Search

The innovations algorithm is a classical recursive forecasting algorithm used in time series analysis. We develop the innovations algorithm for a class of nonnegative regularly varying time series models constructed via transformed-linear arithmetic. In addition to providing the best linear predictor, the algorithm also enables us to estimate parameters of transformed-linear regularly-varying moving average (MA) models, thus providing a tool for modeling. We first 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, from which we develop the transformed-linear innovations algorithm. Turning our attention to the class of transformed linear MA($\infty$) models, we give results on parameter estimation and also show that this class of models is dense in the class of possible tail pairwise dependence functions (TPDFs). We also develop an extremes analogue of the classical Wold decomposition. Simulation study shows that our class of models captures tail dependence for the GARCH(1,1) model and a Markov time series model, both of which are outside our class of models., Comment: 17 pages, 3 figures

We propose a model to flexibly estimate joint tail properties by exploiting the convergence of an appropriately scaled point cloud onto a compact limit set. Characteristics of the shape of the limit set correspond to key tail dependence properties. We directly model the shape of the limit set using Bezier splines, which allow flexible and parsimonious specification of shapes in two dimensions. We then fit the Bezier splines to data in pseudo-polar coordinates using Markov chain Monte Carlo sampling, utilizing a limiting approximation to the conditional likelihood of the radii given angles. By imposing appropriate constraints on the parameters of the Bezier splines, we guarantee that each posterior sample is a valid limit set boundary, allowing direct posterior analysis of any quantity derived from the shape of the curve. Furthermore, we obtain interpretable inference on the asymptotic dependence class by using mixture priors with point masses on the corner of the unit box. Finally, we apply our model to bivariate datasets of extremes of variables related to fire risk and air pollution.

In this paper, we present a computer vision-based approach to measure the sizes and growth rates of apple fruitlets. Measuring the growth rates of apple fruitlets is important because it allows apple growers to determine when to apply chemical thinners to their crops in order to optimize yield. The current practice of obtaining growth rates involves using calipers to record sizes of fruitlets across multiple days. Due to the number of fruitlets needed to be sized, this method is laborious, time-consuming, and prone to human error. With images collected by a hand-held stereo camera, our system, segments, clusters, and fits ellipses to fruitlets to measure their diameters. The growth rates are then calculated by temporally associating clustered fruitlets across days. We provide quantitative results on data collected in an apple orchard, and demonstrate that our system is able to predict abscise rates within 3.5% of the current method with a 6 times improvement in speed, while requiring significantly less manual effort. Moreover, we provide results on images captured by a robotic system in the field, and discuss the next steps required to make the process fully autonomous.

In order to understand structural relationships among sets of variables at extreme levels, we develop an extremes analogue to partial correlation. We begin by developing an inner product space constructed from transformed-linear combinations of independent regularly varying random variables. We define partial tail correlation via the projection theorem for the inner product space. We show that the partial tail correlation can be understood as the inner product of the prediction errors from transformed-linear prediction. We connect partial tail correlation to the inverse of the inner product matrix and show that a zero in this inverse implies a partial tail correlation of zero. We then show that under a modeling assumption that the random variables belong to a sensible subset of the inner product space, the matrix of inner products corresponds to the previously-studied tail pairwise dependence matrix. We develop a hypothesis test for partial tail correlation of zero. We demonstrate the performance in two applications: high nitrogen dioxide levels in Washington DC and extreme river discharges in the upper Danube basin.

We consider the problem of performing prediction when observed values are at their highest levels. We construct an inner product space of nonnegative random variables from transformed-linear combinations of independent regularly varying random variables. The matrix of inner products corresponds to the tail pairwise dependence matrix, which summarizes tail dependence. The projection theorem yields the optimal transformed-linear predictor, which has the same form as the best linear unbiased predictor in non-extreme prediction. We also construct prediction intervals based on the geometry of regular variation. We show that these intervals have good coverage in a simulation study as well as in two applications; prediction of high pollution levels, and prediction of large financial losses.

Hazard event sets, a collection of synthetic extreme events over a given period, are important for catastrophe modelling. This paper addresses the issue of generating event sets of extreme river flow for northern England and southern Scotland, a region which has been particularly affected by severe flooding over the past 20 years. We start by analysing historical extreme river flow across 45 gauges, located within the study region, using methods from extreme value analysis, including the concept of extremal principal components. Our analysis reveals interesting connections between the extremal dependence structure and the region's topography/climate. We then introduce a framework which is based on modelling the distribution of the extremal principal components in order to generate synthetic events of extreme river flow. The generative framework is dimension-reducing in that it distinctly handles the principal components based on their contribution to describing the nature of extreme river flow across the study region. We also detail a data-driven approach to select the optimal dimension. Synthetic flood events are subsequently generated efficiently by sampling from the fitted distribution. Our approach for generating hazard event sets can be easily implemented by practitioners and our results indicate good agreement between the observed and simulated extreme river flow dynamics. For the considered application, we also find that our approach outperforms existing statistical approaches for generating hazard event sets., Comment: This revised version provides new material and results which help to use the presented methodology in applications

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.

Understanding dependence structure among extreme values plays an important role in risk assessment in environmental studies. In this work we propose the $\chi$ network and the annual extremal network for exploring the extremal dependence structure of environmental processes. A $\chi$ network is constructed by connecting pairs whose estimated upper tail dependence coefficient, $\hat \chi$, exceeds a prescribed threshold. We develop an initial $\chi$ network estimator and we use a spatial block bootstrap to assess both the bias and variance of our estimator. We then develop a method to correct the bias of the initial estimator by incorporating the spatial structure in $\chi$. In addition to the $\chi$ network, which assesses spatial extremal dependence over an extended period of time, we further introduce an annual extremal network to explore the year-to-year temporal variation of extremal connections. We illustrate the $\chi$ and the annual extremal networks by analyzing the hurricane season maximum precipitation at the US Gulf Coast and surrounding area. Analysis suggests there exists long distance extremal dependence for precipitation extremes in the study region and the strength of the extremal dependence may depend on some regional scale meteorological conditions, for example, sea surface temperature., Comment: 26 pages, 7 figures, 1 table

Uncertainty in return level estimates for rare events, like the intensity of large rainfall events, makes it difficult to develop strategies to mitigate related hazards, like flooding. Latent spatial extremes models reduce uncertainty by exploiting spatial dependence in statistical characteristics of extreme events to borrow strength across locations. However, these estimates can have poor properties due to model misspecification: many latent spatial extremes models do not account for extremal dependence, which is spatial dependence in the extreme events themselves. We improve estimates from latent spatial extremes models that make conditional independence assumptions by proposing a weighted likelihood that uses the extremal coefficient to incorporate information about extremal dependence during estimation. This approach differs from, and is simpler than, directly modeling the spatial extremal dependence; for example, by fitting a max-stable process, which is challenging to fit to real, large datasets. We adopt a hierarchical Bayesian framework for inference, use simulation to show the weighted model provides improved estimates of high quantiles, and apply our model to improve return level estimates for Colorado rainfall events with 1% annual exceedance probability., Comment: 31 pages, 3 figures, 3 tables

We present a method for drawing isolines indicating regions of equal joint exceedance probability for bivariate data. The method relies on bivariate regular variation, a dependence framework widely used for extremes. This framework enables drawing isolines corresponding to very low exceedance probabilities and these lines may lie beyond the range of the data. The method we utilize for characterizing dependence in the tail is largely nonparametric. Furthermore, we extend this method to the case of asymptotic independence and propose a procedure which smooths the transition from asymptotic independence in the interior to the first-order behavior on the axes. We propose a diagnostic plot for assessing isoline estimate and choice of smoothing, and a bootstrap procedure to visually assess uncertainty.

Employing the framework of regular variation, we propose two decompositions which help to summarize and describel high-dimensional tail dependence. Via transformation, we define a vector space on the positive orthant, yielding the notion of basis. With a suitably-chosen transformation, we show that transformed-linear operations applied to regularly varying random vectors preserve regular variation. Rather than model regular-variation's angular measure, we summarize tail dependence via a matrix of pairwise tail dependence metrics. This matrix is positive semidefinite, and eigendecomposition allows one to interpret tail dependence via the resulting eigenbasis. Additionally this matrix is completely positive, and a resulting decomposition allows one to easily construct regularly varying random vectors which share the same pairwise tail dependencies. We illustrate our methods with Swiss rainfall data and financial return data.

The Brown-Resnick max-stable process has proven to be well-suited for modeling extremes of complex environmental processes, but in many applications its likelihood function is intractable and inference must be based on a composite likelihood, thereby preventing the use of classical Bayesian techniques. In this paper we exploit a case in which the full likelihood of a Brown-Resnick process can be calculated, using componentwise maxima and their partitions in terms of individual events, and we propose two new approaches to inference. The first estimates the partitions using declustering, while the second uses random partitions in a Markov chain Monte Carlo algorithm. We use these approaches to construct a Bayesian hierarchical model for extreme low temperatures in northern Fennoscandia.

This project aims to explore which combinations of meteorological conditions are associated with extreme ground level ozone conditions. Our approach focuses only on the tail by optimizing the tail dependence between the ozone response and functions of meteorological covariates. Since there is a long list of possible meteorological covariates, the space of possible models cannot be explored completely. Consequently, we perform data mining within the model selection context, employing an automated model search procedure. Our study is unique among extremes applications as optimizing tail dependence has not previously been attempted, and it presents new challenges, such as requiring a smooth threshold. We present a simulation study which shows that the method can detect complicated conditions leading to extreme responses and resists overfitting. We apply the method to ozone data for Atlanta and Charlotte and find similar meteorological drivers for these two Southeastern US cities. We identify several covariates which help to differentiate the meteorological conditions which lead to extreme ozone levels from those which lead to merely high levels.