note20 - Chapter 10 Smoothing Methods 2 Scatterplot...

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Unformatted text preview: 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 , or the neighborhood of x * averaging the Y values of observations in the neighborhood as an estimate of Y value at x – 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 or defined only at x 1 , . . ., x n 1 • Bin smoother – cutpoints-∞ = c < c 1 < ··· < c K − 1 < c K = ∞ – the indices of the data points in each region I k = { i : c k ≤ x i < c k +1 } k = 0 , 1...
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This note was uploaded on 07/22/2011 for the course STA 4502 taught by Professor Staff during the Spring '08 term at University of Florida.

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note20 - Chapter 10 Smoothing Methods 2 Scatterplot...

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