GY460 Techniques of Spatial Analysis

1. GY460 Techniques of Spatial Analysis Lecture 5: Exploratory analysis of spatial patterns Steve Gibbons

2. Introduction • Suppose we want to know • A) Are there spatial patterns? • Are there clusters of manufacturing productivity? • Are their crime clusters? • Various global spatial statistics available to answer this • Suppose we want to know • B) Which places generate these spatial patterns? • Where are there clusters of manufacturing productivity? • Where are their crime clusters? • We need local indicators • Descriptive statistics – “Exploratory Spatial Data Analysis”

3. ‘Mean’ v ‘covariance’ v ‘density’ methods • Three general classes of methods • Methods that are based on local means amongst neighbouring events: we’ve looked at these already – see the ‘smoothing’ lecture e.g. • Kernel regression, Interpolation • Methods that are based on local covariances between neighbouring events e.g. • Moran’s I (global) • Local Indicators of Spatial Association (LISA) - Local Moran’s I • Methods that are based on the density of events or things • Kernel density estimates, distance based measures: Ripley’s K, Durnaton and Overman’s K-density

4. Global and Local Indicators of Spatial Association

5. Spatial autocorrelation • Assume places (regions, districts, firms people etc) are fixed • Variable (x) recorded at places s • Is the data x random across space or are there similarities between neighbours? • Does a high value of x tend to be associated with a high value of x in neighbouring places (and low values with low)?

6. Random - no spatial autocorrelation

7. Overly dispersed - negatively autocorrelated

8. Positive spatial autocorrelation

9. Gobal indicators • ‘Is there spatial autocorrelation’? • Global indicators of spatial association provide the answer • E.g. Moran’s I • Where I’ve used ~ to indicate deviations from mean

10. LISA • ‘Where is the spatial autocorrelation’? • Local indicators of spatial association (LISA) provide answer • Anselin (1995) definition: LISA • Indicates spatial clustering of similar values around the observation • Sum of LISAs proportional to a Global indicator

11. Local Moran I • Local Indicator (Local Moran I) • Product of (centred) x and ‘neighbouring’ x at place i • Divided by the variance of x • Note: mean of Local = Global

12. Local Moran I 5 4 1 2 3

13. LISA are map-able: regional convergence Source: Rey, S and B. Montouri, US Regional Economic Convergence: A Spatial Econometric Perspective, Regional Studies, 33 (2) 143-156

14. Moran scatter-plot • See Anselin (1995, Local Indicators of Spatial Association , Geographical Analysis) • This is just a graph of ‘average neighbourhood’ x (Wx) • against x • Or use standardised values • E.g. from Anselin (1995) • Conflict in African countries 1966-78

15. Moran scatter-plot

16. Moran scatter-plot: components of spatial autocorrelation Li - Low-high Li + High-high Wx 0 Li - High-low Li + Low-low 0 x

17. Outliers: boundary areas cause problems (edge effects) Sudan Egypt

18. US Income Convergence Example

19. Example: London crime data • Burglary rates, 2001. Global I = 0.624

20. Local Moran I Map Not-significant High-High Low-Low Low-High High-Low ?

21. Local Moran Significance Map Not-significant P=0.05 P=0.01 P=0.001 P=0.0001 ?

22. Moran Scatter Plot ?

23. LISA – Hypothesis tests • H0: no spatial clustering at point i • Use analytical standard errors • e.g. see Anselin (1995) (map room) • Or Fortheringham chapter on Local Analysis • Or simulate null distribution by random re-assignment • LISA are specific to each observation (place) • Z-statistics or p-values are specific to each observation

24. LISA – Hypothesis tests • If we have n observations we have • n tests, n z-statistics, n p-values • One test statistic has p-value p • Probability of observing a significant test amongst n independent tests is • High probability of ‘Type I’ error • Wrongly reject Null of no clustering

25. LISA – Hypothesis tests • Need Bonferroni correction • Significance level = , z statistic p-value = p • Corrected p-value is n*p • Test at /n • Conservative if data is spatially correlated because the tests are correlated • Probability of observing significant test statistic by chance under H0 is

26. Example: Growth in London crime • Growth in burglary rates, 1999-2002. Global I = 0.328

27. Local Moran I z-scores • Z(0.05) = 1.96, no correction • Z(0.05) = 3.77, bonferroni correction (634 wards) ?

28. Conclusions on LISA • Local Moran’s I (and other LISA) useful for showing places where significant spatial autocorrelation exists • Purely descriptive • Though potential to combine with regression analysis for further analysis • Residuals? • Dependent variable?

29. Spatial point pattern analysis

30. Introduction • Spatial autocorrelation analysis tell us about similarities/dissimilarities in the characteristics of neighbouring places • Typically zonal aggregated data, or characteristics related to objects that are treated as fixed in space • Point pattern analysis looks for patterns in the spatial location of events • “Events” are assigned to points in space • e.g. infection by bird-flu, site where firm operates, place where crime occurs, redwood seedlings • Some parallels: e.g. if we aggregate crime events to zones we get zonal crime rate data • Point pattern analysis has the advantage that it is not directly dependent on zone definitions (MAUP)

31. Spatial point patterns Aggregated Random

32. Spatial point patterns Regular

33. Complete Spatial Randomness • The simplest “null hypothesis” regarding spatial point patterns • The number of events N(A) in any planar region A with area |A| follows a Poisson distribution with mean: • Given N(A)= n, the events in A are an independent random sample from the uniform distribution on A • Poisson process has constant “intensity” • Intensity is the expected number of events per unit area • Also mean = variance See Diggle p.47

34. Grid based approaches • Divide area up in to grids (area and calculate observed “intensity” at each grid square (number of obs divided by area) • Tests for CSR based on grid counts G={g1,g2,…} • Under CSR, independent identically Poisson distributed with mean • Do point counts G follow Poisson distribution? Use mean=variance property • Under CSR E[I]=1; >1 implies aggregation; <1 implies dispersion • But what size grid?

35. Testing for CSR • CSR not particularly useful “null hypothesis” for economic/geographic processes • e.g. we wouldn’t want to test firm localisation against this assumption (why?) • But useful starting point • Other benchmarks preferable – e.g. distribution of manufacturing firms as the ‘null hypothesis’ in Marcon and Pruech (2003), Duranton and Overman (2005)

36. Kernel intensity/density estimates • Space is continuous. • Grid squares approaches give discontinuous estimate based on arbitrary grid • More general approach: kernel intensity estimates • k(.): kernel weighting function (a bivariate probability density function) • h: bandwidth - higher bandwidth increases bias, but reduces variance; somewhat arbitrary though methods available for optimal bandwidth choice • s: grid point • si: data points

37. Kernel intensity/density estimates • A simple kernel intensity estimate using a “uniform” kernel 2 = 0.716

38. Kernel intensity/density • Note: technically the kernel density is • So that adding up over the sample points, the density sums to 1 • Sometimes (e.g. GIS) the intensities are referred to as densities

39. Edge effects R2 x R1 y

40. Correcting edge effects • Intensity estimated lower at point y than at point x • Corrections can be based on • % area of circle within R1 • % circumference of circle within R1 • [circumference easier to calculate] • drawing buffer zones

41. K function • The “K function” is the expected number of events within distance d of an event, divided by mean intensity in the study area (i.e. number of events/ area)

42. Ripley’s K • Ripley’s (1976) estimator of K • Where |A| means area of study area A, and means distance between s_i and s_j • Also need to take care of edge effects • If events uniformly distributed with intensity  then expected number of events within distance d is  d2 • So expected K(d) under uniform distribution (CSR) is d2

43. Ripley’s K d=1m 5m 5m If uniform K(1) = = 3.14

44. Checking for clustering • Under CSR with uniform intensity expect K(d) K d

45. Hypothesis tests • Sampling distribution of these spatial point process statistics is often unknown • Possible to derive analytical point-wise confidence intervals for kernel estimates • But more generally use “monte-carlo”, “bootstrap” and random assignment methods

46. Postscript: Geographically weighted regression

47. Postscript 1: GWR • Sometimes we’d like to know about variation in regression parameters over space • One technique: Geographically Weighted Regression • To get the parameters at place s0, estimate weighted least squares regression, i.e. OLS on: • Where the weight on each observation wi0 decreases with distance from place s0 • See Fotheringham Chapter 5, and/or Brunsdon et al, 1998, Geographically Weighted Regression – Modelling Spatial Non-stationarity, The Statistician, Vol. 47, no. 3.