Geometrical Methods in Multivariate Risk Management: Algorithms and Applications

This thesis builds a framework for evaluating downside risk from multivariate data via a special class of risk measures (RM). The peculiarity of the analysis lies in getting rid of strong data distributional assumptions and in orientation towards the most critical data in risk management: those with asymmetries and heavy tails. At the same time, under typical assumptions, such as the ellipticity of the data probability distribution, the conformity with classical methods is shown. The constructed class of RM is a multivariate generalization of the coherent distortion RM, which possess valuable properties for a risk manager. The design of the framework is twofold. The first part contains new computational geometry methods for the high-dimensional data. The developed algorithms demonstrate computability of geometrical concepts used for constructing the RM. These concepts bring visuality and simplify interpretation of the RM. The second part develops models for applying the framework to actual problems. The spectrum of applications varies from robust portfolio selection up to broader spheres, such as stochastic conic optimization with risk constraints or supervised machine learning.

Combined effects of soil biotic and abiotic factors, influenced by sewage sludge incorporation, on the incidence of corn stalk rot.

Abstract: The objectives of this study were to evaluate the combined effects of soil bioticand abiotic factors on the incidence of Fusarium corn stalk rot, during four annual incorporations of two typesofsewagesludge intosoil ina 5-years field assay under tropical conditions and topredict the effectsof these variables on the disease. For each type of sewage sludge, the following treatments were included: control with mineral fertilization recommended for corn; control without fertilization; sewage sludge based on the nitrogen concentration that provided the same amount of nitrogen as in the mineral fertilizer treatment; and sewage sludge that provided two, four and eight times the nitrogen concentration recommended for corn. Increasing dosages of both types of sewage sludge incorporated into soil resulted in increased corn stalk rot incidence, being negatively correlated with corn yield. A global analysis highlighted the effect of the year of the experiment, followed by the sewage sludge dosages. The type of sewage sludge did not affect the disease incidence. Amultiple logistic model using a stepwise procedure was fitted based on the selection of a model that included the three explanatory parameters for disease incidence: electrical conductivity, magnesium and Fusarium population. In the selected model, the probability of higher disease incidence increased with an increase of these three explanatory parameters. When the explanatory parameters were compared, electrical conductivity presented a dominant effect and was the main variable to predict the probability distribution curves of Fusarium corn stalk rot, after sewage sludge application into the soil.

A Unified Model for XVA, including Interest Rates and Rating

We start in Chapter 2 to investigate linear matrix-valued SDEs and the Itô-stochastic Magnus expansion. The Itô-stochastic Magnus expansion provides an efficient numerical scheme to solve matrix-valued SDEs. We show convergence of the expansion up to a stopping time τ and provide an asymptotic estimate of the cumulative distribution function of τ. Moreover, we show how to apply it to solve SPDEs with one and two spatial dimensions by combining it with the method of lines with high accuracy. We will see that the Magnus expansion allows us to use GPU techniques leading to major performance improvements compared to a standard Euler-Maruyama scheme. In Chapter 3, we study a short-rate model in a Cox-Ingersoll-Ross (CIR) framework for negative interest rates. We define the short rate as the difference of two independent CIR processes and add a deterministic shift to guarantee a perfect fit to the market term structure. We show how to use the Gram-Charlier expansion to efficiently calibrate the model to the market swaption surface and price Bermudan swaptions with good accuracy. We are taking two different perspectives for rating transition modelling. In Section 4.4, we study inhomogeneous continuous-time Markov chains (ICTMC) as a candidate for a rating model with deterministic rating transitions. We extend this model by taking a Lie group perspective in Section 4.5, to allow for stochastic rating transitions. In both cases, we will compare the most popular choices for a change of measure technique and show how to efficiently calibrate both models to the available historical rating data and market default probabilities. At the very end, we apply the techniques shown in this thesis to minimize the collateral-inclusive Credit/ Debit Valuation Adjustments under the constraint of small collateral postings by using a collateral account dependent on rating trigger.

Toward an understanding of the atmospheric forcing uncertainties in the Mediterranean Sea

The thesis has extensively investigated for the first time the statistical distributions of atmospheric surface variables and heat fluxes for the Mediterranean Sea. After retrieving a 30-year atmospheric analysis dataset, we have captured the spatial patterns of the probability distribution of the relevant atmospheric variables for ocean atmospheric forcing: wind components (U,V), wind amplitude, air temperature (T2M), dewpoint temperature (D2M) and mean sea-level pressure (MSL-P). The study reveals that a two-parameter PDF is not a good fit for T2M, D2M, MSL-P and wind components (U,V) and a three parameter skew-normal PDF is better suited. Such distribution captures properly the data asymmetric tails (skewness). After removing the large seasonal cycle, we show the quality of the fit and the geographic structure of the PDF parameters. It is found that the PDF parameters vary between different regions, in particular the shape (connected to the asymmetric tails) and the scale (connected to the spread of the distribution) parameters cluster around two or more values, probably connected to the different dynamics that produces the surface atmospheric fields in the Mediterranean basin. Moreover, using the atmospheric variables, we have computed the air-sea heat fluxes for a 20-years period and estimated the net heat budget over the Mediterranean Sea. Interestingly, the higher resolution analysis dataset provides a negative heat budget of –3 W/m2 which is within the acceptable range for the Mediterranean Sea heat budget closure. The lower resolution atmospheric reanalysis dataset(ERA5) does not satisfy the heat budget closure problem pointing out that a minimal resolution of the atmospheric forcing is crucial for the Mediterranean Sea dynamics. The PDF framework developed in this thesis will be the basis for a future ensemble forecasting system that will use the statistical distributions to create perturbations of the atmospheric ocean forcing.

Modelli di diffusion MRI avanzati per lo studio microstrutturale del cervello

La quantificazione non invasiva delle caratteristiche microstrutturali del cervello, utilizzando la diffusion MRI (dMRI), è diventato un campo sempre più interessante e complesso negli ultimi due decenni. Attualmente la dMRI è l’unica tecnica che permette di sondare le proprietà diffusive dell’acqua, in vivo, grazie alla quale è possibile inferire informazioni su scala mesoscopica, scala in cui si manifestano le prime alterazioni di malattie neurodegenerative, da tale tipo di dettaglio è potenzialmente possibile sviluppare dei biomarcatori specifici per le fasi iniziali di malattie neurodegenerative. L’evoluzione hardware degli scanner clinici, hanno permesso lo sviluppo di modelli di dMRI avanzati basati su acquisizioni multi shell, i quali permettono di ovviare alle limitazioni della Diffusion Tensor Imaging, in particolare tali modelli permettono una migliore ricostruzione trattografica dei fasci di sostanza bianca, grazie ad un’accurata stima della Orientation Distribution Function e la stima quantitativa di parametri che hanno permesso di raggiungere una miglior comprensione della microstruttura della sostanza bianca e delle sue eventuali deviazioni dalla norma. L’identificazione di biomarcatori sensibili alle prime alterazioni microstrutturali delle malattie neurodegenerative è uno degli obbiettivi principali di tali modelli, in quanto consentirebbero una diagnosi precoce e di conseguenza un trattamento terapeutico tempestivo prima di una significante perdità cellulare. La trattazione è suddivisa in una prima parte di descrizione delle nozioni fisiche di base della dMRI, dell’imaging del tensore di diffusione e le relative limitazioni, ed in una seconda parte dove sono analizzati tre modelli avanzati di dMRI: Diffusion Kurtosis Imaging, Neurite Orientation Dispersion and Density Imaging e Multi Shell Multi Tissue Constrained Spherical Deconvolution. L'obiettivo della trattazione è quello di offrire una panoramica sulle potenzialità di tali modelli.

Characterization of Probability Distributions Using the Residual Entropy Function

The present study on the characterization of probability distributions using the residual entropy function. The concept of entropy is extensively used in literature as a quantitative measure of uncertainty associated with a random phenomenon. The commonly used life time models in reliability Theory are exponential distribution, Pareto distribution, Beta distribution, Weibull distribution and gamma distribution. Several characterization theorems are obtained for the above models using reliability concepts such as failure rate, mean residual life function, vitality function, variance residual life function etc. Most of the works on characterization of distributions in the reliability context centers around the failure rate or the residual life function. The important aspect of interest in the study of entropy is that of locating distributions for which the shannon’s entropy is maximum subject to certain restrictions on the underlying random variable. The geometric vitality function and examine its properties. It is established that the geometric vitality function determines the distribution uniquely. The problem of averaging the residual entropy function is examined, and also the truncated form version of entropies of higher order are defined. In this study it is established that the residual entropy function determines the distribution uniquely and that the constancy of the same is characteristics to the geometric distribution

The generalized inverse Weibull distribution

The inverse Weibull distribution has the ability to model failure rates which are quite common in reliability and biological studies. A three-parameter generalized inverse Weibull distribution with decreasing and unimodal failure rate is introduced and studied. We provide a comprehensive treatment of the mathematical properties of the new distribution including expressions for the moment generating function and the rth generalized moment. The mixture model of two generalized inverse Weibull distributions is investigated. The identifiability property of the mixture model is demonstrated. For the first time, we propose a location-scale regression model based on the log-generalized inverse Weibull distribution for modeling lifetime data. In addition, we develop some diagnostic tools for sensitivity analysis. Two applications of real data are given to illustrate the potentiality of the proposed regression model.

A generalized modified Weibull distribution for lifetime modeling

A four parameter generalization of the Weibull distribution capable of modeling a bathtub-shaped hazard rate function is defined and studied. The beauty and importance of this distribution lies in its ability to model monotone as well as non-monotone failure rates, which are quite common in lifetime problems and reliability. The new distribution has a number of well-known lifetime special sub-models, such as the Weibull, extreme value, exponentiated Weibull, generalized Rayleigh and modified Weibull distributions, among others. We derive two infinite sum representations for its moments. The density of the order statistics is obtained. The method of maximum likelihood is used for estimating the model parameters. Also, the observed information matrix is obtained. Two applications are presented to illustrate the proposed distribution. (C) 2008 Elsevier B.V. All rights reserved.

General results for the beta-modified Weibull distribution

We study in detail the so-called beta-modified Weibull distribution, motivated by the wide use of the Weibull distribution in practice, and also for the fact that the generalization provides a continuous crossover towards cases with different shapes. The new distribution is important since it contains as special sub-models some widely-known distributions, such as the generalized modified Weibull, beta Weibull, exponentiated Weibull, beta exponential, modified Weibull and Weibull distributions, among several others. It also provides more flexibility to analyse complex real data. Various mathematical properties of this distribution are derived, including its moments and moment generating function. We examine the asymptotic distributions of the extreme values. Explicit expressions are also derived for the chf, mean deviations, Bonferroni and Lorenz curves, reliability and entropies. The estimation of parameters is approached by two methods: moments and maximum likelihood. We compare by simulation the performances of the estimates from these methods. We obtain the expected information matrix. Two applications are presented to illustrate the proposed distribution.

The exponentiated generalized gamma distribution with application to lifetime data

A four-parameter extension of the generalized gamma distribution capable of modelling a bathtub-shaped hazard rate function is defined and studied. The beauty and importance of this distribution lies in its ability to model monotone and non-monotone failure rate functions, which are quite common in lifetime data analysis and reliability. The new distribution has a number of well-known lifetime special sub-models, such as the exponentiated Weibull, exponentiated generalized half-normal, exponentiated gamma and generalized Rayleigh, among others. We derive two infinite sum representations for its moments. We calculate the density of the order statistics and two expansions for their moments. The method of maximum likelihood is used for estimating the model parameters and the observed information matrix is obtained. Finally, a real data set from the medical area is analysed.

A Log-Linear Regression Model for the Beta-Weibull Distribution

We introduce the log-beta Weibull regression model based on the beta Weibull distribution (Famoye et al., 2005; Lee et al., 2007). We derive expansions for the moment generating function which do not depend on complicated functions. The new regression model represents a parametric family of models that includes as sub-models several widely known regression models that can be applied to censored survival data. We employ a frequentist analysis, a jackknife estimator, and a parametric bootstrap for the parameters of the proposed model. We derive the appropriate matrices for assessing local influences on the parameter estimates under different perturbation schemes and present some ways to assess global influences. Further, for different parameter settings, sample sizes, and censoring percentages, several simulations are performed. In addition, the empirical distribution of some modified residuals are displayed and compared with the standard normal distribution. These studies suggest that the residual analysis usually performed in normal linear regression models can be extended to a modified deviance residual in the proposed regression model applied to censored data. We define martingale and deviance residuals to evaluate the model assumptions. The extended regression model is very useful for the analysis of real data and could give more realistic fits than other special regression models.

A modified Weibull distribution

A new lifetime distribution capable of modeling a bathtub-shaped hazard-rate function is proposed. The proposed model is derived as a limiting case of the Beta Integrated Model and has both the Weibull distribution and Type I extreme value distribution as special cases. The model can be considered as another useful 3-parameter generalization of the Weibull distribution. An advantage of the model is that the model parameters can be estimated easily based on a Weibull probability paper (WPP) plot that serves as a tool for model identification. Model characterization based on the WPP plot is studied. A numerical example is provided and comparison with another Weibull extension, the exponentiated Weibull, is also discussed. The proposed model compares well with other competing models to fit data that exhibits a bathtub-shaped hazard-rate function.

Modified Discrete PSO to Increase the Delivered Energy Probability in Distribution energy Systems

This paper proposes a PSO based approach to increase the probability of delivering power to any load point by identifying new investments in distribution energy systems. The statistical failure and repair data of distribution components is the main basis of the proposed methodology that uses a fuzzyprobabilistic modeling for the components outage parameters. The fuzzy membership functions of the outage parameters of each component are based on statistical records. A Modified Discrete PSO optimization model is developed in order to identify the adequate investments in distribution energy system components which allow increasing the probability of delivering power to any customer in the distribution system at the minimum possible cost for the system operator. To illustrate the application of the proposed methodology, the paper includes a case study that considers a 180 bus distribution network.

DC Fuzzy Multicriteria Approach to Increase the Probability of Delivering Power in Distribution Networks

This paper proposes a methodology to increase the probability of delivering power to any load point through the identification of new investments. The methodology uses a fuzzy set approach to model the uncertainty of outage parameters, load and generation. A DC fuzzy multicriteria optimization model considering the Pareto front and based on mixed integer non-linear optimization programming is developed in order to identify the adequate investments in distribution networks components which allow increasing the probability of delivering power to all customers in the distribution network at the minimum possible cost for the system operator, while minimizing the non supplied energy cost. To illustrate the application of the proposed methodology, the paper includes a case study which considers an 33 bus distribution network.

Increase of the Delivered Power Probability in Distribution Networks using Pareto DC Programming

A methodology to increase the probability of delivering power to any load point through the identification of new investments in distribution network components is proposed in this paper. The method minimizes the investment cost as well as the cost of energy not supplied in the network. A DC optimization model based on mixed integer non-linear programming is developed considering the Pareto front technique in order to identify the adequate investments in distribution networks components which allow increasing the probability of delivering power for any customer in the distribution system at the minimum possible cost for the system operator, while minimizing the energy not supplied cost. Thus, a multi-objective problem is formulated. To illustrate the application of the proposed methodology, the paper includes a case study which considers a 180 bus distribution network