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Sensitivity Analyses for Tumor Growth ModelsMendis, Ruchini Dilinika 01 April 2019 (has links)
This study consists of the sensitivity analysis for two previously developed tumor growth models: Gompertz model and quotient model. The two models are considered in both continuous and discrete time. In continuous time, model parameters are estimated using least-square method, while in discrete time, the partial-sum method is used. Moreover, frequentist and Bayesian methods are used to construct confidence intervals and credible intervals for the model parameters. We apply the Markov Chain Monte Carlo (MCMC) techniques with the Random Walk Metropolis algorithm with Non-informative Prior and the Delayed Rejection Adoptive Metropolis (DRAM) algorithm to construct parameters' posterior distributions and then obtain credible intervals.
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Kernel Selection for Convergence and Efficiency in Markov Chain Monte CarolPotter, Christopher C. J. 24 April 2013 (has links)
Markov Chain Monte Carlo (MCMC) is a technique for sampling from a target probability distribution, and has risen in importance as faster computing hardware has made possible the exploration of hitherto difficult distributions. Unfortunately, this powerful technique is often misapplied by poor selection of transition kernel for the Markov chain that is generated by the simulation.
Some kernels are used without being checked against the convergence requirements for MCMC (total balance and ergodicity), but in this work we prove the existence of a simple proxy for total balance that is not as demanding as detailed balance, the most widely used standard. We show that, for discrete-state MCMC, that if a transition kernel is equivalent when it is “reversed” and applied to data which is also “reversed”, then it satisfies total balance. We go on to prove that the sequential single-variable update Metropolis kernel, where variables are simply updated in order, does indeed satisfy total balance for many discrete target distributions, such as the Ising model with uniform exchange constant.
Also, two well-known papers by Gelman, Roberts, and Gilks (GRG)[1, 2] have proposed the application of the results of an interesting mathematical proof to the realistic optimization of Markov Chain Monte Carlo computer simulations. In particular, they advocated tuning the simulation parameters to select an acceptance ratio of 0.234 .
In this paper, we point out that although the proof is valid, its result’s application to practical computations is not advisable, as the simulation algorithm considered in the proof is so inefficient that it produces very poor results under all circumstances. The algorithm used by Gelman, Roberts, and Gilks is also shown to introduce subtle time-dependent correlations into the simulation of intrinsically independent variables. These correlations are of particular interest since they will be present in all simulations that use multi-dimensional MCMC moves.
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Estudo do resfriamento em um sistema com múltiplos estados fundamentais / A study of cooling in a system with several ground states.Henrique Santos Guidi 29 October 2007 (has links)
Estudamos um sistema de dois níveis acoplados como um modelo que imita o comportamento de líquidos super-resfriados. Em equilíbrio o modelo apresenta uma fase líquida e uma fase cristalina com diversos estados fundamentais. O modelo é definido numa rede quadrada e a cada sítio é associada uma variável estocástica de Ising. A característica que torna este modelo particularmente interessante é que ele apresenta estados metaestáveis duráveis que podem desaparecer dentro do tempo acessível para as simulações numéricas. Para imitar o processo de formação dos vidros, realizamos simulações de Monte Carlo a taxas de resfriamento constante. Apresentamos também simulações para resfriamentos súbitos a temperatura abaixo da temperatura de fusão. / We study a coupled two level systems as a model that imitate the behavior of supercooled liquids that become structural glasses under cooling. In the equilibrium the model shows a liquid phase and a crystalline phase with many grouond states. The model is defined on a square lattice and to each site a stochastic Ising variable is associated. The feature that makes this model particularly interesting is that it display durable metastables states which can vanish within the time available for numerical simulations. In order to imitate the glass former process, we perform Monte Carlo simulations at constant cooling rate. We present also simulations for quenchs to temperatures below the melting temperature.
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Estudo do resfriamento em um sistema com múltiplos estados fundamentais / A study of cooling in a system with several ground states.Guidi, Henrique Santos 29 October 2007 (has links)
Estudamos um sistema de dois níveis acoplados como um modelo que imita o comportamento de líquidos super-resfriados. Em equilíbrio o modelo apresenta uma fase líquida e uma fase cristalina com diversos estados fundamentais. O modelo é definido numa rede quadrada e a cada sítio é associada uma variável estocástica de Ising. A característica que torna este modelo particularmente interessante é que ele apresenta estados metaestáveis duráveis que podem desaparecer dentro do tempo acessível para as simulações numéricas. Para imitar o processo de formação dos vidros, realizamos simulações de Monte Carlo a taxas de resfriamento constante. Apresentamos também simulações para resfriamentos súbitos a temperatura abaixo da temperatura de fusão. / We study a coupled two level systems as a model that imitate the behavior of supercooled liquids that become structural glasses under cooling. In the equilibrium the model shows a liquid phase and a crystalline phase with many grouond states. The model is defined on a square lattice and to each site a stochastic Ising variable is associated. The feature that makes this model particularly interesting is that it display durable metastables states which can vanish within the time available for numerical simulations. In order to imitate the glass former process, we perform Monte Carlo simulations at constant cooling rate. We present also simulations for quenchs to temperatures below the melting temperature.
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Bayesian Parameter Estimation on Three Models of InfluenzaTorrence, Robert Billington 11 May 2017 (has links)
Mathematical models of viral infections have been informing virology research for years. Estimating parameter values for these models can lead to understanding of biological values. This has been successful in HIV modeling for the estimation of values such as the lifetime of infected CD8 T-Cells. However, estimating these values is notoriously difficult, especially for highly complex models. We use Bayesian inference and Monte Carlo Markov Chain methods to estimate the underlying densities of the parameters (assumed to be continuous random variables) for three models of influenza. We discuss the advantages and limitations of parameter estimation using these methods. The data and influenza models used for this project are from the lab of Dr. Amber Smith in Memphis, Tennessee. / Master of Science
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Étude de la performance d’un algorithme Metropolis-Hastings avec ajustement directionnelMireuta, Matei 08 1900 (has links)
Les méthodes de Monte Carlo par chaîne de Markov (MCMC) sont des outils très populaires
pour l’échantillonnage de lois de probabilité complexes et/ou en grandes dimensions.
Étant donné leur facilité d’application, ces méthodes sont largement répandues
dans plusieurs communautés scientifiques et bien certainement en statistique, particulièrement
en analyse bayésienne. Depuis l’apparition de la première méthode MCMC en
1953, le nombre de ces algorithmes a considérablement augmenté et ce sujet continue
d’être une aire de recherche active.
Un nouvel algorithme MCMC avec ajustement directionnel a été récemment développé
par Bédard et al. (IJSS, 9 :2008) et certaines de ses propriétés restent partiellement
méconnues. L’objectif de ce mémoire est de tenter d’établir l’impact d’un paramètre clé
de cette méthode sur la performance globale de l’approche. Un second objectif est de
comparer cet algorithme à d’autres méthodes MCMC plus versatiles afin de juger de sa
performance de façon relative. / Markov Chain Monte Carlo algorithms (MCMC) have become popular tools for sampling
from complex and/or high dimensional probability distributions. Given their relative
ease of implementation, these methods are frequently used in various scientific
areas, particularly in Statistics and Bayesian analysis. The volume of such methods has
risen considerably since the first MCMC algorithm described in 1953 and this area of
research remains extremely active.
A new MCMC algorithm using a directional adjustment has recently been described
by Bédard et al. (IJSS, 9:2008) and some of its properties remain unknown. The objective
of this thesis is to attempt determining the impact of a key parameter on the global
performance of the algorithm. Moreover, another aim is to compare this new method to
existing MCMC algorithms in order to evaluate its performance in a relative fashion.
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Best practice of extracting magnetocaloric properties in magnetic simulationsBylin, Johan January 2019 (has links)
In this thesis, a numerical study of simulating and computing the magnetocaloric properties of magnetic materials is presented. The main objective was to deduce the optimal procedure to obtain the isothermal change in entropy of magnetic systems, by evaluating two different formulas of entropy extraction, one relying on the magnetization of the material and the other on the magnet's heat capacity. The magnetic systems were simulated using two different Monte Carlo algorithms, the Metropolis and Wang-Landau procedures. The two entropy methods proved to be comparably similar to one another. Both approaches produced reliable and consistent results, though finite size effects could occur if the simulated system became too small. Erroneous fluctuations that invalidated the results did not seem stem from discrepancies between the entropy methods but mainly from the computation of the heat capacity itself. Accurate determination of the heat capacity via an internal energy derivative generated excellent results, while a heat capacity obtained from a variance formula of the internal energy rendered the extracted entropy unusable. The results acquired from the Metropolis algorithm were consistent, accurate and dependable, while all of those produced via the Wang-Landau method exhibited intrinsic fluctuations of varying severity. The Wang-Landau method also proved to be computationally ineffective compared to the Metropolis algorithm, rendering the method not suitable for magnetic simulations of this type.
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Simulações numéricas de Monte Carlo aplicadas no estudo das transições de fase do modelo de Ising dipolar bidimensional / Numerical Monte Carlo simulations applied to study of phase transitions in two-dimensional dipolar Ising modelRizzi, Leandro Gutierrez 24 April 2009 (has links)
O modelo de Ising dipolar bidimensional inclui, além da interação ferromagnética entre os primeiros vizinhos, interações de longo alcance entre os momentos de dipolo magnético dos spins. A presença da interação dipolar muda completamente o sistema, apresentando um rico diagrama de fase, cujas características têm originado inúmeros estudos na literatura. Além disso, a possibilidade de explicar fenômenos observados em filmes magnéticos ultrafinos, os quais possuem diversas aplicações em àreas tecnológicas, também motiva o estudo deste modelo. O estado fundamental ferromagnético do modelo de Ising puro é alterado para uma série de fases do tipo faixas, as quais consistem em domínios ferromagnéticos de largura $h$ com magnetizações opostas. A largura das faixas depende da razao $\\delta$ das intensidades dos acoplamentos ferromagnético e dipolar. Através de simulações de Monte Carlo e técnicas de repesagem em histogramas múltiplos identificamos as temperaturas críticas de tamanho finito para as transições de fase quando $\\delta=2$, o que corresponde a $h=2$. Calculamos o calor específico e a susceptibilidade do parâmetro de ordem, no intervalo de temperaturas onde as transições são observadas, para diferentes tamanhos de rede. As técnicas de repesagem permitem-nos explorar e identificar máximos distintos nessas funções da temperatura e, desse modo, estimar as temperaturas críticas de tamanho finito com grande precisão. Apresentamos evidências numéricas da existência de uma fase nemática de Ising para tamanhos grandes de rede. Em nossas simulações, observamos esta fase para tamanhos de rede a partir de $L=48$. Para verificar o quanto a interação dipolar de longo alcance afeta as estimativas físicas, nós calculamos o tempo de autocorrelação integrado nas séries temporais da energia. Inferimos daí quão severo é o critical slowing down (decaimento lento crítico) para esse sistema próximo às transições de fase termodinâmicas. Os resultados obtidos utilizando um algoritmo de atualização local foram comparados com os resultados obtidos utilizando o algoritmo multicanônico. / Two-dimensional spin model with nearest-neighbor ferromagnetic interaction and long-range dipolar interactions exhibit a rich phase diagram, whose characteristics have been exploited by several studies in the recent literature. Furthermore, the possibility of explain observed phenomena in ultrathin magnetic films, which have many technological applications, also motivates the study of this model. The presence of dipolar interaction term changes the ferromagnetic ground state expected for the pure Ising model to a series of striped phases, which consist of ferromagnetic domains of width $h$ with opposite magnetization. The width of the stripes depends on the ratio $\\delta$ of the ferromagnetic and dipolar couplings. Monte Carlo simulations and reweighting multiple histograms techniques allow us to identify the finite-size critical temperatures of the phase transitions when $\\delta=2$, which corresponds to $h=2$. We calculate, for different lattice sizes, the specific heat and susceptibility of the order parameter around the transition temperatures by means of reweighting techniques. This allows us to identify in these observables, as functions of temperature, the distinct maxima and thereby to estimate the finite-size critical temperatures with high precision. We present numerical evidence of the existence of a Ising nematic phase for large lattice sizes. Our results show that simulations need to be performed for lattice sizes at least as large as $L=48$ to clearly observe the Ising nematic phase. To access how the long-range dipolar interaction may affect physical estimates we also evaluate the integrated autocorrelation time in energy time series. This allows us to infer how severe is the critical slowing down for this system with long-range interaction and nearby thermodynamic phase transitions. The results obtained using a local update algorithm are compared with results obtained using the multicanonical algorithm.
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Étude de la performance d’un algorithme Metropolis-Hastings avec ajustement directionnelMireuta, Matei 08 1900 (has links)
Les méthodes de Monte Carlo par chaîne de Markov (MCMC) sont des outils très populaires
pour l’échantillonnage de lois de probabilité complexes et/ou en grandes dimensions.
Étant donné leur facilité d’application, ces méthodes sont largement répandues
dans plusieurs communautés scientifiques et bien certainement en statistique, particulièrement
en analyse bayésienne. Depuis l’apparition de la première méthode MCMC en
1953, le nombre de ces algorithmes a considérablement augmenté et ce sujet continue
d’être une aire de recherche active.
Un nouvel algorithme MCMC avec ajustement directionnel a été récemment développé
par Bédard et al. (IJSS, 9 :2008) et certaines de ses propriétés restent partiellement
méconnues. L’objectif de ce mémoire est de tenter d’établir l’impact d’un paramètre clé
de cette méthode sur la performance globale de l’approche. Un second objectif est de
comparer cet algorithme à d’autres méthodes MCMC plus versatiles afin de juger de sa
performance de façon relative. / Markov Chain Monte Carlo algorithms (MCMC) have become popular tools for sampling
from complex and/or high dimensional probability distributions. Given their relative
ease of implementation, these methods are frequently used in various scientific
areas, particularly in Statistics and Bayesian analysis. The volume of such methods has
risen considerably since the first MCMC algorithm described in 1953 and this area of
research remains extremely active.
A new MCMC algorithm using a directional adjustment has recently been described
by Bédard et al. (IJSS, 9:2008) and some of its properties remain unknown. The objective
of this thesis is to attempt determining the impact of a key parameter on the global
performance of the algorithm. Moreover, another aim is to compare this new method to
existing MCMC algorithms in order to evaluate its performance in a relative fashion.
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Recyclage des candidats dans l'algorithme Metropolis à essais multiplesGroiez, Assia 03 1900 (has links)
Les méthodes de Monte Carlo par chaînes de Markov (MCCM) sont des méthodes
servant à échantillonner à partir de distributions de probabilité. Ces techniques
se basent sur le parcours de chaînes de Markov ayant pour lois stationnaires
les distributions à échantillonner. Étant donné leur facilité d’application, elles
constituent une des approches les plus utilisées dans la communauté statistique,
et tout particulièrement en analyse bayésienne. Ce sont des outils très populaires
pour l’échantillonnage de lois de probabilité complexes et/ou en grandes dimensions.
Depuis l’apparition de la première méthode MCCM en 1953 (la méthode de
Metropolis, voir [10]), l’intérêt pour ces méthodes, ainsi que l’éventail d’algorithmes
disponibles ne cessent de s’accroître d’une année à l’autre.
Bien que l’algorithme Metropolis-Hastings (voir [8]) puisse être considéré
comme l’un des algorithmes de Monte Carlo par chaînes de Markov les plus généraux,
il est aussi l’un des plus simples à comprendre et à expliquer, ce qui en fait
un algorithme idéal pour débuter. Il a été sujet de développement par plusieurs
chercheurs. L’algorithme Metropolis à essais multiples (MTM), introduit dans la
littérature statistique par [9], est considéré comme un développement intéressant
dans ce domaine, mais malheureusement son implémentation est très coûteuse
(en termes de temps).
Récemment, un nouvel algorithme a été développé par [1]. Il s’agit de l’algorithme
Metropolis à essais multiples revisité (MTM revisité), qui définit la méthode
MTM standard mentionnée précédemment dans le cadre de l’algorithme
Metropolis-Hastings sur un espace étendu.
L’objectif de ce travail est, en premier lieu, de présenter les méthodes MCCM,
et par la suite d’étudier et d’analyser les algorithmes Metropolis-Hastings ainsi
que le MTM standard afin de permettre aux lecteurs une meilleure compréhension
de l’implémentation de ces méthodes. Un deuxième objectif est d’étudier les
perspectives ainsi que les inconvénients de l’algorithme MTM revisité afin de voir
s’il répond aux attentes de la communauté statistique. Enfin, nous tentons de combattre le problème de sédentarité de l’algorithme MTM revisité, ce qui donne
lieu à un tout nouvel algorithme. Ce nouvel algorithme performe bien lorsque le
nombre de candidats générés à chaque itérations est petit, mais sa performance
se dégrade à mesure que ce nombre de candidats croît. / Markov Chain Monte Carlo (MCMC) algorithms are methods that are used
for sampling from probability distributions. These tools are based on the path
of a Markov chain whose stationary distribution is the distribution to be sampled.
Given their relative ease of application, they are one of the most popular
approaches in the statistical community, especially in Bayesian analysis. These
methods are very popular for sampling from complex and/or high dimensional
probability distributions.
Since the appearance of the first MCMC method in 1953 (the Metropolis algorithm,
see [10]), the interest for these methods, as well as the range of algorithms
available, continue to increase from one year to another.
Although the Metropolis-Hastings algorithm (see [8]) can be considered as
one of the most general Markov chain Monte Carlo algorithms, it is also one of
the easiest to understand and explain, making it an ideal algorithm for beginners.
As such, it has been studied by several researchers. The multiple-try Metropolis
(MTM) algorithm , proposed by [9], is considered as one interesting development
in this field, but unfortunately its implementation is quite expensive (in terms of
time).
Recently, a new algorithm was developed by [1]. This method is named the revisited
multiple-try Metropolis algorithm (MTM revisited), which is obtained by
expressing the MTM method as a Metropolis-Hastings algorithm on an extended
space.
The objective of this work is to first present MCMC methods, and subsequently
study and analyze the Metropolis-Hastings and standard MTM algorithms
to allow readers a better perspective on the implementation of these methods.
A second objective is to explore the opportunities and disadvantages of
the revisited MTM algorithm to see if it meets the expectations of the statistical
community. We finally attempt to fight the sedentarity of the revisited MTM algorithm,
which leads to a new algorithm. The latter performs efficiently when the
number of generated candidates in a given iteration is small, but the performance of this new algorithm then deteriorates as the number of candidates in a given
iteration increases.
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