Transformed Linear Prediction for Extremes

TL;DR

The paper develops extremal linear prediction for high-level extreme events by constructing an inner product space of nonnegative random variables through transformed-linear combinations of independent regularly varying variables and deriving an optimal predictor via the projection theorem. Tail dependence is summarized by the tail pairwise dependence matrix $\Sigma_{\bm X}$, which is estimable from pairwise data and plays the role of an extreme covariance. Uncertainty in predictions is quantified using angular measures obtained via completely positive decompositions and the polar geometry of regular variation to form prediction intervals that adapt to the predicted extreme level. The approach is demonstrated through simulations and two real-data applications (NO$_2$ and UK precipitation), showing competitive interval coverage and flexibility in high dimensions with relatively simple estimation based on the TPDM.

Abstract

We address the problem of prediction for extreme observations by proposing an extremal linear prediction method. We construct an inner product space of nonnegative random variables derived from transformed-linear combinations of independent regularly varying random variables. Under a reasonable modeling assumption, the matrix of inner products corresponds to the tail pairwise dependence matrix, which can be easily estimated. We derive the optimal transformed-linear predictor via the projection theorem, which yields a predictor with the same form as the best linear unbiased predictor in non-extreme settings. We quantify uncertainty for prediction errors by constructing prediction intervals based on the geometry of regular variation. We demonstrate the effectiveness of our method through a simulation study and its applications to predicting high pollution levels, and extreme precipitation.

Figures (4)

• Figure 1: Maximum NO$_2$ measurements for January 23, 2020. All observations are above the empirical .98 quantile for each location.
• Figure 2: (Left) the scatter plot of $\widehat{X}_{p+1}$ and $X_{p+1}$ with $95\%$ conditional prediction intervals given each large value of $\widehat{X}_{p+1}.$ (Right) The approximate conditional density $f_{X_{p+1}|\widehat{X}_{p+1}}(X_{p+1}|\widehat{X}_{p+1} = 27.07)$. The horizontal segments indicate the $95\%$ conditional prediction interval, and the star denotes the actual value of 40.95. The units of the horizontal axis are the predicted values and the units of the conditional density are omitted.
• Figure 3: (Left) Scatterplot of $\widehat{X}_{5}$ and $X_{5}$ with the 95% prediction intervals on the Pareto scale. (Middle) Scatterplot and 95% prediction intervals after transformation to the original scale of the NO$_2$ data. (Right) Comparison of the point predictions and 95% prediction intervals using the transformed linear approach (solid line) and a Gaussian-based approach (dashed line) for five recent dates when Alexandria is not observed.
• Figure 4: (Left) Estimated TPDM estimates between the target station (marked with a star) and all remaining stations. (Right) Estimated weights based on the estimated TPDM.

Theorems & Definitions (1)

• Theorem 1