THE PAPERS
The average size and scatter of clustered objects can vary with time, as can the direction of migration of the collective, if any such migration occurs.
In the context of mining history, geology determines the richest or most accessible deposits, whilst economics and technological evolution encourage the geographical drift of activity across orefields or coalfields.
Accordingly it would be convenient to represent instantaneous size, scatter, and directional tendency using a single simple diagrammatic figure.
This can be done with a dart-shaped quadrilateral, a belemnoid, whose base is proportional to standard deviation; whose apex radius represents the mean; whose re-entrant vertex is the cluster center of mass; and whose apex points in the direction of cluster migration.
The area of such a belemnoid is one half of the product of the mean and the standard deviation so defined, and a corollary is that belemnoid area is a function of cluster population and collective magnitude. The mathematical behaviour of figure area has extensive technical implications.
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BELEM.BAS is a program to compute the descriptive statistics of the magnitudes in contemporaneous clustered or classified entities located in a planar cartesian grid system. The program also computes the spacial mass centroids and dynamic radii of successive clusters and erects plottable belemnoids that summarise magnitude migration direction and magnitude mean and standard deviation for each cluster. Confidence limits are quoted for the mean of the entities in each cluster. The program was perfected at Bloxwich on 22 July 1996, and written in MicroSoft Qbasic.
The BELPROG paper addresses certain technical features of the automation.
Because very small cluster populations may be anticipated, confidence limits computation invokes the Student ( Gosset ) t-statistic, and Hill's Process is used to compute that. The Hill determination was refined by my composite method of algebraic kedging that enhances the accuracy of the t estimate in difficult circumstances.
The program includes manual and automatic cartographic scaling options. A discussion of requisite batch data is also provided.
Appendices give a program source listing and certain specimen test data.
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Delaunay triangulations require a convex hull. A problem arises in assessing the partial contribution of hull boundary vertices to the areal concentration of all cluster vertices.
Because Delaunay triangulations furnish "locally-equilateral" triangles they can be modelled by the internal triangulation of regular hexagons.
The total angular contribution, T, of hexagon body and boundary points, in terms of complete equilateral triangles, is six times the square of the Characteristic Side Length, L. For a general convex hull, L may be fractional.
T is a function of the total cluster mass.
It was shown that for a hexagon that includes between two and 119,399 points more than 1% of points are boundary points
When the number of boundary points is known their Mass Potential Contribution is proportional to half the boundary point count less two.
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In 1987 Sloan described a scheme for Delaunay Triangulation and published a FORTRAN 77 implementation.
A number of theoretical and practical claims are made for this process, both in terms of "Delaunayness" ( an optimality of triangulation ), and in terms of computational efficiency.
Miles published a set of test statistics for theoretical "Delaunayness" neglective of hull boundary effects because defined for an infinite plane.
In the present SLOANVAL paper thirty-six experiments involving five to 128 random points were run. Clearly these hulls exhibit marked boundary effects, and these can intuitively be apprehended even by inspection of the "facetted" plots of the defined networks.
Nevertheless, Milesian criteria of "Delaunayness" were broadly satisfied for Sloan's process. Statistical inferences of "Delaunayness" variation with point count were contradictory.
Sloan asserts that his process mill time is of order 1.0642, implying a very shallow power law increase with point count. Whilst he employed up to 10000 test points, I could detect no significant departure from linear mill time growth for my experiments of up to 128 points. My equipment precluded larger experiments.
Read this Paper
If we require to assess in detail the association of two point clusters that have been intertriangulated we will need to obtain information about the areas and elongations of the component triangles.
Any triangle has a unique area, but an infinity of centroids, and at least two centroids must be chosen to define an elongation.
An ideal choice would involve a centroid of pure space, and a second centroid that is a mass function.
I chose two simply-definable candidates: The incenter ( center of an inscribed circle ); and the inertial mass centroid familiar from geographical literature.
The ELONGA paper describes the centroids' relationships and certain technical aspects of computation.
The program ELONGA.BAS is appended. It was written in MicroSoft Qbasic, and perfected at Bloxwich on 19 August 2000. The program generates, for a set of pre-existing triangles, the centroids' co-ordinates and separations, and the triangles' areas, perimeters, elongation and Arithmetised elongation. It was validated using the MUNSTER Delaunay test set of twenty-six triangles.
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The Binomial Expansion is exploited to develop a novel principle for the quantitative assessment of the degree of blending between two distinct point clouds in two-dimensional Euclidean space.
Some possible metrics of association are then explored, both in terms of homogeneity coefficients and in terms of triangle elongation statistics.
A brief personal exordium sketches the author's gestation of the method, and a small test illustration of some of the methods is incorporated.
Read this Paper
| NAME OF PAPER: | Characterising the Belemnoid: |
| A Graphical Device for Mapping | |
| Descriptive Statistics | |
| DATE OF WRITING: | 10 June 1996 |
| LENGTH: | 24 A4 Pages at Times New Roman 12 |
| ILLUSTRATIONS: | 5 |
| EQUATIONS: | 44 |
| TABLES: | 4 |
| REFERENCES: | 3 |
| APPENDICES: | All Appendices included in the Internet Version |
| FILENAME: | belemf.pdf |
| ABSTRACT: |
The average size and scatter of clustered objects can vary with time, as can the direction of migration of the collective, if any such migration occurs.
In the context of mining history, geology determines the richest or most accessible deposits, whilst economics and technological evolution encourage the geographical drift of activity across orefields or coalfields.
Accordingly it would be convenient to represent instantaneous size, scatter, and directional tendency using a single simple diagrammatic figure.
This can be done with a dart-shaped quadrilateral, a belemnoid, whose base is proportional to standard deviation; whose apex radius represents the mean; whose re-entrant vertex is the cluster center of mass; and whose apex points in the direction of cluster migration.
The area of such a belemnoid is one half of the product of the mean and the standard deviation so defined, and a corollary is that belemnoid area is a function of cluster population and collective magnitude. The mathematical behaviour of figure area has extensive technical implications.
Read this Paper
| NAME OF PAPER: | A Description of Program BELEM.BAS |
| DATE OF WRITING: | 26 July 1996 |
| LENGTH: | 27 A4 Pages at Times New Roman 12 |
| ILLUSTRATIONS: | 0 |
| EQUATIONS: | 0 |
| TABLES: | 0 |
| REFERENCES: | 6 |
| APPENDICES: | All Appendices included in the Internet Version |
| FILENAME: | belprogc.pdf |
| ABSTRACT: |
BELEM.BAS is a program to compute the descriptive statistics of the magnitudes in contemporaneous clustered or classified entities located in a planar cartesian grid system. The program also computes the spacial mass centroids and dynamic radii of successive clusters and erects plottable belemnoids that summarise magnitude migration direction and magnitude mean and standard deviation for each cluster. Confidence limits are quoted for the mean of the entities in each cluster. The program was perfected at Bloxwich on 22 July 1996, and written in MicroSoft Qbasic.
The BELPROG paper addresses certain technical features of the automation.
Because very small cluster populations may be anticipated, confidence limits computation invokes the Student ( Gosset ) t-statistic, and Hill's Process is used to compute that. The Hill determination was refined by my composite method of algebraic kedging that enhances the accuracy of the t estimate in difficult circumstances.
The program includes manual and automatic cartographic scaling options. A discussion of requisite batch data is also provided.
Appendices give a program source listing and certain specimen test data.
Read this Paper
| NAME OF PAPER: | The Estimation of Vertex Concentrations |
| in Delaunay Triangulations | |
| Whether Limited or Sufficient Data is Available | |
| DATE OF WRITING: | 15 August 1999 |
| LENGTH: | 10 A4 Pages at Times New Roman 12 |
| ILLUSTRATIONS: | 2 |
| EQUATIONS: | 16 |
| TABLES: | 2 |
| REFERENCES: | 1 |
| APPENDICES: | No Appendices |
| FILENAME: | concestf.pdf |
| ABSTRACT: |
Delaunay triangulations require a convex hull. A problem arises in assessing the partial contribution of hull boundary vertices to the areal concentration of all cluster vertices.
Because Delaunay triangulations furnish "locally-equilateral" triangles they can be modelled by the internal triangulation of regular hexagons.
The total angular contribution, T, of hexagon body and boundary points, in terms of complete equilateral triangles, is six times the square of the Characteristic Side Length, L. For a general convex hull, L may be fractional.
T is a function of the total cluster mass.
It was shown that for a hexagon that includes between two and 119,399 points more than 1% of points are boundary points
When the number of boundary points is known their Mass Potential Contribution is proportional to half the boundary point count less two.
Read this Paper
| NAME OF PAPER: | A Statistical Validation of |
| Sloan's Process | |
| For Delaunay Triangulation | |
| DATE OF WRITING: | 8 October 1999 |
| LENGTH: | 61 A4 Pages at Times New Roman 12 |
| ILLUSTRATIONS: | 0 |
| EQUATIONS: | 31 |
| TABLES: | 1 |
| REFERENCES: | 4 |
| APPENDICES: | All Appendices included in the Internet Version |
| FILENAME: | sloanvale.pdf |
| ABSTRACT: |
In 1987 Sloan described a scheme for Delaunay Triangulation and published a FORTRAN 77 implementation.
A number of theoretical and practical claims are made for this process, both in terms of "Delaunayness" ( an optimality of triangulation ), and in terms of computational efficiency.
Miles published a set of test statistics for theoretical "Delaunayness" neglective of hull boundary effects because defined for an infinite plane.
In the present SLOANVAL paper thirty-six experiments involving five to 128 random points were run. Clearly these hulls exhibit marked boundary effects, and these can intuitively be apprehended even by inspection of the "facetted" plots of the defined networks.
Nevertheless, Milesian criteria of "Delaunayness" were broadly satisfied for Sloan's process. Statistical inferences of "Delaunayness" variation with point count were contradictory.
Sloan asserts that his process mill time is of order 1.0642, implying a very shallow power law increase with point count. Whilst he employed up to 10000 test points, I could detect no significant departure from linear mill time growth for my experiments of up to 128 points. My equipment precluded larger experiments.
Read this Paper
| NAME OF PAPER: | A Metric of Elongation |
| With a Program for Computation | |
| DATE OF WRITING: | 19 August 2000 |
| LENGTH: | 21 A4 Pages at Times New Roman 12 |
| ILLUSTRATIONS: | 1 |
| EQUATIONS: | 16 |
| TABLES: | 1 |
| REFERENCES: | 0 |
| APPENDICES: | All Appendices included in the Internet Version |
| FILENAME: | elongaf.pdf |
| ABSTRACT: |
If we require to assess in detail the association of two point clusters that have been intertriangulated we will need to obtain information about the areas and elongations of the component triangles.
Any triangle has a unique area, but an infinity of centroids, and at least two centroids must be chosen to define an elongation.
An ideal choice would involve a centroid of pure space, and a second centroid that is a mass function.
I chose two simply-definable candidates: The incenter ( center of an inscribed circle ); and the inertial mass centroid familiar from geographical literature.
The ELONGA paper describes the centroids' relationships and certain technical aspects of computation.
The program ELONGA.BAS is appended. It was written in MicroSoft Qbasic, and perfected at Bloxwich on 19 August 2000. The program generates, for a set of pre-existing triangles, the centroids' co-ordinates and separations, and the triangles' areas, perimeters, elongation and Arithmetised elongation. It was validated using the MUNSTER Delaunay test set of twenty-six triangles.
Read this Paper
| NAME OF PAPER: | A Measure of Spatial Association By Triangulation |
| DATE OF WRITING: | 1 September 2011 |
| LENGTH: | 28 A4 Pages at Times New Roman 12 |
| ILLUSTRATIONS: | 6 |
| EQUATIONS: | 33 |
| TABLES: | 9 |
| REFERENCES: | 1 |
| APPENDICES: | All Appendices included in the Internet Version |
| FILENAME: | spacasso.pdf |
| ABSTRACT: |
The Binomial Expansion is exploited to develop a novel principle for the quantitative assessment of the degree of blending between two distinct point clouds in two-dimensional Euclidean space.
Some possible metrics of association are then explored, both in terms of homogeneity coefficients and in terms of triangle elongation statistics.
A brief personal exordium sketches the author's gestation of the method, and a small test illustration of some of the methods is incorporated.
Read this Paper
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"Studies in Spacial Association"
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