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Spatial Weights
Luc Anselin
University of Illinois, Urbana-Champaignhttp://geog55.geog.uiuc.edu/sa
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Outline
Connectivity in Space
Spatial Weights
Practical Issues
Spatial Lag Operator
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Spatial Connectivity
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Why Spatial Weights?
Spatial Correlation
Cov[yi.yh] 0, for i h
Structure of Correlation which i, h interact?
N observations to estimate N(N-1)/2 interactions
impose structure in terms of what are theneighbors for each location
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Spatial Arrangement Need to Impose Structure on the Extent of Spatial
Interaction Neighborhood View
define neighborhood set N(i) for each location i
spatial weights matrix Pairs View
order pairs of locations i-j in function of separating
distance semivariogram (geostatistics)
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Neighborhood Set Geographic-Cartographic Contiguity (GIS)
common boundary = contiguity common border, common vertex
distance band = isotropy
interaction border length and distance
Spatial Interaction distance decay, gravity, entropy
scale dependent, identification problems
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Neighborhood Set (continued)
Socio-Economic Distance multidimensional distance based on
socio-economic indicators Euclidean, Mahalanobis
example: income, ethnicity, industrial structure, tradeflows, migration flows
problem with endogeneity variables for distance same as in model
zero distances 1/(zi - zj) when zi = zj
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Spatial Weights
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Example: N=6contiguity = common boundary
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contiguity as a graphlink between nodes = contiguity
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Spatial Weights MatrixDefinition
N by N positive matrix W, withelements wij
Simplest Form: Binary Contiguitywij = 1 for i and j neighbors(e.g. dij < critical distance)
wij = 0 otherwise,wii = 0 by convention
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How to Define Weights Contiguity
common boundary
Distance distance band
k-nearest neighbors
General
social distance complex distance decay functions
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Contiguity Regular Grid Regular Grid
rook 2, 4, 6, 8
bishop
1, 3, 7, 9 queen
both
1 2 3
4 5 6
7 8 9
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Contiguity Irregular Units Irregular Units
common border rook
common vertex
039 and 067 queen
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Contiguity Weights in DynESDA2
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Distance Based Weights
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General Spatial Weights Cliff-Ord Weights
wij to reflect potential spatial interactionbetween i and j
wij
= [dij]-a.[b
ij]b
withdij as distance between i and jbij as share of common boundary between i and
j in perimeter of i
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General Spatial Weights(continued)
Weights May Contain Parameters inverse distance weights
wij = 1 / dij
estimated from data or chosen a priori in practice: second power (gravity model)
identification problems in nonlinear weights
interaction is multiplicative:
. wij =
(1 / dij)
parameters and not separately identified
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General Spatial Weights(continued)
K-Nearest Neighbor Weights k neighbors, irrespective of actual distance
warps space
Economic Weights (Case) block structure, state effect
wij = 1 for all i, j in block
economic distance |ri - rj|, weight = 1/|ri - rj| e.g., r = total employment
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Row-Standardized
Spatial Weights
Motivation averaging of neighboring values
= form of spatial smoothing
spatial parameters comparable
wsij = wij / j wijj w
sij = 1
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Practical Issues
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Characteristics of
Spatial Weights
Measures of Overall Connectedness percent nonzero weights (sparseness) average weight
average number of links
principal eigenvalue
Location-Specific Measures most/least connected observations
unconnected observations = islands
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Weights Characteristics in
DynESDA2
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Higher Order Contiguity Recursive Definition
contiguous of order k is first ordercontiguousto order k-1
2nd is first order contiguous to first
Remove Circularity and Redundancy
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contiguity as a graphlink between nodes = contiguity
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Circularity and Redundancy in
Higher Order Weights
Powering of the Weights Matrix standard approach invalid for contiguity
Removing Circularity and Redundancy
sparse network representation of weights modified Dijkstra algorithm to identify
number of steps between nearest
neighbors (Anselin and Smirnov) number of steps = order of contiguity
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Spatial Lag Operator
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Spatial Shift No Direct Counterpart to Time Series
Shift Operator time series: Lk = yt-k spatial series: which h are shifted by k
from location i? on regular lattice: east, west, north, south
(i - 1, j) (i + 1, j) (i, j - 1) (i, j + 1)
arbitrary for irregular lattice different number of neighbors by observation
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Spatial Lag Operator Distributed Lag
row-standardized weights j wij = 1 spatial lag is weighted average of
neighboring values
j wij.yj, for each i vectorWy
spatial lag does not contain yi
spatial lag is a smoother not a window average
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value yi = $42,3004 neighborsvalues for neighbors: $50,200,$64,600, $45,000, $34,200
spatial lag = (1/4)$50,200 +(1/4)$64,600 + (1/4)$45,000 +(1/4)$34,200 = $48,500
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Interpretation of Spatial Lag Linear Association = Spatial Autocorrelation
comparison of value of y at i to average ofvalues at neighboring locations yi and (Wy)i similar = positive spatial
autocorrelation (high-high, low-low) yi and (Wy)i dissimilar = negative spatial
autocorrelation (low-high, high-low)