XploRe Help : spline

Keywords - Function groups - @ A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

Group:	Statistical Data Analysis
Topic:	Nonparametric Methods
See also:

Function:		spline
Description:		spline fits a cubic spline to input data.

Link:

Smoothing Methods

Usage:	`ynew = spline(x,y,{xnew,lambda,w})`
Input:
	`x`	n x 1 vector of the predictor variable. There should be at least 4 distinct x values. The x values must be ordered x[1] <= x[2] <= ...<= x[n].
	`y`	n x 1 vector of the regressor variable. This vector must be of the same dimensionality as x.
	`xnew`	m x 1 vector of the new ordered points at which the cubic spline must be computed xnew[1] <= xnew[2] <= ...<= xnew[n]. The default value is xnew = x.
	`lambda`	a scalar that controls the choice of a smoothing method. If lambda = 0 the interpolating cubic spline is computed. In the case of lambda > 0 the ordinary cubic smoothing spline with the smoothing parameter lambda is computed. If lambda < 0 then the smoothing parameter is chosen based on the Generalized Cross Validation method. The default value is -1.
	`w`	n x 1 vector of the weights. The default value is w[i]=1.
Output:
	`ynew`	m x 1 vector that contains the smoothed data.

Note:

By means of the parameter lambda, the user can control the trade-off between closeness of fit and smoothness of fit of the approximation. If lambda is too large then the data will be fitted by a straight line. On the other hand if lambda is too small the curve will be too wiggly. In the extrem case lambda = 0 the function returns an interpolating curve. A properly chosen lambda will result in a good compromise between closeness of fit and smoothness of fit. In order to chose the smoothing parameter properly, the user is recommended to make this choice adaptively by setting lambda < 0 or to inspect the fits for different lambda > 0 graphically.

Example:

;cubic spline interpolation ; generate a dataset n = 5 m = 100 t = 1:n tnew = 1:m xnew = tnew/m-1/m x = t/n-1/n y = cos(10*x) yold = cos(10*xnew) func = xnew~yold d = x~y ; interpolate the data y ynew = spline(x,y,xnew,0) ; plot the data ; the uderlying function is shown by a dashed black line ; cubic spline interpolation is a blue line ; linear interpolation is a red line dnew = xnew~ynew results = createdisplay(2,1) setmaskl(d,t',4,1,2) setmaskp(d,4,3,10) setmaskl(dnew,tnew',1,1,3) setmaskp(dnew,1,0,1) setmaskl(func,tnew',0,4,2) setmaskp(func,1,0,1) show(results,1,1,d,dnew,func) setgopt(results,1,1,"title","cubic spline interpolation") ; ;adaptive smoothing ; generate the dataset n = 50 m = 100 vr = 1.0 t = 1:n tnew = 1:m xnew = tnew/m-1/m x = t/n-1/n y = cos(10*x) + 3*x ;add Gaussian white noise to the data randomize(1234567) data = y + vr*normal(n) d = x~y ndata = x~data ; call the adaptive cubic spline smoother(lambda < 0.0) ynew = spline(x,data,xnew,-10) ; plot the data ; the uderlying function is shown by a black dashed line ; adaptive estimator is a blue line ; input data are red crosses dnew = xnew~ynew setmaskl(d,t',0,4,2) setmaskp(d,0,0,10) setmaskl(ndata,t',3,1,1) setmaskp(ndata,4,11,8) setmaskl(dnew,tnew',1,1,4) setmaskp(dnew,1,0,1) show(results,2,1,d,dnew,ndata) setgopt(results,2,1,"title","adaptive spline smoothing")

Result:

n/a