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INTRODUCTION 1 CHAPTER 1. Essential mathematical concepts 5 1.0. Overview 5 1.1. Mathematical Basis 5 1.2. First order sample error, sample data variability sample spacing and the Sample Error Theorem 12 1.3. Maximum error of cells for a tessellation given a positive variability bound for its data cells 26 CHAPTER 2. Extending 1-D results to 2-D. 35 2.0. The development of semi-balanced tessellations in any dimension 35 2.1. Extending one-dimensional results to two dimensions 40 2.2. Symmetries of coordinates 65 CHAPTER 3. Extending 2D Results to K dimensions 67 3.0. Extending to K dimensions 67 3.1. Properties of the transformation ( ) :[ , ] [ , ] K F x a b ? a b 88 3.2. The semi-balanced property of the transformed K-cells 96 CHAPTER 4. Practical methodology 99 4.0. Introduction to practical multidimensional balanced error sampling 99 4.1. Basic summary of mathematical concepts and results 100 4.2. Practical Methodology 108 4.3. Determining a variability bound from sample data sets 110 4.4. Analyzing maximum sample error versus sample size 116 4.5. Determining a semi-balanced tessellation with M 2-cells given the variability Bound B(x) 119 4.6. Interpolation using the semi-balanced pattern of points produced by the transformation 2 F (x) :[a, b]?[a, b] 125 SUMMARY and CONCLUSIONS 131 REFERENCES 133 EXPANDED TABLE OF CONTENTS 135 INDEX 139

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