Is there a theory for piecewise differentiable regression polynomials?

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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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  • One word comes to my mind... spline.
    – N74
    Aug 14 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.
    – Laray
    Aug 14 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).
    – JJacquelin
    Aug 14 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
    – Laray
    Aug 14 at 10:00










  • You said you use Matlab... so I meant www.mathworks.com/help/curvefit/smoothing-splines.html
    – N74
    Aug 14 at 10:11















up vote
0
down vote

favorite












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?







share|cite|improve this question




















  • One word comes to my mind... spline.
    – N74
    Aug 14 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.
    – Laray
    Aug 14 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).
    – JJacquelin
    Aug 14 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
    – Laray
    Aug 14 at 10:00










  • You said you use Matlab... so I meant www.mathworks.com/help/curvefit/smoothing-splines.html
    – N74
    Aug 14 at 10:11













up vote
0
down vote

favorite









up vote
0
down vote

favorite











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?







share|cite|improve this question












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?









share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Aug 14 at 7:24









Laray

1,671413




1,671413











  • One word comes to my mind... spline.
    – N74
    Aug 14 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.
    – Laray
    Aug 14 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).
    – JJacquelin
    Aug 14 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
    – Laray
    Aug 14 at 10:00










  • You said you use Matlab... so I meant www.mathworks.com/help/curvefit/smoothing-splines.html
    – N74
    Aug 14 at 10:11

















  • One word comes to my mind... spline.
    – N74
    Aug 14 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.
    – Laray
    Aug 14 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).
    – JJacquelin
    Aug 14 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
    – Laray
    Aug 14 at 10:00










  • You said you use Matlab... so I meant www.mathworks.com/help/curvefit/smoothing-splines.html
    – N74
    Aug 14 at 10:11
















One word comes to my mind... spline.
– N74
Aug 14 at 8:00




One word comes to my mind... spline.
– N74
Aug 14 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.
– Laray
Aug 14 at 8:38




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.
– Laray
Aug 14 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).
– JJacquelin
Aug 14 at 9:57




@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).
– JJacquelin
Aug 14 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
– Laray
Aug 14 at 10:00




Looks nice, but (sadly) I don't understand a single word of French. I will go with an Optimization with equality constraints
– Laray
Aug 14 at 10:00












You said you use Matlab... so I meant www.mathworks.com/help/curvefit/smoothing-splines.html
– N74
Aug 14 at 10:11





You said you use Matlab... so I meant www.mathworks.com/help/curvefit/smoothing-splines.html
– N74
Aug 14 at 10:11
















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