GWR_WhitePaper

# GWR_WhitePaper - GEOGRAPHICALLY WEIGHTED REGRESSION WHITE...

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GEOGRAPHICALLY WEIGHTED REGRESSION WHITE PAPER MARTIN CHARLTON A STEWART FOTHERINGHAM National Centre for Geocomputation National University of Ireland Maynooth Maynooth, Co Kildare, IRELAND March 3 2009

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2 The authors gratefully acknowledge support from a Strategic Research Cluster grant (07/SRC/I1168) by Science Foundation Ireland under the National Development Plan.
Contents Regression 1 Regression with Spatial Data 3 Geographically Weighted Regression (GWR) 5 Outputs from GWR 9 Interpreting Parameter Estimates 10 Extensions to GWR 12 References 13

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1 Regression Regression encompasses a wide range of methods for modelling the relationship between a dependent variable and a set of one or more independent variables. The dependent variable is sometimes known as the y-variable, the response variable or the regressand. The independent variables are sometimes known as x-variables, predictor variables, or regressors. A regression model is expressed as an equation. In its simplest form a linear regression model can take the form i i i x y ε β β + + = 1 0 for i=1 … n In this equation i y is the response variable, here measured at some location i , x i is the independent variable, i ε is the error term, and 0 β and 1 β are parameters 1 which are to be estimated such that the value of ( ) = n i i i y y 1 2 ˆ is minimised over the n observations in the dataset. The i y ˆ is the predicted or fitted value for the i th observation, given the i th value of x. The term ) ˆ ( i i y y is known as the residual for the i th observation, and the residuals should be both independent and drawn identically from a Normal Distribution with a mean of zero. Such a model is usually fitted using a procedure known as Ordinary Least Squares (OLS). More generally, a multiple linear regression model may be written: i mi m i i i x x x y ε β β β β + + + + + = ... 2 2 1 1 0 for i=1 … n where the predictions of the dependent variable are obtained through a linear combination of the independent variables. The OLS estimator takes the form: y X X X T T 1 ) ( ˆ = β where β ˆ is the vector of estimated parameters, X is the design matrix which contains the values of the independent variables and a column of 1s, y is the vector of observed values, and 1 ) ( X X T is the inverse of the variance-covariance matrix. 1 The term coefficient is sometimes used instead of parameter.
2 Sometimes it is desirable to weight the observations in a regression, for example, different levels of data uncertainty. The weights are placed in the leading diagonal of a square matrix W and the estimator is altered to include the weighting: Wy X WX X T T 1 ) ( ˆ = β The ability of the model to replicate the observed y values is measured by the goodness of fit. This is conveniently expressed by the r 2 value which runs from 0 to 1 and measures the proportion of variation in the observed y which is accounted for (sometimes “explained by”) by variation in the model. The r 2 can often be increased merely by adding variables, so the adjusted r 2

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