Cubic smoothing spline error measure weights

How exactly does the error measure weights work in csaps cubic smoothing spline function?

What does it mean by error measure weights? I have read the documentation through but it is unclear to me.

I would like to construct a smoothing spline based on weights of each spline, meaning that ones with greater weight value should be considered more for the spline. (i.e. something with a weight of 0.1 would be considered more of an outlier compared to something with a weight of 0.9) Is this how it works in csaps?

It would be great if there was some kind of mathematical explanation of how csaps works.

Answer 1

[Abhaya]

Hi MJ,

The MATLAB ‘csaps’ function tries to minimize the error function, which depends on error values and smoothing measure.

In the context of csaps, the error measure weights w_i control how much each data point contributes to the fitting error. A higher weight for a data point means that point has more influence on the spline, while a lower weight means the point has less influence.

The default value for the weight vector w in the error measure is ones(size(x)).

Below is an example to help illustrate how weights affect the spline fitting:

x = [1, 2, 3, 4, 5];
y = [2, 2, 3, 4, 1];
w1 = [1,1,0.5,0.1,1];  
w2 = [1,1,0.5,1,1];

lambda = 0.8;  %smoothing factor

xx=linspace(1,5,100);
pp1 = csaps(x,y,lambda,xx,w1);
pp2 = csaps(x,y,lambda,xx,w2);

figure;
plot(x, y, 'ko', 'MarkerFaceColor', 'k');  
hold on;
plot(linspace(1,5,100),pp1, 'b-', 'LineWidth', 2); 
plot(linspace(1,5,100),pp2, 'r-', 'LineWidth', 2);
ylim([1,6])
legend('Data points','Spline with reduced weight at x=4','Spline with weight at x=4 set to 1');

For more information on MATLAB ‘csaps’ function please refer to the MATLAB documentation linked below.

https://www.mathworks.com/help/curvefit/csaps.html

Hope it resolves your query.