A HIDDEN MARKOV MODEL FOR JOINT ESTIMATION OF GENOTYPE AND COPY NUMBER IN HIGH-THROUGHPUT SNP CHIPS

Amplifications and deletions of chromosomal DNA, as well as copy-neutral loss of heterozygosity have been associated with diseases processes. High-throughput single nucleotide polymorphism (SNP) arrays are useful for making genome-wide estimates of copy number and genotype calls. Because neighboring SNPs in high throughput SNP arrays are likely to have dependent copy number and genotype due to the underlying haplotype structure and linkage disequilibrium, hidden Markov models (HMM) may be useful for improving genotype calls and copy number estimates that do not incorporate information from nearby SNPs. We improve previous approaches that utilize a HMM framework for inference in high throughput SNP arrays by integrating copy number, genotype calls, and the corresponding confidence scores when available. Using simulated data, we demonstrate how confidence scores control smoothing in a probabilistic framework. Software for fitting HMMs to SNP array data is available in the R package ICE.

A Bayesian analysis for spatial processes with application to mapping

In geographical epidemiology, maps of disease rates and disease risk provide a spatial perspective for researching disease etiology. For rare diseases or when the population base is small, the rate and risk estimates may be unstable. Empirical Bayesian (EB) methods have been used to spatially smooth the estimates by permitting an area estimate to "borrow strength" from its neighbors. Such EB methods include the use of a Gamma model, of a James-Stein estimator, and of a conditional autoregressive (CAR) process. A fully Bayesian analysis of the CAR process is proposed. One advantage of this fully Bayesian analysis is that it can be implemented simply by using repeated sampling from the posterior densities. Use of a Markov chain Monte Carlo technique such as Gibbs sampler was not necessary. Direct resampling from the posterior densities provides exact small sample inferences instead of the approximate asymptotic analyses of maximum likelihood methods (Clayton & Kaldor, 1987). Further, the proposed CAR model provides for covariates to be included in the model. A simulation demonstrates the effect of sample size on the fully Bayesian analysis of the CAR process. The methods are applied to lip cancer data from Scotland, and the results are compared. ^

Composition of minerals and rocks from the Markov Deep, Central Atlantic

The paper presents materials on composition and texture of weakly serpentinized ultrabasic rocks from the western and eastern walls of the Markov Deep (5°30.6'-5°32.4'N) in the rift valley of the Mid-Atlantic Ridge. Predominant harzburgites with protogranular and porphyroclastic textures contain two major generations of minerals: the first generation composes the bulk of rocks and consists of Ol_89.8-90.4 + En_90.2-90.8 + Di_91.8 + Chr (Cr#32.3-36.6, Mg#67.2-70.0), while the second generation composes very thin branching veinlets and consists of PlAn_32-47 + Ol_74.3-77.1 + Opx_55.7-71.9 + Cpx_67.5 + Amph_53.7-74.2 + Ilm. Syndeformational olivine neoblasts in recrystallization zones are highly magnesian. Concentrations and covariations of major elements in harzburgites indicate that these rocks are depleted in mantle residues (high Mg# of minerals and whole-rock samples and low in CaO, Al2O3, and TiO2) that are significantly enriched in trace HFSE and REE (Zr, Hf, Y, LREE, and all REE). Mineralogy and geochemistry of harzburgites were formed by interaction of mantle residues with hydrous, strongly fractionated melts that impregnated them. Mineral composition of veinlets in harzburgites and mineralogical-geochemical characteristics of related plagiogranites and gabbronorites suggest that these plagiogranites were produced by melt residuals after crystallization of gabbronorites. Modern characteristics of harzburgites were shaped by the following processes: (i) partial melting of mantle material simultaneously with its subsolidus deformations, (ii) brittle-plastic deformations associated with cataclastic flow and recrystallization, and (iii) melt percolation along zones of maximal stress relief and interaction of this melt with magnesian mantle residue.

Sm-Nd, Rb-Sr, and He isotopic composition and hydrocarbon gases in rocks and minerals from the Markov Deep, Mid-Atlantic Ridge rift valley

The paper presents characteristics of the Nd and Sr isotopic systems of ultrabasic rocks, gabbroids, plagiogranites, and their minerals as well as data on helium and hydrocarbons in fluid inclusions of the same samples. Materials presented in this publication were obtained by studying samples dredged from the MAR crest zone at 5°-6°N (U/Pb zircon dating, geochemical and petrological-mineralogical studies). It was demonstrated that variations in the isotopic composition of He entrapped in rocks and minerals were controlled by variable degrees of mixing of juvenile He, which is typical of basaltic glass for MAR (DM source), and atmospheric He. Increase in the atmospheric He fraction in plutonic rocks and, to a lesser degree, in their minerals reflects involvement of seawater or hydrated material of the oceanic crust in magmatic and postmagmatic processes. This conclusion finds further support in positive correlation between the fraction of mantle He (R ratio) and 87Sr/86Sr ratio. High-temperature hydration of ultrabasic rocks (amphibolization) was associated with increase in the fraction of mantle He, while their low-temperature hydration (serpentinization) was accompanied by drastic decrease in this fraction and significant increase in 87Sr/86Sr ratio. Insignificant variations in 143Nd/144Nd (close to 0.5130) and 87Sr/86Sr (0.7035) in most of gabbroids and plagiogranites as well as the fraction of mantle He in these rocks, amphibolites, and their ore minerals indicate that the melts were derived from the depleted mantle. Similar e-Nd values of gabbroids, plagiogranites, and fresh harzburgites (6.77-8.39) suggest that these rocks were genetically related to a single mantle source. e-Nd value of serpentinized lherzolites (2.62) likely reflects relations of these relatively weakly depleted mantle residues to another source. Aforementioned characteristics of the rocks generally reflect various degrees of mixing of depleted mantle components with crustal components (seawater) during metamorphic and hydrothermal processes that accompanied formation of the oceanic crust.

The cutoff phenomenon in finite Markov chains.

Natural mixing processes modeled by Markov chains often show a sharp cutoff in their convergence to long-time behavior. This paper presents problems where the cutoff can be proved (card shuffling, the Ehrenfests' urn). It shows that chains with polynomial growth (drunkard's walk) do not show cutoffs. The best general understanding of such cutoffs (high multiplicity of second eigenvalues due to symmetry) is explored. Examples are given where the symmetry is broken but the cutoff phenomenon persists.

Aspects of Markov Chains and Particle Systems

Thesis (Ph.D.)--University of Washington, 2016-06

Uniqueness criteria for continuous-time Markov chains with general transition structures

We derive necessary and sufficient conditions for the existence of bounded or summable solutions to systems of linear equations associated with Markov chains. This substantially extends a famous result of G. E. H. Reuter, which provides a convenient means of checking various uniqueness criteria for birth-death processes. Our result allows chains with much more general transition structures to be accommodated. One application is to give a new proof of an important result of M. F. Chen concerning upwardly skip-free processes. We then use our generalization of Reuter's lemma to prove new results for downwardly skip-free chains, such as the Markov branching process and several of its many generalizations. This permits us to establish uniqueness criteria for several models, including the general birth, death, and catastrophe process, extended branching processes, and asymptotic birth-death processes, the latter being neither upwardly skip-free nor downwardly skip-free.

On the existence of uni-instantaneous Q-processes with a given finite mu-invariant measure

Let S be a countable set and let Q = (q(ij), i, j is an element of S) be a conservative q-matrix over S with a single instantaneous state b. Suppose that we are given a real number mu >= 0 and a strictly positive probability measure m = (m(j), j is an element of S) such that Sigma(i is an element of S) m(i)q(ij) = -mu m(j), j 0 b. We prove that there exists a Q-process P(t) = (p(ij) (t), i, j E S) for which m is a mu-invariant measure, that is Sigma(i is an element of s) m(i)p(ij)(t) = e(-mu t)m(j), j is an element of S. We illustrate our results with reference to the Kolmogorov 'K 1' chain and a birth-death process with catastrophes and instantaneous resurrection.

Variational Markov Chain Monte Carlo for Bayesian smoothing of non-linear niffusions

In this paper we develop set of novel Markov chain Monte Carlo algorithms for Bayesian smoothing of partially observed non-linear diffusion processes. The sampling algorithms developed herein use a deterministic approximation to the posterior distribution over paths as the proposal distribution for a mixture of an independence and a random walk sampler. The approximating distribution is sampled by simulating an optimized time-dependent linear diffusion process derived from the recently developed variational Gaussian process approximation method. Flexible blocking strategies are introduced to further improve mixing, and thus the efficiency, of the sampling algorithms. The algorithms are tested on two diffusion processes: one with double-well potential drift and another with SINE drift. The new algorithm's accuracy and efficiency is compared with state-of-the-art hybrid Monte Carlo based path sampling. It is shown that in practical, finite sample, applications the algorithm is accurate except in the presence of large observation errors and low observation densities, which lead to a multi-modal structure in the posterior distribution over paths. More importantly, the variational approximation assisted sampling algorithm outperforms hybrid Monte Carlo in terms of computational efficiency, except when the diffusion process is densely observed with small errors in which case both algorithms are equally efficient.

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.

On the Structure of Spatial Branching Processes

The paper is a contribution to the theory of branching processes with discrete time and a general phase space in the sense of [2]. We characterize the class of regular, i.e. in a sense sufficiently random, branching processes (Φk) k∈Z by almost sure properties of their realizations without making any assumptions about stationarity or existence of moments. This enables us to classify the clans of (Φk) into the regular part and the completely non-regular part. It turns out that the completely non-regular branching processes are built up from single-line processes, whereas the regular ones are mixtures of left-tail trivial processes with a Poisson family structure.

Branching Processes in Autoregressive Random Environment

2000 Mathematics Subject Classification: 60J80, 60K05.

A Stochastic Control Approach to a Parabolic Equation, Reciprocal Processes

2000 Mathematics Subject Classi cation: 49L60, 60J60, 93E20.

A Note on the Extinction Problem for Controlled Multitype Branching Processes

2000 Mathematics Subject Classification: 60J80, 60J10.

On the Equality of Sharp and Germ - Fields for Gaussian Processes and Fields

2000 Mathematics Subject Classification: 60G15, 60G60; secondary 31B15, 31B25, 60H15