Lecture 7: Spatial Autocorrelation (Local)
Quantitative Methods II
Dr. Adams
Moran’s I
Global measure of spatial autocorrelation
Single value calculated across all spatial units Points
Polygons
Measure of the overall clustering of the data. Assumes homogeneity.
Options for Moran’s I
R, spdep::moran.mc()
ArcGIS, Spatial Statistics toolbox -> Analyzing Patterns
toolset GeoDa
Crime
What if you wanted to know if crime hot spots were occuring?
Moran’s I, indicates clustering
G-Statistic
The G statistic distinguishes between hot spots and cold spots.
Identifies spatial concentrations When G is relatively large
high values clusters When G is relatively low
low values clusters
Further reading: Getis, A. and Ord, J.K. (1992) The analysis of spatial association by use of distance statistics Geographical Analysis, 24(3) 189-206
Spatial Weights – Measuring Nearness
Contiguity
Binary (0/1)
Zones that share boundaries = 1 Distance
Continuous
Measure distance between points
Historically, distance between polygon centroids
Weight Matrix Format
Raw Matrix (0/1) Row standardized
Divide each value by the row sum
Adjusts for varying number of borders
Exercise: Weight Matrix
Matrix
Fill in a binary contiguity matrix
## A B C D E F G H row.sum
## A 0
##B ##C ##D ##E ##F ##G ##H
0
0
0
0
0
0
0
Contiguity Issues
Polygons that are close but do not share a border? Non border connections, e.g. bridges & tunnels
Length of border?
Calculate using the length of shared border More computation required
House Value, Moran’s I = 0.64333
Local Measures
Calculate a value for each observation
Motivation
Different spatial patterns may occur across space Spatial Processes
See: Luc Anselin 1995 Local Indicators of Spatial Association-LISA Geographical Analysis 27: 93-115
Often global measures have an associated local measure
House Value, Local Moran’s I
6
5
4
3
2
1
0
−1
Local Moran’s I (LISA)
LISA: Local Indicator of Spatial Association
We can map the LISA value for each polygon.
How does spatial autocorrealation occur across space. Also, we can map out statistically significant polygons
Each observation has a local I value and a test statistic.
Hot Spot Map – LISA
High−High High−Low Low−High Low−Low
Not Significant Missing Data
Local Moran’s I Calculation
Ii =ziwijzj j
where zi is the standardized values of xi
z i = x i − x ̄ σx
wij is the spatial weight for i and j
1 N
σx =N
(xi −μ)2
i=1
Prepare
Poly_ID x z i
A ___ B ___ C ___ D ___ E ___ F ___ G ___ H ___
x ̄ =?
LocalMoran’sIforD,EandC-σx =1.65
Ii =zijwijzj |zi =xi−x ̄ σx
Local Moran’s I Statistical Significance
Statistical significance is calculated through simulations. It may differ slightly each time it is calculated
Available in ArcGIS
Significance Maps in R
High−High High−Low Low−High Low−Low
Not Significant Missing Data
Code in R
## Load plot.local.moran function source(“https://raw.githubusercontent.com/gisUTM/
spatialplots/master/plotlocalmoran.R”) polyNB <- poly2nb(polys)
weights <- nb2listw(polyNB)
LISA <- localmoran(polys@data$houseValue, weights,
na.action = na.exclude)
plot.local.moran(polys, "houseValue", LISA, weights = weights)