Interim analysis of clinical trials: Simulation studies of fractional Brownian motion

Interim clinical trial monitoring procedures were motivated by ethical and economic considerations. Classical Brownian motion (Bm) techniques for statistical monitoring of clinical trials were widely used. Conditional power argument and α-spending function based boundary crossing probabilities are popular statistical hypothesis testing procedures under the assumption of Brownian motion. However, it is not rare that the assumptions of Brownian motion are only partially met for trial data. Therefore, I used a more generalized form of stochastic process, called fractional Brownian motion (fBm), to model the test statistics. Fractional Brownian motion does not hold Markov property and future observations depend not only on the present observations but also on the past ones. In this dissertation, we simulated a wide range of fBm data, e.g., H = 0.5 (that is, classical Bm) vs. 0.5< H <1, with treatment effects vs. without treatment effects. Then the performance of conditional power and boundary-crossing based interim analyses were compared by assuming that the data follow Bm or fBm. Our simulation study suggested that the conditional power or boundaries under fBm assumptions are generally higher than those under Bm assumptions when H > 0.5 and also matches better with the empirical results. ^

Developing the follow-up schedules for women with breast cancer

Background: The follow-up care for women with breast cancer requires an understanding of disease recurrence patterns and the follow-up visit schedule should be determined according to the times when the recurrence are most likely to occur, so that preventive measure can be taken to avoid or minimize the recurrence. Objective: To model breast cancer recurrence through stochastic process with an aim to generate a hazard function for determining a follow-up schedule. Methods: We modeled the process of disease progression as the time transformed Weiner process and the first-hitting-time was used as an approximation of the true failure time. The women's "recurrence-free survival time" or a "not having the recurrence event" is modeled by the time it takes Weiner process to cross a threshold value which represents a woman experiences breast cancer recurrence event. We explored threshold regression model which takes account of covariates that contributed to the prognosis of breast cancer following development of the first-hitting time model. Using real data from SEER-Medicare, we proposed models of follow-up visits schedule on the basis of constant probability of disease recurrence between consecutive visits. Results: We demonstrated that the threshold regression based on first-hitting-time modeling approach can provide useful predictive information about breast cancer recurrence. Our results suggest the surveillance and follow-up schedule can be determined for women based on their prognostic factors such as tumor stage and others. Women with early stage of disease may be seen less frequently for follow-up visits than those women with locally advanced stages. Our results from SEER-Medicare data support the idea of risk-controlled follow-up strategies for groups of women. Conclusion: The methodology we proposed in this study allows one to determine individual follow-up scheduling based on a parametric hazard function that incorporates known prognostic factors.^

Simulation of academic performance rates, estimation of indicators of subjects and of a complete degree

The purpose of this paper is to present a program written in Matlab-Octave for the simulation of the time evolution of student curricula, i.e, how students pass their subjects along time until graduation. The program computes, from the simulations, the academic performance rates for the subjects of the study plan for each semester as well as the overall rates, which are a) the efficiency rate defined as the ratio of the number of students passing the exam to the number of students who registered for it and b) the success rate, defined as the ratio of the number of students passing the exam to the number of students who not only registered for it but also actually took it. Additionally, we compute the rates for the bachelor academic degree which are established for Spain by the National Quality Evaluation and Accreditation Agency (ANECA) and which are the graduation rate (measured as the percentage of students who finish as scheduled in the plan or taking an extra year) and the efficiency rate (measured as the percentage of credits which a student who graduated has really taken). The simulation is done in terms of the probabilities of passing all the subjects in their study plan. The application of the simulator to Polytech students in Madrid, where requirements for passing are specially stiff in first and second year subjects, is particularly relevant to analyze student cohorts and the probabilities of students finishing in the minimum of four years, or taking and extra year or two extra years, and so forth. It is a very useful tool when designing new study plans. The calculation of the probability distribution of the random variable "number of semesters a student has taken to complete the curricula and graduate" is difficult or even unfeasible to obtain analytically, and this is even truer when we incorporate uncertainty in parameter estimation. This is why we apply Monte Carlo simulation which not only provides illustration of the stochastic process but also a method for computation. The stochastic simulator is proving to be a useful tool for identification of the subjects most critical in the distribution of the number of semesters for curriculum vitae (CV) completion and subsequently for a decision making process in terms of CV planning and passing standards in the University. Simulations are performed through a graphical interface where also the results are presented in appropriate figures. The Project has been funded by the Call for Innovation in Education Projects of Universidad Politécnica de Madrid (UPM) through a Project of its school Escuela Técnica Superior de Ingenieros Industriales ETSII during the period September 2010-September 2011.

Maximum likelihood estimation of new state space models for operational modal analysis

The modal analysis of a structural system consists on computing its vibrational modes. The experimental way to estimate these modes requires to excite the system with a measured or known input and then to measure the system output at different points using sensors. Finally, system inputs and outputs are used to compute the modes of vibration. When the system refers to large structures like buildings or bridges, the tests have to be performed in situ, so it is not possible to measure system inputs such as wind, traffic, . . .Even if a known input is applied, the procedure is usually difficult and expensive, and there are still uncontrolled disturbances acting at the time of the test. These facts led to the idea of computing the modes of vibration using only the measured vibrations and regardless of the inputs that originated them, whether they are ambient vibrations (wind, earthquakes, . . . ) or operational loads (traffic, human loading, . . . ). This procedure is usually called Operational Modal Analysis (OMA), and in general consists on to fit a mathematical model to the measured data assuming the unobserved excitations are realizations of a stationary stochastic process (usually white noise processes). Then, the modes of vibration are computed from the estimated model. The first issue investigated in this thesis is the performance of the Expectation- Maximization (EM) algorithm for the maximum likelihood estimation of the state space model in the field of OMA. The algorithm is described in detail and it is analysed how to apply it to vibration data. After that, it is compared to another well known method, the Stochastic Subspace Identification algorithm. The maximum likelihood estimate enjoys some optimal properties from a statistical point of view what makes it very attractive in practice, but the most remarkable property of the EM algorithm is that it can be used to address a wide range of situations in OMA. In this work, three additional state space models are proposed and estimated using the EM algorithm: • The first model is proposed to estimate the modes of vibration when several tests are performed in the same structural system. Instead of analyse record by record and then compute averages, the EM algorithm is extended for the joint estimation of the proposed state space model using all the available data. • The second state space model is used to estimate the modes of vibration when the number of available sensors is lower than the number of points to be tested. In these cases it is usual to perform several tests changing the position of the sensors from one test to the following (multiple setups of sensors). Here, the proposed state space model and the EM algorithm are used to estimate the modal parameters taking into account the data of all setups. • And last, a state space model is proposed to estimate the modes of vibration in the presence of unmeasured inputs that cannot be modelled as white noise processes. In these cases, the frequency components of the inputs cannot be separated from the eigenfrequencies of the system, and spurious modes are obtained in the identification process. The idea is to measure the response of the structure corresponding to different inputs; then, it is assumed that the parameters common to all the data correspond to the structure (modes of vibration), and the parameters found in a specific test correspond to the input in that test. The problem is solved using the proposed state space model and the EM algorithm. Resumen El análisis modal de un sistema estructural consiste en calcular sus modos de vibración. Para estimar estos modos experimentalmente es preciso excitar el sistema con entradas conocidas y registrar las salidas del sistema en diferentes puntos por medio de sensores. Finalmente, los modos de vibración se calculan utilizando las entradas y salidas registradas. Cuando el sistema es una gran estructura como un puente o un edificio, los experimentos tienen que realizarse in situ, por lo que no es posible registrar entradas al sistema tales como viento, tráfico, . . . Incluso si se aplica una entrada conocida, el procedimiento suele ser complicado y caro, y todavía están presentes perturbaciones no controladas que excitan el sistema durante el test. Estos hechos han llevado a la idea de calcular los modos de vibración utilizando sólo las vibraciones registradas en la estructura y sin tener en cuenta las cargas que las originan, ya sean cargas ambientales (viento, terremotos, . . . ) o cargas de explotación (tráfico, cargas humanas, . . . ). Este procedimiento se conoce en la literatura especializada como Análisis Modal Operacional, y en general consiste en ajustar un modelo matemático a los datos registrados adoptando la hipótesis de que las excitaciones no conocidas son realizaciones de un proceso estocástico estacionario (generalmente ruido blanco). Posteriormente, los modos de vibración se calculan a partir del modelo estimado. El primer problema que se ha investigado en esta tesis es la utilización de máxima verosimilitud y el algoritmo EM (Expectation-Maximization) para la estimación del modelo espacio de los estados en el ámbito del Análisis Modal Operacional. El algoritmo se describe en detalle y también se analiza como aplicarlo cuando se dispone de datos de vibraciones de una estructura. A continuación se compara con otro método muy conocido, el método de los Subespacios. Los estimadores máximo verosímiles presentan una serie de propiedades que los hacen óptimos desde un punto de vista estadístico, pero la propiedad más destacable del algoritmo EM es que puede utilizarse para resolver un amplio abanico de situaciones que se presentan en el Análisis Modal Operacional. En este trabajo se proponen y estiman tres modelos en el espacio de los estados: • El primer modelo se utiliza para estimar los modos de vibración cuando se dispone de datos correspondientes a varios experimentos realizados en la misma estructura. En lugar de analizar registro a registro y calcular promedios, se utiliza algoritmo EM para la estimación conjunta del modelo propuesto utilizando todos los datos disponibles. • El segundo modelo en el espacio de los estados propuesto se utiliza para estimar los modos de vibración cuando el número de sensores disponibles es menor que vi Resumen el número de puntos que se quieren analizar en la estructura. En estos casos es usual realizar varios ensayos cambiando la posición de los sensores de un ensayo a otro (múltiples configuraciones de sensores). En este trabajo se utiliza el algoritmo EM para estimar los parámetros modales teniendo en cuenta los datos de todas las configuraciones. • Por último, se propone otro modelo en el espacio de los estados para estimar los modos de vibración en la presencia de entradas al sistema que no pueden modelarse como procesos estocásticos de ruido blanco. En estos casos, las frecuencias de las entradas no se pueden separar de las frecuencias del sistema y se obtienen modos espurios en la fase de identificación. La idea es registrar la respuesta de la estructura correspondiente a diferentes entradas; entonces se adopta la hipótesis de que los parámetros comunes a todos los registros corresponden a la estructura (modos de vibración), y los parámetros encontrados en un registro específico corresponden a la entrada en dicho ensayo. El problema se resuelve utilizando el modelo propuesto y el algoritmo EM.

Estimação do índice de memória em processos estocásticos com memória longa: uma abordagem via ABC

Neste trabalho propomos o uso de um método Bayesiano para estimar o parâmetro de memória de um processo estocástico com memória longa quando sua função de verossimilhança é intratável ou não está disponível. Esta abordagem fornece uma aproximação para a distribuição a posteriori sobre a memória e outros parâmetros e é baseada numa aplicação simples do método conhecido como computação Bayesiana aproximada (ABC). Alguns estimadores populares para o parâmetro de memória serão revisados e comparados com esta abordagem. O emprego de nossa proposta viabiliza a solução de problemas complexos sob o ponto de vista Bayesiano e, embora aproximativa, possui um desempenho muito satisfatório quando comparada com métodos clássicos.

Optimal strategies in political elections

In the Majoritarian Parliamentary System, the government has a constitutional right to call an early election. This right provides the government a control to achieve its objective to remain in power for as long as possible. We model the early election problem mathematically using opinion polls data as a stochastic process to proxy the government's probability of re-election. These data measure the difference in popularity between the government and the opposition. We fit a mean reverting Stochastic Differential Equation to describe the behaviour of the process and consider the possibility for the government to use other control tools, which are termed 'boosts' to induce shocks to the opinion polls by making timely policy announcements or economic actions. These actions improve the government's popularity and have some impact upon the early-election exercise boundary. © Austral. Mathematical Soc. 2005.

Evolving noisy oscillatory dynamics in genetic regulatory networks

We introduce a genetic programming (GP) approach for evolving genetic networks that demonstrate desired dynamics when simulated as a discrete stochastic process. Our representation of genetic networks is based on a biochemical reaction model including key elements such as transcription, translation and post-translational modifications. The stochastic, reaction-based GP system is similar but not identical with algorithmic chemistries. We evolved genetic networks with noisy oscillatory dynamics. The results show the practicality of evolving particular dynamics in gene regulatory networks when modelled with intrinsic noise.

Studies on the generalisation of Gaussian processes and Bayesian neural networks

The assessment of the reliability of systems which learn from data is a key issue to investigate thoroughly before the actual application of information processing techniques to real-world problems. Over the recent years Gaussian processes and Bayesian neural networks have come to the fore and in this thesis their generalisation capabilities are analysed from theoretical and empirical perspectives. Upper and lower bounds on the learning curve of Gaussian processes are investigated in order to estimate the amount of data required to guarantee a certain level of generalisation performance. In this thesis we analyse the effects on the bounds and the learning curve induced by the smoothness of stochastic processes described by four different covariance functions. We also explain the early, linearly-decreasing behaviour of the curves and we investigate the asymptotic behaviour of the upper bounds. The effect of the noise and the characteristic lengthscale of the stochastic process on the tightness of the bounds are also discussed. The analysis is supported by several numerical simulations. The generalisation error of a Gaussian process is affected by the dimension of the input vector and may be decreased by input-variable reduction techniques. In conventional approaches to Gaussian process regression, the positive definite matrix estimating the distance between input points is often taken diagonal. In this thesis we show that a general distance matrix is able to estimate the effective dimensionality of the regression problem as well as to discover the linear transformation from the manifest variables to the hidden-feature space, with a significant reduction of the input dimension. Numerical simulations confirm the significant superiority of the general distance matrix with respect to the diagonal one.In the thesis we also present an empirical investigation of the generalisation errors of neural networks trained by two Bayesian algorithms, the Markov Chain Monte Carlo method and the evidence framework; the neural networks have been trained on the task of labelling segmented outdoor images.

A Cauchy Integral Related to a Robot-safety Device System

We introduce a robot-safety device system attended by two different repairmen. The twin system is characterized by the natural feature of cold standby and by an admissible “risky” state. In order to analyse the random behaviour of the entire system (robot, safety device, repair facility) we employ a stochastic process endowed with probability measures satisfying general Hokstad-type differential equations. The solution procedure is based on the theory of sectionally holomorphic functions, characterized by a Cauchy-type integral defined as a Cauchy principal value in double sense. An application of the Sokhotski-Plemelj formulae determines the long-run availability of the robot-safety device. Finally, we consider the particular but important case of deterministic repair.

Monte Carlo Study of Particle Transport Problem in Air Pollution

2002 Mathematics Subject Classification: 65C05.

Limit Theorems for Non-Critical Branching Processes with Continuous State Space

2000 Mathematics Subject Classification: Primary 60J80, Secondary 60G99.

Intenzitásalapú modellezés és a mértékcsere (Intensity-based modeling and thje change of measure)

A dolgozatban a hitelderivatívák intenzitásalapú modellezésének néhány kérdését vizsgáljuk meg. Megmutatjuk, hogy alkalmas mértékcserével nemcsak a duplán sztochasztikus folyamatok, hanem tetszőleges intenzitással rendelkező pontfolyamat esetén is kiszámolható az összetett kár- és csődfolyamat eloszlásának Laplace-transzformáltja. _____ The paper addresses questions concerning the use of intensity based modeling in the pricing of credit derivatives. As the specification of the distribution of the lossprocess is a non-trivial exercise, the well-know technique for this task utilizes the inversion of the Laplace-transform. A popular choice for the model is the class of doubly stochastic processes given that their Laplace-transforms can be determined easily. Unfortunately these processes lack several key features supported by the empirical observations, e.g. they cannot replicate the self-exciting nature of defaults. The aim of the paper is to show that by using an appropriate change of measure the Laplace-transform can be calculated not only for a doubly stochastic process, but for an arbitrary point process with intensity as well. To support the application of the technique, we investigate the e®ect of the change of measure on the stochastic nature of the underlying process.

Scaling limits of a model for selection at two scales

The dynamics of a population undergoing selection is a central topic in evolutionary biology. This question is particularly intriguing in the case where selective forces act in opposing directions at two population scales. For example, a fast-replicating virus strain outcompetes slower-replicating strains at the within-host scale. However, if the fast-replicating strain causes host morbidity and is less frequently transmitted, it can be outcompeted by slower-replicating strains at the between-host scale. Here we consider a stochastic ball-and-urn process which models this type of phenomenon. We prove the weak convergence of this process under two natural scalings. The first scaling leads to a deterministic nonlinear integro-partial differential equation on the interval $[0,1]$ with dependence on a single parameter, $\lambda$. We show that the fixed points of this differential equation are Beta distributions and that their stability depends on $\lambda$ and the behavior of the initial data around $1$. The second scaling leads to a measure-valued Fleming-Viot process, an infinite dimensional stochastic process that is frequently associated with a population genetics.

Exchange rate dynamics, structural breaks, and central bank interventions in Colombia

We evaluate the effectiveness of the Colombian Central Bank´s interventions in the foreign exchange market during the period 2000 to 2014 -- We examine the stochastic process that describes the exchange rate, with a focus on the detection of structural breaks or unit roots in the data to determine whether the Central Bank´s interventions were effective -- We find that the exchange rate can be described either by a random walk or by a trend-stationary model with multiple breaks -- In neither cases do we find any evidence that the exchange rate was affected by the Central Bank interventions

Discovering characteristics of stochastic collections of process models

Process models in organizational collections are typically modeled by the same team and using the same conventions. As such, these models share many characteristic features like size range, type and frequency of errors. In most cases merely small samples of these collections are available due to e.g. the sensitive information they contain. Because of their sizes, these samples may not provide an accurate representation of the characteristics of the originating collection. This paper deals with the problem of constructing collections of process models, in the form of Petri nets, from small samples of a collection for accurate estimations of the characteristics of this collection. Given a small sample of process models drawn from a real-life collection, we mine a set of generation parameters that we use to generate arbitrary-large collections that feature the same characteristics of the original collection. In this way we can estimate the characteristics of the original collection on the generated collections.We extensively evaluate the quality of our technique on various sample datasets drawn from both research and industry.