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temp_cal: add math explanation for polynomil fit algo

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Siddharth Bharat Purohit
2017-02-16 10:14:59 +05:30
committed by Lorenz Meier
parent 0e34de08fb
commit ab465744f1
1 changed files with 42 additions and 0 deletions
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src/modules/events/temperature_calibration/polyfit.hpp
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@@ -30,6 +30,48 @@
* POSSIBILITY OF SUCH DAMAGE.
*
****************************************************************************/
/*
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
in Vector Form:
Y = V.A + E
where V is Vandermonde matrix in x https://en.wikipedia.org/wiki/Vandermonde_matrix
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
we get A = inv(transpose(V)*V)*(transpose(V)*Y)
we accumulate VTV and VTY as they are of fixed size and can be recursed upon
we can write VTV as
__ __
| 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) |
| . . . . |
| . . . . |
| . . . . |
|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) |
|__ __|
and for VTY
__ __
| ∑(i=0..m)yi |
| ∑(i=0..m)yi*xi |
| . |
| . |
| . |
|∑(i=0..m)yi*xi^n|
|__ __|
for more on expansion refer https://github.com/jgoppert/iekf_analysis/blob/master/Temperature%20Calibration.ipynb
*/
/*
Polygon linear fit
Author: Siddharth Bharat Purohit