Uniform fractional part: a simple fast method for generating continuous random variates

A known theorem in probability is adopted and through a probabilistic approach, it is generalized to develop a method for generating random deviates from the distribution of any continuous random variable. This method, which may be considered as an approximate version of the Inverse Transform algorithm, takes two random numbers to generate a random deviate, while maintaining all the other advantages of the Inverse Transform method, such as the possibility of generating ordered as well as correlated deviates and being applicable to all density functions, regardless of their parameter value

From phenomenological modelling of anomalous diffusion through continuous-time random walks and fractional calculus to correlation analysis of complex systems

This document contains more than one topic, but they are all connected in ei- ther physical analogy, analytic/numerical resemblance or because one is a building block of another. The topics are anomalous diffusion, modelling of stylised facts based on an empirical random walker diffusion model and null-hypothesis tests in time series data-analysis reusing the same diffusion model. Inbetween these topics are interrupted by an introduction of new methods for fast production of random numbers and matrices of certain types. This interruption constitutes the entire chapter on random numbers that is purely algorithmic and was inspired by the need of fast random numbers of special types. The sequence of chapters is chrono- logically meaningful in the sense that fast random numbers are needed in the first topic dealing with continuous-time random walks (CTRWs) and their connection to fractional diffusion. The contents of the last four chapters were indeed produced in this sequence, but with some temporal overlap. While the fast Monte Carlo solution of the time and space fractional diffusion equation is a nice application that sped-up hugely with our new method we were also interested in CTRWs as a model for certain stylised facts. Without knowing economists [80] reinvented what physicists had subconsciously used for decades already. It is the so called stylised fact for which another word can be empirical truth. A simple example: The diffusion equation gives a probability at a certain time to find a certain diffusive particle in some position or indicates concentration of a dye. It is debatable if probability is physical reality. Most importantly, it does not describe the physical system completely. Instead, the equation describes only a certain expectation value of interest, where it does not matter if it is of grains, prices or people which diffuse away. Reality is coded and “averaged” in the diffusion constant. Interpreting a CTRW as an abstract microscopic particle motion model it can solve the time and space fractional diffusion equation. This type of diffusion equation mimics some types of anomalous diffusion, a name usually given to effects that cannot be explained by classic stochastic models. In particular not by the classic diffusion equation. It was recognised only recently, ca. in the mid 1990s, that the random walk model used here is the abstract particle based counterpart for the macroscopic time- and space-fractional diffusion equation, just like the “classic” random walk with regular jumps ±∆x solves the classic diffusion equation. Both equations can be solved in a Monte Carlo fashion with many realisations of walks. Interpreting the CTRW as a time series model it can serve as a possible null- hypothesis scenario in applications with measurements that behave similarly. It may be necessary to simulate many null-hypothesis realisations of the system to give a (probabilistic) answer to what the “outcome” is under the assumption that the particles, stocks, etc. are not correlated. Another topic is (random) correlation matrices. These are partly built on the previously introduced continuous-time random walks and are important in null- hypothesis testing, data analysis and filtering. The main ob jects encountered in dealing with these matrices are eigenvalues and eigenvectors. The latter are car- ried over to the following topic of mode analysis and application in clustering. The presented properties of correlation matrices of correlated measurements seem to be wasted in contemporary methods of clustering with (dis-)similarity measures from time series. Most applications of spectral clustering ignores information and is not able to distinguish between certain cases. The suggested procedure is sup- posed to identify and separate out clusters by using additional information coded in the eigenvectors. In addition, random matrix theory can also serve to analyse microarray data for the extraction of functional genetic groups and it also suggests an error model. Finally, the last topic on synchronisation analysis of electroen- cephalogram (EEG) data resurrects the eigenvalues and eigenvectors as well as the mode analysis, but this time of matrices made of synchronisation coefficients of neurological activity

Methods for generating variates from probability distributions

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.Diverse probabilistic results are used in the design of random univariate generators. General methods based on these are classified and relevant theoretical properties derived. This is followed by a comparative review of specific algorithms currently available for continuous and discrete univariate distributions. A need for a Zeta generator is established, and two new methods, based on inversion and rejection with a truncated Pareto envelope respectively are developed and compared. The paucity of algorithms for multivariate generation motivates a classification of general methods, and in particular, a new method involving envelope rejection with a novel target distribution is proposed. A new method for generating first passage times in a Wiener Process is constructed. This is based on the ratio of two random numbers, and its performance is compared to an existing method for generating inverse Gaussian variates. New "hybrid" algorithms for Poisson and Negative Binomial distributions are constructed, using an Alias implementation, together with a Geometric tail procedure. These are shown to be robust, exact and fast for a wide range of parameter values. Significant modifications are made to Atkinson's Poisson generator (PA), and the resulting algorithm shown to be complementary to the hybrid method. A new method for Von Mises generation via a comparison of random numbers follows, and its performance compared to that of Best and Fisher's Wrapped Cauchy rejection method. Finally new methods are proposed for sampling from distribution tails, using optimally designed Exponential envelopes. Timings are given for Gamma and Normal tails, and in the latter case the performance is shown to be significantly better than Marsaglia's tail generation procedure.Governors of Dundee College of Technolog

Lorenz-based quantitative risk management

In this thesis, we address problems of quantitative risk management using a specific set of tools that go under the name of Lorenz curve and inequality indices, developed to describe the socio-economic variability of a random variable.Quantitative risk management deals with the estimation of the uncertainty that isembedded in the activities of banks and other financial players due, for example, tomarket fluctuations. Since the well-being of such financial players is fundamental for the correct functioning of the economic system, an accurate description and estimation of such uncertainty is crucial.Applied ProbabilityNumerical Analysi

Dependence in Stochastic Simulation Models

There is a growing need for the ability to model and generate samples of dependent random variables as primitive inputs to stochastic models. We consider the case where this dependence is modeled in terms of a partially-specified finite-dimensional random vector. A random vector sampler is commonly required to match a given set of distributions for each of its components (the marginal distributions) and values of their pairwise covariances. The NORTA method, which produces samples via a transformation of a joint-normal random vector sample, is considered the state-of-the-art method for matching this specification. We begin by showing that the NORTA method has certain flaws in its design which limit its applicability. A covariance matrix is said to be feasible for a given set of marginal distributions if a random vector exists with these properties. We develop a computational tool that can establish the feasibility of (almost) any covariance matrix for a fixed set of marginals. This tool is used to rigorously establish that there are feasible combinations of marginals and covariance matrices that the NORTA method cannot match. We further determine that as the dimension of the random vector increases, this problem rapidly becomes acute, in the sense that NORTA becomes increasingly likely to fail to match feasible specifications. As part of this analysis, we propose a random matrix sampling technique that is possibly of wider interest. We extend our study along two natural paths. First, we investigate whether NORTA can be modified to approximately match a desired covariance matrix that the original NORTA procedure fails to match. Results show that simple, elegant modifications to the NORTA procedure can help it achieve close approximations to the desired covariance matrix, and these modifications perform well with increasing dimension. Second, the feasibility testing procedure suggests a random vector sampling technique that can exactly match (almost) any given feasible set of marginals and covariances, i.e., be free of the limitations of NORTA. We develop a strong characterization of the computational effort needed by this new sampling technique. This technique is computationally competitive with NORTA in low to moderate dimensions, while matching the desired covariances exactly

Recent advances on the reduction and analysis of big and high-dimensional data

In an era with remarkable advancements in computer engineering, computational algorithms, and mathematical modeling, data scientists are inevitably faced with the challenge of working with big and high-dimensional data. For many problems, data reduction is a necessary first step; such reduction allows for storage and portability of big data, and enables the computation of expensive downstream quantities. The next step then involves the analysis of big data -- the use of such data for modeling, inference, and prediction. This thesis presents new methods for big data reduction and analysis, with a focus on solving real-world problems in statistics, machine learning and engineering.Ph.D

Advances on Uncertainty Quantification Techniques for Dynamical Systems: Theory and Modelling

[ES] La cuantificación de la incertidumbre está compuesta por una serie de métodos y técnicas computacionales cuyo objetivo principal es describir la aleatoriedad presente en problemas de diversa índole. Estos métodos son de utilidad en la modelización de procesos biológicos, físicos, naturales o sociales, ya que en ellos aparecen ciertos aspectos que no pueden ser determinados de manera exacta. Por ejemplo, la tasa de contagio de una enfermedad epidemiológica o el factor de crecimiento de un volumen tumoral dependen de factores genéticos, ambientales o conductuales. Estos no siempre pueden definirse en su totalidad y por tanto conllevan una aleatoriedad intrínseca que afecta en el desarrollo final. El objetivo principal de esta tesis es extender técnicas para cuantificar la incertidumbre en dos áreas de las matemáticas: el cálculo de ecuaciones diferenciales fraccionarias y la modelización matemática. Las derivadas de orden fraccionario permiten modelizar comportamientos que las derivadas clásicas no pueden, como por ejemplo los efectos de memoria o la viscoelasticidad en algunos materiales. En esta tesis, desde un punto de vista teórico, se extenderá el cálculo fraccionario a un ambiente de incertidumbre, concretamente en el sentido de la media cuadrática. Se presentarán problemas de valores iniciales fraccionarios aleatorios. El cálculo de la solución, la obtención de las aproximaciones de la media y varianza de la solución y la aproximación de la primera función de densidad de probabilidad de la solución son conceptos que se abordarán en los próximos capítulos. Sin embargo, no siempre es sencillo obtener la solución exacta de un problema de valores iniciales fraccionario aleatorio. Por ello en esta tesis también se dedicará un capítulo para describir un procedimiento numérico que aproxime su solución. Por otro lado, desde un punto de vista más aplicado, se desarrollan técnicas computacionales para cuantificar la incertidumbre en modelos matemáticos. Combinando estas técnicas junto con modelos matemáticos apropiados, se estudiarán problemas de dinámica biológica. En primer lugar, se determinará la cantidad de portadores de meningococo en España con un modelo de competencia de Lotka-Volterra fraccionario aleatorio. A continuación, el volumen de un tumor mamario se modelizará mediante un modelo logístico con incertidumbre. Finalmente ayudándonos de un modelo matemático que describe el nivel de glucosa en sangre de un paciente diabético, se pretende dar una recomendación de carbohidratos e insulina que se debe de ingerir para que el nivel de glucosa del paciente esté dentro de una banda de confianza saludable. Es importante subrayar que para poder realizar estos estudios se requieren datos reales, los cuales pueden estar alterados debido a los errores de medición o proceso que se han cometido para obtenerlos. Por este motivo, modelizar correctamente el problema junto con la incertidumbre en los datos es de vital importancia.[CA] La quantificació de la incertesa està composada per una sèrie de mètodes i tècniques computacionals, l'objectiu principal de les quals és descriure l'aleatorietat present en problemes de diversa índole. Aquests mètodes són d'utilitat en la modelització de processos biològics, físics, naturals o socials, ja que en ells apareixen certs aspectes que no poden ser determinats de manera exacta. Per exemple, la taxa de contagi d'una malaltia epidemiològica o el factor de creixement d'un volum tumoral depenen de factors genètics, ambientals o conductuals. Aquests no sempre poden definir-se íntegrament i per tant, comporten una aleatorietat intrínseca que afecta en el desenvolupament final. L'objectiu principal d'aquesta tesi doctoral és estendre tècniques per a quantificar la incertesa en dues àrees de les matemàtiques: el càlcul d'equacions diferencials fraccionàries i la modelització matemàtica. Les derivades d'ordre fraccionari permeten modelitzar comportaments que les derivades clàssiques no poden, com per exemple, els efectes de memòria o la viscoelasticitat en alguns materials. En aquesta tesi, des d'un punt de vista teòric, s'estendrà el càlcul fraccionari a un ambient d'incertesa, concretament en el sentit de la mitjana quadràtica. Es presentaran problemes de valors inicials fraccionaris aleatoris. El càlcul de la solució, l'obtenció de les aproximacions de la mitjana i, la variància de la solució i l'aproximació de la primera funció de densitat de probabilitat de la solució són conceptes que s'abordaran en els pròxims capítols. No obstant això, no sempre és senzill obtindre la solució exacta d'un problema de valors inicials fraccionari aleatori. Per això en aquesta tesi també es dedicarà un capítol per a descriure un procediment numèric que aproxime la seua solució. D'altra banda, des d'un punt de vista més aplicat, es desenvolupen tècniques computacionals per a quantificar la incertesa en models matemàtics. Combinant aquestes tècniques juntament amb models matemàtics apropiats, s'estudiaran problemes de dinàmica biològica. En primer lloc, es determinarà la quantitat de portadors de meningococ a Espanya amb un model de competència de Lotka-Volterra fraccionari aleatori. A continuació, el volum d'un tumor mamari es modelitzará mitjançant un model logístic amb incertesa. Finalment ajudant-nos d'un model matemàtic que descriu el nivell de glucosa en sang d'un pacient diabètic, es pretén donar una recomanació de carbohidrats i insulina que s'ha d'ingerir perquè el nivell de glucosa del pacient estiga dins d'una banda de confiança saludable. És important subratllar que per a poder realitzar aquests estudis es requereixen dades reals, els quals poden estar alterats a causa dels errors de mesurament o per la forma en que s'han obtés. Per aquest motiu, modelitzar correctament el problema juntament amb la incertesa en les dades és de vital importància.[EN] Uncertainty quantification collects different methods and computational techniques aimed at describing the randomness in real phenomena. These methods are useful in the modelling of different processes as biological, physical, natural or social, since they present some aspects that can not be determined exactly. For example, the contagious rate of a epidemiological disease or the growth factor of a tumour volume depend on genetic, environmental or behavioural factors. They may not always be fully described and therefore involve uncertainties that affects on the final result. The main objective of this PhD thesis is to extend techniques to quantify the uncertainty in two mathematical areas: fractional calculus and mathematical modelling. Fractional derivatives allow us to model some behaviours that classical derivatives cannot, such as memory effects or the viscoelasticity of some materials. In this PhD thesis, from a theoretical point of view, fractional calculus is extended into the random framework, concretely in the mean square sense. Initial value problems will be studied. The calculus of the analytic solution, approximations for the mean and for the variance and the computation of the first probability density function are concepts we deal with them thought the following chapters. Nevertheless, it is not always possible to obtain the analytic solution of an initial value problem. Therefore, in this dissertation a chapter is addressed to describe a numerical procedure to approximate the solution for an initial value problem. On the other hand, from a modelling point of view, computational techniques to quantify the uncertainty in mathematical models are developed. Merging these techniques with appropriate mathematical models, problems of biological dynamics are studied. Firstly, the carriers of meningococcus in Spain are determined using a competition Lotka-Volterra random fractional model. Then, the volume of breast tumours is modelled by a random logistic model. Finally, taking advantage of a mathematical model which describes the glucose level of a diabetic patient, a recommendation of insulin shots and carbohydrate intakes is proposed to a patient in order to maintain her/his glucose level in a healthy confidence range. An important observation is that to carry out these studies real data is required and they may include uncertainties contained in the measurements on the process to perform the corresponding study. This it is the reason why it is crucial to properly model the problem taking also into account the randomness of the data.Burgos Simón, C. (2021). Advances on Uncertainty Quantification Techniques for Dynamical Systems: Theory and Modelling [Tesis doctoral]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/166442TESI