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events: update documentation for polyfit algorithm

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This commit is contained in:
Paul Riseborough
2017-02-16 22:17:47 +01:00
committed by Lorenz Meier
parent 7bd8514697
commit 72633ad4c8
1 changed files with 29 additions and 19 deletions
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src/modules/events/temperature_calibration/polyfit.hpp
+29 -19
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@@ -32,34 +32,45 @@
****************************************************************************/
/*
Considering Polynomial Form:
y = a0 + a1.x + a2.x^2 + a3.x^3 + .... + an.x^n
and model for m number of samples will be
yi = a0 + a1.xi + a2.xi^2 + a3.xi^3 + .... + an.xi^n + ei where i =[0,m]
yi are the effect of cause xi and ei random fit error
This algorithm performs a curve fit of m x,y data points using a polynomial
equation of the following form:
in Vector Form:
Y = V.A + E
where V is Vandermonde matrix in x https://en.wikipedia.org/wiki/Vandermonde_matrix
yi = a0 + a1.xi + a2.xi^2 + a3.xi^3 + .... + an.xi^n + ei , where:
so by Ordinary Least Squares derivation by minimising ∑(i=0..m)ei^2 here: http://www2.warwick.ac.uk/fac/soc/economics/staff/vetroeger/teaching/po906_week567.pdf
i = [0,m]
xi is the x coordinate (independant variable) of the i'th measurement
yi is the y coordinate (dependant variable) of the i'th measurement
ei is a random fit error being the difference between the i'th y coordinate
and the value predicted by the polynomial.
we get A = inv(transpose(V)*V)*(transpose(V)*Y)
In vector form this is represented as:
we accumulate VTV and VTY as they are of fixed size and can be recursed upon
we can write VTV as
Y = V.A + E , where:
V is Vandermonde matrix in x -> https://en.wikipedia.org/wiki/Vandermonde_matrix
Y is a vector of length m containing the y measurements
E is a vector of length m containing the fit errors for each measurement
Use an Ordinary Least Squares derivation to minimise ∑(i=0..m)ei^2 -> https://en.wikipedia.org/wiki/Ordinary_least_squares
Note: In the wikipedia reference, the X matrix in reference is equivalent to our V matrix and the Beta matrix is equivalent to our A matrix
A = inv(transpose(V)*V)*(transpose(V)*Y)
We can accumulate VTV and VTY recursively as they are of fixed size, where:
VTV = transpose(V)*V =
__ __
| m x0+x1+...+xn x0^2+x1^2+...+xn^3 .......... x0^m+x1^m+...+xn^m |
|x0+x1+...+xn x0^2+x1^2+...+xn^3 x0^3+x1^3+...+xn^3 .......... x0^(m+1)+x1^(m+1)+...+xn^(m+1) |
| n x0+x1+...+xm x0^2+x1^2+...+xm^3 .......... x0^n+x1^n+...+xn^n |
|x0+x1+...+xn x0^2+x1^2+...+xm^3 x0^3+x1^3+...+xm^3 .......... x0^(n+1)+x1^(n+1)+...+xm^(n+1) |
| . . . . |
| . . . . |
| . . . . |
|x0^m+x1^m+...+xn^m x0^(m+1)+x1^(m+1)+...+xn^(m+1) x0^(m+2)+x1^(m+2)+...+xn^(m+2) .... x0^(2m)+x1^(2m)+...+xn^(2m) |
|x0^n+x1^n+...+xm^n x0^(n+1)+x1^(n+1)+...+xm^(n+1) x0^(n+2)+x1^(n+2)+...+xm^(n+2) .... x0^(2n)+x1^(2n)+...+xm^(2n) |
|__ __|
and for VTY
and VTY = transpose(V)*Y =
__ __
| ∑(i=0..m)yi |
| ∑(i=0..m)yi*xi |
@@ -69,7 +80,6 @@ and for VTY
|∑(i=0..m)yi*xi^n|
|__ __|
for more on expansion refer https://github.com/jgoppert/iekf_analysis/blob/master/Temperature%20Calibration.ipynb
*/
/*
@@ -190,4 +200,4 @@ private:
temp *= x;
}
}
};
};