anselin spatial weights metodos 2006
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Copyright 2004-2006 by Luc Anselin, All Rights Reserved
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Copyright 2004-2006 by Luc Anselin, All Rights Reserved
Spatial Analysis with GeoDa
2. Spatial Weights
Luc AnselinUniversity of Illinois, Urbana-Champaign
http://sal.uiuc.edu/
ERSA Summer School 2006
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Copyright 2004-2006 by Luc Anselin, All Rights Reserved
Outline
Concepts Contiguity Weights Distance-Based Weights Characteristics of Weights Spatially Lagged Variables
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Copyright 2004-2006 by Luc Anselin, All Rights Reserved
Concepts
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Copyright 2004-2006 by Luc Anselin, All Rights Reserved
Spatial Autocorrelation
Match value similarity locational similarity
How to Specify Locational Similarity
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Copyright 2004-2006 by Luc Anselin, All Rights Reserved
Why Spatial Weights
Identification Problem spatial covariance Cov[yi, yj] total number of interactions is N(N-1)/2 only N observations in a cross-section
Incidental Parameter Problem number of parameters increases with
sample size, O(N2)
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Copyright 2004-2006 by Luc Anselin, All Rights Reserved
Solution
Impose Structure on the Problem set some interactions to zero only neighbors interact directly constrain the number of neighbors
Assume a Single Parameter spatial autocorrelation coefficient
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1
4
3
6
2 5
6 Polygon LayoutNeighbor as Having Common Boundary
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1
4
52
3
6
Neighbor Structure as a Graph
NodesLinks
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Spatial Weights Matrix
N by N positive matrix W elements are wij wij nonzero for i,j neighbors, wij = 0 not wii = 0 no self-neighbors
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Binary Contiguity
Weights are 0 or 1 for i, j neighbors, wij = 1
Example, using common boundary
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Types of Spatial Weights
Geographic Weights contiguity distance general weights graph-based weights
Socio-Economic Weights
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Contiguity Weights
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Contiguity
Common Boundary boundary of non-zero length
What is a Non-zero Boundary Three Views rook bishop queen
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1 2 3
4 5 6
7 8 9
Regular Grid Contiguity
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1 2 3
4 5 6
7 8 9
Rook Contiguity
Neighbors for 5: 2, 4, 6, 8Common Border
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Rook Weights
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1 2 3
4 5 6
7 8 9
Bishop Contiguity
Neighbors for 5: 1, 3, 7, 9Common Vertex
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Bishop Weights
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1 2 3
4 5 6
7 8 9
Queen Contiguity
8 Neighbors for 5Both Border and Vertex
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Queen Weights
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Irregular Unit Contiguity
Rook Contiguity: common border onlyQueen Contiguity: border and vertices - 039 and 067
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Contiguity for Points
Thiessen Polygons Contiguity Weights
AA B C
D E
F
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Contiguity Gotchas
Contiguity Derived from GIS checking boundary file for common arcs
and/or points
Regular Grid Polygons lack of precision can cause misalignment artificial neighbors
Street Centerline different conventions can cause islands
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Misaligned Grid Cells
misaligned properly aligned
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Street Centerline Islands
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Distance-Based Weights
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Distance Metrics
For Projected Points planar coordinates
Euclidean Distance dij = [ (xi - yi)2 + (yi - yj)2 ]1/2 as the crow flies
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Distance Metrics (2)
Manhattan Block Distance dij = | xi - xj | + | yi - yj | right angles only
Minkowski Metric dij = ( | xi - xj |p + | yi - yj |p )(1/p) general distance metric
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Great Circle Distance
For Unprojected Points (on a sphere) lat-lon, but note: lat is y and lon is x distance on a sphere (spheroid)
Arc Distance dij = R. arccos[sin(yi).sin(yj)
+cos(xi).cos(xj).cos(xi-xj)]
R = radius of earth 6378 km or 3963 mi
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Distance-Based Weights
Distance Cut-Off critical distance wij = 1 for dij < , wij = 0 otherwise avoid islands maxi(minj dij )
K-Nearest Neighbors no islands, always k neighbors problems with ties weights are asymmetric
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AA B C
D E
F
Threshold
Text
Nearest Neighbor
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Distance Matrix Distance Weights
symmetric
critical distance threshold = 11.2 (radius around D)C is an island -- no neighbors
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Distance Matrixmin neighbor distance
Distance Weights
symmetric
critical distance threshold = 14.1max of nearest neighbor distances
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K-Nearest Neighborsk = 3
Distance Weights
asymmetric
wij = 1 for 3 nearest neighbors
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Characteristics of Weights
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Connectivity
Measures of Overall Connectedness % non-zero weights: sparseness distribution/average number of links average weight
Location-Specific Measures most/least connected observations unconnected observations = islands
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Spatial Weights Characteristics in R
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Connectivity Histogram
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Island Connectivity
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Spatially Lagged Variables
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Spatial Lag
Weighted Average of Neighbors yiL = wi1.y1 + wi2.y2 + ... + wiN.yN yiL = j wijyj, yL = Wy most wij are zero, only non-zero count
Spatial Lag is a Smoother Wy has smaller variance than y
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Spatial Lag Example
Average Neighbor Land Values1/4x50200 + 1/4x45000 + 1/4x34200 + 1/4x64600
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Alexander County and its First Order Neighbors
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Spatial Lag for SIDR74Computed as Average of 4 Neighbors
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