Adaptive Markov chain Monte Carlo and Bayesian filtering for state space models

This thesis is concerned with the state and parameter estimation in state space models. The estimation of states and parameters is an important task when mathematical modeling is applied to many different application areas such as the global positioning systems, target tracking, navigation, brain imaging, spread of infectious diseases, biological processes, telecommunications, audio signal processing, stochastic optimal control, machine learning, and physical systems. In Bayesian settings, the estimation of states or parameters amounts to computation of the posterior probability density function. Except for a very restricted number of models, it is impossible to compute this density function in a closed form. Hence, we need approximation methods. A state estimation problem involves estimating the states (latent variables) that are not directly observed in the output of the system. In this thesis, we use the Kalman filter, extended Kalman filter, Gauss–Hermite filters, and particle filters to estimate the states based on available measurements. Among these filters, particle filters are numerical methods for approximating the filtering distributions of non-linear non-Gaussian state space models via Monte Carlo. The performance of a particle filter heavily depends on the chosen importance distribution. For instance, inappropriate choice of the importance distribution can lead to the failure of convergence of the particle filter algorithm. In this thesis, we analyze the theoretical Lᵖ particle filter convergence with general importance distributions, where p ≥2 is an integer. A parameter estimation problem is considered with inferring the model parameters from measurements. For high-dimensional complex models, estimation of parameters can be done by Markov chain Monte Carlo (MCMC) methods. In its operation, the MCMC method requires the unnormalized posterior distribution of the parameters and a proposal distribution. In this thesis, we show how the posterior density function of the parameters of a state space model can be computed by filtering based methods, where the states are integrated out. This type of computation is then applied to estimate parameters of stochastic differential equations. Furthermore, we compute the partial derivatives of the log-posterior density function and use the hybrid Monte Carlo and scaled conjugate gradient methods to infer the parameters of stochastic differential equations. The computational efficiency of MCMC methods is highly depend on the chosen proposal distribution. A commonly used proposal distribution is Gaussian. In this kind of proposal, the covariance matrix must be well tuned. To tune it, adaptive MCMC methods can be used. In this thesis, we propose a new way of updating the covariance matrix using the variational Bayesian adaptive Kalman filter algorithm.

Operative Treatment of Ovarian Cancer and Borderline Tumors in Different Hospital Categories in Finland

Surgery is the cornerstone of ovarian cancer treatment and maximal cytoreduction is important. In the early 1980’s primary surgical treatment of ovarian cancer was performed in over 80 hospitals in Finland. The significance of the operative volume of the hospital, of the training of the surgeons and of centralization of surgical treatment has been widely discussed. The aim of the present study was to evaluate the outcome of surgical treatment of ovarian cancer in different hospital categories retrospectively and prospectively, and to analyze if any differences are reflected in survival. The retrospective study included 3851 ovarian cancer patients operated between 1983 and 1994 in Finland. The data was analyzed according to hospital category (university, central, and other) and by quartiles of the hospital operative volume. The results showed that patients operated in the highest operative volume hospitals had the best relative survival. When stratifying the analysis by the period of diagnosis (1983-1988 and 1989-1994), the university hospitals improved their performance the most. The prospective part of the thesis was initiated in 1999 and included 307 patients with invasive ovarian cancer and 65 patients with an ovarian borderline tumor. The baseline and 5-year surveys used a questionnaire that was filled in by the operating surgeons. For analysis of the 5-year followup data, the hospitals were divided into three categories (<10, 10-20, or >20 patients operated in 1999). The effect of the surgical volume was analyzed also as a continuous variable (1-47 operations per year). In university hospitals, pelvic lymphadenectomy was performed in 88 %, and para-aortic lymphadenectomy in 73 %, of the patients with stage I disease. The corresponding figures ranged from 11 % to 21 % in the other hospitals. For stage III ovarian cancer patients operated by gynecological oncologists, the estimated odds ratio for no macroscopic residual tumor was 3.0 times higher (95 % CI 1.2-7.5) than for those operated by general gynecologists. In the university and other hospitals 82% of the patients received platinum-based chemotherapy. Platinum + taxane combination was given to 63 % of the patients in the university and in 49 % in the other hospitals (p = 0.0763). Only a minority of the patients with tumors of borderline malignancy were staged according to recommendations, most often multiple peritoneal biopsies and omentectomy were neglected. FIGO stage, patient age, and residual tumor were independent prognostic factors of cancer-specific 5-year survival. A higher hospital operative volume was also a significant prognostic factor for better cancer-specific survival (p = 0.036) and disease-free survival (p = 0.048). In conclusion, ovarian cancer patients operated in high-volume university hospitals were more often optimally debulked and had a significantly better cancer-specific survival than patients operated in other hospitals. These results favor centralization of primary surgical treatment of ovarian cancer.

Minimum error contrast enhancement

Contrast enhancement is an image processing technique where the objective is to preprocess the image so that relevant information can be either seen or further processed more reliably. These techniques are typically applied when the image itself or the device used for image reproduction provides poor visibility and distinguishability of different regions of interest inthe image. In most studies, the emphasis is on the visualization of image data,but this human observer biased goal often results to images which are not optimal for automated processing. The main contribution of this study is to express the contrast enhancement as a mapping from N-channel image data to 1-channel gray-level image, and to devise a projection method which results to an image with minimal error to the correct contrast image. The projection, the minimum-error contrast image, possess the optimal contrast between the regions of interest in the image. The method is based on estimation of the probability density distributions of the region values, and it employs Bayesian inference to establish the minimum error projection.

Statistical Analysis of an SEIR Epidemic Model

This thesis was focussed on statistical analysis methods and proposes the use of Bayesian inference to extract information contained in experimental data by estimating Ebola model parameters. The model is a system of differential equations expressing the behavior and dynamics of Ebola. Two sets of data (onset and death data) were both used to estimate parameters, which has not been done by previous researchers in (Chowell, 2004). To be able to use both data, a new version of the model has been built. Model parameters have been estimated and then used to calculate the basic reproduction number and to study the disease-free equilibrium. Estimates of the parameters were useful to determine how well the model fits the data and how good estimates were, in terms of the information they provided about the possible relationship between variables. The solution showed that Ebola model fits the observed onset data at 98.95% and the observed death data at 93.6%. Since Bayesian inference can not be performed analytically, the Markov chain Monte Carlo approach has been used to generate samples from the posterior distribution over parameters. Samples have been used to check the accuracy of the model and other characteristics of the target posteriors.

Prenatal and perinatal risk factors for bipolar disorder

Bipolar disorder (BPD) is a severe mental disorder associated with considerable morbidity and mortality. Prenatal insults have been shown to be associated with later development of mental disorders and there is a growing interest in the potential role of prenatal and perinatal risk factors in the development of BPD. The aims of this thesis were to describe the overall study design of the Finnish Prenatal Study of Bipolar Disorders (FIPS-B) and demographic characteristics of the sample. Furthermore, it was aimed to examine the association of parental age, parental age difference, perinatal complications and maternal smoking during pregnancy with BPD. This thesis is based on FIPS-B, a nested case-control study using several nationwide registers. The cases included all people born in Finland between January 1st 1983 and December 31st 1998 and diagnosed with BPD according to the Finnish Hospital Discharge Register (FHDR) before December 31st 2008. Controls for this study were people who were without BPD, schizophrenia or diagnoses related to these disorders, identified from the Population Register Centre (PRC), and matched two-fold to the cases on sex, date of birth (+/- 30 days), and residence in Finland on the first day of diagnosis of the matched case. Conditional logistic regression models were used to examine the association between risk factors and BPD. This study included 1887 BPD cases and 3774 matched controls. The mean age at diagnosis was 19.3 years and females accounted for 68% of the cases. Mothers with the lowest educational level had the highest odds of having BPD in offspring. Being born in Eastern and Southern region of Finland increased the odds of having BPD later in life. A U-shaped distribution of odds ratio was observed between paternal age and BPD in the unadjusted analysis. Maternal age and parental age difference was not associated with BPD. Birth by planned caesarean section was associated with increased odd of BPD. Smoking during pregnancy was not associated with BPD in the adjusted analyses. Region of birth and maternal educational level were associated with BPD. Both young and old father’s age was associated with BPD. Most perinatal complications and maternal smoking during pregnancy were not associated with BPD. The findings of this thesis, considered together with previous literature, suggest that the pre- and perinatal risk factor profile varies among different psychiatric disorders.