Delphi - polynominal fit

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theory

A polynominal of nth order : p:=Sum(ai*x^i,i=0..n)=a0+a1*x+a2*x^2+ .. +an*x^n
shall be fitted to a bunch of m points (Xj,Yj). The square error shall be minimal:

error:=Sum(Sqr(p(Xj)-Yj),j=1..m)=minimal
We therefore derive the error to all variables, the aj's.

calculate the square
calculate the sum
derive the terms with a0 in them, omit the others

Diff(error,a0):=Diff(Sum(Sqr(a0+a1*Xj+a2*Xj^2+ .. +an*Xj^n-Yj),j=1..m)=
  Diff(error,a0):=2*Sum(a0+a1*Xj+a2*Xj^2+.. an*Xj^n-Yj,j=1..m)=
  Diff(error,a0):=2*Sum(p(Xj)-Yj,j=1..m)
and equally :
  Diff(error,a1):=2*Sum(Xj*(p(Xj)-Yj),j=1..m)
  Diff(error,a2):=2*Sum(Xj^2*(p(Xj)-Yj),j=1..m)
..
  Diff(error,an):=2*Sum(Xj^n*(p(Xj)-Yj),j=1..m)

For the minimal error, the equations are each set to zero :
Sum(p(Xj)-Yj,j=1..m)=0
Sum(Xj*(p(Xj)-Yj),j=1..m)=0
Sum(Xj^2*(p(Xj)-Yj),j=1..m)=0
..
Sum(Xj^n*(p(Xj)-Yj),j=1..m)=0

Calculating the sums :
Sum(p(Xj)-Yj,j = 1..m)=
   Sum(a0+a1*Xj+a2*Xj^2+ .. +an*Xj^n-Yj,j=1..m)=
   a0*m+a1*Sum(Xj)+a2*Sum(Xj^2)+..+an*Sum(Xj^n)-Sum(Yj)
Sum(Xj*(p(Xj)-Yj),j = 1..m)=
   Sum(a0*Xj+a1*Xj^2+a2*Xj^3+ .. +an*Xj^(n+1)-Xj*Yj,j=1..m)=
   a0*Sum(Xj)+a1*Sum(Xj^2)+a2*Sum(Xj^3)+..+an*Sum(Xj^n+1)-m*Sum(Yj*Xj)
..

Ordering leaves n+1 equations for n+1 unknown. As equation system :

|m         Sum(Xj)       Sum(Xj^2)     .. Sum(Xj^n)    | |a0|    | Sum(Yj)     |
|Sum(Xj)   Sum(Xj^2)     Sum(Xj^3)     .. Sum(Xj^(n+1))| |a1|    | Sum(Xj*Yj)  |
|Sum(Xj^2) Sum(Xj^3)     Sum(Xj^4)     .. Sum(Xj^(n+2))|*|a2| =  | Sum(Xj^2*Yj)|
|..                                                    | |..|    | ..          |
|Sum(Xj^n) Sum(Xj^(n+1)) Sum(Xj^(n+2)) .. Sum(Xj^(2n)) | |an|    | Sum(Xj^n*Yj)|

Which can be solved

notes

There should be more points than the order of the polynominal.
As higher order polynominals lack stability, splines are prefered
The least square linear approximation is the simplest case of a polynominal fit
The calculations are similar when fitting functions other than polynominals

Feedback is welcome

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last updated: 31.july.99

Copyright (99,2000) Ing.Büro R.Tschaggelar