Adjusted Maximum Likelihood and Pseudo-Likelihood Estimation for Noisy Gaussian Markov Random Fields

Dryden, Ian L.; Ippoliti, Luigi; Romagnoli, Luca

doi:10.1198/106186002760180563

Cited by 31 publications

(23 citation statements)

References 20 publications

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“…As an alternative to a maximum likelihood procedure, a pseudo-likelihood estimation procedure, which permits a trade-off between efficiency and simplicity, may also be used for parameter estimation [26,32,33] .…”

Section: Torus Lattice Processesmentioning

confidence: 99%

Inverse Correlations for Multiple Time Series and Gaussian Random Fields and Measures of Their Linear Determinism

Majdi¹

2005

Journal of Mathematics and Statistics

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For a discrete-time vector linear stationary process, {X(t)}, admitting forward and backward autoregressive representations, the variance matrix of an optimal linear interpolator of X(t), based on a knowledge of {X(t-j), j≠0}, is known to be given by Ri (0) The relationship between the matrix A and the corresponding matrix, P, obtained by considering only an optimal one-step linear predictor of X(t) from a knowledge of its infinite past, {X(t-j),j>0}, is also discussed. The possible role the inverse correlation function may have for model specification of a vector ARMA model is explored. Close parallels between the problem of interpolation for a stationary univariate two-dimensional Gaussian random field and time series are examined and an index of linear determinism for the latter class of processes is also defined. An application of this index for model specification and diagnostic testing of a Gaussian Markov Random Field is investigated together with the question of its estimation from observed data. Results are illustrated by a simulation study.

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Section: Torus Lattice Processesmentioning

confidence: 99%

Inverse Correlations for Multiple Time Series and Gaussian Random Fields and Measures of Their Linear Determinism

Majdi¹

2005

Journal of Mathematics and Statistics

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“…Especially in the context of image analysis, several methods have been proposed and most of them assume a Gaussian distributed noise (Close and Whiting, 1996;Lee and Hoppel, 1989;Meer et al, 1990). To illustrate the importance of how the estimation of the signal parameters might be affected by an additive Gaussian error, we present an example of a zero-mean homogeneous Gauss-Markov Random Field (GMRF) (Besag, 1974;Dryden et al, 2002). Here, given an image, and assuming a raster-scan enumeration for the pixels, the distribution of the signal X(s i ) is Gaussian with respective conditional mean and variance…”

Section: Introductionmentioning

confidence: 99%

“…While for a second-order homogeneous GMRF, we have to estimate four spatial interaction parameters, β h , β v , together with β ld and β rd for diagonally adjacent neighbours in North-East and South-East directions. To give a flavour of types of behaviour that the maximum likelihood estimator exhibits under the presence of an additive measurement Gaussian white noise with variance σ 2 ε = 400, we have simulated (Dryden et al, 2002) 100 samples of a second-order GMRF with parameters β h = 0.2, β v = 0, β ld = 0.2, β rd = 0 and τ 2 = 1600. Assuming toroidal boundary conditions (Besag, 1977), Table 1 shows the mean, the standard error and Root Mean Squared Error (RMSE) of the maximum likelihood (ML) parameter estimation for 64 × 64 images.…”

Section: Introductionmentioning

confidence: 99%

“…However, for the noisy case, Dryden et al (2002) show that provided σ 2 ε is known, it is relatively straightforward to obtain adjusted maximum likelihood and pseudo-maximum likelihood estimators which supply very good results in terms of mean squared error. As an example, Table 2 gives the results of estimation from the Adjusted Maximum Likelihood Estimator (AMLE) for the same set of simulations.…”

Section: Introductionmentioning

confidence: 99%

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A noise estimation method for corrupted correlated data

Ippoliti

Romagnoli

Fontanella

2005

JISS

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Data acquisition, both in time and in spatial domains, in many cases yields observations with a measurement error. The identification of such a component, that masks the phenomenon under study (signal), must be carried out before the model of interest is specified. The objective of the paper is to propose an estimator for the parameters of an additive noise and compare it with existing methods by applications to both simulated and real data sets.

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“…Furthermore, we may also construct an MRF over the raw a posteriori probabilities using the results from Comets (1992), or over the z-scores using the Gaussian MRF results in Dryden et al (2002) and Rue and Held (2005); the latter approach can be viewed as a Frequentist alternative to the Bayesian Gaussian MRF mixture model approaches of Wei and Pan (2008) and Wei and Pan (2010). Additionally, we can combine both the noncontexual and MRF phases of estimation into one, in the manner of Ng and McLachlan (2004), Qian and Titterington (1989), and Qian and Titterington (1991) (see Section 13.8 of McLachlan (1992) for details); this approach can also be viewed as a Frequentist version of the Bayesian discrete MRF mixture model of Wei and Pan (2010).…”

Section: Chaptermentioning

confidence: 99%

Finite mixture models for regression problems

Nguyen¹

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Finite mixture models (FMMs) are a ubiquitous tool for the analysis of heterogeneous data across a broad number of fields including agriculture, bioinformatics, botany, cell biology, economics, fisheries research, genetics, genomics, geology, machine learning, medicine, palaeontology, psychology, and zoology, among many others. Due to their flexibility, FMMs can be used to cluster data, classify data, estimate densities, and increasingly, they are also being used to conduct regression analysis and to analyze regression outcomes. There is now an expansive literature on the usage of FMMs for regression, as well as a broad demand for the development of such methods for the analysis of new and complex data.This thesis begins with a summary of the current literature on FMMs and their applications to regression problems. Here, the mixture of regression models (MRMs), cluster-weight models (CWMs), mixtures of experts (MoEs), and mixtures of linear of mixed effects models (MLMMs), as well as other variants of FMMs for regression analysis are introduced. Various properties such as denseness and identifiability, as well as maximum likelihood (ML) estimation techniques such as the expectation-maximization (EM) and minorization-maximization (MM) algorithms are discussed, and a review is presented regarding asymptotic inference and model selection in FMMs. A new result on the characterization of a t linear CWM (LCWM) is also presented. Some new applications of FMMs to regression problems are then discussed.Firstly, a series of models based on FMMs are presented for the clustering and classification of sparsely sampled bivariate functional data. These methods are named mixture of spatial spline regression (MSSR) and MSSR discriminant analysis (MSSRDA). MSSR is constructed using the theory of MLMMs and spatial splines, and an EM algorithm for the ML estimation of the model is presented. MSSRDA is then constructed by combining MSSR with the mixture discriminant analysis framework for classification. The methods are tested on their ability to cluster and classify simulated data. An example application to handwritten digits recognition is then presented. Here, it is shown that MSSR and MSSRDA perform comparably to currently available methods, and outperform said methods in missing data scenarios.Secondly, an FMM is used to produce a false discovery rate (FDR) control procedure for magnetic resonance imaging (MRI) data. In MRI data analysis, millions of hypotheses are often tested simultaneously, resulting in inflated numbers of false positive results. Many of the available FDR techniques for MRI data either do not take into account the spatial structure or rely on difficult to verify assumptions and user-specified parameters. To address these shortcomings, the Markov random field (MRF) FDR (MRF-FDR) technique is presented. MRF-FDR uses a Gaussian mixture model (GMM) to perform FDR control based on empirical-Bayesian principles. An MRF is then used to make the outcome of the GMM spatially coherent. The MRF is fitted using ...

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Adjusted Maximum Likelihood and Pseudo-Likelihood Estimation for Noisy Gaussian Markov Random Fields

Cited by 31 publications

References 20 publications

Inverse Correlations for Multiple Time Series and Gaussian Random Fields and Measures of Their Linear Determinism

Inverse Correlations for Multiple Time Series and Gaussian Random Fields and Measures of Their Linear Determinism

A noise estimation method for corrupted correlated data

Finite mixture models for regression problems

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