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Physics 257
Lecture 6:
Curve Fitting
Prof. Matt Dobbs
Physics Department, McGill U.
Prof. M. Dobbs,
Physics 257
October 25, 2007
2
Admin Items
±
November 8– Last “regular” lecture.
±
November 15 – Last lecture (democratic)
z
We will do 2/3 of the following:
¾
Review and open floor for questions.
¾
Funstuff with matlab (image processing, etc.) – not on exam.
¾
Cosmology lecture.
±
Nov 22 – Study break for final quiz, no lecture.
±
November 29, 12:30pm – Final Quiz
(in class, 90
min). Worth 15%.
Lots of challenge questions
.
Prof. M. Dobbs,
Physics 257
October 25, 2007
3
Curve Fitting
±
Regression analysis – statistical term for
determining the best fit curve.
±
Curve fitting amounts to comparing
data
to a
theoretical function
z
Function might be motivated by a theory
z
… or it might be ad hoc.
±
Curve fitting can be used to:
z
Test a theory.
z
Interpolate
z
Extrapolate
Prof. M. Dobbs,
Physics 257
October 25, 2007
4
Interpolation
±
Some data can be interpolated
by simply connecting the dots.
±
This doesn’t always work well:
z
Some intuition is required to
determine what sort of
interpolation is reasonable.
z
Curve fitting: what sort of curve is
reasonable?
Plots from: http://users.ce.ufl.edu/~kg
url/Classes/Lect3421/NM5_curve_s02.pdf
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View Full DocumentProf. M. Dobbs,
Physics 257
October 25, 2007
5
Curve Fitting
Finding the best fit parameters for any curve is as simple as
minimizing the
χ
2
.
(errr… not always that easy).
±
We derived the equations for this explicitly for a straight line
fit.
±
Similar equations exist for a polynomial of any order,
y= ax
n
+ bx
n1
+ cx
n2
+ dx
n3
+ ex
n4
+ … + z
±
For linear functions,
this is calculable
analytically.
±
For functions that are nonlinear in the fit parameters, it is
more complicated. Sometimes the functions can be linearized,
but often we need to resort to numerical minimization.
∑
=
i
i
i
x
f
a
y
)
(
()
∑
Δ
−
=
i
i
i
i
i
y
a
x
f
y
2
2
2
)
;
(
χ
Prof. M. Dobbs,
Physics 257
October 25, 2007
6
Curve Fitting
±
Where does the
χ
2
come from?
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 Fall '07
 Dobbs
 Physics

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