On Representations of Divergence Measures and Related Quantities in Exponential Families

Within exponential families, which may consist of multi-parameter and multivariate distributions, a variety of divergence measures, such as the Kullback–Leibler divergence, the Cressie–Read divergence, the Rényi divergence, and the Hellinger metric, can be explicitly expressed in terms of the respective cumulant function and mean value function. Moreover, the same applies to related entropy and affinity measures. We compile representations scattered in the literature and present a unified approach to the derivation in exponential families. As a statistical application, we highlight their use in the construction of confidence regions in a multi-sample setup.

Empirical cumulant function based parameter estimation in stable laws

Stable distributions are a subclass of infinitely divisible distributions that form the only family of possible limiting distributions for sums of independent identically distributed random variables. A challenging problem is estimating their parameters because many have densities with no explicit form and infinite moments. To address this problem, a class of closed-form estimators, called cumulant estimators, has been introduced. Cumulant estimators are derived from the logarithm of empirical characteristic function at two arbitrary distinct positive real arguments. This paper extends cumulant estimators in two directions: (i) it is proved that they are asymptotically normal and (ii) a sample based rule for selecting the two arguments is proposed. Extensive simulations show that under the provided selection rule, the closed-form cumulant estimators generally outperform the well-known algorithmic methods.

Detecting spatial patterns with the cumulant function – Part 2: An application to El Niño

The spatial coherence of a measured variable (e.g. temperature or pressure) is often studied to determine the regions of high variability 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 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 Niño/Southern Oscillation and its spatial patterns. We find nonsymmetric temperature patterns corresponding to El Niño and La Niña, 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 Niña episodes is bounded, while the tail during El Niños 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.

Detecting spatial patterns with the cumulant function – Part 1: 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 for this purpose, it provides meaningful results only if the underlying multivariate distribution is Gaussian. Indeed, PCA is based on optimizing second order moments, and the covariance matrix captures the full dependence structure of 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. The 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. Three families of multivariate random vectors, for which explicit computations are obtained, are implemented to illustrate our approach. 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.

Cumulant Based Modal Parameter Extraction of Structures

This paper describes a nonlinear deterministic estimator based on cumulants for the extraction of modal parameters. The signal analysed is composed of multiple exponentially damped real sinusoids in unknown additive noise. Cumulants reduce significantly the effects of noise and are also an efficient way of compressing the sampled data. In modal analysis a sensor may be unable to detect some modes of vibration due to its location. Cumulants estimated from real data sampled at different locations and instances are added directly together. This average cumulant function will contain the eigenvalues for all excited modes of vibration. Finding the frequencies and corresponding damping factors is therefore reduced to solving a single average cumulant function. The performance of the proposed estimator is examined and compared with the Eigensystem Realization Algorithm via simulations.