Desde las contribuciones de Isaac Newton, Gottfried Wilhelm Leibniz, Jacob y Johann Bernoulli en el siglo XVII hasta ahora, las ecuaciones en diferencias y las diferenciales han demostrado su capacidad para modelar satisfactoriamente problemas complejos de gran interés en Ingeniería, Física, Epidemiología, etc. Pero, desde un punto de vista práctico, los parámetros o inputs (condiciones iniciales/frontera, término fuente y/o coeficientes), que aparecen en dichos problemas, son fijados a partir de ciertos datos, los cuales pueden contener un error de medida. Además, pueden existir factores externos que afecten al sistema objeto de estudio, de modo que su complejidad haga que no se conozcan de forma cierta los parámetros de la ecuación que modeliza el problema. Todo ello justifica considerar los parámetros de la ecuación en diferencias o de la ecuación diferencial como variables aleatorias o procesos estocásticos, y no como constantes o funciones deterministas, respectivamente. Bajo esta consideración aparecen las ecuaciones en diferencias y las ecuaciones diferenciales aleatorias. Esta tesis hace un recorrido resolviendo, desde un punto de vista probabilístico, distintos tipos de ecuaciones en diferencias y diferenciales aleatorias, aplicando fundamentalmente el método de Transformación de Variables Aleatorias. Esta técnica es una herramienta útil para la obtención de la función de densidad de probabilidad de un vector aleatorio, que es una transformación de otro vector aleatorio cuya función de densidad de probabilidad es conocida. En definitiva, el objetivo de este trabajo es el cálculo de la primera función de densidad de probabilidad del proceso estocástico solución en diversos problemas basados en ecuaciones en diferencias y diferenciales aleatorias. El interés por determinar la primera función de densidad de probabilidad se justifica porque dicha función determinista caracteriza la información probabilística unidimensional, como media, varianza, asimetría, curtosis, etc., de la solución de la ecuación en diferencias o diferencial correspondiente. También permite determinar la probabilidad de que acontezca un determinado suceso de interés que involucre a la solución. Además, en algunos casos, el estudio teórico realizado se completa mostrando su aplicación a problemas de modelización con datos reales, donde se aborda el problema de la estimación de distribuciones estadísticas paramétricas de los inputs en el contexto de las ecuaciones en diferencias y diferenciales aleatorias. / Ever since the early contributions by Isaac Newton, Gottfried Wilhelm Leibniz, Jacob and Johann Bernoulli in the XVII century until now, difference and differential equations have uninterruptedly demonstrated their capability to model successfully interesting complex problems in Engineering, Physics, Chemistry, Epidemiology, Economics, etc. But, from a practical standpoint, the application of difference or differential equations requires setting their inputs (coefficients, source term, initial and boundary conditions) using sampled data, thus containing uncertainty stemming from measurement errors. In addition, there are some random external factors which can affect to the system under study. Then, it is more advisable to consider input data as random variables or stochastic processes rather than deterministic constants or functions, respectively. Under this consideration random difference and differential equations appear. This thesis makes a trail by solving, from a probabilistic point of view, different types of random difference and differential equations, applying fundamentally the Random Variable Transformation method. This technique is an useful tool to obtain the probability density function of a random vector that results from mapping another random vector whose probability density function is known. Definitely, the goal of this dissertation is the computation of the first probability density function of the solution stochastic process in different problems, which are based on random difference or differential equations. The interest in determining the first probability density function is justified because this deterministic function characterizes the one-dimensional probabilistic information, as mean, variance, asymmetry, kurtosis, etc. of corresponding solution of a random difference or differential equation. It also allows to determine the probability of a certain event of interest that involves the solution. In addition, in some cases, the theoretical study carried out is completed, showing its application to modelling problems with real data, where the problem of parametric statistics distribution estimation is addressed in the context of random difference and differential equations. / Des de les contribucions de Isaac Newton, Gottfried Wilhelm Leibniz, Jacob i Johann Bernoulli al segle XVII fins a l'actualitat, les equacions en diferències i les diferencials han demostrat la seua capacitat per a modelar satisfactòriament problemes complexos de gran interés en Enginyeria, Física, Epidemiologia, etc. Però, des d'un punt de vista pràctic, els paràmetres o inputs (condicions inicials/frontera, terme font i/o coeficients), que apareixen en aquests problemes, són fixats a partir de certes dades, les quals poden contenir errors de mesura. A més, poden existir factors externs que afecten el sistema objecte d'estudi, de manera que, la seua complexitat faça que no es conega de forma certa els inputs de l'equació que modelitza el problema. Tot aço justifica la necessitat de considerar els paràmetres de l'equació en diferències o de la equació diferencial com a variables aleatòries o processos estocàstics, i no com constants o funcions deterministes. Sota aquesta consideració apareixen les equacions en diferències i les equacions diferencials aleatòries. Aquesta tesi fa un recorregut resolent, des d'un punt de vista probabilístic, diferents tipus d'equacions en diferències i diferencials aleatòries, aplicant fonamentalment el mètode de Transformació de Variables Aleatòries. Aquesta tècnica és una eina útil per a l'obtenció de la funció de densitat de probabilitat d'un vector aleatori, que és una transformació d'un altre vector aleatori i la funció de densitat de probabilitat és del qual és coneguda. En definitiva, l'objectiu d'aquesta tesi és el càlcul de la primera funció de densitat de probabilitat del procés estocàstic solució en diversos problemes basats en equacions en diferències i diferencials. L'interés per determinar la primera funció de densitat es justifica perquè aquesta funció determinista caracteritza la informació probabilística unidimensional, com la mitjana, variància, asimetria, curtosis, etc., de la solució de l'equació en diferències o l'equació diferencial aleatòria corresponent. També permet determinar la probabilitat que esdevinga un determinat succés d'interés que involucre la solució. A més, en alguns casos, l'estudi teòric realitzat es completa mostrant la seua aplicació a problemes de modelització amb dades reals, on s'aborda el problema de l'estimació de distribucions estadístiques paramètriques dels inputs en el context de les equacions en diferències i diferencials aleatòries. / Navarro Quiles, A. (2018). COMPUTATIONAL METHODS FOR RANDOM DIFFERENTIAL EQUATIONS: THEORY AND APPLICATIONS [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/98703 / TESIS

We introduce numerical methods for the analysis of random dynamical systems. The subdivision and the continuation algorithm are powerful tools which will be demonstrated for a system from ship dynamics. With our software package we are able to show that the well known safe basin is a moving fractal set. We will also give a numerical approximation of the attracting invariant set (which contains a local attractor) and its evolution.

The art of reconstructing a real-world scene digitally has been on the mind of researchers for decades. Recently, it has attracted more and more attention from companies seeing a chance to bring this kind of technology to the market. Digital reconstruction of buildings in particular is a niche that has both potential and room for improvement. With this background, this thesis will present the design and evaluation of a pipeline made to find and correct approximately flat surfaces in architectural scenes. The scenes are 3D-reconstructed triangle meshes based on RGB images. The thesis will also comprise an evaluation of a few different components available for doing this, leading to a choice of best components. The goal is to improve the visual quality of the reconstruction. The final pipeline is designed with two blocks - one to detect initial plane seeds and one to refine the detected planes. The first block makes use of a multi-label energy formulation on the graph that describes the reconstructed surface. Penalties are assigned to each vertex and each edge of the graph based on the vertex labels, effectively describing a Markov Random Field. The energy is minimized with the help of the alpha-expansion algorithm. The second block uses heuristics for growing the detected plane seeds, merging similar planes together and extracting deviating details. Results on several scenes are presented, showing that the visual quality has been improved while maintaining accuracy compared with ground truth data. / Konsten att digitalt rekonstruera en verklig miljö har länge varit intressant för forskare. Nyligen har området även tilldragit sig mer och mer uppmärksamhet från företag som ser en möjlighet att föra den här typen av teknik till produkter på marknaden. I synnerhet är digital rekonstruktion av byggnader en nisch som har både stor potential och möjlighet till förbättring. Med denna bakgrund så presenterar detta examensarbete designen för och utvärderingen av en pipeline som skapats för att detektera och rätta till approximativt platta regioner i arkitektoniska miljöer. Miljöerna är 3D-rekonstruerade triangelmeshar skapade från RGB-bilder. Examensarbetet omfattar även utvärdering av olika komponenter för att uppnå detta, som avslutas med att de mest lämpliga komponenterna presenteras. Målet i korthet är att förbättra den visuella kvaliteten av en rekonstruerad modell. Den slutgiltiga pipelinen består av två övergripande block - ett för att detektera initiala plan och ett för att förbättra de funna planen. Det första blocket använder en multi-label energiformulering på grafen som beskriver den rekonstruerade ytan. Straffvärden tilldelas varje vertex och varje båge i grafen baserade på varje vertex label. På så sätt beskriver grafen ett Markov Random Field. Energin är sedan minimerad med alpha-expansion-algoritmen. Det andra blocket använder heuristiker för att låta planen växa, slå ihop närliggande plan och för att extrahera avvikande detaljer. Resultat på flera miljöer presenteras också för att påvisa att den visuella kvaliteten har förbättrats utan att rekonstruktionens noggrannhet har försämrats jämfört med ground truth-data.

Random access schemes are primarily used for data transmission in the uplink of cellular networks. Every user in a random access network is programmed to follow a predetermined transmit control policy that is designed to achieve optimal network performance. This approach, however, is not very efficient in cellular networks where channel conditions vary from time to time. Employing a fixed transmission policy may not guarantee optimal performance. To alleviate this problem, recently, channel aware random access schemes have been proposed wherein information available at the physical (PHY) layer is utilized at the higher layers to maximize network throughput. Such a cross-layer approach naturally has its share of challenges and problems. The objective of the proposed research is to study the effect of multicarrier diversity on channel aware random access schemes. First, we describe two generic random access schemes - channel aware multicarrier random access (CAMCRA) and no selection random access (NS-RA) for multicarrier networks. The former is based on judicious carrier selection and exploits multicarrier diversity while the latter does not perform carrier selection. For illustration purposes, we consider the well-known Aloha protocol and study the effect of channel state imperfection on the overall network throughput. We show that networks employing the NS-RA scheme are extremely sensitive to channel measurement errors. More precisely, the asymptotic average throughput of the NS-RA scheme under uncertain channel conditions is zero. The CAMCRA scheme, however, is very robust to channel estimation errors and maintains the same order of throughput.

Complex networks is a vibrant research field and has received much attention over the last decade. Central to this area is the question of how networks around us are constructed. The essential notion of network research is that these systems are assembled in a decentralised way, thus no central agent is planning the network beforehand. Despite this lack of central coordination, many networks present intriguing universalities, such as broad degree distributions, in the form of power-laws. The subject of study in this thesis is a class of networks that are constructed by a node intrinsic variable, called fitness. The way these networks grow could be called a rich-get-richer mechanism. The fitter a node is, the more likely it is to acquire new connections inside the network. Several aspects that are directly connected to these networks are explored in this thesis. In the first part, the properties of growing networks that are driven by fitness are investigated and it is shown that the introduction of growth leads to a topological structure that is different from its static counterpart. In the subsequent chapter, percolation on fitness driven networks is studied. The results give insights into possible mechanisms that can stabilise systems. Furthermore, the theory can be used to identify vulnerable structures around us. In the following chapter, the world trade network is discussed. This numerical investigation highlights possible improvements to the methodology to make statistical analysis more robust. That chapter is followed by an analysis of time-varying networks. Time-varying networks represent an interesting construct that allows a formulation of stochastic processes on the same time-scale as the evolution of the network itself. This possibility is highly relevant to the investigation of epidemics, for instance. In the last chapter, a study of a system of clusters and their self-organised formation is presented.

No description available.

published_or_final_version / abstract / Statistics and Actuarial Science / Doctoral / Doctor of Philosophy

New tools are available for reconstructing the attractor basins of discrete dynamical networks where state-space is linked according the network's dynamics. In this thesis the computer software "Discrete Dynamics Lab" is applied to examine simple networks ranging from cellular automata (CA) to random Boolean networks (RBN), that have been widely applied as idealised models of physical and biological systems, to search for general principles underlying their dynamics. The algorithms and methods for generating pre-images for both CA and RBN, and reconstructing and representing attractor basins are described, and also considered in the mathematical context of random directed graphs. RBN and CA provide contrasting notions of self-organisation. RBN provide models of hierarchical categorisation in biology, for example memory in neural and genomic networks. CA provide models at the lower level of emergent complex pattern. New measures and results are presented on CA attractor basins and how they relate to measures on local dynamics and the Z parameter, characterising ordered to "complex" to chaotic behaviour. A method is described for classifying CA rules by an entropy-variance measure which allows glider rules and related complex rules to be found automatically giving a virtually unlimited sample for further study. The dynamics of RBN and intermediate network architectures are examined in the context of memory, where categorisation occurs at the roots of subtrees as well as at attractors. Learning algorithms are proposed for "sculpting" the basin of attraction field. RBN are proposed as a possible neural network model, and also discussed as a model of genomic regulatory networks, where cell types have been explained as attractors

Jitter is defined as timing uncertainties of digital signals at their intended ideal positions in time. While it undermines valuable clock budget and limits the maximum clock frequency in I/O circuitry, it is one of the most difficult parameters to measure accurately due to the small value and randomness. This thesis proposes a random jitter RMS measurement method using AND and OR operations, which targets BIST applications. This thesis is organized as follows. Chapter 1 introduces the motivation of the proposed work. It includes a comparison between two major approaches to jitter measurement. Chapter 2 explains the proposed random jitter estimation method in detail. Chapter 3 describes circuit implementations with design considerations. Chapter 4 demonstrates estimation results from circuit level simulation runs. Chapter 5 discusses the source of error in the jitter estimation and concludes. / text

摘要 本文主要在探討絕對正則隨機變數序列的隨機修剪L統計量的漸近性，當修剪係系數收斂至a和b時(O<a<b<l)，對它們的分配函數限制並不多；然而當a=0及b=1 時，則限制的條件須更加嚴格，這也就是為什麼我們要做隨機修剪的主要原因。同時，由於大部分的時間序列模式都是絕對正則的隨機變數序列，這也是研究本文的主要動機之一。 本文是想嘗試著把G. R. Shorack (1989)的論文隨機修剪L統計量，推廣，把該文中立相獨立的隨機變數序列換成絕對正則的隨機變數序列。在這同時，我們必需將一些經驗累積分配函數的不等式推廣，推廣過程中將重覆使用Yoshihara (1978) 的機率不等式。 / ABSTRACT We will prove central limit theorem for randomly trimmed L-statistics with absolutely regular random variables. When the fractions trimmed converge to a and l-b, (with 0<a<b<l) there are little restrictions on the df's of the r.v.'s, - but the limiting r.v. has several contributing terms, making the studentization complicated unless the trimming fractions converge fast enough. For a=0 and b=l, the restriction on the rate of convergence of the trimming fractions is more severe, however this is a most reasonable way to trim.