natural cubic spline interpolation of y-values: how to get derivative of the spline wrt the y-values?
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Given a data set with support points x_1,...,x_n and corresponding y-values y_1,...,y_n.
My objective is to create a cubic spline f (with natural boundary conditions) that passes through the y_values. There are, of course, plenty of functions for doing this.
However, for a parameter identification procedure, I have to compute the derivative of the spline f with respect to the y-values -- at arbitrary points within [x1, x_n].
Is there an easy way using built-in functions of Matlab to compute the sensitivities?
2 Kommentare
Torsten
am 8 Mär. 2023
I can understand that you have to compute the derivative of the spline with respect to the parameters, but why with respect to the y-values ?
Akzeptierte Antwort
Bruno Luong
am 8 Mär. 2023
Bearbeitet: Bruno Luong
am 8 Mär. 2023
The derivative f wrt to y_i is the spline interpolate b_i := (0,0,...,1,0...) where 1 is at ith position, since the spline is linear to y values.
Replace spline command with your function that computes natural spline pp form
x=cumsum(rand(1,10));
y=rand(size(x));
xi=linspace(min(x),max(x),500);
f=ppval(spline(x,y),xi)
plot(x,y,'or',xi,f,'b') %
b=eye(length(x));
yd=spline(x,b);
dfdy=ppval(yd,xi); % dfdy(i,j) is the derivative of f(xi(j)) with respect to y(i))
figure
plot(xi,dfdy')
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Bruno Luong
am 8 Mär. 2023
Bearbeitet: Bruno Luong
am 8 Mär. 2023
In this thread https://fr.mathworks.com/matlabcentral/answers/1894800-how-to-remove-noise-from-curves-and-take-their-derivates?s_tid=srchtitle
you can find my function that compute the derivative of a piecewise polynomiall function (pp), inclusing pp form of the spline functions. This function returns the pp form of the derivative, so you can evaluate using MATLAB ppval.
function ppd = ppder(pp)
ppd = pp;
coefs = ppd.coefs;
n = size(coefs,2);
ppd.coefs = coefs(:,1:n-1).*(n-1:-1:1);
ppd.order = ppd.order-1;
end
4 Kommentare
Bruno Luong
am 8 Mär. 2023
Bearbeitet: Bruno Luong
am 8 Mär. 2023
See my answer below, but
"the result should be a single number"
No the result is a scalar function. If we take f at a given point x then it is a scalar.
"...and not a vector b_i := (0,0,...,1,0...)."
I did not tell the derivative is b, the derivative is the spline interpolating b
"why is a cubic spline linear to y values?"
You clearly missunderstand and male confusion betwen being linear and being a lilnear function
Torsten
am 8 Mär. 2023
Verschoben: Torsten
am 8 Mär. 2023
Why do you want to compute the sensitivities manually ?
Usually, the fitting software computes them using a finite-difference approximation, i.e. by calling your function with
y_i
and
y_i+h
getting back
f_j(y_1,...,y_i,...,y_n) and f_j(y_1,...,y_i+h,...,y_n)
and approximating
df_j/dy_i = (f_j(y_1,...,y_i+h,...,y_n)-f_j(y_1,...,y_i,...,y_n))/h
And this would also be my suggestion on how to do it manually if it is really needed.
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