Random Numbers form the heart and soul of a discrete-event simulation system. There are few situations where the actions of the entities in the process being simulated can be completely predicted in advance. The real world processes are more probabilistic than deterministic. Hence, such chances are represented in the system by using various statistical models, like random number generators. These random number generators can be used to represent a various number of factors, such as length of the queue. However, simulations have grown in size and are sometimes required to run on multiple machines, which share the various methods or events in the simulation among themselves. These Machines can be distributed across a LAN or even the internet. In such cases, to keep the validity of the simulation model, we need rollback-able random number generators. This thesis is an effort to develop such rollback able random number generators for the Synchronous Parallel Environment for Emulation and Discrete-Event Simulation (SPEEDES) environment developed by NASA. These rollback-able random number generators will also add several statistical distribution models to the already rich SPEEDES library.

Discrete exponential families have drawn a lot of attention in probability, statistics, and machine learning, both classically and in the recent literature. This thesis studies in depth two discrete exponential families of concrete interest, (i) Exponential Random Graph Models (ERGMs) and (ii) Ising Models. In the ERGM setting, this thesis consider a “degree corrected” version of standard ERGMs, and in the Ising model setting, this thesis focus on Ising models on dense regular graphs, both from the point of view of statistical inference. The first part of the thesis studies the problem of testing for sparse signals present on the vertices of ERGMs. It proposes computably efficient tests for a wide class of ERGMs. Focusing on the two star ERGM, it shows that the tests studied are “asymptotically efficient” in all parameter regimes except one, which is referred to as “critical point”. In the critical regime, it is shown that improved detection is possible. This shows that compared to the standard belief, in this setting dependence is actually beneficial to the inference problem. The main proof idea for analyzing the two star ERGM is a correlations estimate between degrees under local alternatives, which is possibly of independent interest. In the second part of the thesis, we derive the limit of experiments for a class of one parameter Ising models on dense regular graphs. In particular, we show that the limiting experiment is Gaussian in the “low temperature” regime, non Gaussian in the “critical” regime, and an infinite collection of Gaussians in the “high temperature” regime. We also derive the limiting distributions of commonlt studied estimators, and study limiting power for tests of hypothesis against contiguous alternatives (whose scaling changes across the regimes). To the best of our knowledge, this is the first attempt at establishing the classical limits of experiments for Ising models (and more generally, Markov random fields).

In this thesis, we consider stationary random mosaics with a focus on the Poisson-Voronoi mosaic and the Poisson-Delaunay mosaic. We consider properties of stationary random mosaics in R2, such as mean value results of the typical cell. Further, we simulate various mean value results of the typical cell, a random neighbor of the typical cell, and the zero cell for the Poisson-Voronoi mosaic in R2. Some theory of point processes is introduced that is needed for random mosaics, including Palm theory, marked point processes, and the Pois point process. Finally, we consider an incremental flip-based algorithm for generating the Voronoi mosaic.

No description available.

Proliferation of computers with varying word sizes has led to increases in software use where random number generation is required. Several techniques have been developed. Criteria of randomness, portability, period, reproducibility, variety, speed, and storage are used to evaluate developed generation methods. The Tausworthe method is the only method to meet the portability requirement, and is chosen to be implemented. A C language implementation is proposed as a possible implementation and test results are presented to confirm the acceptability of the proposed code. / Master of Science

This thesis presents an algorithm incorporating pattern grammar with random indexing to solve three English synonym benchmarks. A pattern grammar model and a baseline random indexing implementation benchmarked the solution. The results show an significant improvement on the synonym benchmark compared to a baseline random indexing implementation. Most language models today focus on vector space models where the linguistic origins of the information are lost. Even though these algorithms produce good results, it is hard to know where the model learned something. With the help of patterns, we can learn more about how these models work. / Den här uppsatsen presenterar en algoritm som använder sig av mallgrammatik tillsammans med random indexing för att lösa tre synonymtest för engelska. En mallgrammatiksmodell och en referensimplementation av random indexing utvärderades. Resultaten visade en tydlig förbättring på de olika testerna jämfört med referensimplementationen. De flesta språkmodeller idag fokuserar på vektorrepresentationer av språk där det lingvistiska ursprunget hos språket försvinner. Dessa modeller är mycket framgångsrika, men det är svårt att säga något om vad och hur en modell kommit fram till en slutsats. Med hjälp av språkmönster baserade på mallgrammatik kan vi lära oss mer om hur dessa modeller fungerar.

Neste trabalho estudamos um passeio aleatório, unidimensional com ramificação em Z+ em um meio aleatório não identicamente distribuído. Definimos recorrência e transiência para este processo e apresentamos um critério de classificação. / We study a \"supercritical\" branching random walk on Z+ in a one-dimensional non i.i.d. random environment, which considers both the branching mechanism and the step transition. Criteria of (strong) recurrence and transience are presented for this model.

Neste trabalho estudamos um passeio aleatório, unidimensional com ramificação em Z+ em um meio aleatório não identicamente distribuído. Definimos recorrência e transiência para este processo e apresentamos um critério de classificação. / We study a \"supercritical\" branching random walk on Z+ in a one-dimensional non i.i.d. random environment, which considers both the branching mechanism and the step transition. Criteria of (strong) recurrence and transience are presented for this model.

Missing data is one of the challenges we are facing today in modeling valid statistical models. It reduces the representativeness of the data samples. Hence, population estimates, and model parameters estimated from such data are likely to be biased. However, the missing data problem is an area under study, and alternative better statistical procedures have been presented to mitigate its shortcomings. In this paper, we review causes of missing data, and various methods of handling missing data. Our main focus is evaluating various multiple imputation (MI) methods from the multiple imputation of chained equation (MICE) package in the statistical software R. We assess how these MI methods perform with different percentages of missing data. A multiple regression model was fit on the imputed data sets and the complete data set. Statistical comparisons of the regression coefficients are made between the models using the imputed data and the complete data.

HTML clipboard Statistical physics seeks to explain macroscopic properties of matter in terms of microscopic interactions. Of particular interest is the phenomenon of phase transition: the sudden changes in macroscopic properties as external conditions are varied. Two models in particular are of great interest to mathematicians, namely the Ising model of a magnet and the percolation model of a porous solid. These models in turn are part of the unifying framework of the random-cluster representation, a model for random graphs which was first studied by Fortuin and Kasteleyn in the 1970’s. The random-cluster representation has proved extremely useful in proving important facts about the Ising model and similar models. In this work we study the corresponding graphical framework for two related models. The first model is the transverse field quantum Ising model, an extension of the original Ising model which was introduced by Lieb, Schultz and Mattis in the 1960’s. The second model is the space–time percolation process, which is closely related to the contact model for the spread of disease. In Chapter 2 we define the appropriate space–time random-cluster model and explore a range of useful probabilistic techniques for studying it. The space– time Potts model emerges as a natural generalization of the quantum Ising model. The basic properties of the phase transitions in these models are treated in this chapter, such as the fact that there is at most one unbounded fk-cluster, and the resulting lower bound on the critical value in <img src="http://upload.wikimedia.org/math/a/b/8/ab820da891078a8245d7f4f3252aee4f.png" />. In Chapter 3 we develop an alternative graphical representation of the quantum Ising model, called the random-parity representation. This representation is based on the random-current representation of the classical Ising model, and allows us to study in much greater detail the phase transition and critical behaviour. A major aim of this chapter is to prove sharpness of the phase transition in the quantum Ising model—a central issue in the theory— and to establish bounds on some critical exponents. We address these issues by using the random-parity representation to establish certain differential inequalities, integration of which gives the results. In Chapter 4 we explore some consequences and possible extensions of the results established in Chapters 2 and 3. For example, we determine the critical point for the quantum Ising model in <img src="http://upload.wikimedia.org/math/a/b/8/ab820da891078a8245d7f4f3252aee4f.png" /> and in ‘star-like’ geometries. / HTML clipboard Statistisk fysik syftar till att förklara ett materials makroskopiska egenskaper i termer av dess mikroskopiska struktur. En särskilt intressant egenskap är är fenomenet fasövergång, det vill säga en plötslig förändring i de makroskopiska egenskaperna när externa förutsättningar varieras. Två modeller är särskilt intressanta för en matematiker, nämligen Ising-modellen av en magnet och perkolationsmodellen av ett poröst material. Dessa två modeller sammanförs av den så-kallade fk-modellen, en slumpgrafsmodell som först studerades av Fortuin och Kasteleyn på 1970-talet. fk-modellen har sedermera visat sig vara extremt användbar för att bevisa viktiga resultat om Ising-modellen och liknande modeller. I den här avhandlingen studeras den motsvarande grafiska strukturen hos två näraliggande modeller. Den första av dessa är den kvantteoretiska Isingmodellen med transverst fält, vilken är en utveckling av den klassiska Isingmodellen och först studerades av Lieb, Schultz och Mattis på 1960-talet. Den andra modellen är rumtid-perkolation, som är nära besläktad med kontaktmodellen av infektionsspridning. I Kapitel 2 definieras rumtid-fk-modellen, och flera probabilistiska verktyg utforskas för att studera dess grundläggande egenskaper. Vi möter rumtid-Potts-modellen, som uppenbarar sig som en naturlig generalisering av den kvantteoretiska Ising-modellen. De viktigaste egenskaperna hos fasövergången i dessa modeller behandlas i detta kapitel, exempelvis det faktum att det i fk-modellen finns högst en obegränsad komponent, samt den undre gräns för det kritiska värdet som detta innebär. I Kapitel 3 utvecklas en alternativ grafisk framställning av den kvantteoretiska Ising-modellen, den så-kallade slumpparitetsframställningen. Denna är baserad på slumpflödesframställningen av den klassiska Ising-modellen, och är ett verktyg som låter oss studera fasövergången och gränsbeteendet mycket närmare. Huvudsyftet med detta kapitel är att bevisa att fasövergången är skarp—en central egenskap—samt att fastslå olikheter för vissa kritiska exponenter. Metoden består i att använda slumpparitetsframställningen för att härleda vissa differentialolikheter, vilka sedan kan integreras för att lägga fast att gränsen är skarp. I Kapitel 4 utforskas några konsekvenser, samt möjliga vidareutvecklingar, av resultaten i de tidigare kapitlen. Exempelvis bestäms det kritiska värdet hos den kvantteoretiska Ising-modellen på <img src="http://upload.wikimedia.org/math/a/b/8/ab820da891078a8245d7f4f3252aee4f.png" /> , samt i ‘stjärnliknankde’ geometrier. / QC 20100705