There exists a need to estimate the potential financial, epidemiological and societal impact that diseases, and the treatment thereof, can have on society. Markov processes are often used to model diseases to estimate these quantities of interest and have an advantage over standard survival analysis techniques in that multiple events can be studied simultaneously. The theory of Markov processes is well established for processes for which the process parameters are known but not as much of the literature has focussed on the estimation of these transition parameters. This dissertation investigates and implements maximum likelihood estimators for Markov processes based on longitudinal data. The methods are described based on processes that are observed such that all transitions are recorded exactly, processes of which the state of the process is recorded at equidistant time points, at irregular time points and processes for which each process is observed at a possibly different irregular time point. Methods for handling right censoring and estimating the effect of covariates on parameters are described. The estimation methods are implemented by simulating Markov processes and estimating the parameters based on the simulated data so that the accuracy of the estimators can be investigated. We show that the estimators can provide accurate estimates of state prevalence if the process is stationary, even with relatively small sample sizes. Furthermore, we indicate that the estimators lack good accuracy in estimating the effect of covariates on parameters unless state transitions are recorded exactly. The methods are discussed with reference to the msm package for R which is freely available and a popular tool for estimating and implementing Markov processes in disease modelling. Methods are mentioned for the treatment of aggregate data, diseases where the state of patients are not known with complete certainty at every observation and diseases where patient interaction plays a role. / Dissertation (MSc)--University of Pretoria, 2010. / Statistics / unrestricted

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No description available.

Parameter estimation is a pivotal task across various domains such as system identification, statistics, and machine learning. The literature presents numerous estimation procedures, many of which are backed by well-studied asymptotic properties. In the contemporary landscape, highly advanced digital twins (DTs) offer the capability to faithfully replicate real systems through proper tuning. Leveraging these DTs, data-driven estimators can alleviate challenges inherent in traditional methods, notably their computational cost and sensitivity to initializations. Furthermore, traditional estimators often rely on sensitive data, necessitating protective measures. In this thesis, we consider data-driven and privacy-preserving approaches to parameter estimation that overcome many of these challenges. The first part of the thesis delves into an exploration of modern data-driven estimation techniques, focusing on the two-stage (TS) approach. Operating under the paradigm of inverse supervised learning, the TS approach simulates numerous samples across parameter variations and employs supervised learning methods to predict parameter values. Divided into two stages, the approach involves compressing data into a smaller set of samples and the second stage utilizes these samples to predict parameter values. The simplicity of the TS estimator underscores its interpretability, necessitating theoretical justification, which forms the core motivation for this thesis. We establish statistical frameworks for the TS estimator, yielding its Bayes and minimax versions, alongside developing an improved minimax TS variant that excels in computational efficiency and robustness to distributional shifts. Finally, we conduct an asymptotic analysis of the TS estimator. The second part of the thesis introduces an application of data-driven estimation methods, that includes the TS and neural network based approaches, in the design of tuning rules for PI controllers. Leveraging synthetic datasets generated from DTs, we train machine learning algorithms to meta-learn tuning rules, streamlining the calibration process without manual intervention. In the final part of the thesis, we tackle scenarios where estimation procedures must handle sensitive data. Here, we introduce differential privacy constraints into the Bayes point estimation problem to protect sensitive information. Proposing a unified approach, we integrate the estimation problem and differential privacy constraints into a single convex optimization objective, thereby optimizing the accuracy-privacy trade-off. In cases where both observations and parameter spaces are finite, this approach reduces to a tractable linear program which is solvable using off-the-shelf solvers. In essence, this thesis endeavors to address computational and privacy concerns within the realm of parameter estimation. / Skattning av parametrar utgör en fundamental uppgift inom en mängd fält, såsom systemidentifiering, statistik och maskininlärning. I litteraturen finns otaliga skattningsmetoder, utav vilka många understödjs av välstuderade asymptotiska egenskaper. Inom dagens forskning erbjuder noggrant kalibrerade digital twins (DTs) möjligheten att naturtroget återskapa verkliga system. Genom att utnyttja dessa DTs kan data-drivna skattningsmetoder minska problem som vanligtvis drabbar traditionella skattningsmetoder, i synnerhet problem med beräkningsbörda och känslighet för initialiseringvillkor. Traditionella skattningsmetoder kräver dessutom ofta känslig data, vilket leder till ett behov av skyddsåtgärder. I den här uppsatsen, undersöker vi data-drivna och integritetsbevarande parameterskattningmetoder som övervinner många av de nämnda problemen.  Första delen av uppsatsen är en undersökning av moderna data-drivna skattningtekniker, med fokus på två-stegs-metoden (TS). Som metod inom omvänd övervakad maskininlärning, simulerar TS en stor mängd data med ett stort urval av parametrar och tillämpar sedan metoder från övervakad inlärning för att förutsäga parametervärden. De två stegen innefattar datakomprimering till en mindre mängd, varefter den mindre mängden data används för parameterskattning. Tack vare sin enkelhet och tydbarhet lämpar sig två-stegs-metoden väl för teoretisk analys, vilket är uppsatsens motivering. Vi utvecklar ett statistiskt ramverk för två-stegsmetoden, vilket ger Bayes och minimax-varianterna, samtidigt som vi vidareutvecklar minimax-TS genom en variant med hög beräkningseffektivitet och robusthet gentemot skiftade fördelningar. Slutligen analyserar vi två-stegs-metodens asymptotiska egenskaper.  Andra delen av uppsatsen introducerar en tillämpning av data-drivna skattningsmetoder, vilket innefattar TS och neurala nätverk, i designen och kalibreringen av PI-regulatorer. Med hjälp av syntetisk data från DTs tränar vi maskininlärningsalgoritmer att meta-lära sig regler för kalibrering, vilket effektiverar kalibreringsprocessen utan manuellt ingripande.  I sista delen av uppsatsen behandlar vi scenarion då skattningsprocessen innefattar känslig data. Vi introducerar differential-privacy-begränsningar i Bayes-punktskattningsproblemet för att skydda känslig information. Vi kombinerar skattningsproblemet och differential-privacy-begränsningarna i en gemensam konvex målfunktion, och optimerar således avvägningen mellan noggrannhet och integritet. Ifall både observations- och parameterrummen är ändliga, så reduceras problemet till ett lätthanterligt linjärt optimeringsproblem, vilket löses utan vidare med välkända metoder.  Sammanfattningsvis behandlar uppsatsen beräkningsmässiga och integritets-angelägenheter inom ramen för parameterskattning. / <p>QC 20240306</p>

This thesis provides insight into methods for estimating blood perfusion, emphasizing the need for accurate modeling in dynamic physiological environments. The thesis critically examines conventional error function solutions used in steady state or gradually changing blood flow scenarios, revealing their shortcomings in accurately reflecting more rapid changes in blood perfusion. To address this limitation, this study introduces a novel prediction model based on the finite-difference method (FDM) specifically designed to produce accurate results under different blood flow perfusion conditions. A comparative analysis concludes that the FDM-based model is consistent with traditional error function methods under constant blood perfusion conditions, thus establishing its validity under dynamic and steady blood flow conditions. In addition, the study attempts to determine whether analytical solutions exist that are suitable for changing perfusion conditions. Three alternative analytical estimation methods were explored, each exposing the common thread of inadequate responsiveness to sudden changes in blood perfusion. Based on the advantages and disadvantages of the error function and FDM estimation, a combination of these two methods was developed. Utilizing the simplicity and efficiency of the error function, the prediction of contact resistance and core temperature along with the initial blood perfusion was first made at the beginning of the data. Then the subsequent blood perfusion values were predicted using the FDM, as the FDM can effectively respond to changing blood perfusion values. / Master of Science / Blood perfusion, the process of blood flowing through our body's tissues, is crucial for our health. It's like monitoring traffic flow on roads, which is especially important during rapid changes, such as during exercise or medical treatments. Traditional methods for estimating blood perfusion, akin to older traffic monitoring techniques, struggle to keep up with these rapid changes. This research introduces a new approach, using a method often found in engineering and physics, called the finite-difference method (FDM), to create more accurate models of blood flow in various conditions. This study puts this new method to the test against the old standards. We discover that while both are effective under steady conditions, the FDM shines when blood flow changes quickly. We also examined three other methods, but they, too, fell short in these fast-changing scenarios. This work is more than just numbers and models; it's about potentially transforming how we understand and manage health. By combining the simplicity of traditional methods for initial blood flow estimates with the dynamic capabilities of the FDM, we're paving the way for more precise medical diagnostics and treatments.

The STREAMC model is based on the same algorithm as implemented by the Steady Riverine Environmental Assessment Model (STREAM), a mathematical model for the dissolved oxygen (DO) distribution in freshwater streams used by Mississippi Department of Environmental Quality (MDEQ). Typically the water quality models are calibrated manually. In some cases where some objective criterion can be identified to quantify a successful calibration, an auto calibration may be preferable to the manual calibration approach. The auto calibration may be particularly applicable to relatively simple analytical models such as the steady-state STREAMC model. Various techniques of parameter estimation were identified for the model. The model was then subjected to various techniques of parameter estimation identified and/or developed. The parameter estimates obtained by different techniques were tabulated and compared. A final recommendation regarding a preferable parameter estimation technique leading to the auto calibration of the STREAMC model was made.

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Circadian rhythms are the endogenous, roughly 24-hour rhythms that coordinate an organism's interaction with its cycling environment. The molecular mechanism underlying this physiological process is a cell-autonomous oscillator comprised of a complex regulatory network of interacting DNA, RNA and proteins that is surprisingly conserved across many different species. It is not a trivial task to understand how the positive and negative feedback loops interact to generate an oscillator capable of a) maintaining a 24-hour rhythm in constant conditions; b) entraining to external light and temperature signals; c) responding to pulses of light in a rather particular, predictable manner; and d) compensating itself so that the period is relatively constant over a large range of temperatures, even for mutations that affect the basal period of oscillation. Mathematical modeling is a useful tool for dealing with such complexity, because it gives us an object that can be quickly probed and tested in lieu of the experiment or actual biological system. If we do a good job designing the model, it will help us to understand the biology better by predicting the outcome of future experiments. The difficulty lies in properly designing a model, a task that is made even more difficult by an acute lack of quantitative data. Thankfully, our qualitative understanding of a particular phenomenon, i.e. the observed physiology of the cell, can often be directly related to certain mathematical structures. Bifurcation analysis gives us a glimpse of these structures, and we can use these glimpses to build our models with greater confidence. In this dissertation, I will discuss the particular problem of the circadian clock and describe a number of new methods and tools related to bifurcation analysis. These tools can effectively be applied during the modeling process to build detailed models of biological regulatory with greater ease. / Ph. D.

Fertilized frog eggs (and cell-free extracts) undergo periodic oscillations in the activity of "M-phase promoting factor" (MPF), the crucial triggering enzyme for mitosis (nuclear division) and cell division. MPF activity is regulated by a complex network of biochemical reactions. Novak and Tyson, and their collaborators, have been studying the qualitative and quantitative properties of a large system of nonlinear ordinary differential equations that describe the molecular details of this system as currently known. Important clues to the behavior of the model are provided by bifurcation theory, especially characterization of the codimension-1 and -2 bifurcation sets of the differential equations. To illustrate this method, I have been studying a system of 9 ordinary differential equations that describe the frog egg cell cycle with some fidelity. I will describe the bifurcation diagram of this system in a parameter space spanned by the rate constants for cyclin synthesis and cycling degradation. My results suggest either that the cell cycle control system should show dynamical behavior considerably more complex than the limit cycles and steady states reported so far, or that the biochemical rate constants of the system are constrained to avoid regions of parameter space where complex bifurcation points unfold. / Ph. D.

We examine a model of solid-solid phase transitions that includes thermo-elastic effects and an order parameter. The model is derived as a special case of the Gurtin-Fried model posed in one space dimension with a symmetric triple-well free energy in which the relative heights of the wells vary with temperature. We examine the temperature independent case, showing existence of a unique classical solution of a regularized system of partial differential equations using semigroup theory. This is followed by numerical study of a finite element algorithm for the temperature independent model. Finally, we present computational material concerning the temperature dependent model. / Ph. D.