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www.spatialanalysisonline.com Chapter 4 Part A: Geometric & related operations

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3 rd editionwww.spatialanalysisonline.com2 Length & Area – vector datasets Polygon area in the plane

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3 rd editionwww.spatialanalysisonline.com3 Length & Area – vector datasets Polyline length Distance formulas Plane Spherical

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3 rd editionwww.spatialanalysisonline.com4 Length & Area – raster datasets Definition of areas and lines in grid models Membership functions Grid orientation Grid metrics Grid resolution – size, shape and attribute assignment

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3 rd editionwww.spatialanalysisonline.com5 Surface Area – TIN datasets Planimetric area, A Surface area, A'

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3 rd editionwww.spatialanalysisonline.com6 Geometric & related operations Surface Area – Raster datasets Grid model: average of 8 Triangular components

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3 rd editionwww.spatialanalysisonline.com7 Geometric & related operations Surface Area: Unprojected vector datasets Terrestrial quadrangle Terrestrial polygon Great circles, Spherical triangles, Spherical excess and Azimuths

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3 rd editionwww.spatialanalysisonline.com8 Point weeding and line smoothing

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3 rd editionwww.spatialanalysisonline.com9 Centroids and centres 1a Polygons M3 x = [max(x)-min(x)]/2 M3 y = [max(y)-min(y)]/2

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3 rd editionwww.spatialanalysisonline.com10 Centroids and centres 1b Polygons Data source: US Census Bureau

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3 rd editionwww.spatialanalysisonline.com11 Centroids and centres 1c Polygons Maximum inscribed circle Minimum bounding circle

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3 rd editionwww.spatialanalysisonline.com12 Centroids and centres Point sets (weighted or unweighted) M6 (MAT point) – iteration formula

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3 rd editionwww.spatialanalysisonline.com13 Point (object) in polygon MBR screening Semi-line algorithm Standard cases Special cases Winding number (wn) algorithm

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3 rd editionwww.spatialanalysisonline.com14 Polygon decomposition Convex parts Triangulations Skeletonisation/medial axis transforms Object labelling Assignment of attributes Clipping/cookie cutters

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3 rd editionwww.spatialanalysisonline.com15 Polygon shape – alternative measures Dimensionless global measures Perimeter 2 /Area ratio (P2A): Shape Index or Compactness ratio (C) Related bounding figure (RBF) Measures for point and line/network sets

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3 rd editionwww.spatialanalysisonline.com16 Layers & overlay operations - vector Intersection/OR; Union/AND; Not/Difference; Exclusive OR/Symmetric difference OGC OpenGIS Simple Features Specification: Spatial Analysis MethodDescription Note: a and b are two geometries (one or more geometric objects or features points, line objects, polygons, surfaces including their boundaries); I(x) is the interior of x; dim(x) is the dimension of x, or maximum dimension if x is the result of a relational operation Spatial analysis Distancethe shortest distance between any two points in the two geometries as calculated in the spatial reference system of this geometry Bufferall points whose distance from this geometry is less than or equal to a specified distance value Convex Hullthe convex hull of this geometry Intersectionthe point set intersection of the current geometry with another selected geometry Unionthe point set union of the current geometry with another selected geometry Differencethe point set difference of the current geometry with another selected geometry Symmetric difference the point set symmetric difference of the current geometry with another selected geometry (logical XOR)

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3 rd editionwww.spatialanalysisonline.com17 MethodDescription Note: a and b are two geometries (one or more geometric objects or features points, line objects, polygons, surfaces including their boundaries); I(x) is the interior of x; dim(x) is the dimension of x, or maximum dimension if x is the result of a relational operation Spatial relations Equalsspatially equal to: a=b Disjoint spatial disjoint: equivalent to a b= Intersects spatially intersects: [a b] is equivalent to [not a disjoint(b)] Touches spatially touches: equivalent to [a b and I(a) I(b)= ]; does not apply if a and b are points Crosses spatially crosses: equivalent to [dim(I(a) I(b))<max{dim(I(a)),dim(I(b))} and a b a and a b b] Within spatially within: within(b) is equivalent to [a b=a and I(a) I(b) ] Containsspatially contains: [a contains(b)] is equivalent to [b within(a)] Overlaps spatially overlaps: equivalent to [dim(I(a) I(b))=dim(I(a))=dim(I(b)) and a b a and a b b] Relatespatially relates, tested by checking for intersections between the interior, boundary and exterior of the two components OGC OpenGIS Simple Features Specification: Spatial Relations Layers & overlay operations - vector

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3 rd editionwww.spatialanalysisonline.com18 Layers & overlay operations – vector Terminologies/operations often mentioned: Spatial join Slivers/sliver handling Clipping/cookie cutters Dissolving and merging Concatenation and conflation Transformation (attribute assignment rules)

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3 rd editionwww.spatialanalysisonline.com19 Areal interpolation Assignment of attributes Using smallest available zonings By proportion of area intersected (new zonings) By surface modelling Utilising ancilliary data (e.g. road networks) Volume preserving/pycnophylactic assignment

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3 rd editionwww.spatialanalysisonline.com20 Districting/re-districting Spatial constraints Contiguity Compactness Attribute constraints/objectives Mean size/spread Attribute values Attribute mix Statistical (scale) effects Spatial (arrangement) effects

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3 rd editionwww.spatialanalysisonline.com21 Districting/re-districting Statistical (scale) effects - example Employed (000s)Unemployed (000s)Total (000s) (Unemployed %) Area A European81990 (10%) Asian9110 (10%) Total9010100 (10%) Area B European401050 (20%) Asian401050 (20%) Total8020100 (20%) A and B European12119140 (13.6%) Asian491160 (18.3%) Total17030200 (15%)

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3 rd editionwww.spatialanalysisonline.com22 Districting/re-districting Spatial (arrangement) effects - example

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3 rd editionwww.spatialanalysisonline.com23 Classification – univariate schemes:1 Classification schemeDescription/application Unique valuesEach value is treated separately, for example mapped as a distinct colour Manual classificationThe analyst specifies the boundaries between classes required as a list, or specifies a lower bound and interval or lower and upper bound plus number of intervals required Equal interval, SliceThe attribute values are divided into n classes with each interval having the same width=Range/n. For raster maps this operation is often called slice Defined intervalA variant of manual and equal interval, in which the user defines each of the intervals required Exponential intervalIntervals are selected so that the number of observations in each subsequent interval increases exponentially Equal count or quantileIntervals are selected so that the number of observations in each interval is the same. If each interval contains 25% of the observations the result is known as a quartile classification. Ideally the procedure should indicate the exact numbers assigned to each class, since they will rarely be exactly equal. PercentilePercentile plots are a variant of equal count or quantile plots. In the standard version equal percentages (percentiles) are included in each class. In GeoDas implementation of percentile plots unequal numbers are assigned to provide classes that contain 6 intervals: 1%, 1% to <10%, 10% to <50%, 50% to <90%, 90% to <99% and 99%

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3 rd editionwww.spatialanalysisonline.com24 Classification – univariate schemes:2 Classification schemeDescription/application Natural breaksWidely used within GIS packages, these are forms of variance-minimisation classification. Breaks are typically uneven, and are selected to separate values where large changes in value occur. May be significantly affected by the number of classes selected and tends to have unusual class boundaries. Standard deviationThe mean and standard deviation of the attribute values are calculated, and values classified according to their deviation from the mean (z-transform). The transformed values are then mapped, usually at intervals of 1.0 or 0.5 standard deviations. Note that this results in no central class, only classes either side of the mean BoxA variant of quartile classification designed to highlight outliers. Typically six classes are defined, these being the 4 quartiles, plus two further classifications based on outliers that may exist within the lower and upper quartiles. These outliers are defined as being data items (if any) that are more than 1.5 times the inter-quartile range (IQR) from the median. An even more restrictive set is defined by 3.0 the IQR. A slightly different formulation is sometimes used to determine these box ends or hinge values. Box plots are implemented in GeoDa and STARS, but are not generally found in mainstream GIS software. They are commonly implemented in statistics packages, including the MATLab Statistics Toolbox

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3 rd editionwww.spatialanalysisonline.com25 Classification – Natural breaks Jenks Natural Breaks algorithm Step 1: The user selects the attribute, x, to be classified and specifies the number of classes required, k Step 2: A set of k 1 random or uniform values are generated in the range [min{x},max{x}]. These are used as initial class boundaries Step 3: The mean values for each initial class are computed and the sum of squared deviations of class members from the mean values is computed. The total sum of squared deviations (TSSD) is recorded Step 4: Individual values in each class are then systematically assigned to adjacent classes by adjusting the class boundaries to see if the TSSD can be reduced. This is an iterative process, which ends when improvement in TSSD falls below a threshold level, i.e. when the within class variance is as small as possible and between class variance is as large as possible. True optimisation is not assured. The entire process can be optionally repeated from Step 1 and TSSD values compared

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3 rd editionwww.spatialanalysisonline.com26 Classification – Multivariate methods Dimensional analysis/reduction methods Factor analysis Principal Components Analysis - PCA Multi-dimensional scaling – MDS Cluster analysis Non-hierarchical Hierarchical Aggregation vs disaggregation methods Assignment methods Discriminant analysis

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3 rd editionwww.spatialanalysisonline.com27 Classification – Multivariate clustering:1 MethodDescription – Unsupervised methods Simple one- class clustering A technique that generates up to M clusters by assigning each input cell to the nearest cluster if its Euclidean distance is less than a given threshold. If not the cell becomes a new cluster centre. It principal merit is speed, but its quality of assignment may not be acceptable K-meansPartition-based algorithm. K-means clustering attempts to partition a multivariate dataset into K distinct (non-overlapping) clusters such that points within a cluster are as close as possible in multi- dimensional space, and as far away as possible from points in other clusters. Fuzzy c-means (FCM) Similar to the K-means procedure but uses weighted distances rather than unweighted distances. Weights are computed from prior analysis of sample data for a specified number of classes. These cluster centres then define the classes and all cells are assigned a membership weight for each cluster. The process then proceeds as for K-means but with distances weighted by the prior assigned membership coefficients Minimum distribution angle An iterative procedure similar to K-means but instead of computing the distance from points to selected centres this method treats cell centres and data points as directed vectors from the origin. The angle between the data point and the cluster centre vector provides a measure of similarity of attribute mix (ignoring magnitude). This concept is similar to considering mixes of red and blue paint to produce purple. It is the proportions that matter rather than the amounts of paint used ISODATA/ ISOCluster (Iterative Self- Organising) Again, similar to the K-means procedure but at each iteration the various clusters are examined to see if they would benefit from being combined or split, based on a number of criteria: (i) combination if two cluster centres are closer than a pre-defined tolerance they are combined and a new mean of means calculated as the cluster centre; if the number of members in a cluster is below a given level the cluster is discarded and the members re-assigned to the closest cluster; and (ii) separation if the number of members, or the standard deviation, or the average distance from the cluster centre exceed pre-defined values than the cluster may be split

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3 rd editionwww.spatialanalysisonline.com28 Classification – Multivariate clustering:2 MethodDescription – Supervised methods Minimum distance to meanEssentially the same as Simple one-pass clustering but cluster centres are pre- determined by analysis of a training dataset. Fast but subject to similar problems as the Simple method Maximum likelihoodA method that uses statistical analysis (variance and covariance) of a training dataset, whose contents are assumed to be Normally distributed. It seeks to determine the probability (or likelihood) that a cell should be assigned to a particular cluster, with assignment being based on the Maximum Likelihood value computed. Stepwise linear/FisherThis is essentially a Discriminant Analysis method, which attempts to compute linear functions of the dataset variables that best explain or discriminate between values in a training dataset. New linear functions are added incrementally, orthogonal to each other, and then these functions are used to assign all data points to the classes. The criterion function minimised in such methods is usually Mahalanobis distance, or the D 2 function. Classified tree analysisA univariate hierarchical data splitting procedure, that progressively divides the training dataset pixels into two classes based on a splitting rule, and then further subdivides these two classes

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3 rd editionwww.spatialanalysisonline.com29 Boundaries and zone membership Convex hulls Point sets Object sets Boundary definition issues Applications MBRs/MERs MBR/MER Convex hull

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3 rd editionwww.spatialanalysisonline.com30 Boundaries and zone membership Non-convex hulls Example criteria Polygonal form (or non-polygonal alpha hulls) Compactness Convex-like Inclusion of all objects/points Reflects the density of objects Procedures Expansion Contraction Density contouring

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3 rd editionwww.spatialanalysisonline.com31 Boundaries and zone membership Non-convex hulls – alpha hulls >0 <0 >>0 <<0

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3 rd editionwww.spatialanalysisonline.com32 Boundaries & zone membership Fuzzy boundaries Use of fuzzy membership functions Sigmoidal, j-shaped, linear, user-defined Use of fuzzy classification procedures Wombling Confusion index Classification entropy

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3 rd editionwww.spatialanalysisonline.com33 Boundaries & zone membership Breaklines and natural boundaries Hard breaklines Soft breaklines Faults/breaklines with areal extent Barriers Applications

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3 rd editionwww.spatialanalysisonline.com34 Tessellations & triangulations Delaunay triangulation Three points form a Delaunay Triangulation if and only if (iff) a circle which passes through all three points contains no other points in the set unique (broadly) best (fewest thin triangles) one of many possible triangulations of irregular point sets (TINs)

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3 rd editionwww.spatialanalysisonline.com35 Tessellations & triangulations TINs Delaunay triangulation is one example of a TIN Designed – engineering Derived – fixed point sets Derived fixed plus rule-based point sets Adaptive densification

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3 rd editionwww.spatialanalysisonline.com36 Tessellations & triangulations Tessellations (of the plane) Simple regular grids Intersections at regular spacings Cells: square, rectangular, triangular, hexagonal Fill or partially fill study region Typically square cells and square or rectangular grids Irregular grids Typically triangulations Dual of triangulation defines a set of regions

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3 rd editionwww.spatialanalysisonline.com37 Tessellations & triangulations Voronoi regions (proximity polygons) AKA: Thiessen polygons, Dirichlet cells Dual of Delaunay triangulation Given a set of points, {S}, in the plane, every location in a Voronoi polygon is closer to one member of {S} than to any other member (ignoring ties) Uniquely partitions the plane (tessellates) – edge regions are treated in a variety of ways One of many possible planar enforced partitions of the plane

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3 rd editionwww.spatialanalysisonline.com38 Tessellations & triangulations Voronoi regions (proximity polygons) - Many applications e.g. assignment of point data attributes to nearest zones Simple forms of interpolation (nearest and natural neighbour) Spatial partitioning Analysis of zones of influence, growth etc… Grid based Voronoi regions Generated by spread (ACS) or distance transform (DT) algorithms Shape and boundary affected by grid size, orientation and algorithm Can be applied uniform or variable cost surfaces Related to optimum routing problems

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3 rd editionwww.spatialanalysisonline.com39 Tessellations & triangulations Voronoi region – network based

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