Outline ESDA

Exploratory Spatial Data Analysis ESDA

Luc Anselin

University of Illinois, Urbana-Champaign http://www.spacestat.com

Outline

ESDA Exploring Spatial Patterns Global Spatial Autocorrelation Local Spatial Autocorrelation

ESDA

1 Classes of Spatial Data (Cressie)

Point Patterns points on a map Geostatistical Data points as sample locations Lattice/Regional Data polygons or points (centroids)

Lattice or Regional Data

Spatial Process index set D is fixed collection of countably many points in Rd finite, discrete spatial units Data fixed points or discrete locations (regions) » examples: county tax rates, state unemployment Research Question interest focuses on statistical inference patterns, estimation, specification tests

WV Housing Values (1990) counties, county centroids, Thiessen polygons

2 EDA and Space

EDA = Discover Potentially Explicable Patterns (Good) Data Visualization (Buja) Interactive View Manipulation » focusing individual views » linking multiple views » arranging many views No Role for Explicit Treatment of Space

Exploratory Spatial Data Analysis

EDA + Describe Spatial Distributions spatial trends, spatial means Identify Atypical Observations spatial outliers Discover Patterns of Spatial Association spatial clusters Suggest Spatial Regimes spatial non-stationarity

ESDA Functionality

Dynamic Graphics linking and brushing statistical plots and map Visualizing Spatial Distributions spatial box map smoothing rates Visualizing Spatial Autocorrelation spatial lag pie charts and bar charts Moran scatterplot and map, LISA maps

3 Exploring Spatial Patterns

Dynamically Linked Windows

Dynamic Graphics different “views” of data: histogram, box plot, scatterplot, list views dynamically linked: click on one, corresponding points (areas) on others highlighted geographic brushing: map as a view of data

4 Global Spatial Autocorrelation

5 Spatial Association

Null Hypothesis: No Spatial Association values observed at a location do not depend on values observed at neighboring locations observed spatial pattern of values is equally likely as any other spatial pattern the location of values may be altered without affecting the information content of the data

Observed (left) and randomized (right) distribution for Columbus Crime

Randomization polyid 1 became 14 polyid 2 became 20 polyid 3 became 48 ...

Observed (left) and randomized (right) distribution for Columbus Crime

Moran’s I = 0.486 Moran’s I = -0.003

6 Alternative Hypotheses of SA

Positive Spatial Association like values tend to cluster in space neighbors are similar Negative Spatial Association neighbors are dissimilar checkerboard pattern

Moran’s I Spatial Autocorrelation Statistic

Moran’s I cross-product statistic Σ Σ Σ 2 I = (N/S0) i j wij.zi.zj / i zi µ= Σ Σ with zi = xi - and S0 = i j wij Inference normal distribution randomization permutation

Interpretation of Moran’s I

Positive Spatial Autocorrelation I > -1/(n-1), or z > 0 spatial clustering of high and/or low values » no distinction between high or low Negative Spatial Autocorrelation I < -1/(n-1), or z < 0 checkerboard pattern, “competition”

7 Spatial Lag Chart

Spatial Lag Visualization value at i compared to weighted average

of neighbors: xi relative to (Wx)i similar values = positive SA dissimilar values = negative SA Spatial Lag Pie Chart xi and (Wx)i as proportions of “pie” (x > 0 only) Spatial Lag Bar Chart xi and (Wx)i as bars

Ww_hoval

Hoval

8 Moran Scatterplot

Linear Spatial Association linear association between value at i and weighted average of neighbors: Σ j wij yj vs. yi , or Wy vs y four quadrants » high-high, low-low = spatial clusters » high-low, low-high = spatial outliers Moran’s I slope of linear scatterplot smoother I = z’Wz / z’z

9 Use of Moran Scatterplot

Classification of Spatial Association Local Nonstationarity outliers high leverage points sensitivity to boundary values Regimes nonlinear association » different slopes in subsets of the data

10 Local Spatial Autocorrelation

LISA Definition (Anselin 1995) Local Indicators of Spatial Association

LISA satisfies two requirements indicate significant spatial clustering for each location sum of LISA proportional to a global indicator of spatial association LISA Forms of Global Statistics local Moran, local Geary, local Gamma

Use of LISA

Identify Hot Spots significant local clusters in the absence of global association significant local outliers » high surrounded by low and vice versa Indicate Local Instability local deviations from global pattern of spatial association

11 Local Moran

Local Moran Statistic Σ Ii = (zi/ m2) j wij.zj Σ i Ii = N.I Inference randomization assumption conditional permutation local dependence or heterogeneity? Visualization LISA map and Moran Significance Map