Geographically Weighted Regression

Use and application of spatially weighted

regression for environmental data

analysis

Ken Sheehan - April 5, 2010

-John Muir in “My first summer in the Sierra”

• Marveled that “one could run down the boulder field at full speed and the rocks were perfectly spaced for such an endeavor”

•Some data has inherent spatial qualities

•Ignore, or address?

Progression of ideas at WVU

• Spatial analysis for resource management

• Advanced spatial analysis• Can’t find the fish? Study it’s

habitat…– Important because– stream habitat dictates stream biota– Principle of “What’s there” is dictated

by “what’s there” (which goes for many systems, not just environmental).

Spatial Data and Streams

• Likely to be autocorrelated• Geology

– Substrate

• Flow• Depth• Sheehan and Welsh (2009)

Most Recently• Research on Grayling and Wapiti Creeks,

Greater Yellowstone ecosystem.

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Before Delving into GWR…

• Background on linear regression• Fitting a line to a data set

– Assumes homoskedacity• Static (flat variance)

– Great for predicting relationships– Heavily used, perhaps most dominant type

of statistical analysis in environmental and other fields

• Classic examination of observed versus expected

• Spatial autocorrelation (Legendre 1993)

• Red herring (Diniz 2003)

• or sweet new tool ?

– Yes and no

Progression of ideas

Background Continued..

• Fotheringham and Brunsden (1998)

• Modification of linear regression formula to include spatial attributes of data.

Standard regression formula

GWR regression formula

Concepts

– Different than adding x,y coordinates to ordinary linear regression analysis datasets

– Creates a moving variance for data with non-stationarity (regional variation).

– Not all data is appropriate for Geographically Weighted Regression.

– Still a work in progress- econometrics

Demonstration of GWR

• Wapiti and Grayling

• Deceptively complex process