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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?

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  • $\begingroup$ One word comes to my mind... spline. $\endgroup$
    – N74
    Aug 14, 2018 at 8:00
  • $\begingroup$ 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. $\endgroup$
    – Laray
    Aug 14, 2018 at 8:38
  • $\begingroup$ @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). $\endgroup$
    – JJacquelin
    Aug 14, 2018 at 9:57
  • $\begingroup$ Looks nice, but (sadly) I don't understand a single word of French. I will go with an Optimization with equality constraints $\endgroup$
    – Laray
    Aug 14, 2018 at 10:00
  • $\begingroup$ You said you use Matlab... so I meant www.mathworks.com/help/curvefit/smoothing-splines.html $\endgroup$
    – N74
    Aug 14, 2018 at 10:11

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