Principal Component Analysis for Extremes and Application to U.S. Precipitation

We propose a method for analyzing extremal behavior through the lens of a most efficient basis of vectors. The method is analogous to principal component analysis, but is based on methods from extreme value analysis. Specifically, rather than decomposing a covariance or correlation matrix, we obtain our basis vectors by performing an eigendecomposition of a matrix that describes pairwise extremal dependence. We apply the method to precipitation observations over the contiguous United States. We find that the time series of large coefficients associated with the leading eigenvector shows very strong evidence of a positive trend, and there is evidence that large coefficients of other eigenvectors have relationships with El Niño–Southern Oscillation.

Regional frequency analysis of extreme storm surges using the extremogram approach

To withstand coastal flooding, protection of coastal facilities and structures must be designed with the most accurate estimate of extreme storm surge return levels (SSRLs). However, because of the paucity of data, local statistical analyses often lead to poor frequency estimations. The regional frequency analysis (RFA) reduces the uncertainties associated with these estimations by extending the dataset from local (only available data at the target site) to regional (data at all the neighboring sites including the target site) and by assuming, at the scale of a region, a similar extremal behavior. In this work, the empirical spatial extremogram (ESE) approach is used. This is a graph representing all the coefficients of extremal dependence between a given target site and all the other sites in the whole region. It allows quantifying the pairwise closeness between sites based on the extremal dependence. The ESE approach, which should help with have more confidence in the physical homogeneity of the region of interest, is applied on a database of extreme skew storm surges (SSSs) and used to perform a RFA.

Regional frequency analysis of extreme storm surges using the extremogram approach

To resist marine submersion, coastal protection must be designed by taking into account the most accurate estimate of the return levels of extreme events, such as storm surges. However, because of the paucity of data, local statistical analyses often lead to poor frequency estimations. Regional Frequency Analysis (RFA) reduces the uncertainties associated with these estimations, by extending the dataset from local (only available data at the target site) to regional (data at all the neighboring sites including the target site) and by assuming, at the scale of a region, a similar extremal behavior. RFA, based on the index flood method, assumes that, in a homogeneous region, observations at sites, normalized by a local index, follow the same probability distribution. In this work, the spatial extremogram approach is used to form a physically homogeneous region centered on the target site. The approach is applied on a database of extreme skew storm surges and used to carry out a RFA.

Extremal behavior in sectional matrices

In this paper, we recall the object sectional matrix which encodes the Hilbert functions of successive hyperplane sections of a homogeneous ideal. We translate and/or reprove recent results in this language. Moreover, some new results are shown about their maximal growth, in particular, a new generalization of Gotzmann’s Persistence Theorem, the presence of a GCD for a truncation of the ideal, and applications to saturated ideals.

Risk Analysis of Cumulative Intraday Return Curves

Motivated by the risk inherent in intraday investing, we propose several ways of quantifying extremal behavior of a time series of curves. A curve can be extreme if it has shape and/or magnitude much different than the bulk of observed curves. Our approach is at the nexus of functional data analysis and extreme value theory. The risk measures we propose allow us to assess probabilities of observing extreme curves not seen in a historical record. These measures complement risk measures based on point-to-point returns, but have different interpretation and information content. Using our approach, we study how the financial crisis of 2008 impacted the extreme behavior of intraday cumulative return curves. We discover different impacts on shares in important sectors of the US economy. The information our analysis provides is in some cases different from the conclusions based on the extreme value analysis of daily closing price returns.

EXTREMAL BEHAVIOR OF LONG-TERM INVESTORS WITH POWER UTILITY

We consider a Bayesian financial market with one bond and one stock where the aim is to maximize the expected power utility from terminal wealth. The solution of this problem is known, however there are some conjectures in the literature about the long-term behavior of the optimal strategy. In this paper, we prove that for positive coefficient in the power utility the long-term investor is very optimistic and behaves as if the best drift has been realized. In case the coefficient in the power utility is negative the long-term investor is very pessimistic and behaves as if the worst drift has been realized.

Homogenous regions based on extremogram for regional frequency analysis of extreme skew storm surges

To resist marine submersion, coastal protection must be designed by taking into account the most accurate estimate of the return levels of extreme events, such as storm surges. However, because of the paucity of data, local statistical analyses often lead to poor frequency estimations. Regional Frequency Analysis (RFA) reduces the uncertainties associated with these estimations, by extending the dataset from local (only available data at the target site) to regional (data at all the neighboring sites including the target site) and by assuming, at the scale of a region, a similar extremal behavior. RFA, based on the index flood method, assumes that, in a homogeneous region, observations at sites, normalized by a local index, follow the same probability distribution. In this work, the spatial extremogram approach is used to form a physically homogeneous region centered on the target site. The approach is applied on a database of extreme skew storm surges and used to carry out a RFA.