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note20 - Chapter 10 Smoothing Methods 2 Scatterplot...

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Chapter 10. Smoothing Methods 2. Scatterplot Smoothers What is a smoother a tool for summarizing the trend of a response Y as a function of predictors X = ( X 1 , . . . , X p ) produces an estimate of the trend that is less variable than Y not assume a rigid form of the dependence of Y on X a tool for nonparametric regression Categorical Predictor: a scatterplot smooth * X ∈ { 1 , 2 , . . . , K } and Y is continuous response * for each k ( k = 1 , . . . , K ), ˆ f ( k ) = 1 | I k | summationdisplay i I k y i where I k = { i | x i = k } is the index set and | I k | is the size of I k Continuous Predictor: local averaging * locate the values close to x 0 , or the neighborhood of x 0 * averaging the Y values of observations in the neighborhood as an estimate of Y value at x 0 Two main problems in smoothing * how to average the response values in each neighborhood how to choose smoother * how big to take the neighborhoods in terms of an adjustable parameter the smoothing parameter governs the fundamental bias-variance trade- off * choose the smoothing parameter based on the data * choose the smoothing parameter in an optimal way for trade bias against variance Scatterplot smoother: the smoother for one-dimensional predictor data: y = ( y 1 , . . . , y n ) and x = ( x 1 , . . . , x n ) with no tie smoother s ( x ) is defined for all x 0 or defined only at x 1 , . . . , x n 1

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Bin smoother cutpoints -∞ = c 0 < c 1 < · · · < c K 1 < c K =
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note20 - Chapter 10 Smoothing Methods 2 Scatterplot...

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