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In this paper, spatial data specified by auto-beta models is analysed by considering a supervised classification problem of classifying feature observation into one of two populations. Two classification rules based on conditional Bayes discriminant function (BDF) and linear discriminant function (LDF) are proposed. These classification rules are critically compared by the values of the actual error rates through the simulation study.

Spatial Classification Rule with Distance (SCRD) method is used in two dimensional coordinate system, which limits the usage of existing spatial information in MRI, CT and other three dimensional (layered) images. The SCRD method is extended to be applied in three dimensional coordinate space. Artificial experiment is performed in order to show ability to use SCRD method with three dimensional medical or other similar images where training sample is available.

When using Spatial Correlation Rule with Distance (SCRD) the selection of the neighborhood scheme influences classification accuracy. Spatial dependency in different situations remains at various distances, so, according to this, in applications it is important to choose a suitable neighborhood scheme. In the earlier papers of the authors, the nearest neighbor scheme was used. In this paper, several different neighborhood schemes are examined by large experiment.

In this paper the remote sensed image classification example using spacial classification rule with distance (SCRD) is examined. This supervised classification method was first presented in paper [11]. This method is improved version of earlier method PBDF [4, 10, 9], during the classification it incorporates more spatial information. The advantage of this method is its ability to classify data which is corrupted by Gaussian random field and it is typical to remotely sensed images classified in this letter which are corrupted by clouds. Classification accuracy is compared with earlier method and with other commonly used supervised classification methods.

In this paper supervised classification method is proposed. It is based on Bayes discriminant functions (BDF) and it deals with the problem of optimal classification for images, which are corrupted by natural phenomenon such as cloud, smoke or fog. Solving such a problem is very important when we have remotely sensed information, which very often is corrupted by clouds. For example, the remotely sensed images from the territory of Lithuania are very often corrupted by clouds. The idea of classification, using BDF with incorporated spatial dependency between the observation to be classified and the training sample is presented in earlier works of the author. The novelty of this paper is the method how to use these methods for the real situation, i.e. for the remotely sensed image which is naturally covered by clouds. Visual and numerical results are presented in this paper, which show the advantage of this method against BDF ignoring spatial dependency between training sample and observation to be classified and against the method using grey level cooccurrence matrices.

In image classification often occur such situations, when images in some level are corrupted by additive noise. Such noise in image classification can be modeled by Gaussian random fields (GRF). In image classification supervised and unsupervised methods are used. In this paper we compare our proposed supervised classification methods based on plugin Bayes discriminant functions (PBDF) (see [6] and [11]) with unsupervised classification method based on grey level co-occurrence matrix (GLCM) (see e.g. [8] and [1]). The remotely sensed image is used for classification (USGS Earth Explorer). Also GRF with different spatial correlation range are generated and added to the original remotely sensed image. Such situation can naturally occur during forest fire, when smoke covers some territory. These images are used for classification accuracy examination.

In spatial classification it is usually assumed that features observations given labels are independently distributed. We have retracted this assumption by proposing stationary Gaussian random field model for features observations. The label are assumed to follow Disrete Random Field (DRF) model. Formula for exact error rate based on Bayes discriminant function (BDF) is derived. In the case of partial parametric uncertainty (mean parameters and variance are unknown), the approximation of the expected error rate associated with plug-in BDF is also derived. The dependence of considered error rates on the values of range and clustering parameters is investigated numerically for training locations being second-order neighbors to location of observation to be classified.