Estructuras estocásticas notables en procesos puntuales espacio-temporales y medidas de riesgo bivariantes

Abstract

Statistical modeling is a set of mathematical formulations which provide a better understanding of phenomena, especially those that underlie relationships between variables that in turn have random influence. In this thesis work, the statistical modeling of criminal phenomena is carried out through the development of stochastic structures in spatio-temporal point processes, in order to reproduce events as accurately as possible to reality in terms of their evolution in space, time and space-time. First, the spatio-temporal dependence of spatially correlated self-excited processes is structured, considering a stochastic difference equation for the intensity of the spatio-temporal process, which captures both the dependence due to self-excitation and the dependence on an underlying spatial process. The reasoning of Clark and Dixon (2021) and Reinhart (2018) is followed, but capturing the nonlinearity of the spatio-temporal covariates by means of a generalized additive structure (GAM) with B-splines. Second, a latent spatio-temporal process model, i.e. a LGCP, is structured with a separable first order intensity function. The deterministic time component is the result of a generalized linear model (GLM) with meteorological covariate effects, days and months of the year, harmonic regression parameters, annual periodicity of incidence rates and global trend. For the spatial deterministic component a generalized additive model (GAM) is considered, following the reasoning, of Diggle et al. (2005), Taylor et al. (2013) and González and Mateu (2021), but with integrated smoothness estimation with univariate B-splines. And for the stochastic component, we consider a Gaussian field that models the spatio-temporal dependence and variation of the events. Thirdly, selfexcitation mechanisms are identified among the crime serie in a continuous time by means of Hawkes point processes. The background rate of each component is estimated with a nonparametric stochastic reconstruction; it includes a temporal periodicity, a separable spatial component and a long-term trend. The semi-parametric maximum likelihood estimation of the relaxation coefficients to stabilize and secure the estimation process, and the iterative algorithm of Zhuang and Mateu (2019) and Zhuang (2006) are followed. Spatio-temporal point process models are good mathematical tools for data analysis, and therefore, the described models provide reliable predictions, as shown throughout this thesis in crime data. In order to go beyond modeling and predictions of point patterns, a procedure is formulated to identify the spatial exceedances in bivariate LGCP and their associated regions, as well as the quantification of the hazard in terms of probabilities. For this, we model the spatial decomposition of thefts (crime without the use of violence) and robberies (crime with the use of violence or intimidation) to represent them in a variation associated with a particular type of crime, which is possible by means of a stochastic structure composed of a within-flow component and anotherone between-flows. For this modeling, we use log-Gaussian Cox processes along the same line of Taylor et al. (2015). And a focus on risk measures such as Value-at-Risk and Expeted Shorfall (Malevergne and Sornette, 2006) is incorporated. These measures are based on percentiles of the distribution of offenses exceeding a high threshold. A dependency structure using the extreme Gumbel-Hougaard copula is considered (Salvadori et al., 2007), and generalized Pareto marginal distributions (Castellanos and Cabras, 2007; Abad et al., 2014). In general, a Bayesian framework with MCMC-MALA and maximum likelihood is adopted for the inference on the parameters of the different models. A case study is developed with real data of crimes registered in the city of Riobamba-Ecuador and some temporal, spatial and spatio-temporal covariates. The results obtained provide relevant information on the modeling, prediction and extreme localized behavior of criminal events; however, they can be very useful to be applied to data sets of different areas. Finally, conclusions are drawn and some open ideas are put forward.