Transcription of Extending Linear Regression: Weighted Least Squares ...
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Extending Linear Regression: Weighted Least Squares ...
Extending Linear Regression: Weighted LeastSquares, Heteroskedasticity, Local PolynomialRegression36-350, Data Mining23 October 2009 Contents1 Weighted Least Squares12 Weighted Least Squares as a Solution to Heteroskedasticity ..53 Local Linear Regression104 Exercises151 Weighted Least SquaresInstead of minimizing the residual sum of Squares ,RSS( ) =n i=1(yi ~xi )2(1)we could minimize theweightedsum of Squares ,WSS( , ~w) =n i=1wi(yi ~xi )2(2)This includes ordinary Least Squares as the special case where all the weightswi= 1. We can solve it by the same kind of algebra we used to solve theordinary Linear Least Squares problem. But why would we want to solve it? Forthree may care very strongly about predicting the re-sponse for certain values of the input ones we expect to see often again,ones where mistakes are especially costly or embarrassing or painful, than others.
Suppose we visit the Oracle of Regression (Figure 4), who tells us that the noise has a standard deviation that goes as 1 + x2=2. We can then use this to improve our regression, by solving the weighted least squares problem rather than ordinary least squares (Figure 5). This not only looks better, it is better: the estimated line is now 3:09
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