Modeling Spatial Extremes via Ensemble-of-Trees of Pairwise Copulas

Abstract

Assessing the risk of extreme events in a spatial domain, such as hurricanes, floods and droughts, presents unique significance in practice. Unfortunately, the existing extreme-value statistical models are typically not feasible for practical largescale problems. Graphical models, on the other hand, are capable of handling sizable number of variables, but have yet to be explored in the realm of extreme-value analysis. To bridge the gap, an extreme-value graphical model is introduced in this paper, i.e., an ensemble-of-trees of pairwise copulas (ETPC). In the proposed graphical model, extreme-value marginal distributions are stitched together by means of pairwise copulas, which in turn are the building blocks of the ensemble of trees. Novel linear-complexity stochastic gradient based algorithms are then developed for learning the ETPC model and inferring missing data. As a result, the ETPC model is applicable to extreme-value problems with thousands of variables. It can be proven that, under mild conditions, the ETPC model exhibits the favorable property of tail-dependence between an arbitrary pair of sites (variables); consequently, the model is able to reliably capture statistical dependence between extreme values at different sites. Experimental results for both synthetic and real data demonstrate the advantages of the ETPC model in modeling fitting, imputation, and computational efficiency.

Graphical Models for Extremes

“…However, to retain computational tractability, a promising approach might be to consider finite mixtures of tree‐based multivariate Pareto distributions similarly to Yu et al . () and Vettori et al . ().…”

“…Such models have small parameter dimension but they rely on strong assumptions on stationarity and cannot be applied to multivariate, non‐spatial data without information on an underlying space. Other approaches include constructions through factor copulas (Lee and Joe, ), ensembles of trees combining bivariate copulas (Yu et al ., ), graphical models for large censored observations (Hitz and Evans, ) and eigendecompositions (Cooley and Thibaud, ). Closely related to our concept of conditional independence is the work of Coles and Tawn () and Smith et al .…”

Visual and lidar-based SLAM by variational bayesian methods

“…In the implementation, the diagonal entries are typically initialized with large values, and then decreases progressively. This process can be viewed as a simulated annealing [1,68] or smoothing-based optimization [202], which contributes to finding a maximum globally [101,202]. To minimize the effect of inaccurate noise pre-estimates, we jointly estimate the noise variances together with the posterior distribution.…”

“…Stochastic gradients in Algorithm 1 can help to escape from poor solutions [5,15,129]. In addition, the uncertainty matrix simulates an annealing process [1,68] or smoothing-based optimization [202], contributing to a superior optimum [101,202]. Since we estimate α 1:K , β 1:K in an adaptive manner, the resulting algorithm is robust to inaccurate noise priors.…”

“…The variational distribution is Gaussian. This inference can be viewed as a smoothing-based optimization and can escape from shallow local extrema easily [101,202]. In addition, the employed stochastic gradients can also help to avoid poor solutions [5,15,129].…”