Transformed-Linear Models for Time Series Extremes

TL;DR

This paper introduces transformed-linear models for time series extremes that effectively capture tail dependence, providing a new approach that outperforms traditional models in estimating upper tail quantities, with applications to windspeed data.

Contribution

The paper develops a novel class of non-negative, regularly-varying time series models using transformed-linear operations, and introduces the concept of weak tail stationarity and the tail pairwise dependence function (TPDF).

Findings

Models better estimate upper tail quantities than ARMA.

Fitted models successfully applied to windspeed data.

Transformed-linear MA(∞) models form an inner product space.

Abstract

In order to capture the dependence in the upper tail of a time series, we develop non-negative regularly-varying time series models that are constructed similarly to classical non-extreme ARMA models. Rather than fully characterizing tail dependence of the time series, we define the concept of weak tail stationarity which allows us to describe a regularly-varying time series through the tail pairwise dependence function (TPDF) which is a measure of pairwise extremal dependencies. We state consistency requirements among the finite-dimensional collections of the elements of a regularly-varying time series and show that the TPDF's value does not depend on the dimension being considered. So that our models take nonnegative values, we use transformed-linear operations. We show existence and stationarity of these models, and develop their properties such as the model TPDF's. Additionally, we show the class of transformed-linear MA($\infty$) models forms an inner product space. Motivated by investigating conditions conducive to the spread of wildfires, we fit models to hourly windspeed data and find that the fitted transformed-linear models produce better estimates of upper tail quantities than traditional ARMA models or than classical linear regularly varying models.