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| Linear Fitting: More equations, greater error ? http://forum.alglib.net/viewtopic.php?f=2&t=308 |
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| Author: | marcs [ Wed Feb 23, 2011 3:58 pm ] |
| Post subject: | Linear Fitting: More equations, greater error ? |
Dear Sergey, first I want to take the opportunity an gratulate you on ALGLIB, a great library, extremely clean coding style, very efficient. Great work, a good programmer :) I have only one simple question: when using linear interpolation (it is in fact polynomial regression, most of the values in the function matrix are powers of >some x<), I encounter the issue that when increasing the number of equations (degrees of freedom in my model) over a certain limit, often about 5 * 20 equations (means 5 sub-matrices with x^0 - x^20), the overall error starts to increase again. Is it likely to be related to floating point / IEEE754 issues? I would guess so. Best Regards :) |
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| Author: | Sergey.Bochkanov [ Wed Feb 23, 2011 4:50 pm ] |
| Post subject: | Re: Linear Fitting: More equations, greater error ? |
Almost surely, it is floating point issue. Polynomial basis is ill-conditioned, and having x^20 as basis function will give you very badly conditioned matrix. You can check fitting report, in particular - rep.taskrcond variable. If it is lower than 10^(-10), than you definitely have problems with condition number. I recommend you to switch to better basis - for example, to replace x^i with centered Chebyshev polynomials. |
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| Author: | marcs [ Wed Feb 23, 2011 8:18 pm ] |
| Post subject: | Re: Linear Fitting: More equations, greater error ? |
Thank You for your time! |
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| Author: | Doug Jenkins [ Thu Feb 24, 2011 11:08 am ] |
| Post subject: | Re: Linear Fitting: More equations, greater error ? |
The polyfit function seems to give much better results for fitting high order polynomials than the least squares functions. See an Excel implementation and examples here: http://newtonexcelbach.wordpress.com/20 ... functions/ |
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| Author: | marcs [ Fri Feb 25, 2011 4:11 pm ] |
| Post subject: | Re: Linear Fitting: More equations, greater error ? |
Doug, I'm aware. The polynomial fitting uses a barycentric method. However, not all equations in my model are polynomials. |
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| Author: | Sergey.Bochkanov [ Fri Feb 25, 2011 6:59 pm ] |
| Post subject: | Re: Linear Fitting: More equations, greater error ? |
Quote: The polynomial fitting uses a barycentric method. Polynomial fitting uses Chebyshev basis internally, it just converts results into barycentric form before output. Quote: However, not all equations in my model are polynomials. I've recommended you to use Chebyshev polynomials as basis because you can use it with non-polynomial basis functions. |
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| Author: | marcs [ Sat Feb 26, 2011 8:35 pm ] |
| Post subject: | Re: Linear Fitting: More equations, greater error ? |
Thanks again, I have looked at the matrix condition and I'm using now Chebychev Polynomials with -1 <= x <= +1 |
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