont aussi modélisées par un processus stochastique similaire, de «shot noise» filtré. Nous avons analysé les propriétés statistiques de ces processus dans le cadre des transformations de processus ponctuels de Poisson. Des propriétés de ces transformations sont dérivées les statistiques non-stationnaires du processus. Nous obtenons ainsi des expressions analytiques exactes pour les moments et cumulants du processus filtré dans le cas général des taux d'entrée variables. Ce travail ouvre de nombreuses perspectives pour l'analyse de neurones dans les conditions in vivo, en présence d'entrées synaptiques intenses et bruitées. / Neurons interact through their membrane potential that generally has a complex time evolution due to numerous irregular synaptic inputs received. This complex time evolution is best described in probabilistic terms due to this irregular or "noisy" activity. The time evolution of the membrane potential is therefore both stochastic and deterministic: it is stochastic since it is driven by random input arrival times, but also deterministic, since subjecting a biological neuron to the same sequence of input arrival times often results in very similar membrane potential traces. In this thesis, we investigated key statistical properties of a simplified neuron model under nonstationary input from other neurons that results in nonstationary evolution of membrane potential statistics. We considered a passive neuron model without spiking mechanism that is driven by input currents or conductances in the form of shot noise processes. Under such input, membrane potential fluctuations can be modeled as filtered shot noise currents or conductances. We analyzed the statistical properties of these filtered processes in the framework of Poisson Point Processes transformations. The key idea is to express filtered shot noise as a transformation of random input arrival times and to apply the properties of these transformations to derive its nonstationary statistics. Using this formalism we derive exact analytical expressions, and useful approximations, for the mean and joint cumulants of the filtered process in the general case of variable input rate. This work opens many perspectives for analyzing neurons under in vivo conditions, in the presence of intense and noisy synaptic inputs.

<p>Temporal point processes (TPP) are mathematical approaches for modeling asynchronous event sequences by considering the temporal dependency of each event on past events and its instantaneous rate. Temporal point processes can model various problems, from earthquake aftershocks, trade orders, gang violence, and reported crime patterns, to network analysis, infectious disease transmissions, and virus spread forecasting. In each of these cases, the entity's behavior with the corresponding information is noted over time as an asynchronous event sequence, and the analysis is done using temporal point processes, which provides a means to define the generative mechanism of the sequence of events and ultimately predict events and investigate causality.</p> <p><br></p> <p>Among point processes, Hawkes process as a stochastic point process is able to model a wide range of contagious and self-exciting patterns. One of Hawkes process's well-known applications is predicting the evolution of viral processes on networks, which is an important problem in biology, the social sciences, and the study of the Internet. In existing works, mean-field analysis based upon degree distribution is used to predict viral spreading across networks of different types. However, it has been shown that degree distribution alone fails to predict the behavior of viruses on some real-world networks. Recent attempts have been made to use assortativity to address this shortcoming. This thesis illustrates how the evolution of such a viral process is sensitive to the underlying network's structure. </p> <p><br></p> <p>In Chapter 3, we show that adding assortativity does not fully explain the variance in the spread of viruses for a number of real-world networks. We propose using the graphlet frequency distribution combined with assortativity to explain variations in the evolution of viral processes across networks with identical degree distribution. Using a data-driven approach, by coupling predictive modeling with viral process simulation on real-world networks, we show that simple regression models based on graphlet frequency distribution can explain over 95\% of the variance in virality on networks with the same degree distribution but different network topologies. Our results highlight the importance of graphlets and identify a small collection of graphlets that may have the most significant influence over the viral processes on a network.</p> <p><br></p> <p>Due to the flexibility and expressiveness of deep learning techniques, several neural network-based approaches have recently shown promise for modeling point process intensities. However, there is a lack of research on the possible adversarial attacks and the robustness of such models regarding adversarial attacks and natural shocks to systems. Furthermore, while neural point processes may outperform simpler parametric models on in-sample tests, how these models perform when encountering adversarial examples or sharp non-stationary trends remains unknown. </p> <p><br></p> <p>In Chapter 4, we propose several white-box and black-box adversarial attacks against deep temporal point processes. Additionally, we investigate the transferability of white-box adversarial attacks against point processes modeled by deep neural networks, which are considered a more elevated risk. Extensive experiments confirm that neural point processes are vulnerable to adversarial attacks. Such a vulnerability is illustrated both in terms of predictive metrics and the effect of attacks on the underlying point process's parameters. Expressly, adversarial attacks successfully transform the temporal Hawkes process regime from sub-critical to into a super-critical and manipulate the modeled parameters that is considered a risk against parametric modeling approaches. Additionally, we evaluate the vulnerability and performance of these models in the presence of non-stationary abrupt changes, using the crimes and Covid-19 pandemic dataset as an example.</p> <p><br></p> <p> Considering the security vulnerability of deep-learning models, including deep temporal point processes, to adversarial attacks, it is essential to ensure the robustness of the deployed algorithms that is despite the success of deep learning techniques in modeling temporal point processes.</p> <p> </p> <p>In Chapter 5, we study the robustness of deep temporal point processes against several proposed adversarial attacks from the adversarial defense viewpoint. Specifically, we investigate the effectiveness of adversarial training using universal adversarial samples in improving the robustness of the deep point processes. Additionally, we propose a general point process domain-adopted (GPDA) regularization, which is strictly applicable to temporal point processes, to reduce the effect of adversarial attacks and acquire an empirically robust model. In this approach, unlike other computationally expensive approaches, there is no need for additional back-propagation in the training step, and no further network is required. Ultimately, we propose an adversarial detection framework that has been trained in the Generative Adversarial Network (GAN) manner and solely on clean training data. </p> <p><br></p> <p>Finally, in Chapter 6, we discuss implications of the research and future research directions.</p>

Stochastic point processes and associated methodology provide a means for the statistical analysis and modeling of the spatial point pattern formed from forest tree stem locations. Stochastic Gibbs point processes were explored as models that could simulate short-range clustering arising from reproduction of trees by stump sprouting, and intermediate-range inhibition of trees that may result from competition for light and growing space. This study developed and compared three pairwise interaction processes with parametric models for 2nd-order potentials and three triplets processes with models for 2nd- and 3rd-order potentials applied to a mixed-species hardwood forest in the Southern Appalachian Mountains of western North Carolina. Although the 2nd-order potentials of both the pairwise interaction and triplets processes were allowed to be purely or partially attractive, the proposed Gibbs point process models were demonstrated to be locally stable. The proposed Gibbs point processes were simulated using Markov Chain Monte Carlo (MCMC) methods; in particular, a reversible-jump Metropolis-Hastings algorithm with birth, death, and shift proposals was utilized. Parameters for the models were estimated by a Bayesian inferential procedure that utilizes MCMC methods to draw samples from the Gibbs posterior density. Two Metropolis-Hastings algorithms that do this sampling were compared; one that estimated ratios of intractable normalizing constants of the Gibbs likelihood by importance sampling and another that introduced an auxiliary variable to cancel the normalizing constants with those in the auxiliary variable's proposal distribution. Results from this research indicated that attractive pairwise interaction models easily degenerate into excessively clustered patterns, whereas triplets processes with attractive 2nd-order and repulsive 3rd-order interactions are more robust against excessive clustering. Bayesian inference for the proposed triplets models was found to be very computationally expensive. Slow mixing of both algorithms used for the inference combined with the long iteration times limited the practicality of the Bayesian approach. However the results obtained here indicate that triplets processes can be used to draw inference for and simulate patterns of mixed-species Appalachian hardwood forests. / Ph. D.

Extreme Value Theory nds application in problems concerning low probability but high consequence events. In hydrology the study of heavy rainfall is important in regional ood risk assessment. In particular, the N-year return level is a key output of an extreme value analysis, hence care needs to be taken to ensure that the model is accurate and that the level of imprecision in the parameter estimates is made explicit. Rainfall is a process that evolves over time and space. Therefore, it is anticipated that at extreme levels the process would continue to show temporal and spatial correlation. In this study interest is in whether any trends in heavy rainfall can be detected for the Western Cape. The focus is on obtaining the 50-year daily winter rainfall return level and investigating whether this quantity is homogenous over the study area. The study is carried out in two stages. In the rst stage, the point process approach to extreme value theory is applied to arrive at the return level estimates at each of the fteen sites. Stationarity is assumed for the series at each station, thus an issue to deal with is that of short-range temporal correlation of threshold exceedances. The proportion of exceedances is found to be smaller (approximately 0.01) for stations towards the east such as Jonkersberg, Plettenbergbay and Tygerhoek. This can be attributed to rainfall values being mostly low, with few instances where large amounts of rainfall were observed. Looking at the parameters of the point process extreme value model, the location parameter estimate appears stable over the region in contrast to the scale parameter estimate which shows an increase towards in a south easterly direction. While the model is shown to t exceedances at each station adequately, the degree of uncertainty is large for stations such as Tygerhoek, where the maximum observed rainfall value is approximately twice as large as the high rainfall values. This situation was also observed at other stations and in such cases removal of these high rainfall values was avoided to minimize the risk of obtaining inaccurate return level estimates. The key result is an N-year rainfall return level estimate at each site. Interest is in mapping an estimate of the 50-year daily winter rainfall return level, however to evaluate the adequacy of the model at each site the 25-year return level is considered since a 25 year return period is well within the range of the observed data. The 25-year daily winter rainfall return level estimate for Ladismith is the smallest at 22:42 mm. This can be attributed to the station's generally low observed winter rainfall values. In contrast, the return level estimate for Tygerhoek is high, almost six times larger than that of Ladismith at 119:16 mm. Visually design values show di erences between sites, therefore it is of interest to investigate whether these di erences can be modelled. The second stage is the geostatistical analysis of the 50-year 24-hour rainfall return level The aim here is to quantify the degree of spatial variation in the 50-year 24-hour rainfall return level estimates and to use that association to predict values at unobserved sites within the study region. A tool for quantifying spatial variation is the variogram model. Estimation of the parameters of this model require a su ciently large sample, which is a challenge in this study since there is only fteen stations and therefore only fteen observations for the geostatistical analysis. To address this challenge, observations are expanded in space and time and then standardized and to create a larger pool of data from which the variogram is estimated. The obtained estimates are used in ordinary and universal kriging to derive the 50-year 24-hour winter rainfall return level maps. It is shown that 50-year daily winter design rainfall over most of the Western Cape lies between 40 mm and 80 mm, but rises sharply as one moves towards the east coast of the region. This is largely due to the in uence of large design values obtained for Tygerhoek. In ordinary kriging prediction uncertainty is lowest around observed values and is large if the distance from these points increases. Overall, prediction uncertainty maps show that ordinary kriging performs better than universal kriging where a linear regional trend in design values is included.