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C++ API

Function File: [yhat, lambda] = regdatasmooth (x, y, [options])

Smooths the y vs. x values of 1D data by Tikhonov regularization. The smooth y-values are returned as yhat. The regularization parameter lambda that was used for the smoothing may also be returned.

Note: the options have changed! Currently supported input options are (multiple options are allowed):

"d", value
the smoothing derivative to use (default = 2)
"lambda", value
the regularization paramater to use
"stdev", value
the standard deviation of the measurement of y; an optimal value for lambda will be determined by matching the provided value with the standard devation of yhat-y; if the option "relative" is also used, then a relative standard deviation is inferred
"gcv"
use generalized cross-validation to determine the optimal value for lambda; if neither "lambda" nor "stdev" options are given, this option is implied
"lguess", value
the initial value for lambda to use in the iterative minimization algorithm to find the optimal value (default = 1)
"xhat", vector
A vector of x-values to use for the smooth curve; must be monotonically increasing and must at least span the data
"weights", vector
A vector of weighting values for fitting each point in the data.
"relative"
use relative differences for the goodnes of fit term. Conflicts with the "weights" option.
"midpointrule"
use the midpoint rule for the integration terms rather than a direct sum; this option conflicts with the option "xhat"

Please run the demos for example usage.

References: Anal. Chem. (2003) 75, 3631; AIChE J. (2006) 52, 325

See also: rgdtsmcorewrap rgdtsmcore

Demonstration 1

The following code

 npts = 100;
 x = linspace(0,2*pi,npts)';
 x = x + 2*pi/npts*(rand(npts,1)-0.5);
 y = sin(x);
 y = y + 1e-1*randn(npts,1);
 yp = ddmat(x,1)*y;
 y2p = ddmat(x,2)*y;
 [yh, lambda] = regdatasmooth (x, y, "d",4,"stdev",1e-1,"midpointrule");
 lambda
 yhp = ddmat(x,1)*yh;  
 yh2p = ddmat(x,2)*yh;
 clf
 subplot(221)
 plot(x,y,'o','markersize',5,x,yh,x,sin(x))
 title("y(x)")
 legend("noisy","smoothed","sin(x)","location","northeast");
 subplot(222)
 plot(x(1:end-1),[yp,yhp,cos(x(1:end-1))])
 axis([min(x),max(x),min(yhp)-abs(min(yhp)),max(yhp)*2])
 title("y'(x)")
 legend("noisy","smoothed","cos(x)","location","southeast");
 subplot(223)
 plot(x(2:end-1),[y2p,yh2p,-sin(x(2:end-1))])
 axis([min(x),max(x),min(yh2p)-abs(min(yh2p)),max(yh2p)*2])
 title("y''(x)")
 legend("noisy","smoothed","-sin(x)","location","southeast");
 %--------------------------------------------------------
 % smoothing of monotonic data, using "stdev" to determine the optimal lambda

Produces the following output

lambda =  1

and the following figure

Figure 1

Demonstration 2

The following code

 npts = 20;
 x = rand(npts,1)*2*pi;
 y = sin(x);
 y = y + 1e-1*randn(npts,1);
 xh = linspace(0,2*pi,200)';
 [yh, lambda] = regdatasmooth (x, y, "d", 3, "xhat", xh);
 lambda
 clf
 figure(1);
 plot(x,y,'o','markersize',10,xh,yh,xh,sin(xh))
 title("y(x)")
 legend("noisy","smoothed","sin(x)","location","northeast");
 %--------------------------------------------------------
 % smoothing of scattered data, using "gcv" to determine the optimal lambda

Produces the following output

lambda =  0.0011140

and the following figure

Figure 1

Package: data-smoothing