Surface Shape Regions as Manifestations of a Socio-economic Phenomenon

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Surface Shape Regions as Manifestations of a Socio-economic Phenomenon a solution to the choropleth mapping problem by John (Jack) Sirles Massey a thesis submitted for the

1 Surface Shape Regions as Manifestations of a Socio-economic Phenomenon a solution to the choropleth mapping problem by John (Jack) Sirles Massey a thesis submitted for the degree of Master of Science in Mathematical & Computer Sciences in the Discipline of Pure Mathematics of the School of Mathematical Sciences The University of Adelaide 25 November 2011

2 dedicated to my wife, partner and friend... Kathleen (Kathie) Egan Massey ii

3 ABSTRACT A choropleth map is a cartographic document. It shows a geographic study area tessellated by a set of polygons that differ in shape and size. Each polygon is depicted by a uniform symbol representing the manifestation of some phenomenon. This thesis focuses on socio-economic phenomena. We want to delineate a set of socio-economic regions within a study area. These regions are used for decision making about the delivery of specific goods and services and/or the provision of specific community infrastructure. However, we have identified three fundamental weaknesses associated with the use of choropleth maps for socio-economic regionalisation. Therefore, as an alternative to the choropleth map if we think explicitly in R 3, then the best representation of the spatial distribution of a socioeconomic phenomenon is a smooth surface. The socio-economic data we use are collected during a national census of population and are summarised for areas, i.e., polygons. To accommodate these data we have developed and applied a method for gridding and smoothing termed regularisation in order to build a smooth surface. We apply Green s theorem and use path integrals with much simplification to compute a smoothed datum for each intersection of a, say, 100 by 100 grid that describes a surface. Mathematically, surface shape is interpreted through the comparison of curvatures. Surface shape analysis involves the measurement of the Gaussian and mean curvatures at the internal intersections of the grid. Curvature measurement requires at least a twice differentiable function. We have invented such a function based on Lagrange interpolation. It is called a Lagrange polynomial in x y. Each internal intersection of the grid is the (2, 2) element of a 3 3 matrix extracted from the grid. We compute a Lagrange polynomial in x y for each 3 3 matrix. Then we use this polynomial to measure the curvatures and classify the shape. Contiguous grid intersections of the same shape class comprise a shape neighbourhoods region interpreted as a specific manifestation of a socio-economic phenomenon. Hence, we have the basis for describing the spatial distribution of the phenomenon. Three investigations into the construction of quadratic polynomials as alternative functions are described. Two of these quadratic polynomials are called exact fit in the sense that the polynomial returns the exact z-datum associated with each xy-pair used in its construction. Construction of a best fit quadratic polynomial based on least squares interpolation comprises the third investigation. We compare the four different types of polynomials and of these we choose the Lagrange polynomial in x y as most appropriate. Given a relatively high density grid, e.g., 250 by 250, regardless of the polynomial used the resulting maps of shape neighbourhoods regions are virtually identical. This surprising convergence in R 2 is explained. Is a map of shape neighbourhoods regions an accurate description of the spatial distribution of a socio-economic phenomenon? We effect an indirect evaluation of a known phenomenon represented by the spatial distribution of f(x, y) = sin x sin y. We compute the true map of shape neighbourhoods regions of this phenomenon. An approximate map of shape neighbourhoods regions is computed by sampling with 100 randomly generated polygons. Comparison implies that the approximate map is an accurate representation of the true map. This conclusion is supported strongly by the results of a study of a nonperiodic nonrandom known phenomenon, based on a combination of exponential functions in x and y. This has a surface similar to that of a socio-economic phenomenon. We review selected geographic studies in which mathematical tools have been used for analytical purposes. Mathematical analysis is gaining broader acceptance in geography. The innovative, high quality Surpop work of British geographers is described, and we comment on the strongly complementary nature of the research presented in this thesis to the Surpop work. We describe 18 future research directions and themes; suggestions are made on how each may be undertaken. Next, we summarise each of the ten results of the research presented in this thesis. The thesis concludes with a statement of the medium-term research directions of the researcher and his acknowledgements. iii

4 CONTENTS Dedication Abstract Contents Statement of Originality and Consent Chapter 1 Justification Introduction An Example Using Choropleth Maps: Weaknesses and Difficulties 6 1) Data collection and summary data geometries 6 2) Complexity of data interpretation 8 3) Homogeneity and discontinuity The Nature of a Socio-Economic Region Thinking in R 3 The Design Principle for a New Mapping Methodology Structure and Content of this Thesis 22 Chapter 2 Regularisation Introduction Gridding and Smoothing Using Green s Theorem 28 1) Description 29 2) Derivation 30 3) Example 31 4) Data representation geometry 34 5) Burnside revisited Epilogue 39 Chapter 3 Classification Introduction Lagrange Polynomial in x y 43 1) A Lagrange Polynomial in x 43 2) A Lagrange Polynomial in x y Shape Neighbourhoods Engine 48 1) Calculating the first and second partial derivatives 48 2) Measuring the Gaussian and mean curvatures 48 3) Classifying the shape in the neighbourhood of an internal grid intersection 49 4) Mapping the shape neighbourhoods regions Simplification of a Lagrange Polynomial in x y 56 iv ii iii iv vii

5 Chapter 4 Approximation Introduction An Exact Fit Quadratic Polynomial A Hybrid Exact Fit Quadratic Polynomial A Best Fit Quadratic Polynomial Approximating Polynomial Functions 79 1) Which function should we use? 79 2) Convergence in R ) Spatial autocorrelation and extraction of overlapping 3 3 matrices 84 Chapter 5 Evaluation Introduction Evaluation Method Terminology and Overview 88 1) Terminology 88 2) Overview Evaluation Method Implementation Nature of the Spatial Distribution and Selection of the Level of Smoothing Evaluation of an Arbitrary Surface 105 1) Introduction 105 2) A mathematical function that describes a nonperiodic nonrandom surface 106 3) Point in polygon calculation of volume, area and average datum of a Voronoi polygon 110 4) Computation of an approximate surface and an approximate shape neighbourhoods map 111 5) Conclusion 111 Chapter 6 Conclusion Introduction Some Studies of Surfaces A Personal Overview 113 1) Some physical geographic studies of surfaces 114 2) Some human geographic studies of surfaces Research Directions 120 1) Markets for socio-economic regionalisation 120 2) Use of socio-economic regionalisation 120 3) Best practice socio-economic regionalisation 120 4) Data collection geometry 121 5) What is a socio-economic region? 121 v

6 6) Spatial cognition thinking in R 2 and R ) Smoothing by regularisation 121 8) Positioning of data collection centroids 121 9) Shape classification using functions describing polygons ) Shape classification using numerical values of Gaussian and mean curvatures ) Grid geometry ) Extraction of matrices from a grid and building of a function for a matrix element ) Polynomials and other functions ) Grid density and amount of smoothing ) Convergence in R ) Value of surfaces ) Evaluation of shape neighbourhoods regions ) Software Summary of Results of the Research 124 1) Weaknesses and difficulties associated with the use of choropleth maps 124 2) The nature of a socio-economic region 124 3) Spatial cognition in R 2 and R ) Gridding and smoothing using Green s theorem regularisation 125 5) Surface shape classification 125 6) Construction of a Lagrange polynomial in x y 125 7) The shape neighbourhoods engine 125 8) Alternative polynomial functions 126 9) Convergence in R ) Accuracy of the maps of shape neighbourhoods regions Conclusion and Acknowledgements 126 Appendices Appendix A 128 Appendix B 129 Appendix C 136 Appendix D 137 References 141 vi

7 STATEMENT OF ORIGINALITY AND CONSENT This work contains no material that has been accepted for the award of any other degree or diploma in any university or tertiary institution to John (Jack) Sirles Massey and, to the best of my knowledge and belief, contains no material previously published or written by another person, except where due reference has been made in the text. I give consent to this copy of my thesis, when deposited in the University Library, being available for loan and photocopying, subject to the provisions of the Copyright Act I also give permission for the digital version of my thesis to be made available on the web, via the University s digital research repository, the Library catalogue, the Australian Digital Theses Program (ADTP) and also through web search engines, unless permission has been granted by the University to restrict access for a period of time. SIGNED: DATE: John (Jack) Sirles Massey vii

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