GWR_WhitePaper - GEOGRAPHICALLY WEIGHTED REGRESSION WHITE...

Info icon This preview shows pages 1–6. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
2 The authors gratefully acknowledge support from a Strategic Research Cluster grant (07/SRC/I1168) by Science Foundation Ireland under the National Development Plan.
Image of page 2
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
Image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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.
Image of page 4
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
Image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 6
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern