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Curve Fitting Problem
by A. Kalten
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Hello,
I need some advice on solving a curve fitting problem. Ordinarily, a curve fitting algorithm, such as polyfit or wpolyfit in octave, will determine the coefficients of a polynomial that best fits the empirical data. If the model is a second degree polynomial, a x^2 + b x + c, the algorithm will return the best coefficients a, b, and c. The trouble I am having is that I need to fit a second degree polynomial where some of the coefficients are either already known or include another known parameter as a factor. For example, I need to find the best value for k in this equation: k^2/4 x + k p x + p^2 where p is a known quantity. Although there are three coefficients, there is really only one unknown parameter, that is k. Using polyfit from octave will return a, b, and c where a = k^2/4, b = k p, and c = p^2. Solving the two equations for k gives values that are significantly different (they differ in the second decimal place). Is there an algorithm available that allows some of the parameters of the model to be predetermined or to include some predetermined factor? I suspect that the only solution would be to write a custom least squares program. AK _______________________________________________ Help-octave mailing list Help-octave@... https://www.cae.wisc.edu/mailman/listinfo/help-octave |
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Re: Curve Fitting Problem
by Ben Abbott
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On Jun 26, 2008, at 6:57 PM, A. Kalten wrote: > Hello, > > I need some advice on solving a curve fitting problem. > > Ordinarily, a curve fitting algorithm, such as polyfit > or wpolyfit in octave, will determine the coefficients of a > polynomial that best fits the empirical data. If the model > is a second degree polynomial, a x^2 + b x + c, the algorithm > will return the best coefficients a, b, and c. > > The trouble I am having is that I need to fit a second > degree polynomial where some of the coefficients are > either already known or include another known parameter > as a factor. For example, I need to find the best value > for k in this equation: > > k^2/4 x + k p x + p^2 > > where p is a known quantity. > > Although there are three coefficients, there is really > only one unknown parameter, that is k. Using polyfit from octave > will return a, b, and c where a = k^2/4, b = k p, and c = p^2. > > Solving the two equations for k gives values that > are significantly different (they differ in the second > decimal place). > > Is there an algorithm available that allows some of the > parameters of the model to be predetermined or to include > some predetermined factor? > > I suspect that the only solution would be to write a > custom least squares program. > > AK I assume you intended your example to be: k^2/4 * x^2 + k * p * x + p^2 ? On first inspection, it appears your example has non-linear constraints. Is this example the specific problem you'd like to solve, or are you looking for something a general solution? Ben _______________________________________________ Help-octave mailing list Help-octave@... https://www.cae.wisc.edu/mailman/listinfo/help-octave |
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Re: Curve Fitting Problem
by A. Kalten
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On Thu, 26 Jun 2008 21:14:12 -0400
Ben Abbott <bpabbott@...> wrote: > > On first inspection, it appears your example has non-linear constraints. > > Is this example the specific problem you'd like to solve, or are you > looking for something a general solution? > This is a specific problem, but there are/will be others of a similar nature. I am asking if there are off-the-shelf curve-fitting algorithms, maybe for octave, that will allow constraints on some of the parameters, or is custom programming the only way to go. AK _______________________________________________ Help-octave mailing list Help-octave@... https://www.cae.wisc.edu/mailman/listinfo/help-octave |
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Re: Curve Fitting Problem
by Ben Abbott
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On Jun 26, 2008, at 9:52 PM, A. Kalten wrote: > On Thu, 26 Jun 2008 21:14:12 -0400 > Ben Abbott <bpabbott@...> wrote: > >> >> On first inspection, it appears your example has non-linear >> constraints. >> >> Is this example the specific problem you'd like to solve, or are you >> looking for something a general solution? >> > > This is a specific problem, but there are/will be others of a similar > nature. > > I am asking if there are off-the-shelf curve-fitting algorithms, > maybe for octave, that will allow constraints on some of the > parameters, > or is custom programming the only way to go. > > AK If the constraints are linear, then a general solution is possible. In fact there is a free version available for Matlab (although I don't know what sort of license is involved). If the constraints are non-linear (as in your example) then the difficulty of a solution will depend upon whether or not the variables are pure real or complex valued. Ben _______________________________________________ Help-octave mailing list Help-octave@... https://www.cae.wisc.edu/mailman/listinfo/help-octave |
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Re: Curve Fitting Problem
by Sergei Steshenko-2
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Applications From Scratch: http://appsfromscratch.berlios.de/ --- On Thu, 6/26/08, A. Kalten <akalten@...> wrote: > From: A. Kalten <akalten@...> > Subject: Curve Fitting Problem > To: help-octave@... > Date: Thursday, June 26, 2008, 3:57 PM [snip] > I suspect that the only solution would be to write a > custom least squares program. > > AK > > Maybe not. If I understand your problem correctly, is more or less like this; 1) there is f(x) to be approximated; 2) the desired approximation is: a * x^2 + b * x + c; 3) for example, 'a' above is for whatever reason predetermined. Then, I think, you need to approximate f(x) - a * x^2 with b * x + c , i.e. you can use the existing curve fitting function on a different input function. Regards, Sergei. _______________________________________________ Help-octave mailing list Help-octave@... https://www.cae.wisc.edu/mailman/listinfo/help-octave |
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Re: Curve Fitting Problem
by A. Kalten
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On Thu, 26 Jun 2008 19:52:32 -0700 (PDT)
Sergei Steshenko <sergstesh@...> wrote: > > If I understand your problem correctly, is more or less like this; > > 1) there is f(x) to be approximated; > 2) the desired approximation is: > > a * x^2 + b * x + c; > In this case there is no function to be approximated. There is only data that fits a theoretical model. I was inquiring about any off-the-shelf algorithms that can fit a function to a data set where some of the parameters of the function need to be constrained in a certain way. AK _______________________________________________ Help-octave mailing list Help-octave@... https://www.cae.wisc.edu/mailman/listinfo/help-octave |
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Octave and Image Processing
by Abdul Rahman Riza-2
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Dear Friends,
Anybody can figure out how powerffull octave to tacke image processing problem? Riza No virus found in this outgoing message. Checked by AVG. Version: 7.5.524 / Virus Database: 270.4.1/1521 - Release Date: 6/26/2008 11:20 AM _______________________________________________ Help-octave mailing list Help-octave@... https://www.cae.wisc.edu/mailman/listinfo/help-octave |
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Re: Curve Fitting Problem
by Jaroslav Hajek-2
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On Fri, Jun 27, 2008 at 6:37 AM, A. Kalten <akalten@...> wrote:
> On Thu, 26 Jun 2008 19:52:32 -0700 (PDT) > Sergei Steshenko <sergstesh@...> wrote: > >> >> If I understand your problem correctly, is more or less like this; >> >> 1) there is f(x) to be approximated; >> 2) the desired approximation is: >> >> a * x^2 + b * x + c; >> > > In this case there is no function to be approximated. There > is only data that fits a theoretical model. I was inquiring > about any off-the-shelf algorithms that can fit a function > to a data set where some of the parameters of the function > need to be constrained in a certain way. > I think there are at least two implementations of the levenberg-marquardt algorithm in OctaveForge's optim package. Having a matlab-compatible lsqnonlin in Octave would be good. In general, making the optimization stuff more systematic and matlab-compatible is a worthy long-term goal. It's on my TODO list, too :) A couple of months ago I have proposed to include also other parts of MINPACK (which is employed for fsolve) to implement lsqnonlin. I was ready to volunteer for this project (which would benefit me as well). However, JWE suggested that Octave could employ GSL in the future, which has the MINPACK algorithms rewritten in C, in a reentrant and safe way. On the contrary, making MINPACK reentrant (which I think is desirable) would involve modifying the code. I have thus decided to postpone any work on the optimization routines until the future of GSL in Octave is clear. GSL would probably allow us to get rid of much (if not all) of libcruft, however, it has its problems as well. One thing is that GSL does not support floats (single precision) almost anywhere; now that Octave supports them that could cause problems. Also, GSL cannot (not even optionally) employ LAPACK for linear algebra (which is faster and more elaborate than GSL's own LA stuff). I am not aware of the status of GSL in Octave at this moment. regards > AK > > _______________________________________________ > Help-octave mailing list > Help-octave@... > https://www.cae.wisc.edu/mailman/listinfo/help-octave > -- RNDr. Jaroslav Hajek computing expert Aeronautical Research and Test Institute (VZLU) Prague, Czech Republic url: www.highegg.matfyz.cz _______________________________________________ Help-octave mailing list Help-octave@... https://www.cae.wisc.edu/mailman/listinfo/help-octave |
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Re: Octave and Image Processing
by Søren Hauberg
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fre, 27 06 2008 kl. 12:55 -0700, skrev Abdul Rahman Riza:
> Anybody can figure out how powerffull octave to tacke image processing > problem? A lot of people use Octave for image processing, so I think the answer is yes :-) If you want help you need to ask more specific questions, as nobody can answer a question that haven't been asked. Søren _______________________________________________ Help-octave mailing list Help-octave@... https://www.cae.wisc.edu/mailman/listinfo/help-octave |
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Re: Curve Fitting Problem
by Peter Cloetens
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My suggestions would be:
either: treat this as a non-linear fitting problem with 1 unknown and use for example "leasqr" from the octave-forge package optim or: solve explicitly for your unknown "k" and you end up with an explicit solution. Something like: Minimize the cost function cost(k) = sum |y_i - k^2/4*x_i^2 - k*p*x_i - p^2|^2 with respect to k Derivation leads to a cubic (third order) equation in k: a*k^3 + b*k^2 +c*k + d = 0 with a = 1/8*sum(x_i^4) b = 3/4*p*sum(x_i^3) c = 3/2*p^2*sum(x_i^2) - 1/2*sum(y_i*x_i^2) d = -p*sum(y_i*x_i) + p^3*sum(x_i) The cubic equation can be solved with the octave function fzero or explicitly with one of the methods described here: http://en.wikipedia.org/wiki/Cubic_equation Peter A. Kalten wrote: > Hello, > > I need some advice on solving a curve fitting problem. > > Ordinarily, a curve fitting algorithm, such as polyfit > or wpolyfit in octave, will determine the coefficients of a > polynomial that best fits the empirical data. If the model > is a second degree polynomial, a x^2 + b x + c, the algorithm > will return the best coefficients a, b, and c. > > The trouble I am having is that I need to fit a second > degree polynomial where some of the coefficients are > either already known or include another known parameter > as a factor. For example, I need to find the best value > for k in this equation: > > k^2/4 x + k p x + p^2 > > where p is a known quantity. > > Although there are three coefficients, there is really > only one unknown parameter, that is k. Using polyfit from octave > will return a, b, and c where a = k^2/4, b = k p, and c = p^2. > > Solving the two equations for k gives values that > are significantly different (they differ in the second > decimal place). > > Is there an algorithm available that allows some of the > parameters of the model to be predetermined or to include > some predetermined factor? > > I suspect that the only solution would be to write a > custom least squares program. > > AK > > _______________________________________________ > Help-octave mailing list > Help-octave@... > https://www.cae.wisc.edu/mailman/listinfo/help-octave Help-octave mailing list Help-octave@... https://www.cae.wisc.edu/mailman/listinfo/help-octave |
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Re: Curve Fitting Problem
by Sergei Steshenko-2
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Applications From Scratch: http://appsfromscratch.berlios.de/ --- On Thu, 6/26/08, A. Kalten <akalten@...> wrote: > From: A. Kalten <akalten@...> > Subject: Re: Curve Fitting Problem > To: help-octave@... > Date: Thursday, June 26, 2008, 9:37 PM > On Thu, 26 Jun 2008 19:52:32 -0700 (PDT) > Sergei Steshenko <sergstesh@...> wrote: > > > > > If I understand your problem correctly, is more or > less like this; > > > > 1) there is f(x) to be approximated; > > 2) the desired approximation is: > > > > a * x^2 + b * x + c; > > > > In this case there is no function to be approximated. > There > is only data that fits a theoretical model. I was > inquiring > about any off-the-shelf algorithms that can fit a function > to a data set where some of the parameters of the function > need to be constrained in a certain way. > > AK > When I said "function to be approximated" I meant tabular function, so if you have tabular f(x) for a set of given xs, you can as well approximate f(x) - a * x ^ 2 for the same set of given xs. Regards, Sergei. _______________________________________________ Help-octave mailing list Help-octave@... https://www.cae.wisc.edu/mailman/listinfo/help-octave |
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Re: Curve Fitting Problem
by Fredrik Lingvall
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A. Kalten wrote:
> Hello, > > I need some advice on solving a curve fitting problem. > > Ordinarily, a curve fitting algorithm, such as polyfit > or wpolyfit in octave, will determine the coefficients of a > polynomial that best fits the empirical data. If the model > is a second degree polynomial, a x^2 + b x + c, the algorithm > will return the best coefficients a, b, and c. > > The trouble I am having is that I need to fit a second > degree polynomial where some of the coefficients are > either already known or include another known parameter > as a factor. For example, I need to find the best value > for k in this equation: > > k^2/4 x + k p x + p^2 > > where p is a known quantity. > > Although there are three coefficients, there is really > only one unknown parameter, that is k. Using polyfit from octave > will return a, b, and c where a = k^2/4, b = k p, and c = p^2. > > Solving the two equations for k gives values that > are significantly different (they differ in the second > decimal place). > > Is there an algorithm available that allows some of the > parameters of the model to be predetermined or to include > some predetermined factor? > > I suspect that the only solution would be to write a > custom least squares program. > > AK > > Do you have any prior information of the parameter k? Do you, for example, know if k is positive, or do you have any idea how much it may vary (variance of k), mean value, if k is bound to some interval, etc? An idea is then to formulate the problem as a parameter estimation problem, using the model y = k^2/4 x^2 + k p x + p^2 + e where e "describes" the misfit of the model. Say, for example, that you know that k is in the interval [a,b] and by assuming Gassian errors, e, you can find the most likely k (i.e, the maximum a posteriori estimate) by finding the k which maximizes \hat{k} = arg max { exp (-1/ (2 sigma_e^2) ( Y-k^2/4 X.^2 + k p X + p^2)^T ( Y-k^2/4 X.^2 + k p X + p^2) ) } k for k in [a,b] where Y is the data vector Y =[y(1) y(2) ... y(N)]^T. If you instead have knowledge of the mean and variance of k then you can assign a Gaussian prior for k which will result in a different (unbounded) optimization problem. /Fredrik -- Fredrik Lingvall, PhD E-mail: fl@..., fredrik.lingvall@... Web: http://folk.uio.no/fl/ _______________________________________________ Help-octave mailing list Help-octave@... https://www.cae.wisc.edu/mailman/listinfo/help-octave |
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Re: Curve Fitting Problem
by A. Kalten
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On Fri, 27 Jun 2008 07:34:35 +0200
"Jaroslav Hajek" <highegg@...> wrote: > > Having a > matlab-compatible lsqnonlin in Octave would be good. In general, > making the optimization stuff more systematic and matlab-compatible is > a worthy long-term goal. > The Matlab lsqnonlin did the job for me. It allows some or all of the coefficients of the model to be constrained. If very tight bounds (e.g. +/- 1e-6) are placed on a coefficient, it is the same as fixing the coefficient to that value. But thanks to everyone who responded. I will certainly keep all the suggestions in mind. That is the value of this list. One can learn a great deal by encountering other perspectives. AK _______________________________________________ Help-octave mailing list Help-octave@... https://www.cae.wisc.edu/mailman/listinfo/help-octave |
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Re: Octave and Image Processing
by Abdul Rahman Riza-2
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Dear Friends,
I follow instruction from octave tutorial as following octave:1> I = loadimage ("default.img"); octave:2> S = conv2 (I, ones (5, 5) / 25, "same"); octave:3> [Dx, Dy] = gradient (S); octave:4> what next..?? I am a newbie here but I want to input an image, do a simple opreration, display its result and save it. Anybody can give me a simple example? Riza
2008/6/27 Søren Hauberg <soren@...>: fre, 27 06 2008 kl. 12:55 -0700, skrev Abdul Rahman Riza: _______________________________________________ Help-octave mailing list Help-octave@... https://www.cae.wisc.edu/mailman/listinfo/help-octave |
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Re: Octave and Image Processing
by Søren Hauberg
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Quoting Abdul Rahman Riza <abdulrahmanriza@...>:
> I follow instruction from octave tutorial as following > > octave:1> I = loadimage ("default.img"); > octave:2> S = conv2 (I, ones (5, 5) / 25, "same"); > octave:3> [Dx, Dy] = gradient (S); > octave:4> > > what next..?? > > I am a newbie here but I want to input an image, do a simple opreration, > display its result and save it. If you then do grad_len = sqrt (Dx.^2 + Dy.^2); imshow (grad_len, []) then you should get a gradient image (i.e. something that highlights edges). If you install the 'image' package from OctaveForge (http://octave.sf.net) then you'll get a bunch of functions that'll allow you to do more interesting things. One example would be somthing like E = edge (I); imshow (E) which will detect edges and display them. But unless you mention a specific problem you want to solve I can't give you better examples. If you install the 'image' packagem, you'll be able to save your results using imwrite ("test.png", E); Hope that helps, Søren > > Anybody can give me a simple example? > > Riza > 2008/6/27 Søren Hauberg <soren@...>: > >> fre, 27 06 2008 kl. 12:55 -0700, skrev Abdul Rahman Riza: >> > Anybody can figure out how powerfull octave to tacke image processing >> > problem? >> >> A lot of people use Octave for image processing, so I think the answer >> is yes :-) >> If you want help you need to ask more specific questions, as nobody >> can answer a question that haven't been asked. >> >> Søren >> >> > _______________________________________________ Help-octave mailing list Help-octave@... https://www.cae.wisc.edu/mailman/listinfo/help-octave |
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