Modelling micro-texture

There are many instances when texture contains valuable information in images, and various methods have been used for texture analysis. We distinguish between micro-textures and macro-textures. The paper models micro-texture using the general spin Ising model from statistical mechanics. This model allows for any number of grey levels and any set of pair interactions. For a given texture, we select an appropriate set of pair interactions and estimate the correspomding parameter values, using linked cluster expansions of the auto-covariances and the partition function. The series expansions are valid for parameters smaller than the critical parameters for which an infinite system would exhibit a phase transition. Hence, sufficiently small-grained micro-textures may be modelled. To ensure that the data meet this requirement, we simulate the model using the Markov chain Meet Carlo method and estimate its critical parameters using the series expansions. We demonstrate these methods on both real and simulated images.     

Modelling micro-texture

There are many instances when texture contains valuable information in images, and various methods have been used for texture analysis. We distinguish between micro-textures and macro-textures. The paper models micro-texture using the general spin Ising model from statistical mechanics. This model allows for any number of grey levels and any set of pair interactions. For a given texture, we select an appropriate set of pair interactions and estimate the correspomding parameter values, using linked cluster expansions of the auto-covariances and the partition function. The series expansions are valid for parameters smaller than the critical parameters for which an infinite system would exhibit a phase transition. Hence, sufficiently small-grained micro-textures may be modelled. To ensure that the data meet this requirement, we simulate the model using the Markov chain Meet Carlo method and estimate its critical parameters using the series expansions. We demonstrate these methods on both real and simulated images.     

Approximate computations for binary Markov random fields and their use in Bayesian models

Discrete Markov random fields form a natural class of models to represent images and spatial datasets. The use of such models is, however, hampered by a computationally intractable normalising constant. This makes parameter estimation and a fully Bayesian treatment of discrete Markov random fields difficult. We apply approximation theory for pseudo-Boolean functions to binary Markov random fields and construct approximations and upper and lower bounds for the associated computationally intractable normalising constant. As a by-product of this process we also get a partially ordered Markov model approximation of the binary Markov random field. We present numerical examples with both the pairwise interaction Ising model and with higher-order interaction models, showing the quality of our approximations and bounds. We also present simulation examples and one real data example demonstrating how the approximations and bounds can be applied for parameter estimation and to handle a fully Bayesian model computationally.     

Surveillance of the interaction parameter of the ising model

Surveillance to detect changes of spatial patterns is of interest in many areas such as environmental control and regional analysis. Here the interaction parameter of the Ising model, is considered. A minimal sufficient statistic and its asymptotic distribution are used. It is demonstrated that the convergence to normal, distribution is rapid. The main result is that when the lattice is large, all approximations are better in several respects. It is shown that, for large lattice sizes, earlier results on surveillance of a normally distributed random variable can be used in cases of most interest. The expected delay of alarm at a fixed level of false alarm probability is examined for some examples.     

Model selection for infinite variance time series

In this we consider the problem of model selection for infinite variance time series. We introduce a group of model selection critera based on a general loss function Ψ. This family includes various generalizations of predictive least square and AIC Parameter estimation is carried out using Ψ. We use two loss functions commonly used in robust estimation and show that certain criteria out perform the conventional approach based on least squares or Yule-Walker estima­tion for heavy tailed innovations. Our conclusions are based on a comprehensive study of the performance of competing criteria for a wide selection of AR(2) models. We also consider the performance of these techniques when the ‘true’ model is not contained in the family of candidate models.     

Bayesian inference for bivariate generalized linear models in diagnosing renal arterial obstruction

Generalized linear models are well-established generalizations of the linear models used for regression and analysis of variance. They allow flexible mean structures and general distributions, other than the linear link and normal response assumed in regression. Further enhancements using ideas from multivariate analysis improve power and precision by modelling dependencies between response variables. This paper focuses on the specific case of regression models for bivariate Bernoulli responses and investigates their analysis using a Bayesian approach. The important problem of renal arterial obstruction is considered, as a medical application of these models.     

A Monte Carlo approach to quantifying discrepancies between intractable posterior distributions

The computational demand required to perform inference using Markov chain Monte Carlo methods often obstructs a Bayesian analysis. This may be a result of large datasets, complex dependence structures, or expensive computer models. In these instances, the posterior distribution is replaced by a computationally tractable approximation, and inference is based on this working model. However, the error that is introduced by this practice is not well studied. In this paper, we propose a methodology that allows one to examine the impact on statistical inference by quantifying the discrepancy between the intractable and working posterior distributions. This work provides a structure to analyse model approximations with regard to the reliability of inference and computational efficiency. We illustrate our approach through a spatial analysis of yearly total precipitation anomalies where covariance tapering approximations are used to alleviate the computational demand associated with inverting a large, dense covariance matrix.     

On two approximations to the F-distribution: Application to testing for intraclass correlation in family studies

Two approximations to the F-distribution are evaluated in the context of testing for intraclass correlation in the analysis of family data. The evaluation is based on a computation of empirical significance levels and a comparison between p-values associated with these approximations and the corresponding exact p-values. It is found that the approximate methods may give very unsatisfactory results, and exact methods are therefore recommended for general use.     

The Touchard distribution

We present a novel model, which is a two-parameter extension of the Poisson distribution. Its normalizing constant is related to the Touchard polynomials, hence the name of this model. It is a flexible distribution that can account for both under- or overdispersion and concentration of zeros that are frequently found in non-Poisson count data. In contrast to some other generalizations, the Hessian matrix for maximum likelihood estimation of the Touchard parameters has a simple form. We exemplify with three data sets, showing that our suggested model is a competitive candidate for fitting non-Poisson counts.     

Bayesian inference of three-parameter bathtub-shaped lifetime distribution

We investigate a Bayesian inference in the three-parameter bathtub-shaped lifetime distribution which is obtained by adding a power parameter to the two-parameter bathtub-shaped lifetime distribution suggested by Chen (2000). The Bayes estimators under the balanced squared error loss function are derived for three parameters. Then, we have used Lindley's and Tierney–Kadane approximations (see Lindley 1980; Tierney and Kadane 1986) for computing these Bayes estimators. In particular, we propose the explicit form of Lindley's approximation for the model with three parameters. We also give applications with a simulated data set and two real data sets to show the use of discussed computing methods. Finally, concluding remarks are mentioned.     

General saddlepoint approximations and normalizing transformations for multivariate statistics

The use of general saddlepoint approximations is investigated for the problem of approximating the tail probabilities of statistics in multivariate analysis. A method based on normalizing transformations is proposed to prevent po¬tential deficiencies in general saddlepoint approximations. The efficiency of the proposed method is illustrated through examples of the sample correlation     

Approximate Bayesian inference for latent Gaussian models by using integrated nested Laplace approximations

Summary.  Structured additive regression models are perhaps the most commonly used class of models in statistical applications. It includes, among others, (generalized) linear models, (generalized) additive models, smoothing spline models, state space models, semiparametric regression, spatial and spatiotemporal models, log-Gaussian Cox processes and geostatistical and geoadditive models. We consider approximate Bayesian inference in a popular subset of structured additive regression models, latent Gaussian models , where the latent field is Gaussian, controlled by a few hyperparameters and with non-Gaussian response variables. The posterior marginals are not available in closed form owing to the non-Gaussian response variables. For such models, Markov chain Monte Carlo methods can be implemented, but they are not without problems, in terms of both convergence and computational time. In some practical applications, the extent of these problems is such that Markov chain Monte Carlo sampling is simply not an appropriate tool for routine analysis. We show that, by using an integrated nested Laplace approximation and its simplified version, we can directly compute very accurate approximations to the posterior marginals. The main benefit of these approximations is computational: where Markov chain Monte Carlo algorithms need hours or days to run, our approximations provide more precise estimates in seconds or minutes. Another advantage with our approach is its generality, which makes it possible to perform Bayesian analysis in an automatic, streamlined way, and to compute model comparison criteria and various predictive measures so that models can be compared and the model under study can be challenged.     

Exact Goodness‐of‐Fit Testing for the Ising Model

The Ising model is one of the simplest and most famous models of interacting systems. It was originally proposed to model ferromagnetic interactions in statistical physics and is now widely used to model spatial processes in many areas such as ecology, sociology, and genetics, usually without testing its goodness of fit. Here, we propose various test statistics and an exact goodness‐of‐fit test for the finite‐lattice Ising model. The theory of Markov bases has been developed in algebraic statistics for exact goodness‐of‐fit testing using a Monte Carlo approach. However, finding a Markov basis is often computationally intractable. Thus, we develop a Monte Carlo method for exact goodness‐of‐fit testing for the Ising model that avoids computing a Markov basis and also leads to a better connectivity of the Markov chain and hence to a faster convergence. We show how this method can be applied to analyze the spatial organization of receptors on the cell membrane.     

Local sensitivity approximations for selectivity bias

Observational data analysis is often based on tacit assumptions of ignorability or randomness. The paper develops a general approach to local sensitivity analysis for selectivity bias, which aims to study the sensitivity of inference to small departures from such assumptions. If M is a model assuming ignorability, we surround M by a small neighbourhood N defined in the sense of Kullback–Leibler divergence and then compare the inference for models in N with that for M . Interpretable bounds for such differences are developed. Applications to missing data and to observational comparisons are discussed. Local approximations to sensitivity analysis are model robust and can be applied to a wide range of statistical problems.     

Power and Sample Size for Fixed-Effects Inference in Reversible Linear Mixed Models

ABSTRACT

Despite the popularity of the general linear mixed model for data analysis, power and sample size methods and software are not generally available for commonly used test statistics and reference distributions. Statisticians resort to simulations with homegrown and uncertified programs or rough approximations which are misaligned with the data analysis. For a wide range of designs with longitudinal and clustering features, we provide accurate power and sample size approximations for inference about fixed effects in the linear models we call reversible. We show that under widely applicable conditions, the general linear mixed-model Wald test has noncentral distributions equivalent to well-studied multivariate tests. In turn, exact and approximate power and sample size results for the multivariate Hotelling–Lawley test provide exact and approximate power and sample size results for the mixed-model Wald test. The calculations are easily computed with a free, open-source product that requires only a web browser to use. Commercial software can be used for a smaller range of reversible models. Simple approximations allow accounting for modest amounts of missing data. A real-world example illustrates the methods. Sample size results are presented for a multicenter study on pregnancy. The proposed study, an extension of a funded project, has clustering within clinic. Exchangeability among the participants allows averaging across them to remove the clustering structure. The resulting simplified design is a single-level longitudinal study. Multivariate methods for power provide an approximate sample size. All proofs and inputs for the example are in the supplementary materials (available online).     

Exact and approximate distributions for linear contrasts of means and variances

A Bayesian analysis is presented for the K-group Behrens-Fisher problem. Both exact posterior distributions and approximations were developed for both a general linear contrast of the K means and the K variances, given either proper diffuse or informative conjugate priors. The contrast of variances is a unique feature of the heterogeneous variance model that enables investigators to test specific effects of experimental manipulations on variance. Finally, important-differences were observed between the heterogeneous variance model and the homogeneous model.     

Simple least squares estimation versus best linear unbiased prediction

Necessary and sufficient conditions are developed for the simple least squares estimator to coincide with the best linear unbiased predictor. The conditions obtained are valid for a general linear model and are generalizations of the condition given by Watson (1972). Also, as a preliminary result, a new representation of the best linear unbiased predictor is established.     

Local model uncertainty and incomplete-data bias (with discussion)

Summary.  Problems of the analysis of data with incomplete observations are all too familiar in statistics. They are doubly difficult if we are also uncertain about the choice of model. We propose a general formulation for the discussion of such problems and develop approximations to the resulting bias of maximum likelihood estimates on the assumption that model departures are small. Loss of efficiency in parameter estimation due to incompleteness in the data has a dual interpretation: the increase in variance when an assumed model is correct; the bias in estimation when the model is incorrect. Examples include non-ignorable missing data, hidden confounders in observational studies and publication bias in meta-analysis. Doubling variances before calculating confidence intervals or test statistics is suggested as a crude way of addressing the possibility of undetectably small departures from the model. The problem of assessing the risk of lung cancer from passive smoking is used as a motivating example.     

The complex Watson distribution and shape analysis

The complex Watson distribution is an important simple distribution on the complex sphere which is used in statistical shape analysis. We describe the density, obtain the integrating constant and provide large sample approximations. Maximum likelihood estimation and hypothesis testing procedures for one and two samples are described. The particular connection with shape analysis is discussed and we consider an application examining shape differences between normal and schizophrenic brains. We make some observations about Bayesian shape inference and finally we describe a more general rotationally symmetric family of distributions.