GAUSS ERROR INTEGRAL
For approximation of functions or operators integrating least squaresl is a
very powerfiul tool.
Its very easy to understand but often hard to compute. Many mathematical methods
are based
on this Error Integral. Especially most Integral Transformations, like Fourier,
La Place, Z..
In the following it will be described, to approximate the function f(x) by
an synthese function
s(x), using this method. s(x) depending on n free parameters a(p)
. p=1..n .
Therfore s(x) is s( x, a1, a2, a3,
.... an).
Its to find parameters a1, a2 a3 ... an, that s(x) is a good appoximation for
f(x).
GAUSS does it like this :
Just measure the distances squarts between s(x) and f(x) for all x.
Thats easy: | s(x),f(x ) |2
= ( f(x) - s(x) )2
This are the errors for each x.
Now just integrate alle erros in an intervall x = alpha to betha,
and thats a value for the quality of the approximation.
Therefore the least square Integral can be interpreted as an Error-Integral.
(You can chooses the interval alpha..beta for your taste ! )
| b | |||
| J error = | ∫ | [ f(x) - s ( x, a p) ] 2 dx | |
| a |
Of course we want a good approximation with minimal error.
So lets minimize this error Integral !
Looking for a minimum of the error Integral can be made, setting the partial
differentation for each
parameter a(p) to zero:
δJerror
/ δa(p)
= 0
For every p, that means every parameter a(p) .
Its convinient to make differentation under the Integral !
| b | |||
| ∫ | 2 * [ f(x)- s ( x, ap )] * δs(ap)/δap dx | = 0 | p = 0 ..n |
| a |
| b | b | ||
| ∫ | s ( x, ap )] * δs(ap)/δap dx= | ∫ | f(x)* δs(ap)/δap dx , p=0..n |
| a |
(1)
|
a |
(2)
|
For each parameter a(p) you ll get an new equation. Equation (1) = (2) stands
for an system
of Equations. On the left (1) still s(x,a(p)) is present. So we can expect this
side leading to
an Matrix of Integrals. Right side (2) contains Integral of f(x)*ds(a(p))/da(p).
In most cases
not containing any parameter a(p) any more.
Uhh i dont understand !
With following easy example you
will see, its very easy to do !
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