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Fit Analyses

Nonparametric Smoothing Spline

Two criteria can be used to select an estimator {\hat{f_\lambda}} for the function f:

A standard measure of goodness of fit is the mean residual sum of squares

\frac{1}n\sum_{i=1}^n{( y_{i}- \hat{f_\lambda}( x_{i}))^2}

A measure of the smoothness of a fit is the integrated squared second derivative

\int_{-{\infty}}^{{\infty}}{( {\hat{f_\lambda}"}(x))^2 dx}

A single criterion that combines the two criteria is then given by

S(\lambda) = \frac{1}n \sum_{i=1}^n{( y_{i}- \hat{f_\lambda}( x_{i}))^2} + \lambda\int_{-{\infty}}^{{\infty}}{( {\hat{f_\lambda}"}(x) )^2 dx}
where {\hat{f_\lambda}} belongs to the set of all continuously differentiable functions with square integrable second derivatives, and \lambda is a positive constant.

The estimator that results from minimizing S(\lambda)is called the smoothing spline estimator. This estimator fits a cubic polynomial in each interval between points. At each point xi, the curve and its first two derivatives are continuous (Reinsch 1967).

The smoothing parameter \lambda controls the amount of smoothing; that is, it controls the trade-off between the goodness of fit to the data and the smoothness of the fit. You select a smoothing parameter \lambda by specifying a constant c in the formula

\lambda = (Q / 10)^3 c
where Q is the interquartile range of the explanatory variable. This formulation makes c independent of the units of X.

After choosing Curves:Spline, you specify a smoothing parameter selection method in the Spline Fit dialog.

fit38.gif (3910 bytes)

Figure 39.40: Spline Fit Dialog

The default Method:GCV uses a c value that minimizes the generalized cross validation mean squared error { \rm{MSE}_{GCV}(\lambda)}.Figure 39.41 displays smoothing spline estimates with c values of 0.0017 (the GCV value) and 15.2219 (DF=3). Use the slider in the table to change the c value of the spline fit.

fit39.gif (14218 bytes)

Figure 39.41: Smoothing Spline Estimates

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