Resumen de Analysis of fractality characteristics in spatio-temporal processes. Applications to geophysical data
Francisco Javier Esquivel Sánchez 
Over the last decades, the consequences of earthquakes on the inhabitants and the environment where they occur have led to a considerable effort in their study. The increasing interest in understanding earthquakes, jointly with the close relationship of seismicity analysis with Statistics (see, as a reference, the works about Statistical Seismology by Vere-Jones 2000, 2001, 2010) have motivated us to center our attention in this research on this geophysical phenomenon.
In this thesis work, we focus on the statistical exploitation of information from regional space-time series of seismic events. We are interested in the intrinsic structural complexity of the spatio-temporal evolution patterns of seismic occurrences within a certain region and a certain period of time. Two main objectives are addressed: the study of the dimensional interaction and the analysis of the structural features at different scales. In particular, we focus on the following aspects: study of scaling behaviour, analysis of dimensional interaction, assessment of evolutionary changes, incorporation of effects, application to relevant real seismic data, and application to random fields excursion sets.
With this purpose, we propose various technical extensions in the context of information, complexity and multifractal analysis, combined with some methodological complementary aspects. Specifically, we formulate a new version of generalized dimensions based on Tsallis non-extensive statistics; we establish a limiting connection between some well-known complexity measures and generalized dimensions, leading to a concept of `multifractal complexity'; we construct a dependence coefficient in the multifractal domain, useful for dimensional interaction assessment; and we propose a formal approach to incorporate local effects, such as those of the magnitude or potential information from covariates. Some transformations on the temporal component, combined with the former techniques, allow the evaluation of the effect of the occurrence times on the structural complexity displayed by the data. Further, the sliding windows technique provides a useful tool for assessing evolutionary changes (relevant, in particular, to describe temporal structural heterogeneities, as well as for detection of possible precursory elements). The fitting of the non-extensive frequency-magnitude distribution of earthquakes by means of a weighted non-linear regression, considering weights according to the released energy, provides improved results in terms of goodness of fit.
Three real seismic series are studied, namely corresponding to the areas of Agrón, El Hierro and Torreperogil, in Spain. These applications are developed not only for illustration of the usefulness of the technical and methodological aspects introduced to describe seismicity, but also considering their intrinsic interest in relation to relevant aspects underlaying their different nature.
In particular, regarding Agrón series the study performed combining the techniques with temporal transformations has been found to be very practical for assessment of the spatio-temporal association. As for El Hierro series, the techniques applied have detected temporal changes in the structural characteristics which could be seen as a precursory indicator to the volcanic eruption. Finally, the study of the Torreperogil series has shown some results not present in the other data sets, which may suggest the existence of external factors. However, such an assertion must be further investigated based on comparative analysis with similar data sets.
As for the application to random field excursion sets defined by threshold exceedances, we perform a preliminary study using the above techniques on the variations of structural characteristics with respect to increasing threshold, and depending on the local variability and dependence range properties, based on the spatial point patterns defined by centroid of connected components and by A-exit points. The results show that multifractal analysis and complexity measures constitute a useful complementary approach for studying structural characteristics of spatial threshold exceedances.
Various open lines and further directions for continuing research are pointed out from this thesis work. These involve, in particular, further analytical aspects and generalization, incorporation of errors and data extensions, directionality assessment, and sensitivity analysis.
