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Mathematics-Online course: Linear Algebra - Linear Systems of Equations - Approximation Problems

Best Fitting Line

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A best fitting line,

$\displaystyle p(t) = u + vt\, , $

to a set of points $ (t_i,f_i)$, $ i=1,\ldots,n$, can be determined by minimizing the sum of the squares of the errors

$\displaystyle \sum_{i=1}^n (f_i - p(t_i))^2\, . $

\includegraphics[width=.7\moimagesize]{a_ausgleichsgerade}

If we have at least two different $ t$-coordinates, we obtain the following formulas for the axis intercept $ u$ and the slope $ v$:

$\displaystyle u$ $\displaystyle = \frac{(\sum t_i^2)(\sum f_i)-(\sum t_i)(\sum t_if_i)} {n(\sum t_i^2)-(\sum t_i)^2}$    
$\displaystyle v$ $\displaystyle =\frac{n(\sum t_i f_i)-(\sum t_i)(\sum f_i)} {n(\sum t_i^2)-(\sum t_i)^2}\, .$    

A necessary condition for a minimum is that the derivatives of the sum of the squared errors with respect to both $ u$ and $ v$ equal zero:

0 $\displaystyle = 2\sum_i (u+vt_i-f_i)$    
0 $\displaystyle = 2\sum_i t_i(u+vt_i-f_i)\, .$    

The matrix form of these two equations is

$\displaystyle \left(\begin{array}{cc} n & \sum t_i \\ \sum t_i & \sum t_i^2 \... ...) = \left(\begin{array}{c} \sum f_i \\ \sum t_i f_i \end{array}\right)\,. $

If at least two of the $ t_i$ are different, then the determinant is

$\displaystyle (\underbrace{1+\cdots+1}_{n\,\text{times}}) \vert\left(t_1,\ldot... ...\right)^{\operatorname t}\vert _2^2 - \left(\sum_i 1\cdot t_i\right)^2 > 0\,, $

and we obtain the given solution by Cramer's rule.
  automatically generated 4/21/2005