Linear prediction and partial tail correlation for extremes

2022 Summer.Includes bibliographical references.This dissertation consists of three main studies for extreme value analyses: linear prediction for extremes, uncertainty quantification for predictions, and investigating conditional relationships between variables at their extreme levels. We employ multivariate regular variation to provide a framework for modeling dependence in the upper tail, which is assumed to be a direction of interest. Cooley and Thibaud [2019] consider transformed-linear operations to define a vector space on the nonnegative orthant and show regular variation is preserved by these transformed-linear operations. Extending the approach of Cooley and Thibaud [2019], we first consider the problem of performing prediction when observed values are at extreme levels. This linear approach is motivated by the limitation that traditional extreme value models have difficulties fitting a high dimensional extreme value model. We construct an inner product space of nonnegative random variables from transformed-linear combinations of independent regularly varying random variables. Rather than fully characterizing extremal dependence in high dimensions, we summarize tail behavior via a matrix of pairwise tail dependencies. The projection theorem yields the optimal transformed-linear predictor, which has a similar form to the best linear unbiased predictor in non-extreme prediction. We then quantify uncertainty for the prediction of extremes by using information contained in the tail pairwise dependence matrix. We create the 95% prediction interval based on the geometry of regular variation. We show that the prediction intervals have good coverage in a simulation study as well as in two applications: prediction of high NO2 air pollution levels, and prediction of large financial losses. We also compare prediction intervals with a linear approach to ones with a parametric approach. Lastly, we develop the novel notion of partial tail correlation via projection theorem in the inner product space. Partial tail correlations are the analogue of partial correlations in non-extreme statistics but focus on extremal dependence. Partial tail correlation can be represented by the inner product of prediction errors associated with the previously defined best transformed-linear prediction for extremes. We find a connection between the partial tail correlation and the inverse matrix of tail pairwise dependencies. We then develop a hypothesis test for zero elements in the inverse extremal matrix. We apply the idea of partial tail correlation to assess flood risk in application to extreme river discharges in the upper Danube River basin. We compare the extremal graph constructed from the idea of the partial tail correlation to physical flow connections on the Danube

Transformed-linear prediction for extremes

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

Partial Tail Correlation for Extremes

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

Formation of Artificial Lipid Membrane and their Photolysis in Mineral Water including Germanium

We have attempted to determine the Germanium ion (Ge 4+ ) effect on the human body by observing the formation of artificial lipid membrane and photolysis in the mineral water containing Ge 4+ ion. The artificial lipid membrane is prepared by using the phospholipid in the Germanium water and the formation efficiency of the liposomes is compared with those obtained in the plain mineral water without Ge 4+ ion. This work shows that the liposomes are formed in the Germanium water better than in the non-Germanium water. The liposomes can be photolyzed by superoxide anion (O 2 -. ) produced in the presence of some peptide such as NAT (N-acethyl-L-tryptophan). However, this is inhibited by superoxide dismutase (SOD), and it was found that the activity of SOD on the inhibition of the liposomes oxidative damage is higher in the Germanium water than in the non-Germanium water. It is concluded that the Ge 4+ ion in mineral water helps the formation of new cell as well as elevation of SOD activity for the lipid oxidation