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Calculate spline using points, with error bars

I have a set of points and I want to draw a curve through it.
I can use a spline (y = ax^3 + by^2 + cx + d) to do that.

Each point has a error bar associated with it (+/- some value).
Can this (or should this) be used in calculation of the spline?
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allelopath
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allelopath
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2 Solutions
 
d-glitchCommented:
You should probably just plot the spline curve and include the error bars.
This is pretty standard practice.

You might plot three curves:  nominal, max, and min.
This is not standard.

What is your data?  And what are you trying to prove?
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allelopathAuthor Commented:
A question about including the error bar with the spline.

Suppose a data point of (3,4) with error of +/- 1
(so a valid point is anywhere from (3,3) to (3,5) )
Then the spline is estimated and at that point the curve goes through (3,4.5)
So then (3,4.5) +/- 1 is between (3,3.5) and (3,5.5)
This doesn't seem right in that the 5.5 is beyond the 5
or is this just the way it works?
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d-glitchCommented:
What is your data?  Not the numbers, but the meaning of the numbers.

Where do the error bars come from?  

Do you really have the same +/- 1 errors on both the x- and y-axis??
That is a little unusual.  
Wouldn't that give you a unit circle around (3, 4) rather than a square?

A spline should actually go through what ever points you specify -- (3, 4) not (3, 4.5).
Your error bars (or circle) should apply to the original data, not the fitted curve.
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allelopathAuthor Commented:
>>A spline should actually go through what ever points you specify -- (3, 4) not (3, 4.5).
Oh, you're right. duh.
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d-glitchCommented:
You might try to fit a line (or some other curve) to your data using a least squares regression.

In that case, it is possible for the curve to miss the data point and the error bar entirely.  

This would suggest

EITHER  
Your error bars are too small.  ==>  You don't understand the limits of your measurement.

OR
You have chose the wrong type of fit.  ==>  You don't understand the underlying process.
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allelopathAuthor Commented:
Ok, so I think there is smoothing and there is interpolation. With interpolation (as in a cubic spline) the curve will go through the data points, with smoothing, the curve may miss the data point. Correct?
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d-glitchCommented:
Smoothing implies some sort of fitting or averaging of the data.
And yes, the resultant curve may not hit any of the data points.

Curve fitting implies something else.  Perhaps assuming some kind of relationship among the data points.
And a fitted curve need not pass though any of the data either.

With interpolation (linear or cubic), the curve hits all the data points and you make assumptions between points.
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allelopathAuthor Commented:
Another semantics/terminology question
Does 'spline' imply interpolation (curve hits all data points)
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ozoCommented:
this is what I understand "spline" to mean
http://en.wikipedia.org/wiki/Spline_(mathematics)
for
y = ax^3 + by^2 + cx + d
I would use
http://mathworld.wolfram.com/LeastSquaresFittingPolynomial.html
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d-glitchCommented:
>>  Another semantics/terminology question
 >>  Does 'spline' imply interpolation (curve hits all data points)

A cubic spline hits all the data points AND matches the slopes between segments.

http://www.physics.utah.edu/~detar/phys6720/handouts/cubic_spline/cubic_spline/node1.html
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