Variable modeling of spatially distributed random interval observation

by Njeri Wabiri Often, data is not observed exactly, but rather an interval range or fuzzy measure of data is observed. In the spatial domain, observed intervals in the same neighborhood influence each other. The assimilation of imprecise data in spatial models is not only appealing but is also becoming an important research topic. However, the development of models, for spatial imprecise (fuzzy) data is still an emerging field in environmental applications. ~ & part of this emerging field, this thesis addresses the dual of randomness (imprecise obtJerving of data as an interval) and the spatial dependence by (i) extending kriging to handle iDtervaJs, (ii) a non-parametric approach using keme1s, (iii) Bi-vuiable copula bi«ing approach that encompass the modeling of UDeeriaiBty (dependence) between the observed lower aad upper interval values and (iv) a hazy -credibility kriging approach that utilizes a fuzzy variable, the equivalent of obsened interval data. First, we use the support function concept for embedding" interval~valued data into Hilbert space. This mirrors the pair wise dissimilarities among the intenraJs, and generates a UDivariate rodguration of random intervals, interpreted as 2'D location in the domain of a suitable Hilbert space. The approach opens an interface between iD~va1ued data analysis and support function-based data analysis. Each random interval corresponds to a random support function, and the interval space assumes a coordinate system with intervals as bivariable random vectors of lower-upper random variables. A generalized weighted interval metric and corresponding interval kernel is defined. A kriging approach to model spatial random intervals is proposed. The aPproach focuses on stationary isotropic random interval models and utilizes the center-radius variable representation of random intervals. Two tecbniques:a) composite kriging and b) component-wise kriging are defined based on proposed composite and component-wise (co ) variance models. The proposed non-parametric approach relaxes the stationarity assumptions in kriging, focusing on non-stationary random interval models. The approach allows the observed random spatial-interval data to define the optimal functional form of the spatial random interval process. The problem of providing spatial interval-valued estimates of the spatial interval regression function based on some kernel function defined in a Hilbert space is Un ive rsi ty of C ap e T ow n Njeri Wabiri addressed via a Kernel interval regression technique in a joint spatial interval domain. Our approach combines a spatial adaptive kernel and a robust interval uncertainty kernel. The spatial kernel ensures that only samples sharing similar intensity and gradient information are gathered for local approximation. The robust uncertainty kernel adapts to the interval structure and minimizes the influence of outliers. This kernel-based technique is found to be efficient when data is mapped to a high dimensional feature space where algorithms as simple as their linear counterparts in input space are used. As an alternative to indicator kriging for spatial random interval/vector data, we pr.r pose two scalar variable modeling techniques. In additional to spatial interval prediction, a unique contribution to the literature of the spatial data analysis and GIS modeling is made. The two techniques a) bivariable copula grade kriging, and b) credibility distribution grade kriging of spatially distributed random intervals exploit the joint distribution of the spatial random intervals. The copula approach combines analytical distribution (marginal) models and an empirical copUla. The credibility approach is based a scalar fuzzy variable, the equivalent of the random interval set, with maximum entropy. This leads to a tractable analytical model for the uncertain phenomena allowing for easy prediction and mapping.