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Wavelet theory constitutes one of the most significant mathematical advances for signal processing, and thus presents a great interest in point process analysis. For instance, numerous approaches have been explored to estimate the first-order intensity of a point process with wavelets. In this thesis, we intend to consolidate existing results in wavelet-based linear estimation and investigate further applications of wavelets on point processes in the context of multiresolution analysis. We initially take this wavelet-based approach to estimate the first and higher-order intensities of a point process in any finite dimension and under a continuous spatial setting. We perform a statistical study of wavelet linear estimators when the observed events are located in a hyperrectangle of R^d. It is notably shown that the linear estimator of the complete k-th order intensity is the product of k linear estimators of the first-order intensity. Such wavelet modelling also motivates the construction of a first-order multiresolution analysis, through the definition of properties at different scales, termed J-th level homogeneity and L-th level innovation. Likelihood ratio tests for these properties are provided and studied under Poisson processes and Haar wavelets. A key result is that the applicability of these tests is linked to the product of the number of realizations and the expectation measure of the process. This means that one can use the asymptotic distributions of the test statistics from a single point pattern if the expectation measure is itself sufficiently high. The hypothesis test for L-th level innovation is then used to design new data-driven thresholding strategies, each based on a different grouping of wavelet coefficients. Our thresholding methods are studied through extensive simulations and applied to NetFlow data to exhibit the differences between human and automated behaviour. Since wavelet estimation of Cox processes has received very little treatment to this day, we provide new developments in this topic essentially through the wavelet-based estimation of the pointwise probability density or mass function for the intensity field. The behaviour of this estimator is studied with example Cox process models for different wavelets and resolutions, followed by an application to firing patterns from virtual reality military training. We eventually extend the idea of J-th level homogeneity to Cox processes by re-defining it through the mean intensity field, which we then test with a method based on Hotelling's t-squared statistic. ; Open Access

This survey is devoted to the statistical analysis of duration models and point processes. The first section introduces specific concepts and definitions for single-spell duration models. Section two is devoted to the presentation of conditional duration models which incorporate the effects of explanatory variables. Competing risks models are presented in the third section. The fourth section is concerned with statistical inference, with a special emphasis on non- and semi- parametric estimation of single-spell duration models. Section 5 sets forth the main definitions for point and counting processes. Section 6 presents important elementary examples of point processes, namely Poisson, Markov and semi-Markov processes. The last section presents a general semi-parametric framework for studying point processes with explanatory variables.

We propose new summary statistics for intensity‐reweighted moment stationary point processes, that is, point processes with translation invariant n‐point correlation functions for all , that generalise the well known J‐, empty space, and spherical Palm contact distribution functions. We represent these statistics in terms of generating functionals and relate the inhomogeneous J‐function to the inhomogeneous reduced second moment function. Extensions to space time and marked point processes are briefly discussed.

The paper introduces new methods for inference with count data registered on a set of aggregation units. Such data are omnipresent in epidemiology because of confidentiality issues: it is much more common to know the county in which an individual resides, say, than to know their exact location in space. Inference for aggregated data has traditionally made use of models for discrete spatial variation, e.g. conditional auto-regressive models. We argue that such discrete models can be improved from both a scientific and an inferential perspective by using spatiotemporally continuous models to model the aggregated counts directly. We introduce methods for delivering (limiting) continuous inference with spatiotemporal aggregated count data in which the aggregation units might change over time and are subject to uncertainty. We illustrate our methods by using two examples: from epidemiology, spatial prediction of malaria incidence in Namibia, and, from politics, forecasting voting under the proposed changes to parliamentary boundaries in the UK. © 2018 Royal Statistical Society

Abstract. A simplified stochastic model for earthquake occurrence focusing on the spatio-temporal interactions between earthquakes is presented. The model is a marked point process model in which each earthquake is represented by its magnitude and coordinates in space and time. The model incorporates the occurrence of aftershocks as well as the build-up and subsequent release of strain. The parameters of the model are estimated from a maximum likelihood calculation.

1 Introduction -- 1.1 Arrivals in time -- 1.2 Reliability -- 1.3 Safety assessment -- 1.4 Random stress and strength -- Notes on the literature -- Problems -- 2 Point processes -- 2.1 The probabilistic context -- 2.2 Two methods of representation -- 2.3 Parameters of point processes -- 2.4 Transformation to a process with constant arrival rate -- 2.5 Time between arrivals -- Notes on the literature -- Problems -- 3 Homogeneous Poisson processes -- 3.1 Definition -- 3.2 Characterization -- 3.3 Time between arrivals for the hP process -- 3.4 Relations to the uniform distribution -- 3.5 A process with simultaneous arrivals -- Notes on the literature -- Problems -- 4 Application of point processes to a theory of safety assessment -- 4.1 The Reactor Safety Study -- 4.2 The annual probability of a reactor accident -- 4.3 A stochastic consequence model -- 4.4 A concept of rare events -- 4.5 Common mode failures -- 4.6 Conclusion -- Notes on the literature -- Problems -- 5 Renewal processes -- 5.1 Probabilistic theory -- 5.2 The renewal process cannot model equipment wearout -- Notes on the literature -- Problems -- 6 Poisson processes -- 6.1 The Poisson model -- 6.2 Characterization of regular Poisson processes -- 6.3 Time between arrivals for Poisson processes -- 6.4 Further observations on software error detection -- Notes on the literature -- Problems -- 7 Superimposed processes -- Notes on the literature -- Problems -- 8 Markov point processes -- 8.1 Theory -- 8.2 The Poisson process -- 8.3 Facilitation and hindrance -- Notes on the literature -- Problems -- 9 Applications of Markov point processes -- 9.1 Egg-laying dispersal of the bean weevil -- 9.2 Application of facilitation — hindrance to the spatial distribution of benthic invertebrates -- 9.3 The Luria-Delbrück model -- 9.4 Chance placement of balls in cells -- 9.5 A model for multiple vehicle automobile accidents -- 9.6 Engels' model -- Notes on the literature -- Problems -- 10 The order statistics process -- 10.1 The sampling of lifetimes -- 10.2 Derivation from the Poisson process -- 10.3 A Poisson model of equipment wearout -- Notes on the literature -- Problems -- 11 Competing risk theory -- 11.1 Markov chain model -- 11.2 Classical competing risks -- 11.3 Competing risk presentation of reactor safety studies -- 11.4 Delayed fatalities -- 11.5 Proportional hazard rates -- Notes on the literature -- Problems -- Further reading -- Appendix 1 Probability background -- A1.1 Probability distributions -- A1.2 Expectation -- A1.3 Transformation of variables -- A1.4 The distribution of order statistics -- A1.5 Conditional probability -- A1.6 Operational methods in probability -- A1.7 Convergence concepts and results in the theory of probability -- Notes on the literature -- Appendix 2 Technical topics -- A2.1 Existence of point process parameters -- A2.2 No simultaneous arrivals -- Solutions to a few of the problems -- References -- Author index.

Second‐order orientation methods provide a natural tool for the analysis of spatial point process data. In this paper, we extend to the spatiotemporal setting the spatial point pair orientation distribution function. The new space–time orientation distribution function is used to detect space–time anisotropic configurations. An edge‐corrected estimator is defined and illustrated through a simulation study. We apply the resulting estimator to data on the spatiotemporal distribution of fire ignition events caused by humans in a square area of 30 × 30 km2 for 4 years. Our results confirm that our approach is able to detect directional components at distinct spatiotemporal scales. © 2014 The Authors. Statistica Neerlandica © 2014 VVS.