Interpretable statistics for complex modelling: quantile and topological learning

As the complexity of our data increased exponentially in the last decades, so has our need for interpretable features. This thesis revolves around two paradigms to approach this quest for insights. In the first part we focus on parametric models, where the problem of interpretability can be seen as a “parametrization selection”. We introduce a quantile-centric parametrization and we show the advantages of our proposal in the context of regression, where it allows to bridge the gap between classical generalized linear (mixed) models and increasingly popular quantile methods. The second part of the thesis, concerned with topological learning, tackles the problem from a non-parametric perspective. As topology can be thought of as a way of characterizing data in terms of their connectivity structure, it allows to represent complex and possibly high dimensional through few features, such as the number of connected components, loops and voids. We illustrate how the emerging branch of statistics devoted to recovering topological structures in the data, Topological Data Analysis, can be exploited both for exploratory and inferential purposes with a special emphasis on kernels that preserve the topological information in the data. Finally, we show with an application how these two approaches can borrow strength from one another in the identification and description of brain activity through fMRI data from the ABIDE project

The persistence landscape and some of its properties

Persistence landscapes map persistence diagrams into a function space, which may often be taken to be a Banach space or even a Hilbert space. In the latter case, it is a feature map and there is an associated kernel. The main advantage of this summary is that it allows one to apply tools from statistics and machine learning. Furthermore, the mapping from persistence diagrams to persistence landscapes is stable and invertible. We introduce a weighted version of the persistence landscape and define a one-parameter family of Poisson-weighted persistence landscape kernels that may be useful for learning. We also demonstrate some additional properties of the persistence landscape. First, the persistence landscape may be viewed as a tropical rational function. Second, in many cases it is possible to exactly reconstruct all of the component persistence diagrams from an average persistence landscape. It follows that the persistence landscape kernel is characteristic for certain generic empirical measures. Finally, the persistence landscape distance may be arbitrarily small compared to the interleaving distance.Comment: 18 pages, to appear in the Proceedings of the 2018 Abel Symposiu

The World-Trade Web: Topological Properties, Dynamics, and Evolution

This paper studies the statistical properties of the web of import-export relationships among world countries using a weighted-network approach. We analyze how the distributions of the most important network statistics measuring connectivity, assortativity, clustering and centrality have co-evolved over time. We show that all node-statistic distributions and their correlation structure have remained surprisingly stable in the last 20 years -- and are likely to do so in the future. Conversely, the distribution of (positive) link weights is slowly moving from a log-normal density towards a power law. We also characterize the autoregressive properties of network-statistics dynamics. We find that network-statistics growth rates are well-proxied by fat-tailed densities like the Laplace or the asymmetric exponential-power. Finally, we find that all our results are reasonably robust to a few alternative, economically-meaningful, weighting schemes.Comment: 44 pages, 39 eps figure