Centered parameterizations and dependence limitations in Markov random field models

Markov random field models incorporate terms representing local statistical dependence among variables in a discrete-index random field. Traditional parameterizations for models based on one-parameter exponential family conditional distributions contain components that would appear to reflect large-scale and small-scale model behaviors, and it is natural to attempt to match these structures with large-scale and small-scale patterns in a set of data. Traditional manners of parameterizing Markov random field models do not allow such correspondence, however. We propose an alternative centered parameterization that, while not leading to different models, allows a correspondence between model structures and data structures to be successfully accomplished. The ability to make these connections is important when incorporating covariate information into a model or if a sequence of models is fit over time to investigate and interpret possible changes in data structure. We demonstrate the improved interpretation that results from use of centered parameterizations. Centered parameterizations also lend themselves to computation of an interpretable decomposition of mean squared error, and this is demonstrated both analytically and through a simulated example. A breakdown in model behavior occurs even with centered parameterizations if dependence parameters in Markov random field models are allowed to become too large. This phenomenon is discussed and illustrated using an auto-logistic model.     

Wavelet goodness-of-fit test for dependent data

We introduce a new goodness-of-fit test which can be applied to hypothesis testing about the marginal distribution of dependent data. We derive a new test for the equivalent hypothesis in the space of wavelet coefficients. Such properties of the wavelet transform as orthogonality, localisation and sparsity make the hypothesis testing in wavelet domain easier than in the domain of distribution functions. We propose to test the null hypothesis separately at each wavelet decomposition level to overcome the problem of bi-dimensionality of wavelet indices and to be able to find the frequency where the empirical distribution function differs from the null in case the null hypothesis is rejected. We suggest a test statistic and state its asymptotic distribution under the null and under some of the alternative hypotheses.     

Linear and nonlinear causality tests in an LSTAR model: wavelet decomposition in a nonlinear environment

In this paper, we use simulated data to investigate the power of different causality tests in a two-dimensional vector autoregressive (VAR) model. The data are presented in a nonlinear environment that is modelled using a logistic smooth transition autoregressive function. We use both linear and nonlinear causality tests to investigate the unidirection causality relationship and compare the power of these tests. The linear test is the commonly used Granger causality F test. The nonlinear test is a non-parametric test based on Baek and Brock [A general test for non-linear Granger causality: Bivariate model. Tech. Rep., Iowa State University and University of Wisconsin, Madison, WI, 1992] and Hiemstra and Jones [Testing for linear and non-linear Granger causality in the stock price–volume relation, J. Finance 49(5) (1994), pp. 1639–1664]. When implementing the nonlinear test, we use separately the original data, the linear VAR filtered residuals, and the wavelet decomposed series based on wavelet multiresolution analysis. The VAR filtered residuals and the wavelet decomposition series are used to extract the nonlinear structure of the original data. The simulation results show that the non-parametric test based on the wavelet decomposition series (which is a model-free approach) has the highest power to explore the causality relationship in nonlinear models.     

Federal Statistics in a Complex Environment: The Case of the 1980 Census

The taking and the interpretation of something as big and as complicated as the national census is more than an exercise in statistical thinking. It involves other diverse fields such as ethics, epistemology, law, and politics. This article shows that a national census is more akin to so-called ill-structured problems. Unlike well-structured problems, the formulation of an ill-structured problem varies from field to field and from person to person, and the various aspects of an ill-structured problem (i.e., ethics, epistemology, etc.) cannot be clearly separated from one another. The 1980 census is discussed as an ill-structured problem, and a method for treating such problems is presented, within which statistical information is only one component.     

Hidden Markov mesh random field models in image analysis

This paper addresses the image modeling problem under the assumption that images can be represented by third-order, hidden Markov mesh random field models. The range of applications of the techniques described hereafter comprises the restoration of binary images, the modeling and compression of image data, as well as the segmentation of gray-level or multi-spectral images, and image sequences under the short-range motion hypothesis. We outline coherent approaches to both the problems of image modeling (pixel labeling) and estimation of model parameters (learning). We derive a real-time labeling algorithm-based on a maximum, marginal a posteriori probability criterion-for a hidden third-order Markov mesh random field model. Our algorithm achieves minimum time and space complexities simultaneously, and we describe what we believe to be the most appropriate data structures to implement it. Critical aspects of the computer simulation of a real-time implementation are discussed, down to the computer code level. We develop an (unsupervised) learning technique by which the model parameters can be estimated without ground truth information. We lay bare the conditions under which our approach can be made time-adaptive in order to be able to cope with short-range motion in dynamic image sequences. We present extensive experimental results for both static and dynamic images from a wide variety of sources. They comprise standard, infra-red and aerial images, as well as a sequence of ultrasound images of a fetus and a series of frames from a motion picture sequence. These experiments demonstrate that the method is subjectively relevant to the problems of image restoration, segmentation and modeling.     

Equivalence of Gaussian measures for some nonstationary random fields

Gaussian random fields whose covariance structures are described by a power law model provide a simple and flexible class of models for isotropic random fields. This class includes fractional Brownian fields as a special case. Because these random fields are nonstationary, the extensive results available on equivalence of Gaussian measures for stationary models do not apply to them. This work shows that results on equivalence for two stationary Gaussian random field models extend in a natural way to the equivalence of a stationary model and a power law model. This result is used to show that if we use a power law model for predicting a random field at unobserved locations when in fact the random field is stationary, we can obtain asymptotically optimal predictions as long as the high frequency behavior of the true spectral density is sufficiently close to the high frequency behavior of the spectral density of the power law model.     

Fully Bayesian Binary Markov Random Field Models: Prior Specification and Posterior Simulation

We propose a flexible prior model for the parameters of binary Markov random fields (MRF), defined on rectangular lattices and with maximal cliques defined from a template maximal clique. The prior model allows higher‐order interactions to be included. We also define a reversible jump Markov chain Monte Carlo algorithm to sample from the associated posterior distribution. The number of possible parameters for a higher‐order MRF becomes high, even for small template maximal cliques. We define a flexible parametric form where the parameters have interpretation as potentials for clique configurations, and limit the effective number of parameters by assigning apriori discrete probabilities for events where groups of parameter values are equal. To cope with the computationally intractable normalising constant of MRFs, we adopt a previously defined approximation of binary MRFs. We demonstrate the flexibility of our prior formulation with simulated and real data examples.     

Wavelet thresholding via a Bayesian approach

We discuss a Bayesian formalism which gives rise to a type of wavelet threshold estimation in nonparametric regression. A prior distribution is imposed on the wavelet coefficients of the unknown response function, designed to capture the sparseness of wavelet expansion that is common to most applications. For the prior specified, the posterior median yields a thresholding procedure. Our prior model for the underlying function can be adjusted to give functions falling in any specific Besov space. We establish a relationship between the hyperparameters of the prior model and the parameters of those Besov spaces within which realizations from the prior will fall. Such a relationship gives insight into the meaning of the Besov space parameters. Moreover, the relationship established makes it possible in principle to incorporate prior knowledge about the function's regularity properties into the prior model for its wavelet coefficients. However, prior knowledge about a function's regularity properties might be difficult to elicit; with this in mind, we propose a standard choice of prior hyperparameters that works well in our examples. Several simulated examples are used to illustrate our method, and comparisons are made with other thresholding methods. We also present an application to a data set that was collected in an anaesthesiological study.     

Wavelet fuzzy hybrid model for physico-financial signals

In the present paper, a fuzzy logic-based method is combined with wavelet decomposition to develop a step-by-step dynamic hybrid model to analyze and approximate one-dimensional physico-financial signals characterized by fuzzy values. Computational tests based on a well-known signal and conducted with the pure fuzzy model, the wavelet one and the new hybrid model, are developed and result in an efficient hybrid one.     

Bayesian texture segmentation of weed and crop images using reversible jump Markov chain Monte Carlo methods

Summary. A Bayesian method for segmenting weed and crop textures is described and implemented. The work forms part of a project to identify weeds and crops in images so that selective crop spraying can be carried out. An image is subdivided into blocks and each block is modelled as a single texture. The number of different textures in the image is assumed unknown. A hierarchical Bayesian procedure is used where the texture labels have a Potts model (colour Ising Markov random field) prior and the pixels within a block are distributed according to a Gaussian Markov random field, with the parameters dependent on the type of texture. We simulate from the posterior distribution by using a reversible jump Metropolis–Hastings algorithm, where the number of different texture components is allowed to vary. The methodology is applied to a simulated image and then we carry out texture segmentation on the weed and crop images that motivated the work.     

具有GARCH（1，1）-正态误差项的非对称单位根检验研究

摘　　要:理论研究表明许多经济变量呈现出非对称的门限自回归（TAR）或动态门限自回归（M-TAR）数据生成机制,因而非对称单位根检验就成为该领域的主要研究方向之一。本文对非对称单位根检验Enders-Granger方法在GARCH（1,1）-正态误差项下的检验水平与检验势作了系统的仿真研究。研究表明：GARCH（1,1）-正态误差项的TAR或M-TAR模型会对该方法的检验水平和检验势产生重要影响。     

On the use of different metrics for assessing complex pattern reproductions

Nowadays, there is an increasing interest in multi-point models and their applications in Earth sciences. However, users not only ask for multi-point methods able to capture the uncertainties of complex structures and to reproduce the properties of a training image, but also they need quantitative tools for assessing whether a set of realizations have the properties required. Moreover, it is crucial to study the sensitivity of the realizations to the size of the data template and to analyze how fast realization-based statistics converge on average toward training-based statistics. In this paper, some similarity measures and convergence indexes, based on some physically measurable quantities and cumulants of high-order, are presented. In the case study, multi-point simulations of the spatial distribution of coarse-grained limestone and calcareous rock, generated by using three templates of different sizes, are compared and convergence toward training-based statistics is analyzed by taking into account increasing numbers of realizations.     

Bayesian image analysis with Markov chain Monte Carlo and coloured continuum triangulation models

It is now possible to carry out Bayesian image segmentation from a continuum parametric model with an unknown number of regions. However, few suitable parametric models exist. We set out to model processes which have realizations that are naturally described by coloured planar triangulations. Triangulations are already used, to represent image structure in machine vision, and in finite element analysis, for domain decomposition. However, no normalizable parametric model, with realizations that are coloured triangulations, has been specified to date. We show how this must be done, and in particular we prove that a normalizable measure on the space of triangulations in the interior of a fixed simple polygon derives from a Poisson point process of vertices. We show how such models may be analysed by using Markov chain Monte Carlo methods and we present two case-studies, including convergence analysis.     

Application of wavelets to the pre-processing of underwater sounds

In this paper we consider data on underwater sounds of differing types. Our objective is to filter background noise and achieve an acceptable level of reduction in the raw data, whilst at the same time maintaining the main features of the original signal. In particular, we consider data compression through the use of wavelet analysis followed by a thresholding of small coefficients in the resulting multiresolution decomposition. Various methods to threshold the wavelet representation are discussed and compared using recordings of dolphin sounds. An empirical modification to one of them is also proposed which shows promise in better preserving certain structures in our particular sound data.     

Wavelet Improvement of the Over-Rejection of Unit Root Test Under GARCH Errors: An Application to Swedish Immigration Data

In this article, we use the wavelet technique to improve the over-rejection problem of the traditional Dickey–Fuller tests for unit root when the data is associated with volatility like the GARCH(1, 1) effect. The logic of this technique is based on the idea that the wavelet spectrum decomposition can separate out information of different frequencies in the data series. We prove that the asymptotic distribution of the test in the wavelet environment is still the same as the traditional Dickey–Fuller type of tests. The finite sample property is improved when the data suffers from GARCH error. The investigation of the size property and the finite sample distribution of the test is carried out by Monte Carlo experiment. An empirical example with data on the net immigration to Sweden during the period 1950–2000 is used to illustrate the performance of the wavelet improved test under GARCH errors. The results reveal that using the traditional Dickey–Fuller type of tests, the unit root hypothesis is rejected while our wavelet improved test do not reject as it is more robust to GARCH errors in finite samples.     

Multiscale Bayesian state-space model for Granger causality analysis of brain signal

Modelling time-varying and frequency-specific relationships between two brain signals is becoming an essential methodological tool to answer theoretical questions in experimental neuroscience. In this article, we propose to estimate a frequency Granger causality statistic that may vary in time in order to evaluate the functional connections between two brain regions during a task. We use for that purpose an adaptive Kalman filter type of estimator of a linear Gaussian vector autoregressive model with coefficients evolving over time. The estimation procedure is achieved through variational Bayesian approximation and is extended for multiple trials. This Bayesian State Space (BSS) model provides a dynamical Granger-causality statistic that is quite natural. We propose to extend the BSS model to include the à trous Haar decomposition. This wavelet-based forecasting method is based on a multiscale resolution decomposition of the signal using the redundant à trous wavelet transform and allows us to capture short- and long-range dependencies between signals. Equally importantly it allows us to derive the desired dynamical and frequency-specific Granger-causality statistic. The application of these models to intracranial local field potential data recorded during a psychological experimental task shows the complex frequency-based cross-talk between amygdala and medial orbito-frontal cortex.     

A frequency domain comparison of estimation methods for gaussian fields

This paper demonstrates that well-known parameter estimation methods for Gaussian fields place different emphasis on the high and low frequency components of the data. As a consequence, the relative importance of the frequencies under the objective of the analysis should be taken into account when selecting an estimation method, in addition to other considerations such as statistical and computational efficiency. The paper also shows that when noise is added to the Gaussian field, maximum pseudolikelihood automatically sets the smoothing parameter of the model equal to one. A simulation study then indicates that generalised cross-validation is more robust than maximum likelihood un-

der model misspecification in smoothing and image restoration problems. This has implications for Bayesian procedures since these use the same weightings of the frequencies as the likelihood.     

基于数据汇总的普查调查框误差研究

作为一种全面调查,普查数据的生产过程可以视为由个体数据汇总为总量数据的过程.为了开展普查数据质量评估与控制研究,从中国普查调查实施过程共性出发,构建普查数据汇总模型的一般形式,并以此为基础界定普查调查框及其作用,将普查划分为两种类型;同时从普查数据汇总的角度论述普查调查框误差的量化形式,进一步完善单位清查(清查摸底)环节在普查数据汇总中的理论意义.     

Denoising ozone concentration measurements with BAMS filtering

We propose a method for filtering self-similar geophysical signals infected by an autoregressive noise using a combination of non-decimated wavelet transform and a Bayesian model. In the application part, we consider separating the instrumentation noise from high frequency ozone concentration measurements sampled in the atmospheric boundary layer. The elicitation of priors needed to specify the statistical model in this application is guided by the well-known Kolmogorov K41-theory, which describes the statistical structure of turbulent high frequency scalar concentration fluctuations.