Model misspecification in peaks over threshold analysis

Classical peaks over threshold analysis is widely used for statistical modeling of sample extremes, and can be supplemented by a model for the sizes of clusters of exceedances. Under mild conditions a compound Poisson process model allows the estimation of the marginal distribution of threshold exceedances and of the mean cluster size, but requires the choice of a threshold and of a run parameter, $K$, that determines how exceedances are declustered. We extend a class of estimators of the reciprocal mean cluster size, known as the extremal index, establish consistency and asymptotic normality, and use the compound Poisson process to derive misspecification tests of model validity and of the choice of run parameter and threshold. Simulated examples and real data on temperatures and rainfall illustrate the ideas, both for estimating the extremal index in nonstandard situations and for assessing the validity of extremal models.Comment: Published in at http://dx.doi.org/10.1214/09-AOAS292 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

Scaling and Multiscaling Exponents in Networks and Flows

The main focus of this paper is on mathematical theory and methods which have a direct bearing on problems involving multiscale phenomena. Modern technology is refining measurement and data collection to spatio-temporal scales on which observed geophysical phenomena are displayed as intrinsically highly variable and intermittant heirarchical structures,e.g. rainfall, turbulence, etc. The heirarchical structure is reflected in the occurence of a natural separation of scales which collectively manifest at some basic unit scale. Thus proper data analysis and inference require a mathematical framework which couples the variability over multiple decades of scale in which basic theoretical benchmarks can be identified and calculated. This continues the main theme of the research in this area of applied probability over the past twenty years

Tail asymptotics for Shepp-statistics of Brownian motion in ℝ d

Let $X(t)$, $t\in R$, be a $d$-dimensional vector-valued Brownian motion, $d\ge 1$. For all $b\in R^d\setminus (-\infty,0]^d$ we derive exact asymptotics of P(X(t+s)-X(t) >u b\mbox{ for some }t\in[0,T],\ s\in[0,1]}\mbox{as }u\to\infty, that is the asymptotical behavior of tail distribution of vector-valued analog of Shepp-statistics for $X$; we cover not only the case of a fixed time-horizon $T>0$ but also cases where $T\to 0$ or $T\to\infty$. Results for high excursion probabilities of vector-valued processes are rare in the literature, with currently no available approach suitable for our problem. Our proof exploits some distributional properties of vector-valued Brownian motion, and results from quadratic programming problems. As a by-product we derive a new inequality for the `supremum' of vector-valued Brownian motions

Tracing diffusion in porous media with fractal properties

This work is concerned with conditional averaging methods which can be used for modeling of transport in porous media with volume reactions in the fluid phase and surface reactions at the fluid/solid interface. The model under consideration takes into account convection, diffusion within the pores and on larger scales, and homogeneous and heterogeneous reactions. Near the interface with fractal properties, the fluid flow is slow, and diffusion, as a transport mechanism, dominates over convection. Following the conditional moment closure paradigm, we employ a diffusion tracer as a reference scalar field that makes the conditional averaging sensitive to the proximity of a point to the interface. The resulting conditionally averaged reactive transport equations are governed by the probability density function (PDF) of the diffusion tracer, and this makes the study of its behavior an important problem. We consider a hitting time stochastic interpretation of the diffusion tracer, establish integral equations relating it to a subsidiary distance tracer, and obtain distance-diffusion inequalities. Assuming that the fluid/solid interface and pores themselves possess fractal properties which are quantified, in particular, by a variant of the Minkowski-Bouligand fractal dimension, we investigate the interplay between the interface and network scenarios of fractality in the scaling laws of the diffusion tracer PDF. We also discuss and employ several hypotheses, including a lognormal cascade hypothesis on the behavior of the diffusion tracer at different length scales

Data-based analysis of extreme events: inference, numerics and applications

The concept of extreme events describes the above average behavior of a process, for instance, heat waves in climate or weather research, earthquakes in geology and financial crashes in economics. It is significant to study the behavior of extremes, in order to reduce their negative impacts. Key objectives include the identification of the appropriate mathematical/statistical model, description of the underlying dependence structure in the multivariate or the spatial case, and the investigation of the most relevant external factors. Extreme value analysis (EVA), based on Extreme Value Theory, provides the necessary statistical tools. Assuming that all relevant covariates are known and observed, EVA often deploys statistical regression analysis to study the changes in the model parameters. Modeling of the dependence structure implies a priori assumptions such as Gaussian, locally stationary or isotropic behavior. Based on EVA and advanced time-series analysis methodology, this thesis introduces a semiparametric, nonstationary and non- homogenous framework for statistical regression analysis of spatio-temporal extremes. The involved regression analysis accounts explicitly for systematically missing covariates; their influence was reduced to an additive nonstationary offset. The nonstationarity was resolved by the Finite Element Time Series Analysis Methodology (FEM). FEM approximates the underlying nonstationarity by a set of locally stationary models and a nonstationary hidden switching process with bounded variation (BV). The resulting FEM-BV- EVA approach goes beyond a priori assumptions of standard methods based, for instance, on Bayesian statistics, Hidden Markov Models or Local Kernel Smoothing. The multivariate/spatial extension of FEM-BV-EVA describes the underlying spatial variability by the model parameters, referring to hierarchical modeling. The spatio-temporal behavior of the model parameters was approximated by locally stationary models and a spatial nonstationary switching process. Further, it was shown that the resulting spatial FEM-BV-EVA formulation is consistent with the max-stability postulate and describes the underlying dependence structure in a nonparametric way. The proposed FEM-BV-EVA methodology was integrated into the existent FEM MATLAB toolbox. The FEM-BV-EVA framework is computationally efficient as it deploys gradient free MCMC based optimization methods and numerical solvers for constrained, large, structured quadratic and linear problems. In order to demonstrate its performance, FEM-BV-EVA was applied to various test-cases and real-data and compared to standard methods. It was shown that parametric approaches lead to biased results if significant covariates are unresolved. Comparison to nonparametric methods based on smoothing regression revealed their weakness, the locality property and the inability to resolve discontinuous functions. Spatial FEM-BV-EVA was applied to study the dynamics of extreme precipitation over Switzerland. The analysis identified among others three major spatially dependent regions

Ruin problems of a two-dimensional fractional Brownian motion risk process

This paper investigates ruin probability and ruin time of a two-dimensional fractional Brownian motion risk process. The net loss process of an insurance company is modeled by a fractional Brownian motion. The two-dimensional fractional Brownian motion risk process models the surplus processes of an insurance and a reinsurance company, where the net loss is divided between them in some specified proportions. The ruin problem considered is that of the two-dimensional risk process first entering the negative quadrant, that is, the simultaneous ruin problem. We derive both asymptotics of the ruin probability and approximations of the scaled conditional ruin time as the initial capital tends to infinity

Max-stable processes for threshold exceedances in spatial extremes

The analysis of spatial extremes requires the joint modeling of a spatial process at a large number of stations. Multivariate extreme value theory can be used to model the joint extremal behavior of environmental data such as precipitation, snow depths or daily temperatures. Max-stable processes are the natural generalization of extremal dependence structures to infinite dimensions arising from the extension of multivariate extreme value theory. However, there have been few works on the threshold approach of max-stable processes. Padoan, Ribatet and Sisson proposed the maximum composite likelihood approach for fitting max-stable processes to avoid the complexity and unavailability of the multivariate density function. We propose the threshold version of max-stable process estimation and we apply the pairwise composite likelihood method to it. We assume a strict form of condition, so called the second-order regular variation condition, for the distribution satisfying the domain of attraction. To obtain the limit behavior, we also consider the increasing domain structure with stochastic sampling design based on the setting and conditions in Lahiri and we then establish consistency and asymptotic normality of the estimator for dependence parameter in the threshold method of max-stable processes. The method is studied by simulation and illustrated by the application of temperature data in North Carolina, United States

Systematic development of coarse-grained polymer models

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Chemical Engineering, 2006.Includes bibliographical references (p. [159]-163).The coupling between polymer models and experiments has improved our understanding of polymer behavior both in terms of rheology and dynamics of single molecules. Developing these polymer models is challenging because of the wide range of time and length scales. Mechanical models of polymers have been used to understand average heological properties as well as the deviation a single polymer molecule has from the average response. This leads to more physically significant constitutive relations, which can be coupled with fluid mechanic simulations to predict and understand the theological response of polymer solutions and melts. These models have also been used in conjunction with single molecule polymer experiments. While these have provided insight into the dynamics of polymers in rheological flows, they have also helped to design single molecule manipulation experiments. Promising research in this area includes DNA separation and stretching devices. A typical atomic bond has a length of 10-10m and vibration time scale of 10-14s. A typical experiment in a microfluidic device has lengths of order 10-5m and times of order 102s. It is not possible to capture these larger length and time scales of interest while capturing exactly the behavior at the smaller length and time scales.(cont.) This necessitates a process of coarse-graining which sacrifices the details at the small scale which are not necessary while retaining the important features that do affect the response at the larger scales. This thesis focuses on the coarse-graining of polymers into a series of beads connected by springs. The function which gives the retractive force in the spring as a function of the extension is called the spring force law. In many new microfluidic applications the previously used spring force laws produce significant errors in the model. We have systematically analyzed the coarse-graining and development of the spring force law to understand why these force laws fail. In particular, we analyzed the force-extension behavior which quantifies how much the polymer extends under application of an external force. We identified the key dimensionless group that governs the response and found that it is important to understand the different constraints under which the polymer is placed. This understanding led to the development of new spring force laws which are accurate coarse-grained models by construction. We also examined the response in other situations such as weak and strong flows.(cont.) This further illustrated the disadvantages of the previous force laws which were eliminated by using the new force laws. This thesis will have practical impact because the new spring force laws can easily be implemented in current polymer models. This will improve the accuracy of the models and place the models on firmer theoretical footing. Because the spring force law has been developed independently of other coarse-grained interactions, this thesis will also help in determining the best parameters for other interactions because they will not need to compensate for an error in the spring force law. These new spring force laws will help form the framework of coarse-grained models which can help understand a wide range of situations in which the behavior at a small scale affects the large time and length scale behavior.by Patrick Theodore Underhill.Ph.D

A PDF closure model for compressible turbulent chemically reacting flows

The objective of the proposed research project was the analysis of single point closures based on probability density function (pdf) and characteristic functions and the development of a prediction method for the joint velocity-scalar pdf in turbulent reacting flows. Turbulent flows of boundary layer type and stagnation point flows with and without chemical reactions were be calculated as principal applications. Pdf methods for compressible reacting flows were developed and tested in comparison with available experimental data. The research work carried in this project was concentrated on the closure of pdf equations for incompressible and compressible turbulent flows with and without chemical reactions