Smoothers

The GAM Procedure

Smoothers

A smoother is a tool for summarizing the trend of a response measurement Y as a function of one or more predictor measurements X₁, ... , X_p. It produces an estimate of the trend that is less variable than Y itself. An important property of a smoother is its nonparametric nature. It doesn't assume a rigid form for the dependence of Y on X₁, ... , X_p. This section gives a brief overview of the smoothers that can be used with the GAM procedure.

Cubic Smoothing Spline

A smoothing spline is the solution to the following optimization problem: among all functions $\eta(x)$ with two continuous derivatives, find one that minimizes the penalized least square

$\sum_{i=1}^n(y_i - \eta(x_i))^2 + \lambda \int^b_a (\eta^{''}(t))^2 dt$

where $\lambda$ is a fixed constant, and $a \le x_1 \le ... \le x_n \le b$ . The first term measures closeness to the data while the second term penalizes curvature in the function. It can be shown that there exists an explicit, unique minimizer, and that minimizer is a natural cubic spline with knots at the unique values of x_i.

The parameter $\lambda$ is the smoothing parameter. Large values of $\lambda$ produce smoother curves while smaller values produce wiggly curves.

Thin-Plate Smoothing Spline

The theoretical foundations for the thin-plate smoothing spline are described in Duchon (1976, 1977) and Meinguet (1979). Further results and applications are given in Wahba and Wendelberger (1980). Refer to "The TPSPLINE Procedure" in SAS/STAT User's Guide, Version 8 for more details.

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