Search

The novel approach for classifying spatial Gaussian data based on Bayes discriminant functions in terms of semivariograms has been developed. We derived the closed-form expressions for the Maximum ...

The problem of supervised classifying of a multivariate Gaussian Markov random field (GMRF) observation into one of two populations specified by different regression mean models is considered. We focus on a multivariate conditionally autoregressive model, a subclass of GMRF with parametrical structure proposed by Pettitt et al. (2002) and generalized by Jin et al. (2005). For complete parametric certainty and with fixed training sample locations, the formula of the Bayes error rate is derived. Plug-in Bayes discriminant function obtained by replacing the regression and scale parameters with their ML estimators in the Bayes discriminant function is investigated. The novel formulas for the actual error rate and the approximation of the expected error rate (AER) associated with plug-in Bayes discriminant function are derived. These results are multivariate generalizations of the univariate ones. A numerical analysis of the derived formulas is implemented for the bivariate conditionally autoregressive model sampling on a regular 2-dimensional unit spacing lattice with the neighborhood structure based on the Euclidean distance. A comparison of the proposed classifier performance for different parametric structures of populations is done by studying the accuracy of the derived AER. Finally, the influence of the neighborhood order on the accuracy of the derived AER is examined.

Given the spatial lattice endowed with particular neighborhood structure, the problem of classifying a scalar Gaussian Markov random field (GMRF) observation into one of two populations specified by different regression coefficients and special parametric covariance (precision) matrix is considered. Classification rule based on the plug-in Bayes discriminant function with inserted ML estimators of regression coefficients, spatial dependence and scale parameters is studied. The novel closed-form expression for the actual risk and the approximation of the expected risk (AER) associated with the aforementioned classifier are derived. This is the extension of the previous study of GMRF classification to the case of complete parametric uncertainty. Derived AER is used as the main performance measure for the considered classifier. GMRF sampled on a regular 2-dimensional unit spacing lattice endowed with neighborhood structure based on the Euclidean distance between sites is used for a simulation experiment. The sampling properties of ML estimators and the accuracy of the derived AER for various values of spatial dependence parameters and Mahalanobis distance are studied. The influence of the neighborhood size on the accuracy of the proposed AER is examined as well.

In this article we focus on the problem of supervised classifying of the spatio-temporal Gaussian random field observation into one of two classes, specified by different mean parameters. The main distinctive feature of the proposed approach is allowing the class label to depend on spatial location as well as on time moment. It is assumed that the spatio-temporal covariance structure factors into a purely spatial component and a purely temporal component following AR(p) model. In numerical illustrations with simulated data, the influence of the values of spatial and temporal covariance parameters to the derived error rates for several prior probabilities models are studied.

Relying on the results of multivariate analysis of the re-analysis case studies from the BaltCoast project, specific features of integrated coastal management (ICM) approaches in Estonia, Latvia, Lithuania, and the Kaliningrad Oblast of the Russian Federation are highlighted in this paper. Eleven Eastern Baltic ICM case studies have been re-analyzed in-depth, which was the main focus of the present paper, covering a wide range of coastal landscapes, themes, policy issues, and ICM approaches. Five principal components explaining 84.86% of the total variance of ICM factor scores have been elicited by calculating rotation sums of squared loadings: (1) Stakeholder Involvement, (2) Research Base, (3) Planning Consistency, (4) Policy Environment, and (5) Development of Plans/Strategies. A truly dedicated involvement of a wide range of stakeholders throughout all process stages showed to be critical for further successful advance of ICM processes and principles based on the systems approach in the Baltic States and the Russian Federation.

Visual distinctiveness of the series of successive coastal landscapes featuring the Curonian Spit, a UNESCO World Heritage site, has been investigated using black and white landscape photos as visual stimuli. Based on the results of the chi-square test it was confirmed with strong statistical evidence that lay visitors indeed are capable of visual distinguishing different coastal landscapes and habitats which occur in the succession series from the shifting dunes to the mature forest. In all investigated 45 different cases of coastal landscapes the ‘null hypothesis’ was rejected with p

In this paper, a novel approach to comparison and selection of spatial linear mixed models based on a hybrid estimator of actual correct classification rates is considered. We focus on the method of statistical classification based on Bayes discriminant function (BDF) by providing modified well-known methods for estimation of correct classification rates. The proposed approach is illustrated on ecological, spatially distributed data and allows comparing various spatial linear mixed models. Assuming that spatial data follow Gaussian random field (GRF) model, the problem of classifying its observation into one of two populations is considered. The populations are specified by the common linear regressors but different values of regression parameters. Authors concern with classification procedures associated with BDF and its sample version (SDF). SDF is built by replacing the unknown parameters in BDF with their maximum likelihood estimators obtained from the training sample. Two estimators of the actual correct classification rates, i.e., the plug-in and the apparent correct classification rates are considered as estimators of performance measures of classifier based on SDF. A hybrid estimator of the actual correct classification rate, obtained by averaging the mentioned above estimators, is used as the main criterion for model selection. Different types of spatial data models for invasive species (zebra mussels), distributed in the Curonian Lagoon, are compared by the defined criterion. The advantage of the proposed approach against indicator kriging method is shown. Selected spatial models can aid in the mapping of the presence and absence of zebra mussels in the Curonian Lagoon.

The paper addresses the possibilities of spatio-temporal statistical modelling of basic hydrophysical and meteorological parameters of sea surface layer in the south-eastern Baltic Sea, Curonian Lagoon. The aim of the paper is to compare two methods (multivariate linear regression and regression kriging) for the analysis of changes and trends of ice phenomena, their dependence on changes in the air temperature, sea surface temperature and water salinity. The prediction of ice conditions for several locations at different distances from the reference sites shows that spatial information is an extremely important factor in making forecasts. The application of the regression kriging is more efficient than the multivariate linear regression for predicting the ice phenomena in semi-enclosed basins and lagoons.

Given training sample, the problem of classifying the scalar Gaussian random field observation into one of several classes specified by different regression mean models and common parametric covariance function is considered. The classifier based on the plug-in Bayes classification rule formed by replacing unknown parameters in Bayes classification rule with their ML estimators is investigated. This is the extension of the previous one from the two-class case to the multiclass case. The novel close form expressions for the actual error rate and approximation of the expected error rate incurred by proposed classifier are derived. These error rates are suggested as performance measures for the proposed classifier. The three-class case with feature modelled by scalar stationary Gaussian random field on regular lattice with exponential covariance function is used for the numerical analysis of the proposed classifier performance. The accuracy of the obtained approximation is checked through a simulation study for various parametric structure cases.

The problem of classifying a spatial multivariate Gaussian data into one of several categories specified by different regression mean models is considered. The classifier based on plug-in Bayes classification rule (PBCR) formed by replacing unknown parameters in Bayes classification rule (BCR) with category parameters estimators is investigated. This is the extension of the previous one from the two category case to the multiple category case. The novel close-form expressions for the Bayes misclassification probability and actual error rate associated with PBCR are derived. These error rates are suggested as performance measures for the classifications procedure. The three-category case with feature modelled by bivariate stationary Gaussian random field on regular lattice with exponential covariance function is used for the numerical analysis. Dependence of the derived error rates on category parameters is studied.

This paper discusses the problem of classifying a multivariate Gaussian random field observation into one of the several categories specified by different parametric mean models. Investigation is conducted on the classifier based on plug-in Bayes classification rule (PBCR) formed by replacing unknown parameters in Bayes classification rule (BCR) with category parameters estimators. This is the extension of the previous one from the two category cases to the multi-category case. The novel closed-form expressions for the Bayes classification probability and actual correct classification rate associated with PBCR are derived. These correct classification rates are suggested as performance measures for the classifications procedure. An empirical study has been carried out to analyze the dependence of derived classification rates on category parameters.

In current paper a problem of classification of T-distributed random field observation into one of two populations specified by common scaling function is considered. The ML and LS estimators of themean parameters are plugged into the linear discriminant function. The closed form expressions for the Bayes error rate and the actual error rate associated with the aforementioned discriminant functions are derived. This is the extension of one for the Gaussian case. The actual error rates are used to evaluate and compare the performance of the plug-in discriminant function by means of Monte Carlo study.

In this paper spatial classification rules based on Bayes discriminant functions are considered. The novelty of this work is that the statistical supervised classification method is improved by extending the influence of spatial correlation between observation to be classified and training sample. Such methods are used for data containing spatially correlated noise. Method accuracy is tested experimentally on artificially corrupted images. This classification rule with distance based conditional distribution for class label shows advantage against other classification rule ignoring such influence and against other commonly used supervised classification methods.

We consider the problem of supervised classifying the multivariate Gaussian random field (GRF) single observation into one of two populations in case of given training sample. The populations are specified by different regression mean models and by common factorized covariance function. For completely specified populations, we derive a formula for Bayes error rate. In the case of unknown regression parameters and feature covariance matrix, the plug-in Bayes discriminant function based on ML estimators of parameters is used for classification. We derive the actual error rate and the asymptotic expansion of the expected error rate associated with plug-in Bayes discriminant function. These results are multivariate generalizations of previous ones. Numerical analysis of the derived formulas is implemented for the bivariate GRF observations at locations belonging to the two-dimensional lattice with unit spacing.

In statistical image classification, it is usually assumed that feature observations given class labels are independently distributed [1] . Even in the case when training sample is formed by dependent feature observations, the feature observations to be classified are often assumed to be independent from training sample [2] , [3] . Proposed approach is based on the retraction of this assumption by considering the stationary Gaussian random field (GRF) model for features [4] . The conditional distribution of class label at unclassified location (pixel) is assumed to be dependent on its location spatial adjacency with spatial framework of training sample. For given training sample, the Bayes discriminant function (BDF) and plug-in BDF (PBDF) are proposed for classification. Performance of the proposed BDF and PBDF are tested and compared with ones ignoring spatial correlation among feature observations to be classified and training sample. In the first example, the restoration of the image of figure corrupted by the GRF is performed by pixels classification. In the numerous parametric structure cases, the advantage of proposed BDF against competing one is shown visually and is confirmed numerically. In the second example, three spatial sampling designs for training data are compared on the basis of actual error rate values of proposed PBDF.

In the usual statistical approach of spatial classification, it is assumed that the feature observations are independent conditionally on class labels (conditional independence). Discarding this popular assumption, we consider the problem of statistical classification by using multivariate stationary Gaussian Random Field (GRF) for modeling the conditional distribution given class labels of feature observations. The classes are specified by multivariate regression model for means and by common factorized covariance function. In the two-class case and for the class labels modeled by Random Field (RF) based on 0–1 divergence, the formula of the Expected Bayes Error Rate (EBER) is derived. The effect of training sample size on the EBER and the influence of statistical parameters to the values of EBER are numerically evaluated in the case when the spatial framework of data is the subset of the 2-dimensional rectangular lattice with unit spacing.

For given training sample, the problem of supervised classifying the multivariate stationary Gaussian random field (GRF) observations into one of two populations is considered. The populations are specified by different regression mean models and by common factorized covariance function. For completely specified populations the formula of Bayes error rate is derived. In the case of unknown regression parameters and covariance function parameters, the plug-in Bayes discriminant function based on ML estimators of parameters is used for spatial classification. The actual error rate and the asymptotic approximation of the expected error rate (AER) associated with plug-in Bayes discriminant function are derived. These results are multivariate generalization of previous one [1] for complete parametric uncertainty case. Numerical analysis of the derived formulas is implemented for the bivariate stationary GRF observations at locations belonging to the 2-dimensional lattice with unit spacing and exponential spatial correlation. Numerical comparison of two training label configurations (TLC) by the values of AER as optimality criterion is implemented.