Identification of patterns for space-time event networks

This paper provides new tools for analyzing spatio-temporal event networks. We build time series of directed event networks for a set of spatial distances, and based on scan-statistics, the spatial distance that generates the strongest change of event network connections is chosen. In addition, we propose an empirical random network event generator to detect significant motifs throughout time. This generator preserves the spatial configuration but randomizes the order of the occurrence of events. To prevent the large number of links from masking the count of motifs, we propose using standardized counts of motifs at each time slot. Our methodology is able to detect interaction radius in space, build time series of networks, and describe changes in its topology over time, by means of identification of different types of motifs that allows for the understanding of the spatio-temporal dynamics of the phenomena. We illustrate our methodology by analyzing thefts occurred in Medellín (Colombia) between the years 2003 and 2015.Work supported by Red de Violencia y Criminalidad - Universidad Nacional Abierta y a Distancia UNAD, Bogotá Colombia and Universidad Nacional de Colombia sede Bogotá

Optimal sampling design for functional and spatio-temporal random fields

Esta tesis extiende los diseños de muestreo óptimos a la predicción espacial univariada y multivariada de datos funcionales. En ambos casos, se presentan predictores insesgados con sus respectivas varianzas. En el caso univariado, se propone usar cokriging simple sobre el campo aleatorio escalar formado por los puntajes asociados con la representación de los datos funcionales en términos de sus componentes principales funcionales empíricos. En el caso multivariado, se desarrolla la predicción espacial de una variable funcional en sitios no muestreados, usando covariables funcionales, es decir, se presenta el cokriging funcional. Se demuestra que a través de la representación de cada función en términos de sus componentes principales funcionales empíricos, el cokriging funcional solo depende de la auto-covarianza y de la covarianza cruzada de los vectores de puntajes asociados, los cuales son campos aleatorios escalares. Se proponen criterios de diseño para todos los predictores desarrollados en esta tesis. Adicionalmente, se construye una metodología para diseños de muestreo espacial dinámicos que permitan encontrar la estimación óptima de la media espacial y la predicción espacial óptima en un tiempo futuro, basados en la variación temporal de la estructura de dependencia espacial. Las metodologías son aplicadas a las redes de calidad del aire de Bogotá y México.Abstract. We extend the framework of optimal sampling designs to the spatial prediction of univariate and multivariate functional data. In both cases, we derive unbiased predictors and their variances. In the univariate case, we propose to use a simple cokriging predictor with the scalar random fields resulting from the scores associated with the representation of the functional data with the empirical functional principal components. In the multivariate case, we develop spatial prediction of a functional variable at unsampled sites, using functional covariates, that is, we present a functional cokriging method. We show that through the representation of each function in terms of its empirical functional principal components, the functional cokriging only depends on the auto-covariance and cross-covariance of the associated score vectors, which are scalar random fields. Design criteria are given for all predictors derived in this thesis. In addition, we develop a methodology for dynamic spatial sampling designs to find the optimal spatial mean estimation and the optimal spatial prediction at some future time points, based on the temporal variation of the spatial dependence structure. The methodologies are applied to the networks of air quality of Bogot\'a and M\'exico.Doctorad