Asymptotic and structural properties of special cases of the Wright function arising in probability theory

This analysis paper presents previously unknown properties of some special cases of the Wright function whose consideration is necessitated by our work on probability theory and the theory of stochastic processes. Specifically, we establish new asymptotic properties of the particular Wright function 1Ψ1(ρ, k; ρ, 0; x) = X∞ n=0 Γ(k + ρn) Γ(ρn) x n n! (|x| < ∞) when the parameter ρ ∈ (−1, 0)∪(0, ∞) and the argument x is real. In the probability theory applications, which are focused on studies of the Poisson-Tweedie mixtures, the parameter k is a non-negative integer. Several representations involving well-known special functions are given for certain particular values of ρ. The asymptotics of 1Ψ1(ρ, k; ρ, 0; x) are obtained under numerous assumptions on the behavior of the arguments k and x when the parameter ρ is both positive and negative. We also provide some integral representations and structural properties involving the ‘reduced’ Wright function 0Ψ1(−−; ρ, 0; x) with ρ ∈ (−1, 0) ∪ (0, ∞), which might be useful for the derivation of new properties of members of the power-variance family of distributions. Some of these imply a reflection principle that connects the functions 0Ψ1(−−;±ρ, 0; ·) and certain Bessel functions. Several asymptotic relationships for both particular cases of this function are also given. A few of these follow under additional constraints from probability theory results which, although previously available, were unknown to analysts.

Fine mapping and replication of QTL in outbred chicken advanced intercross lines

Background: Linkage mapping is used to identify genomic regions affecting the expression of complex traits. However, when experimental crosses such as F2 populations or backcrosses are used to map regions containing a Quantitative Trait Locus (QTL), the size of the regions identified remains quite large, i.e. 10 or more Mb. Thus, other experimental strategies are needed to refine the QTL locations. Advanced Intercross Lines (AIL) are produced by repeated intercrossing of F2 animals and successive generations, which decrease linkage disequilibrium in a controlled manner. Although this approach is seen as promising, both to replicate QTL analyses and fine-map QTL, only a few AIL datasets, all originating from inbred founders, have been reported in the literature. Methods: We have produced a nine-generation AIL pedigree (n = 1529) from two outbred chicken lines divergently selected for body weight at eight weeks of age. All animals were weighed at eight weeks of age and genotyped for SNP located in nine genomic regions where significant or suggestive QTL had previously been detected in the F2 population. In parallel, we have developed a novel strategy to analyse the data that uses both genotype and pedigree information of all AIL individuals to replicate the detection of and fine-map QTL affecting juvenile body weight. Results: Five of the nine QTL detected with the original F2 population were confirmed and fine-mapped with the AIL, while for the remaining four, only suggestive evidence of their existence was obtained. All original QTL were confirmed as a single locus, except for one, which split into two linked QTL. Conclusions: Our results indicate that many of the QTL, which are genome-wide significant or suggestive in the analyses of large intercross populations, are true effects that can be replicated and fine-mapped using AIL. Key factors for success are the use of large populations and powerful statistical tools. Moreover, we believe that the statistical methods we have developed to efficiently study outbred AIL populations will increase the number of organisms for which in-depth complex traits can be analyzed.

Increasing the power of genome wide association studies in natural populations using repeated measures : evaluation and implementation

1. Genomewide association studies (GWAS) enable detailed dissections of the genetic basis for organisms' ability to adapt to a changing environment. In long-term studies of natural populations, individuals are often marked at one point in their life and then repeatedly recaptured. It is therefore essential that a method for GWAS includes the process of repeated sampling. In a GWAS, the effects of thousands of single-nucleotide polymorphisms (SNPs) need to be fitted and any model development is constrained by the computational requirements. A method is therefore required that can fit a highly hierarchical model and at the same time is computationally fast enough to be useful. 2. Our method fits fixed SNP effects in a linear mixed model that can include both random polygenic effects and permanent environmental effects. In this way, the model can correct for population structure and model repeated measures. The covariance structure of the linear mixed model is first estimated and subsequently used in a generalized least squares setting to fit the SNP effects. The method was evaluated in a simulation study based on observed genotypes from a long-term study of collared flycatchers in Sweden. 3. The method we present here was successful in estimating permanent environmental effects from simulated repeated measures data. Additionally, we found that especially for variable phenotypes having large variation between years, the repeated measurements model has a substantial increase in power compared to a model using average phenotypes as a response. 4. The method is available in the R package RepeatABEL. It increases the power in GWAS having repeated measures, especially for long-term studies of natural populations, and the R implementation is expected to facilitate modelling of longitudinal data for studies of both animal and human populations.

On statistical bounds of heuristic solutions to location problems

Solutions to combinatorial optimization problems, such as problems of locating facilities, frequently rely on heuristics to minimize the objective function. The optimum is sought iteratively and a criterion is needed to decide when the procedure (almost) attains it. Pre-setting the number of iterations dominates in OR applications, which implies that the quality of the solution cannot be ascertained. A small, almost dormant, branch of the literature suggests using statistical principles to estimate the minimum and its bounds as a tool to decide upon stopping and evaluating the quality of the solution. In this paper we examine the functioning of statistical bounds obtained from four different estimators by using simulated annealing on p-median test problems taken from Beasley’s OR-library. We find the Weibull estimator and the 2nd order Jackknife estimator preferable and the requirement of sample size to be about 10 being much less than the current recommendation. However, reliable statistical bounds are found to depend critically on a sample of heuristic solutions of high quality and we give a simple statistic useful for checking the quality. We end the paper with an illustration on using statistical bounds in a problem of locating some 70 distribution centers of the Swedish Post in one Swedish region. 

MAPfastR : quantitative trait loci mapping in outbred line crosses

MAPfastR is a software package developed to analyze QTL data from inbred and outbred line-crosses. The package includes a number of modules for fast and accurate QTL analyses. It has been developed in the R language for fast and comprehensive analyses of large datasets. MAPfastR is freely available at: http://www.computationalgenetics.se/?page_id=7.

Hglm: A package for fitting hierarchical generalized linear models

We present the hglm package for fitting hierarchical generalized linear models. It can be used for linear mixed models and generalized linear mixed models with random effects for a variety of links and a variety of distributions for both the outcomes and the random effects. Fixed effects can also be fitted in the dispersion part of the model.

Genetic heterogeneity of residual variance : estimation of variance components using double hierarchical generalized linear models

Background: The sensitivity to microenvironmental changes varies among animals and may be under genetic control. It is essential to take this element into account when aiming at breeding robust farm animals. Here, linear mixed models with genetic effects in the residual variance part of the model can be used. Such models have previously been fitted using EM and MCMC algorithms. Results: We propose the use of double hierarchical generalized linear models (DHGLM), where the squared residuals are assumed to be gamma distributed and the residual variance is fitted using a generalized linear model. The algorithm iterates between two sets of mixed model equations, one on the level of observations and one on the level of variances. The method was validated using simulations and also by re-analyzing a data set on pig litter size that was previously analyzed using a Bayesian approach. The pig litter size data contained 10,060 records from 4,149 sows. The DHGLM was implemented using the ASReml software and the algorithm converged within three minutes on a Linux server. The estimates were similar to those previously obtained using Bayesian methodology, especially the variance components in the residual variance part of the model. Conclusions: We have shown that variance components in the residual variance part of a linear mixed model can be estimated using a DHGLM approach. The method enables analyses of animal models with large numbers of observations. An important future development of the DHGLM methodology is to include the genetic correlation between the random effects in the mean and residual variance parts of the model as a parameter of the DHGLM.

Likelihood prediction for generalized linear mixed models under covariate uncertainty

This paper presents the techniques of likelihood prediction for the generalized linear mixed models. Methods of likelihood prediction is explained through a series of examples; from a classical one to more complicated ones. The examples show, in simple cases, that the likelihood prediction (LP) coincides with already known best frequentist practice such as the best linear unbiased predictor. The paper outlines a way to deal with the covariate uncertainty while producing predictive inference. Using a Poisson error-in-variable generalized linear model, it has been shown that in complicated cases LP produces better results than already know methods.

How does the use of different road networks effect the optimal location of facilities in rural areas?

The p-median problem is often used to locate P service facilities in a geographically distributed population. Important for the performance of such a model is the distance measure. Distance measure can vary if the accuracy of the road network varies. The rst aim in this study is to analyze how the optimal location solutions vary, using the p-median model, when the road network is alternated. It is hard to nd an exact optimal solution for p-median problems. Therefore, in this study two heuristic solutions are applied, simulating annealing and a classic heuristic. The secondary aim is to compare the optimal location solutions using dierent algorithms for large p-median problem. The investigation is conducted by the means of a case study in a rural region with an asymmetrically distributed population, Dalecarlia. The study shows that the use of more accurate road networks gives better solutions for optimal location, regardless what algorithm that is used and regardless how many service facilities that is optimized for. It is also shown that the simulated annealing algorithm not just is much faster than the classic heuristic used here, but also in most cases gives better location solutions.

Parameter estimation of Poisson generalized linear mixed models based on three different statistical principles : a simulation study

Generalized linear mixed models are flexible tools for modeling non-normal data and are useful for accommodating overdispersion in Poisson regression models with random effects. Their main difficulty resides in the parameter estimation because there is no analytic solution for the maximization of the marginal likelihood. Many methods have been proposed for this purpose and many of them are implemented in software packages. The purpose of this study is to compare the performance of three different statistical principles - marginal likelihood, extended likelihood, Bayesian analysis-via simulation studies. Real data on contact wrestling are used for illustration.

GTM: the generative topographic mapping

This thesis describes the Generative Topographic Mapping (GTM) --- a non-linear latent variable model, intended for modelling continuous, intrinsically low-dimensional probability distributions, embedded in high-dimensional spaces. It can be seen as a non-linear form of principal component analysis or factor analysis. It also provides a principled alternative to the self-organizing map --- a widely established neural network model for unsupervised learning --- resolving many of its associated theoretical problems. An important, potential application of the GTM is visualization of high-dimensional data. Since the GTM is non-linear, the relationship between data and its visual representation may be far from trivial, but a better understanding of this relationship can be gained by computing the so-called magnification factor. In essence, the magnification factor relates the distances between data points, as they appear when visualized, to the actual distances between those data points. There are two principal limitations of the basic GTM model. The computational effort required will grow exponentially with the intrinsic dimensionality of the density model. However, if the intended application is visualization, this will typically not be a problem. The other limitation is the inherent structure of the GTM, which makes it most suitable for modelling moderately curved probability distributions of approximately rectangular shape. When the target distribution is very different to that, theaim of maintaining an `interpretable' structure, suitable for visualizing data, may come in conflict with the aim of providing a good density model. The fact that the GTM is a probabilistic model means that results from probability theory and statistics can be used to address problems such as model complexity. Furthermore, this framework provides solid ground for extending the GTM to wider contexts than that of this thesis.

Spatial modeling of data with excessive zeros applied to reindeer pellet-group counts

We analyze a real data set pertaining to reindeer fecal pellet-group counts obtained from a survey conducted in a forest area in northern Sweden. In the data set, over 70% of counts are zeros, and there is high spatial correlation. We use conditionally autoregressive random effects for modeling of spatial correlation in a Poisson generalized linear mixed model (GLMM), quasi-Poisson hierarchical generalized linear model (HGLM), zero-inflated Poisson (ZIP), and hurdle models. The quasi-Poisson HGLM allows for both under- and overdispersion with excessive zeros, while the ZIP and hurdle models allow only for overdispersion. In analyzing the real data set, we see that the quasi-Poisson HGLMs can perform better than the other commonly used models, for example, ordinary Poisson HGLMs, spatial ZIP, and spatial hurdle models, and that the underdispersed Poisson HGLMs with spatial correlation fit the reindeer data best. We develop R codes for fitting these models using a unified algorithm for the HGLMs. Spatial count response with an extremely high proportion of zeros, and underdispersion can be successfully modeled using the quasi-Poisson HGLM with spatial random effects.

A Microdata Analysis Approach to Transport Infrastructure Maintenance

Maintenance of transport infrastructure assets is widely advocated as the key in minimizing current and future costs of the transportation network. While effective maintenance decisions are often a result of engineering skills and practical knowledge, efficient decisions must also account for the net result over an asset's life-cycle. One essential aspect in the long term perspective of transport infrastructure maintenance is to proactively estimate maintenance needs. In dealing with immediate maintenance actions, support tools that can prioritize potential maintenance candidates are important to obtain an efficient maintenance strategy. This dissertation consists of five individual research papers presenting a microdata analysis approach to transport infrastructure maintenance. Microdata analysis is a multidisciplinary field in which large quantities of data is collected, analyzed, and interpreted to improve decision-making. Increased access to transport infrastructure data enables a deeper understanding of causal effects and a possibility to make predictions of future outcomes. The microdata analysis approach covers the complete process from data collection to actual decisions and is therefore well suited for the task of improving efficiency in transport infrastructure maintenance. Statistical modeling was the selected analysis method in this dissertation and provided solutions to the different problems presented in each of the five papers. In Paper I, a time-to-event model was used to estimate remaining road pavement lifetimes in Sweden. In Paper II, an extension of the model in Paper I assessed the impact of latent variables on road lifetimes; displaying the sections in a road network that are weaker due to e.g. subsoil conditions or undetected heavy traffic. The study in Paper III incorporated a probabilistic parametric distribution as a representation of road lifetimes into an equation for the marginal cost of road wear. Differentiated road wear marginal costs for heavy and light vehicles are an important information basis for decisions regarding vehicle miles traveled (VMT) taxation policies. In Paper IV, a distribution based clustering method was used to distinguish between road segments that are deteriorating and road segments that have a stationary road condition. Within railway networks, temporary speed restrictions are often imposed because of maintenance and must be addressed in order to keep punctuality. The study in Paper V evaluated the empirical effect on running time of speed restrictions on a Norwegian railway line using a generalized linear mixed model.

Probability Theory. Stochastic Processes And Their Applications Probabilistic Analysis Of Some Queueing And Inventory Models

In this thesis we attempt to make a probabilistic analysis of some physically realizable, though complex, storage and queueing models. It is essentially a mathematical study of the stochastic processes underlying these models. Our aim is to have an improved understanding of the behaviour of such models, that may widen their applicability. Different inventory systems with randon1 lead times, vacation to the server, bulk demands, varying ordering levels, etc. are considered. Also we study some finite and infinite capacity queueing systems with bulk service and vacation to the server and obtain the transient solution in certain cases. Each chapter in the thesis is provided with self introduction and some important references