Transformed-linear models for time series extremes

2022 Summer.Includes bibliographical references.In order to capture the dependence in the upper tail of a time series, we develop nonnegative 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. Motivated by investigating conditions conducive to the spread of wildfires, we fit models to hourly windspeed data using a preliminary estimation method 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. The innovations algorithm is a classical recursive algorithm used in time series analysis. We develop an analogous transformed-linear innovations algorithm for our time series models that allows us to perform prediction which is fundamental to any time series analysis. The transformed-linear innovations algorithm also enables us to estimate parameters of the transformed-linear regularly-varying moving average models, thus providing a tool for modeling. We construct an inner product space of transformed-linear combinations of nonnegative regularly-varying random variables and prove its link to a Hilbert space which allows us to employ the projection theorem. We develop the transformed-linear innovations algorithm using the properties of the projection theorem. Turning our attention to the class of MA(∞) models, we talk about estimation and also show that this class of models is dense in the class of possible TPDFs. We also develop an extremes analogue of the classical Wold decomposition. Simulation study shows that our class of models provides adequate models for the GARCH and another model outside our class of models. The transformed-linear innovations algorithm gives us the best prediction and we also develop prediction intervals based on the geometry of regular variation. Simulation study shows that we obtain good coverage rates for prediction errors. We perform modeling and prediction for the hourly windspeed data by applying the innovations algorithm to the estimated TPDF