arXiv subjects

Philippe Naveau

Publications and source records attributed to Philippe Naveau.

39 records · Page 3Linked to original sources

arXiv2013

Approximating the conditional density given large observed values via a multivariate extremes framework, with application to environmental data

Phenomena such as air pollution levels are of greatest interest when observations are large, but standard prediction methods are not specifically designed for large observations. We propose a method, rooted in extreme value theory, which approximates the conditional distribution of an unobserved component of a random vector given large observed values. Specifically, for $\mathbf{Z}=(Z_1,...,Z_d)^T$ and $\mathbf{Z}_{-d}=(Z_1,...,Z_{d-1})^T$, the method approximates the conditional distribution of $[Z_d|\mathbf{Z}_{-d}=\mathbf{z}_{-d}]$ when $|\mathbf{z}_{-d}|>r_*$. The approach is based on the assumption that $\mathbf{Z}$ is a multivariate regularly varying random vector of dimension $d$. The conditional distribution approximation relies on knowledge of the angular measure of $\mathbf{Z}$, which provides explicit structure for dependence in the distribution's tail. As the method produces a predictive distribution rather than just a point predictor, one can answer any question posed about the quantity being predicted, and, in particular, one can assess how well the extreme behavior is represented. Using a fitted model for the angular measure, we apply our method to nitrogen dioxide measurements in metropolitan Washington DC. We obtain a predictive distribution for the air pollutant at a location given the air pollutant's measurements at four nearby locations and given that the norm of the vector of the observed measurements is large.

stat.AP↗

arXiv2007

Detecting spatial patterns with the cumulant function. Part II: An application to El Nino

The spatial coherence of a measured variable (e.g. temperature or pressure) is often studied to determine the regions where this variable varies the most or to find teleconnections, i.e. correlations between specific regions. While usual methods to find spatial patterns, such as Principal Components Analysis (PCA), are constrained by linear symmetries, the dependence of variables such as temperature or pressure at different locations is generally nonlinear. In particular, large deviations from the sample mean are expected to be strongly affected by such nonlinearities. Here we apply a newly developed nonlinear technique (Maxima of Cumulant Function, MCF) for the detection of typical spatial patterns that largely deviate from the mean. In order to test the technique and to introduce the methodology, we focus on the El Nino/Southern Oscillation and its spatial patterns. We find nonsymmetric temperature patterns corresponding to El Nino and La Nina, and we compare the results of MCF with other techniques, such as the symmetric solutions of PCA, and the nonsymmetric solutions of Nonlinear PCA (NLPCA). We found that MCF solutions are more reliable than the NLPCA fits, and can capture mixtures of principal components. Finally, we apply Extreme Value Theory on the temporal variations extracted from our methodology. We find that the tails of the distribution of extreme temperatures during La Nina episodes is bounded, while the tail during El Ninos is less likely to be bounded. This implies that the mean spatial patterns of the two phases are asymmetric, as well as the behaviour of their extremes.

physics.data-an↗

arXiv2007

Detecting spatial patterns with the cumulant function. Part I: The theory

In climate studies, detecting spatial patterns that largely deviate from the sample mean still remains a statistical challenge. Although a Principal Component Analysis (PCA), or equivalently a Empirical Orthogonal Functions (EOF) decomposition, is often applied on this purpose, it can only provide meaningful results if the underlying multivariate distribution is Gaussian. Indeed, PCA is based on optimizing second order moments quantities and the covariance matrix can only capture the full dependence structure for multivariate Gaussian vectors. Whenever the application at hand can not satisfy this normality hypothesis (e.g. precipitation data), alternatives and/or improvements to PCA have to be developed and studied. To go beyond this second order statistics constraint that limits the applicability of the PCA, we take advantage of the cumulant function that can produce higher order moments information. This cumulant function, well-known in the statistical literature, allows us to propose a new, simple and fast procedure to identify spatial patterns for non-Gaussian data. Our algorithm consists in maximizing the cumulant function. To illustrate our approach, its implementation for which explicit computations are obtained is performed on three family of of multivariate random vectors. In addition, we show that our algorithm corresponds to selecting the directions along which projected data display the largest spread over the marginal probability density tails.

math.ST↗