# Is there a theory for piecewise differentiable regression polynomials?

I have an interesting question, I would like to have answered...

I have a very noisy signal $f$, that I want to smoothen out. Using a global regression cannot work, as I don't have a model of the signal (being a measurement of the solution of an ODE). I would like to use multiple regressions $p_1$, $p_2$, etc. to have $$\min_{p_1} \int_{x_0}^{x_1} (f(x)-p_1(x))^2 dx$$ and $$\min_{p_2} \int_{x_1}^{x_2} (f(x)-p_2(x))^2 dx$$ This however is very easy (use Matlab polyfit on each interval) and not a problem. But I want to go further... I want to enforce $p_1(x_1)=p_2(x_1)$ and $p_1'(x_1) = p_2'(x_1)$ to have a continous and smooth result. This will ofcourse increase the regression error.

Is there a theory for this? Or do I need to make my own algorithm via optimization toolboxes?

• One word comes to my mind... spline.
– N74
Aug 14, 2018 at 8:00
• Spline interpolation uses either all my many points for interpolation not reducing all my noise. If I made a spline with only a few parts, I need interpolation points, that I don't have (as $f(x_1)$ is noisy). I would also ignore all other points between the break-points. Aug 14, 2018 at 8:38
• @Laray : I doubt that a general theory is presently available for piecewise polynomial regression. For piecewise linear regression a method based on integral equation fitting is shown with examples in : fr.scribd.com/document/380941024/… . But this theory isn't advanced enough for general piecewise polynomial regression. It could be used in some very simple particular cases (not published in the above referenced paper). Aug 14, 2018 at 9:57
• Looks nice, but (sadly) I don't understand a single word of French. I will go with an Optimization with equality constraints Aug 14, 2018 at 10:00
• You said you use Matlab... so I meant www.mathworks.com/help/curvefit/smoothing-splines.html
– N74
Aug 14, 2018 at 10:11