Cautionary Tales on Spatial Weights

Jerry Platt

Statistical independence is a fundamental property underlying much of inferential theory, yet seems rather the exception in the practical world around us. Just as many time series exhibit temporal dependencies, so also many cross-sectional data sets exhibit spatial dependencies.

There is a developed literature on the identification and weighting of spatial neighbors, and these measures form a basis for capturing the effects of spatial dependencies and clustering. Unfortunately, the choice among weighting schemes often is rather arbitrary and ad hoc, even though the consequences of different choices can be substantial.

Some SPATIAL Issues

• Who are Spatial NEIGHBORS ???– Definition– Yes/No v. Partial

• What are Spatial WEIGHTS ???– Symmetry– Standardized

• How are Spatial ASSOCIATIONS Measured ???– Correlation– Significance

A Toy Example (N=8)ID, latitude, longitude, name, desc

1 34.061486, -117.163815, “RD", "1200 E Colton Ave, Redlands, CA 92374“

2 34.077754, -117.575821, “RC", “9680 Haven Ave, Rancho Cucamonga, CA 91730“

3 33.856433, -118.291351, “LA", "19191 S Vermont Ave, Torrance, CA 90502“

4 33.696904, -117.866482, “OC", "200 Sandpointe Ave, Santa Ana, CA 92707“

5 33.953456, -117.391616, “RV", "3610 Central Ave, Riverside, CA 92506“

6 33.525420, -117.166856, “TM", "27270 Madison Ave, Temecula, CA 92590“

7 34.185022, -118.308760, “BK", "333 N Glenoaks Blvd, Burbank, CA 91502“

8 32.779862, -117.135839, “SD", "9040 Friars Rd, San Diego, CA 92108"

From Points to “Boundaries”

Contiguity Weights1

RD

Median Residential Value @ Zip Code

LowMediumHigh

QUESTION:Is There Any

Spatial Patternin This Map ?

Well, R/B or B/R = 5R/R or B/B = 2

So, seems aNEGATIVE

ASSOCIATION,but too few

points to be sure

Q1 (Connected) Spatial Association

A Measure of Spatial Association PolyCtE1N = ( 8 / 28 ) * Correlation[ Res_Val, Conn_R_V ] = ( 8 / 28 ) * ( -0.61 ) = -0.17

Spatial Association: [-1…….0……+1] Weight Definition Spatial Association

PolyCtE1N Connected, No Adj. -0.17PolyCtE1R Connected, Rows Adj. -0.25QLE2 … or 1 Over -0.15NN1AD Nearest Neighbor -0.49ID1 1 / Distance -0.31ID2 1 / (Distance)^2 -0.31ZIED Zone of Indifference -0.17FDB Fixed Distance Bound -0.30

Interpretations

• There DOES seem to be a spatial association

• The relationship rather clearly IS negative

• The MAGNITUDE of association is # -0.30

• Measures DEVIATE, up to -0.49, down to -0.15

• In any event, N is too small to trust these measures

A Cautionary Tale:Inland Empire Job Centers

WPSM = Workers per Square Mile Apparent High Positive Spatial Correlation…

Spatial Association: [-1…….0……+1] Weight Definition Spatial Association

PolyCtE1N Connected, No Adj. +0.39PolyCtE1R Connected, Rows Adj. +0.43QLE2 … or 1 Over +0.18NN1AD Nearest Neighbor +0.03ID1 1 / Distance +0.04ID2 1 / (Distance)^2 +0.02ZIED Zone of Indifference 0.00FDB Fixed Distance Bound +0.02

Another Cautionary Tale:Southern California School Districts

PCTProfic = % Students Proficient in English Apparent High Positive Spatial Correlation…

Spatial Association: [-1…….0……+1] Weight Definition Spatial Association

PolyCtE1N Connected, No Adj. +0.29PolyCtE1R Connected, Rows Adj. +0.30QLE2 … or 1 Over +0.31NN1AD Nearest Neighbor -0.01ID1 1 / Distance +0.23ID2 1 / (Distance)^2 +0.33ZIED Zone of Indifference, No Adj. +0.06FDB Fixed Distance Bound +0.05

Comments and Recommendations• Spatial Dependencies Matter

• The Choice of Spatial Weights is Critical

• Different Choices Can Yield VERY Different Results

• Try Several Methods Before Settling on 1 (or More)

• Report the Sensitivity of Results to Your Choice