Shape Metrics Jason Parent jason parentuconn edu Academic

Shape Metrics Jason Parent jason. parent@uconn. edu Academic Assistant – GIS Analyst Daniel Civco Professor of Geomatics Center for Land Use Education And Research (CLEAR) Natural Resources and the Environment University of Connecticut Shlomo Angel Adjunct Professor of Urban Planning Robert F. Wagner School of Public Service, New York University Woodrow Wilson School of Public and International Affairs, Princeton University 1

Background ► Metrics that quantify aspects of shape have widespread applications § Pattern analysis in landscape ecology and geography § Identify suitability of a given area for a particular purpose ► Hundreds of metrics exist for measuring characteristics of shapes. § The relevancy and appropriate use is often not clear 2

Why measure characteristics of shape? ► Patch shape and area influences the viability of forest patches for certain forest species. ► Compactness of the urban footprint can be a measure of sprawl in cities. ► Compactness of election districts may indicate gerrymandering. ► Patch shape and area may determine the suitability of a patch of land for a particular purpose 3

Objectives ► Present 10 metrics for measuring various aspects of shape that can be applied to contiguous patches. ► Present a normalized version of each metric that is not affected by shape area § Measures shape compactness § Values range between 0 and 1 with higher values indicating greater compactness ► Present a framework for determining the appropriate metric(s) for a given analysis. ► Present a script tool that can calculate each of the metrics for polygons in a feature class. 4

Defining a polygon in terms of points Interior points Vertex points Perimeter points ► Points defining ► Points evenly ► Points equally perimeter vertices distributed spaced along (inflexion points) throughout shape perimeter 20, 000 points 100 points 5

Normalizing shape metrics - the Equal Area Circle ►A circle is the most compact shape possible for a given area (Angel et al. 2009). ► The Equal Area Circle (EAC) is a circle with an area equal the area of the shape. ► The normalized metrics presented are normalized using the EAC § Creates a measurement of compactness § Metrics normalized with the EAC are highly correlated ► Normalized metrics are appropriate when the influence of shape area is irrelevant or misleading or when a measure of compactness is needed. 6

Shape characteristics ► Distribution of the polygon around a central point ► Distribution of points within the polygon ► Characterizing the polygon interior and perimeter ► Characterizing the polygon as an object to traverse or circumvent 7

Distribution of the shape around a central point 8

Proximity index - definition ► The average Euclidean distance from all interior points to the centroid* d 1 + d 2 +…dn Proximity = n d 4 d 3 d 1 d 2 9 * The average XY coordinates for all vertices that define the shape

Normalized proximity index Proximity. EAC normalized proximity (n. PI) = Proximity. Shape Proximity. EAC = 2 * radius 3 EAC index I = 5776 n. I = 0. 98 I = 4968 n. I = 1. 00 normalized index I = 4944 n. I = 0. 69 I = 7267 n. I = 0. 55 I = 6177 n. I = 0. 44 10

Proximity index - comments ► All points in shape are given equal weight. ► Relatively quick calculation time ► Basic index appropriate for use when distance to the shape’s center is needed… § i. e. Proximity calculated for an urban footprint gives an estimate of the travel distance for residents commuting to the urban center – used to infer travel costs (time, fuel, pollution, etc. ) ► Normalized index appropriate for measuring compactness 11

Spin index - definition The average of the square of the Euclidean distances between all interior points and the centroid. ► Also known as Moment of Inertia in the literature. ► spin = d 12 + d 22 +…dn 2 # of points d 4 d 3 d 1 d 2 12

Normalized spin index spin. EAC normalized spin = spin. Shape spin. EAC = 0. 5 * radius 2 EAC index I = 37990552 n. I =0. 95 I = 27768701 n. I = 1. 00 normalized index I = 27634615 n. I = 0. 48 I = 54370006 I = 50360516 n. I = 0. 33 n. I = 0. 17 13

Spin index - comments ► Similar to Proximity except more weight is given to the polygon’s extremities. ► Relatively quick calculation time. ► Basic metric results not very intuitive. ► Normalized metric appropriate for measuring compactness when focus is on shape extremities… n. Proximity = 0. 44 n. Spin = 0. 17 i. e. Normalized spin calculated for an urban footprint gives a measure of compactness that is more sensitive to the outlying parts of the footprint. This index is more capable of identifying footprints that have “tendril-like” projections (often perceived as an indicator of sprawl). 14

Dispersion - definition The average distance from the centroid to all points on the shape perimeter. ► Based on the Boyce-Clark Index (Boyce and Clarke 1964). ► d 1 + d 2 +…dn dispersion = n d 4 d 3 d 1 d 2 15

Normalized dispersion index circle with dispersion equal to shape dispersion d 3 d 4 d 1 normalized = dispersion d 2 dx is the distance between shape perimeter and the circle perimeter along a radial emanating from the centroid. d 1 + d 2 +…dn deviation = n dispersion – deviation dispersion 16

Dispersion index - examples index I = 8664 n. I = 0. 90 I = 7451 n. I = 1. 00 I = 4539 n. I = 0. 56 I = 7469 n. I = 0. 81 I = 6802 n. I = 0. 53 normalized index ► Normalized values close to 1 indicate equal dispersion in all directions. 17

Dispersion index - comments ► Only points in perimeter are used to calculate compactness. n. I = 0. 90 § Gaps in a shape do not affect the metric ► Relatively quick calculation time. ► Use basic metric when average spread of a phenomena is of interest. ► Normalized metric appropriate for measuring shape compactness when gaps in the shape should be ignored… § i. e. Normalized dispersion can indicate whether a phenomena (i. e. invasive species spread) is propagating from an epicenter equally in all directions. This can give an idea of the effectiveness of containment efforts. 18

Distribution of points within the shape 19

Cohesion index - definition ► The average distance between all pairs of interior points. cohesion = d 1 + d 2 +…dn # of point pairs d 3 d 4 d 5 d 1 d 6 d 2 20

Normalized cohesion index cohesion. EAC normalized cohesion = cohesion. Shape cohesion. EAC = 0. 9054 * radius. EAC index I = 7881 n. I = 0. 98 I = 6739 n. I = 1. 00 I = 6719 n. I = 0. 69 I = 9386 n. I = 0. 58 I = 8282 n. I = 0. 45 normalized index 21

Cohesion index - comments ► All points in shape given equal weight ► Computationally intensive for large numbers of points § Only calculated for a sample of points to improve calculation time. ► Appropriate when the average distance between points in a shape is needed or when distribution of the shape about the center is not relevant. § i. e. Cohesion calculated for an urban footprint gives an estimate of the travel distance for residents commuting within the city – does not assume residents predominantly travel to the urban center. Can be used to infer travel costs (time, fuel, pollution, etc. ). 22

Characterizing the shape interior and perimeter 23

Depth index - definition ► The average distance from the shape’s interior points to the nearest point on the perimeter. 24

Normalized depth index depth. Shape normalized depth = depth. EAC = 1 * radius. EAC 3 index I = 2530 n. I = 0. 89 I = 2487 n. I = 1. 00 I = 563 n. I = 0. 33 I = 711 n. I = 0. 35 I = 552 n. I = 0. 41 normalized index 25

Depth index - comments ► Measures average distance from the interior of a polygon to the edge of the polygon. ► Indicates how susceptible the patch may be to disturbances outside the shape perimeter. § Larger distances indicate greater insulation of the shape’s interior to external events. ► Appropriate when the insulation of a patch’s interior from the surrounding environment is important. § i. e. The depth of a forest patch can indicate the suitability of the patch for species that do not tolerate close proximity to development, open fields, or other land cover types. 26

Viable interior index - definition ► The area of the shape that is beyond the depth of the edge-effect. Edge-width 27

The edge-effect Occurs when a patch can be influenced or degraded by the surrounding environment. § i. e. The edge of a forest patch can be affected by increased exposure to wind, light, invasive species, etc. ► The distance over which the “edge-effect” can occur depends on the issue or species or study § In ecological literature, distances ranges from 25 meters to several hundred meters depending on species and land cover type. § Timber harvest may not be practical within some distance from developed areas. ► We typically assume a 100 meter edge-width for general purposes studies. ► 28

Normalized viable interior index interior. EAC normalized interior = interior. Shape interior. EAC = Π * (radius. EAC – edge)2 Index (ha) I = 17, 127 n. I = 0. 97 I = 13, 068 n. I = 1. 00 I = 353 n. I = 0. 07 I = 2, 176 n. I = 0. 27 I = 339 n. I = 0. 11 normalized index 29

Viable interior index - comments ► Measures area of shape that is not susceptible to influence from the surrounding environment. ► An appropriate edge-width distance must be used for meaningful results. ► Easy and quick to calculate. ► Appropriate when the insulation of a patch’s interior from the surrounding environment is important and an appropriate edge-width is known. § i. e. The interior index of a forest patch can indicate whether the patch contains enough suitable area to support the desired diversity of interior forest species 30

Girth index - definition ► The radius of the largest circle that can be inscribed in the shape d 31

Normalized girth index girth. Shape normalized girth = radius. EAC index I = 7417 n. I = 0. 87 I = 7344 n. I = 1. 00 I = 1461 n. I = 0. 28 I = 1700 n. I = 0. 28 I = 1200 n. I = 0. 29 normalized index 32

Girth index - comments ► Measures the largest circular area that can be fully contained within a shape. ► Can be used to determine if a polygon can accommodate the footprint of a feature such as a proposed development. § i. e. A suitability analysis indicates patches of area that are suitable for development. The girth index can be used to determine which patches contains a contiguous area large enough to contain the footprint of the proposed development. Shape area Circle area A = 1. 19 a = 0. 21 A = 1. 01 a = 0. 74 33

Perimeter index - definition ► The perimeter of the shape 34

Normalized perimeter index normalized perimeter = perimeter. EAC perimeter. Shape index I = 60, 357 n. I = 0. 89 I = 46, 918 n. I = 1. 00 I = 134, 879 n. I = 0. 24 I = 97, 017 n. I = 0. 39 I = 53, 447 n. I = 0. 48 normalized index 35

Normalized perimeter index - comments ► Measures the length of the perimeter of a given shape. ► Very quick and easy to calculate ► Metric gives an indication of the shape’s exposure to external conditions. § i. e. The normalized perimeter index for a forest patch will indicate the exposure of the patch to the surrounding environment. Patches for which the normalized index is maximized will have less exposure. 36

Characterizing the shape as an object to traverse or circumvent 37

Detour Index ► The perimeter of the shape’s convex hull * Convex hull * The convex hull is the convex polygon with the shortest possible perimeter that fully encompasses it. 38

Normalized detour index normalized detour = perimeter. EAC perimeter. Convex Hull ► Larger normalized values indicate that a relatively shorter path is needed to circumvent the shape index I = 66, 373 n. I = 0. 81 I = 48, 349 n. I = 1. 00 normalized index I = 48, 118 n. I = 0. 67 I = 65, 447 n. I = 0. 58 I = 75, 747 n. I = 0. 34 39

Detour index - comments ► The convex hull is the shortest path needed to circumvent a shape. ► The normalized index indicates how large an obstacle a shape presents relative to its area. ► Relatively quick to calculate. ► Appropriate to use when the shape is an obstacle to passage and cannot be traversed. § i. e. The normalized detour index can be used quantify the degree to which a highway obstructs wildlife movement. The effectiveness of wildlife crossings in reducing the obstruction can be analyzed. 40

Traversal Index ► The average distance of the shortest paths connecting any two points on the shape perimeter. § The paths must remain inside the shape. traversal = d 1 + d 2 +…dn # of point pairs d 1 d 2 d 3 41

Normalized traversal index normalized traversal = traversal. EAC traversal. Shape 4 * radius. EAC Π index I = 11, 019 n. I = 0. 98 I = 9397 n. I = 1. 00 normalized index I = 7679 n. I = 0. 85 I = 13, 357 n. I = 0. 57 I = 9039 n. I = 0. 58 42

Traversal index - comments ► The traversal index is the shortest average distance between two points on the perimeter – the paths between points cannot intersect the shape boundary. ► Very long calculation time § 10 -15 minutes per feature ► Appropriate to use when the shortest interior distance between any two points on the patch’s perimeter is relevant. § i. e. The traversal index can estimate the distance required to cross a lake from any direction. 43

Choosing the appropriate metric ► The basic metrics each provide different information about shape characteristics. § The metric used should make logical sense for the analysis. ► The normalized metrics tend to be highly correlated with each other. § One metric may be a good proxy for another to measure compactness. 44

Summary: shape aspect and suggested metrics ► Distribution of the shape around a central point… ► § Proximity § Spin § Dispersion ► § § The shape as an object to traverse or circumvent § Traversal § Detour Characterizing the shape interior and exposure to external conditions ► Perimeter Girth Depth Viable interior Distribution of points within the shape… § Cohesion 45

The Shape Metrics Tool A Python script has been developed to calculate the metrics presented ► Script will run out of Arc. Toolbox for Arc. GIS 9. 3 ► Will be available through Center for Land use Education And Research (CLEAR) website ► § http: //clear. uconn. edu/tools/Sh ape_Metrics/index. html 46

Conclusions ► The basic form of the metrics are influenced by shape area § The appropriate metric to use depends on the application – the metric should make logical sense. ► The normalized version of the metrics provides a measure of compactness. § Normalized metrics tend to be highly correlated with each other and with people’s perception of compactness. ► The script tool will facilitate calculation of shape metrics for polygon feature class data. 47

References ► Angel, S. , J. Parent, and D. L. Civco. 2009. Ten Compactness Properties of Circles: A Unified Theoretical Foundation for the Practical Measurement of Compactness. Canadian Geographer. (in press) ► Angel, S and GM Hyman (2009). Ten Theorems Concerning the Compactness of Circles. (forthcoming). 48

QUESTIONS? Jason Parent jason. parent@uconn. edu Academic Assistant – GIS Analyst Daniel Civco Professor of Geomatics Center for Land Use Education And Research (CLEAR) Natural Resources and the Environment University of Connecticut Shlomo Angel Adjunct Professor of Urban Planning Robert F. Wagner School of Public Service, New York University Woodrow Wilson School of Public and International Affairs, Princeton University Download script at: http: //clear. uconn. edu/tools/Shape_Metrics/index. html 49