1. University of Wisconsin-Milwaukee Geographic Information Science Geography 625 Intermediate Geographic Information Science Instructor: Changshan Wu Department of Geography The University of Wisconsin-Milwaukee Fall 2006 Week4: Point Pattern Analysis

2. University of Wisconsin-Milwaukee Geographic Information Science Outline 1.Revisit IRP/CSR 2.First- and second order effects 3.Introduction to point pattern analysis 4.Describing a point pattern 5.Density-based point pattern measures 6.Distance-based point pattern measures 7.Assessing point patterns statistically

3. University of Wisconsin-Milwaukee Geographic Information Science 1. Revisit IRP/CSR Independent random process (IRP) Complete spatial randomness (CSR) 1.Equal probability: any point has equal probability of being in any position or, equivalently, each small sub-area of the map has an equal chance of receiving a point. 2.Independence: the positioning of any point is independent of the positioning of any other point. and

4. University of Wisconsin-Milwaukee Geographic Information Science 2. First- and second order effects  The independent random process is mathematically elegant and forms a useful starting point for spatial analysis, but its use is often exceedingly naive and unrealistic.  If real-world spatial patterns were indeed generated by unconstrained randomness, geography would have little meaning or interest, and most GIS operations would be pointless. IRP/CSR is not realistic

5. University of Wisconsin-Milwaukee Geographic Information Science 2. First- and second order effects 1. First-order effect  The assumption of Equal probability cannot be satisfied  The locations of disease cases tends to cluster in more densely populated areas  Plants are always clustered in the areas with favored soils. From (http://www.crimereduction.gov.uk/toolkits/fa020203.htm)

6. University of Wisconsin-Milwaukee Geographic Information Science 2. First- and second order effects 2. Second-order effect  The assumption of Independence cannot be satisfied  New developed residential areas tend to near to existing residential areas  Stores of McDonald tend to be far away from each other.

7. University of Wisconsin-Milwaukee Geographic Information Science 2. First- and second order effects In a point process the basic properties of the process are set by a single parameter, the probability that any small area will receive a point – the intensity of the process. First-order stationary: no variation in its intensity over space. Second-order stationary: no interaction between events.

8. University of Wisconsin-Milwaukee Geographic Information Science 3. Introduction to point pattern analysis Examples 1)Hot-spot analysis for crime locations 2)Disease analysis (patterns and environmental relations) 3)Freeway accident pattern analysis Point patterns, where the only data are the locations of a set of point objects, represent the simplest possible spatial data.

9. University of Wisconsin-Milwaukee Geographic Information Science 3. Introduction to point pattern analysis Requirements for a set of events to constitute a point pattern 1)The pattern should be mapped on the plane (prefer to preserve distance between points) 2)The study area should be determined objectively. 3)The pattern should be an enumeration or census of the entities of interest, not a sample 4)There should be a one-to-one correspondence between objects in the study area and events in the pattern 5)Event locations must be proper (should not be the centroids of polygons)

10. University of Wisconsin-Milwaukee Geographic Information Science 4. Describing a Point Pattern Point density (first-order or second-order?) Point separation (first-order or second-order?) When first-order effects are marked, absolute location is an important determinant of observations, and in a point pattern clear variations across space in the number of events per unit area are observed. When second-order effects are strong, there is interaction between locations, depending on the distance between them, and relative location is important.

11. University of Wisconsin-Milwaukee Geographic Information Science 4. Describing a Point Pattern First-order or second order?

12. University of Wisconsin-Milwaukee Geographic Information Science 4. Describing a Point Pattern A set of locations S with n events The study region A has an area a. s 1 (x 1, y 1 ) Mean Center Standard Distance: a measure of how dispersed the events are around their mean center

13. University of Wisconsin-Milwaukee Geographic Information Science 4. Describing a Point Pattern A summary circle can be plotted for the point pattern, centered at with radius d If the standard distance is computed separately for each axis, a summary ellipse can be obtained. Summary circleSummary ellipse

14. University of Wisconsin-Milwaukee Geographic Information Science 5. Density-based point pattern measures Crude density/Overall intensity The crude density changes depending on the study area

15. University of Wisconsin-Milwaukee Geographic Information Science 5. Density-based point pattern measures -Quadrat Count Methods 1.Exhaustive census of quadrats that completely fill the study region with no overlaps The choice of origin, quadrat orientation, and quadrat size affects the observed frequency distribution If quadrat size is too large, then ? If quadrat size is too small, then?

16. University of Wisconsin-Milwaukee Geographic Information Science 5. Density-based point pattern measures -Quadrat Count Methods 2. Random sampling approach is more frequently applied in fieldwork. It is possible to increase the sample size simply by adding more quadrats (for sparse patterns) May describe a point pattern without having complete data on the entire pattern.

17. University of Wisconsin-Milwaukee Geographic Information Science 5. Density-based point pattern measures -Quadrat Count Methods Other shapes of quadrats

18. University of Wisconsin-Milwaukee Geographic Information Science 5. Density-based point pattern measures -Density Estimation The pattern has a density at any location in the study region, not just locations where there is an event This density is estimated by counting the number of events in a region, or kernel, centered at the location where the estimate is to be made. C(p,r) is a circle of radius r centered at the location of interest p Simple density estimation

19. University of Wisconsin-Milwaukee Geographic Information Science 5. Density-based point pattern measures -Density Estimation Bandwidth r If r is too large, then ? If r is too small, then?

20. University of Wisconsin-Milwaukee Geographic Information Science 5. Density-based point pattern measures -Density Estimation Density transformation 1) visualize a point pattern to detect hot spots 2) check whether or not that process is first-order stationary from the local intensity variations 3) Link point objects to other geographic data (e.g. disease and pollution)

21. University of Wisconsin-Milwaukee Geographic Information Science 5. Density-based point pattern measures -Density Estimation Kernel-density estimation (KDE) Kernel functions: weight nearby events more heavily than distant ones in estimating the local density  IDW  Spline  Kriging

22. University of Wisconsin-Milwaukee Geographic Information Science 6. Distance-based point pattern measures  Look at the distances between events in a point pattern  More direct description of the second-order properties

23. University of Wisconsin-Milwaukee Geographic Information Science 6. Distance-based point pattern measures -Nearest-Neighbor Distance Euclidean distance

24. University of Wisconsin-Milwaukee Geographic Information Science 6. Distance-based point pattern measures -Nearest-Neighbor Distance If clustered, has a higher or lower value?

25. University of Wisconsin-Milwaukee Geographic Information Science 6. Distance-based point pattern measures -Distance Functions: G function

26. University of Wisconsin-Milwaukee Geographic Information Science 6. Distance-based point pattern measures -Distance Functions: G function

27. University of Wisconsin-Milwaukee Geographic Information Science 6. Distance-based point pattern measures -Distance Functions: G function  If events are closely clustered together, G increases rapidly at short distance  If events tend to evenly spaced, then G increases slowly up to the distance at which most events are spaced, and only then increases rapidly. The shape of G-function can tell us the way the events are spaced in a point pattern.

28. University of Wisconsin-Milwaukee Geographic Information Science 6. Distance-based point pattern measures -Distance Functions: F function Three steps 1)Randomly select m locations {p 1, p 2, …, p m } 2)Calculate d min (p i, s) as the minimum distance from location p i to any event in the point pattern s 3) Calculate F(d)

29. University of Wisconsin-Milwaukee Geographic Information Science 6. Distance-based point pattern measures -Distance Functions: F function  For clustered events, F function rises slowly at first, but more rapidly at longer distances, because a good proportion of the study area is fairly empty.  For evenly distributed events, F functions rises rapidly at first, then slowly at longer distances.

30. University of Wisconsin-Milwaukee Geographic Information Science 6. Distance-based point pattern measures -Comparisons between G and F functions

31. University of Wisconsin-Milwaukee Geographic Information Science 6. Distance-based point pattern measures -Comparisons between G and F functions

32. University of Wisconsin-Milwaukee Geographic Information Science 6. Distance-based point pattern measures -Distance Functions: K Function The nearest-neighbor distance, and the G and F functions only make use of the nearest neighbor for each event or point in a pattern This can be a major drawback, especially with clustered patterns where nearest-neighbor distances are very short relative to other distances in the pattern. K functions (Ripley 1976) are based on all the distances between events in S.

33. University of Wisconsin-Milwaukee Geographic Information Science 6. Distance-based point pattern measures -Distance Functions: K Function Four steps 1)For a particular event, draw a circle centered at the event (s i ) and with a radius of d 2)Count the number of other events within the circle 3)Calculate the mean count of all events 4)This mean count is divided by the overall study area event density

34. University of Wisconsin-Milwaukee Geographic Information Science 6. Distance-based point pattern measures -Distance Functions: K Function is the study area event density

35. University of Wisconsin-Milwaukee Geographic Information Science 6. Distance-based point pattern measures -Distance Functions: K Function Clustered? Evenly distributed?

36. University of Wisconsin-Milwaukee Geographic Information Science 6. Distance-based point pattern measures -Edge effects Edge effects arise from the fact that events near the edge of the study area tend to have higher nearest-neighbor distances, even though they might have neighbors outside of the study area that are closer than any inside it.

37. University of Wisconsin-Milwaukee Geographic Information Science 7. Assessing Point Patterns Statistically A clustered pattern is likely to have a peaky density pattern, which will be evident in either the quadrat counts or in strong peaks on a kernel-density estimated surface. An evenly distributed pattern exhibits the opposite, an even distribution of quadrat counts or a flat kernel-density estimated surface and relatively long nearest-neighbor distances. But, how cluster? How dispersed?

38. University of Wisconsin-Milwaukee Geographic Information Science 7. Assessing Point Patterns Statistically

39. University of Wisconsin-Milwaukee Geographic Information Science 7. Assessing Point Patterns Statistically -Quadrat Counts Independent random process (IRP) Complete spatial randomness (CSR) and Mean Variance How about mean > variance? mean < variance? A B The variance/mean (VMR) is expected to be 1.0 if the distribution is Poisson.

40. University of Wisconsin-Milwaukee Geographic Information Science 7. Assessing Point Patterns Statistically -Quadrat Counts For a particular observation Mean = number of events / study area A B n is the number of events x is the number of quadrats

41. University of Wisconsin-Milwaukee Geographic Information Science 7. Assessing Point Patterns Statistically -Quadrat Counts Variance A B k = 0: 2 * (0 – 1.25) 2 = 3.125 k = 1: 3 * (1 – 1.25) 2 = 0.1875 k = 2:2 * (2 – 1.25) 2 = 1.125 k = 3:1 * (3 – 1.25) 2 = 3.0625

42. University of Wisconsin-Milwaukee Geographic Information Science 7. Assessing Point Patterns Statistically -Quadrat Counts A B VMR = Variance/Mean = 0.9375/1.25 = 0.75 Clustered? Random? Dispersed?

43. University of Wisconsin-Milwaukee Geographic Information Science 7. Assessing Point Patterns Statistically -Nearest Neighbor Distances The expected value for mean nearest- neighbor distance for a IRP/CSR is The ratio R between observed nearest-neighbor distance to this value is used to assess the pattern If R > 1 then dispersed, else if R < 1 then clustered?

44. University of Wisconsin-Milwaukee Geographic Information Science 7. Assessing Point Patterns Statistically -G and F Functions Clustered Evenly Spaced

45. University of Wisconsin-Milwaukee Geographic Information Science 7. Assessing Point Patterns Statistically K Functions IRP/CSR