GWR_overview - Geographically weighted regression Danlin Yu...

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Geographically weighted regression Danlin Yu Yehua Dennis Wei Dept. of Geog., UWM with modifications by T. Fik (Univ. of FL)
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Outline of the presentation 1. Spatial non-stationarity: an example 2. GWR – some definitions 3. 6 good reasons using GWR 4. Calibration and tests of GWR 5. An example: housing hedonic model in Milwaukee
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1. Stationary v.s non- stationary y i = β 0 + 1 x 1i e3 e2 e1 e4 Stationary process e3 e2 e1 e4 Non-stationary process y i = i0 + i1 x 1i Assumed More realistic
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Simpson’s paradox House density H o u s e P r i c Spatially aggregated data Spatially disaggregated data House density
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Stationary v.s. non-stationary If non-stationarity is modeled by stationary models Possible wrong conclusions might be drawn Residuals of the model might be highly spatial autocorrelated
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Why do relationships vary spatially? Sampling variation Nuisance variation, not real spatial non- stationarity Relationships intrinsically different across space Real spatial non-stationarity Model misspecification Can significant local variations be removed?
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2. Some definitions Spatial non-stationarity: the same stimulus provokes a different response in different parts of the study region Global models: statements about processes which are assumed to be stationary and as such are location independent
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Some definitions Local models: spatial decompositions of global models, the results of local models are location dependent – a characteristic we usually anticipate from geographic (spatial) data
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Linear Regression Regression establishes relationship among a dependent variable and a set of independent variable(s) A typical linear regression model looks like: y i = β 0 + 1 x 1i + 2 x 2i +……+ n x ni + ε i With y i the dependent variable, x ji ( j from 1 to n ) the set of independent variables, and i the residual, all at location i
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When Regression is applied to spatial data it assumes a stationary spatial process The same stimulus provokes the same response in all parts of the study region Highly untenable for spatial process
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Geographically weighted regression Local statistical technique to analyze spatial variations in relationships Spatial non-stationarity is assumed and will be tested Based on Tobler’s stated “First Law of Geography”: everything is related with everything else, but closer things are more related.
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GWR Addresses the non-stationarity directly Allows the relationships to vary over space, i.e., β s do not need to be everywhere the same This is the essence of GWR, in the linear form: y i = i0 + i1 x 1i + i2 x 2i +……+ in x ni + ε i Instead of remaining the same everywhere, s now vary in terms of locations ( i )
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3. Six good reasons to use
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GWR_overview - Geographically weighted regression Danlin Yu...

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