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Publication# Generalized Spatial Regression with Differential Penalization

Abstract

We propose a novel method for the analysis of spatially distributed data from an exponential family distribution, able to efficiently treat data occurring over irregularly shaped domains. We consider a generalized linear framework and extend the work of Sangalli et al. (2013) to distributions other than the Gaussian. In particular, we can handle all distributions within the exponential family, including binomial, Poisson and Gamma outcomes, hence leading to a very broad applicability of the proposed model. We maximize a penalized log-likelihood function. The roughness penalty term involves a suitable differential operator of the spatial field over the domain of interest. This maximization is done via a penalized iterative least square approach (see Wood (2006)). Covariate information can also be included in the model in a semi-parametric setting. The proposed models exploit advanced scientific computing techniques and specifically make use of the Finite Element Method, that provides a basis for piecewise polynomial surfaces and allows to impose boundary conditions on the space distribution of the probability. Finally, we extend theoretically the model to deal with data occurring on a two dimensional manifold.

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Related concepts (11)

Related publications (5)

Related MOOCs (12)

Exponential family

In probability and statistics, an exponential family is a parametric set of probability distributions of a certain form, specified below. This special form is chosen for mathematical convenience, including the enabling of the user to calculate expectations, covariances using differentiation based on some useful algebraic properties, as well as for generality, as exponential families are in a sense very natural sets of distributions to consider. The term exponential class is sometimes used in place of "exponential family", or the older term Koopman–Darmois family.

Data

In common usage and statistics, data (USˈdætə; UKˈdeɪtə) is a collection of discrete or continuous values that convey information, describing the quantity, quality, fact, statistics, other basic units of meaning, or simply sequences of symbols that may be further interpreted formally. A datum is an individual value in a collection of data. Data is usually organized into structures such as tables that provide additional context and meaning, and which may themselves be used as data in larger structures.

Linear regression

In statistics, linear regression is a linear approach for modelling the relationship between a scalar response and one or more explanatory variables (also known as dependent and independent variables). The case of one explanatory variable is called simple linear regression; for more than one, the process is called multiple linear regression. This term is distinct from multivariate linear regression, where multiple correlated dependent variables are predicted, rather than a single scalar variable.

Digital Signal Processing I

Basic signal processing concepts, Fourier analysis and filters. This module can
be used as a starting point or a basic refresher in elementary DSP

Digital Signal Processing II

Adaptive signal processing, A/D and D/A. This module provides the basic
tools for adaptive filtering and a solid mathematical framework for sampling and
quantization

Digital Signal Processing III

Advanced topics: this module covers real-time audio processing (with
examples on a hardware board), image processing and communication system design.

Claire Marianne Charlotte Capelo

The explosive growth of machine learning in the age of data has led to a new probabilistic and data-driven approach to solving very different types of problems. In this paper we study the feasibility of using such data-driven algorithms to solve classic physical and mathematical problems. In particular, we try to model the solution of an inverse continuum mechanics problem in the context of linear elasticity using deep neural networks. To better address the inverse function, we start first by studying the simplest related task,consisting of a building block of the actual composite problem. By empirically proving the learnability of simpler functions, we aim to draw conclusions with respect to the initial problem.The basic inverse problem that motivates this paper is that of a 2D plate with inclusion under specific loading and boundary conditions. From measurements at static equilibrium,we wish to recover the position of the hole. Although some analytical solutions have been formulated for 3D-infinite solids - most notably Eshelby’s inclusion problems - finite problems with particular geometries, material inhomogeneities, loading and boundary conditions require the use of numerical methods which are most often efficient solutions to the forward problem, the mapping from the parameter space to the measurement/signal space, i.e. in our case computing displacements and stresses knowing the size and position of the inclusion. Using numerical data generated from the well-defined forward problem,we train a neural network to approximate the inverse function relating displacements and stresses to the position of the inclusion. The preliminary results on the 2D-finite problem are promising, but the black-box nature of neural networks is a huge issue when it comes to understanding the solution.For this reason, we study a 3D-infinite continuous isotropic medium with unique concentrated load, for which the Green’s function gives an analytical mathematical expression relating relative position of the point force and the displacements in the solid. After de-riving the expression of the inverse, namely recovering the relative position of the force from the Green’s matrix computed at a given point in the medium, we are able to study the sensitivity of the inverse function. From both the expression of the Green’s function and its inverse, we highlight what issues might arise when training neural networks to solve the inverse problem. As the Green’s function is not bijective, bijection must been forced when training for regression. Moreover, due to displacements growing to infinity as we approach the singularity at zero, the training domain must be constrained to be some distance away from the singularity. As we train a neural network to fit the inverse of the Green’s function, we show that the input parameters should include the least possible redundant information to ensure the most efficient training.We then extend our analysis to two point forces. As more loads are added, bijection is harder to enforce as permutations of forces must be taken into account and more collisions may arise, i.e. multiple specific combinations of forces might yield the same measurements.One obvious solution is to increase the number of nodes where displacements are measured to limit the possibility of collision. Through new experiments, we show again that the best training is achieved for the least possible amount of nodes, as long as the training data generated is indeed bijective. As the medium is elastic, we propose a neural network architecture that matches the composite nature of the inverse problem. We also present another formulation of the problem which is invariant to permutations of the forces,namely multilabel classification, and yields good performance in the two-load case.Finally, we study the composite inverse function for 2, 3, 4 and 5 forces. By comparing training and accuracy for different neural network architectures, we expose the model easiest to train. Moreover, the evolution of the final accuracy as more loads are added indicates that deep-neural networks (DNNs) are not well suited to fit a non-linear mapping from and to a high-dimensional space. The results are more convincing for multilabel classification.

2020xtreme value analysis is concerned with the modelling of extreme events such as floods and heatwaves, which can have large impacts. Statistical modelling can be useful to better assess risks even if, due to scarcity of measurements, there is inherently very large residual uncertainty in any analysis. Driven by the increase in environmental databases, spatial modelling of extremes has expanded rapidly in the last decade. This thesis presents contributions to such analysis.
The first chapter is about likelihood-based inference in the univariate setting and investigates the use of bias-correction and higher-order asymptotic methods for extremes, highlighting through examples and illustrations the unique challenge posed by data scarcity. We focus on parametric modelling of extreme values, which relies on limiting distributional results and for which, as a result, uncertainty quantification is complicated. We find that, in certain cases, small-sample asymptotic methods can give improved inference by reducing the error rate of confidence intervals. Two data illustrations, linked to assessment of the frequency of extreme rainfall episodes in Venezuela and the analysis of survival of supercentenarians, illustrate the methods developed.
In the second chapter, we review the major methods for the analysis of spatial extremes models. We highlight the similarities and provide a thorough literature review along with novel simulation algorithms. The methods described therein are made available through a statistical software package.
The last chapter focuses on estimation for a Bayesian hierarchical model derived from a multivariate generalized Pareto process. We review approaches for the estimation of censored components in models derived from (log)-elliptical distributions, paying particular attention to the estimation of a high-dimensional Gaussian distribution function via Monte Carlo methods. The impacts of model misspecification and of censoring are explored through extensive simulations and we conclude with a case study of rainfall extremes in Eastern Switzerland.

The increasing interest in using statistical extreme value theory to analyse environmental data is mainly driven by the large impact extreme events can have. A difficulty with spatial data is that most existing inference methods for asymptotically justified models for extremes are computationally intractable for data at several hundreds of sites, a number easily attained or surpassed by the output of physical climate models or satellite-based data sets. This thesis does not directly tackle this problem, but it provides some elements that might be useful in doing so. The first part of the thesis contains a pointwise marginal analysis of satellite-based measurements of total column ozone in the northern and southern mid-latitudes. At each grid cell, the r-largest order statistics method is used to analyse extremely low and high values of total ozone, and an autoregressive moving average time series model is used for an analogous analysis of mean values. Both models include the same set of global covariates describing the dynamical and chemical state of the atmosphere. The results show that influence of the covariates is captured in both the ``bulk'' and the tails of the statistical distribution of ozone. For some covariates, our results are in good agreement with findings of earlier studies, whereas unprecedented influences are retrieved for two dynamical covariates. The second part concerns the frameworks of multivariate and spatial modelling of extremes. We review one class of multivariate extreme value distributions, the so-called Hüsler--Reiss model, as well as its spatial extension, the Brown--Resnick process. For the former, we provide a detailed discussion of its parameter matrix, including the case of degeneracy, which arises if the correlation matrices of underlying multivariate Gaussian distributions are singular. We establish a simplification for computing the partial derivatives of the exponent function of these two models. As consequence of the considerably reduced number of terms in each partial derivative, computation time for the multivariate joint density of these models can be reduced, which could be helpful for (composite) likelihood inference. Finally, we propose a new variant of the Brown--Resnick process based on the Karhunen--Loève expansion of its underlying Gaussian process. As an illustration, we use composite likelihood to fit a simplified version of our model to a hindcast data set of wave heights that shows highly dependent extremes.