| Re: poly_fit for less number of points [message #84198] |
Mon, 06 May 2013 13:26  |
Craig Markwardt
Messages: 1869
Registered: November 1996
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Senior Member |
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On Sunday, May 5, 2013 6:03:25 PM UTC-4, David Fanning wrote:
> Craig Markwardt writes:
>
>> This fit,
>> res = poly_fit(x-mean(x), y, 2)
>> gives a smooth fit.
>
>
>
> What is the principle here that made you think of this solution and
> caused it to work?
Well, not much. He asked for a smooth function and I gave it to him. Jeremy is right, there is a round-off problem that makes the original POLY_FIT(X,Y,2) not work. Subtracting the mean of X (and/or Y) is a classic solution to the problem of round-off.
As HeinzS intimates, there's really a lot more to the question, like, how smooth should the function be? what degree of polynomial is permissible? how close of an approximation should the function be? what is the tolerance for outliers? why are there no error bars on the data? what is the definition of a "good fit?"
Without any of that information, I felt it worth to just provide *an* answer, even if it wasn't *the* answer to the original implicit question.
Craig
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| Re: poly_fit for less number of points [message #84202 is a reply to message #84198] |
Mon, 06 May 2013 11:02  |
Heinz Stege
Messages: 189
Registered: January 2003
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Senior Member |
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On Sat, 4 May 2013 07:02:57 -0700 (PDT), sid wrote:
> ... But can you please suggest me is there any other fit available to fit this kind of dataset. Because I want to get a smoothed fit.
I'm not sure, what you mean with "smoothed fit". I fully agree with
Craig, that a polynomial fit of degree 2 is very smooth.
Putting a more detailed look onto your data, I see that the function
has to be vary asymmetric to match the data. I guess you tried degree
4 or higher and don't like that wiggles, which are typical for
polynomal fits of higher degree.
It would be very helpful to know, if you are looking for a least
square fit (which compensates uncertainties within the y values) or a
function which interpolates the given points (and goes exactly through
the given points).
In any case I think, that it is very risky to draw an arbitrary curve
through the points on the right side of the diagram. It would be very
helpful, to have a model explaining y vs. x. Can you tell something
about the origin of the data?
A "quick and dirty" solution may be a hyperbolic function like
y(t) = (p0 + p1*t + p2*t^2) / (t - q0)
where p0, p1, p2 and q0 are the (fit-)constants and t=x-mean(x).
Such a non-linear function can be fitted by IDL's curvefit() or Craig
Markwardt's mpfit(). If you want to see the result for your example
before doing the fitting stuff, here are my results:
q0=0.311793, p0=-0.0241749, p1=0.0906013, p2=-0.0602339
But again, this is very speculative. This is one function, that fits
your data. And there may be other functions with other results!
Furthermore I would expect, that you have other data examples, where
the function above does not work!
Heinz
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| Re: poly_fit for less number of points [message #84208 is a reply to message #84202] |
Mon, 06 May 2013 06:48 |
Jeremy Bailin
Messages: 618
Registered: April 2008
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Senior Member |
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On 5/5/13 6:03 PM, David Fanning wrote:
> Craig Markwardt writes:
>
>> This fit,
>> res = poly_fit(x-mean(x), y, 2)
>> gives a smooth fit.
>
> What is the principle here that made you think of this solution and
> caused it to work?
>
> Cheers,
>
> David
>
Roundoff error in the typical fitting routines. Look at the OP's
original x values:
x = [3932.9321,3933.0452,3933.1162,3933.2514,3933.3517,3933.4271 ]
By switching to x-mean(x):
IDL> print, x-Mean(x)
-0.255371 -0.142334 -0.0712891 0.0639648 0.164307
0.239502
The effective precision of the values is 4 extra digits. Within the
polynomial fitting code, they're going to be squared, so roundoff error
in the original data is going to get bad, but the extra 4 digits keep it
under control.
-Jeremy.
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| Re: poly_fit for less number of points [message #84215 is a reply to message #84214] |
Sun, 05 May 2013 12:59 |
Craig Markwardt
Messages: 1869
Registered: November 1996
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Senior Member |
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On Saturday, May 4, 2013 6:44:28 AM UTC-4, sid wrote:
> Hello everyone,
>
> I have only 6 x values and y values, for example
>
> my x = [3932.9321,3933.0452,3933.1162,3933.2514,3933.3517,3933.4271 ]
>
> y = [0.090313701,0.084354165,0.080794218,0.075102273,0.074025219 ,0.082037902]
>
> I have to fit this with polynomial, I tried svdfit and poly_fit. But are not giving good fit. Please suggest me a way to fit these data points.
This fit,
res = poly_fit(x-mean(x), y, 2)
gives a smooth fit.
Craig
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| Re: poly_fit for less number of points [message #84225 is a reply to message #84215] |
Sat, 04 May 2013 07:02 |
sid
Messages: 50
Registered: January 1995
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Member |
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On Saturday, May 4, 2013 4:14:28 PM UTC+5:30, sid wrote:
> Hello everyone,
>
> I have only 6 x values and y values, for example
>
> my x = [3932.9321,3933.0452,3933.1162,3933.2514,3933.3517,3933.4271 ]
>
> y = [0.090313701,0.084354165,0.080794218,0.075102273,0.074025219 ,0.082037902]
>
> I have to fit this with polynomial, I tried svdfit and poly_fit. But are not giving good fit. Please suggest me a way to fit these data points.
>
>
>
> thanking you in advance
>
> sid
Thank you very much sir it works. But can you please suggest me is there any other fit available to fit this kind of dataset. Because I want to get a smoothed fit.
thanking you in advance
sid
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| Re: poly_fit for less number of points [message #84226 is a reply to message #84225] |
Sat, 04 May 2013 05:00 |
Heinz Stege
Messages: 189
Registered: January 2003
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Senior Member |
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On Sat, 4 May 2013 03:44:28 -0700 (PDT), sid wrote:
> Hello everyone,
> I have only 6 x values and y values, for example
> my x = [3932.9321,3933.0452,3933.1162,3933.2514,3933.3517,3933.4271 ]
> y = [0.090313701,0.084354165,0.080794218,0.075102273,0.074025219 ,0.082037902]
> I have to fit this with polynomial, I tried svdfit and poly_fit. But are not giving good fit. Please suggest me a way to fit these data points.
>
> thanking you in advance
> sid
You could try the following: Calculate the average xc of the x values.
Then transform the x values to xt=x-xc and do the fitting for the
tranformed values.
Heinz
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| Re: poly_fit for less number of points [message #84291 is a reply to message #84198] |
Tue, 07 May 2013 01:20 |
Yngvar Larsen
Messages: 134
Registered: January 2010
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Senior Member |
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On Monday, 6 May 2013 22:26:51 UTC+2, Craig Markwardt wrote:
> On Sunday, May 5, 2013 6:03:25 PM UTC-4, David Fanning wrote:
>
>> Craig Markwardt writes:
>
>>> This fit,
>>> res = poly_fit(x-mean(x), y, 2)
>>> gives a smooth fit.
>
>> What is the principle here that made you think of this solution and
>> caused it to work?
>
> Well, not much. He asked for a smooth function and I gave it to him. Jeremy is right, there is a round-off problem that makes the original POLY_FIT(X,Y,2) not work. Subtracting the mean of X (and/or Y) is a classic solution to the problem of round-off.
Often you also want to _normalize_ the variation of X (and possibly also Y), such that X is bounded within [-1,1], e.g.
Xmid = 0.5*(min(X)+max(X))
Xinterval = max(X)-min(X)
Xnorm = 2*(X-Xmid)/Xinterval
This makes all powers of Xnorm to also be bounded within [-1,1], which is good for preservation of precision. In the particular example given by OP, all X values are bounded in an interval with a width of around 0.5, so normalization is not really important, just centering.
[Another option is the statistical normalization: Xnorm = (X-mean(X))/stddev(X)]
> As HeinzS intimates, there's really a lot more to the question, like, how smooth should the function be? what degree of polynomial is permissible? how close of an approximation should the function be? what is the tolerance for outliers? why are there no error bars on the data? what is the definition of a "good fit?"
>
> Without any of that information, I felt it worth to just provide *an* answer, even if it wasn't *the* answer to the original implicit question.
... but of course, what Heinz&Craig write here is the real issue.
--
Yngvar
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| Re: poly_fit for less number of points [message #84726 is a reply to message #84225] |
Sun, 05 May 2013 12:55 |
Craig Markwardt
Messages: 1869
Registered: November 1996
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Senior Member |
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On Saturday, May 4, 2013 10:02:57 AM UTC-4, sid wrote:
> On Saturday, May 4, 2013 4:14:28 PM UTC+5:30, sid wrote:
>
>> Hello everyone,
>
>>
>
>> I have only 6 x values and y values, for example
>
>>
>
>> my x = [3932.9321,3933.0452,3933.1162,3933.2514,3933.3517,3933.4271 ]
>
>>
>
>> y = [0.090313701,0.084354165,0.080794218,0.075102273,0.074025219 ,0.082037902]
>
>>
>
>> I have to fit this with polynomial, I tried svdfit and poly_fit. But are not giving good fit. Please suggest me a way to fit these data points.
>
>>
>
>>
>
>>
>
>> thanking you in advance
>
>>
>
>> sid
>
>
>
> Thank you very much sir it works. But can you please suggest me is there any other fit available to fit this kind of dataset. Because I want to get a smoothed fit.
>
This fit,
res = poly_fit(x, y, 2)
gives a smooth fit.
Craig
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