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'Linear aproximation' of a set of points?

  1. Jul 15, 2006 #1

    radou

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    Let there be a set of points [tex]T=\left\{T_{i}=(x_{i}, y_{i}) \in \textbf{R}^2, i=1, \dots, n : x > 0 \wedge y >0 \right\}[/tex], a line [tex]Ax + By + C = 0[/tex], and another set [tex]D=\left\{d_{i}=\frac{\left|Ax_{i}+By_{i}+C\right|}{\sqrt{A^2+B^2}} \in \textbf{R}, i=1, \dots, n\right\}[/tex]. Next, let's form a sum [tex]S = \sum_{i=1}^n \frac{\left|Ax_{i}+By_{i}+C\right|}{\sqrt{A^2+B^2}}[/tex] which is required to have a minimum. Are [tex]\frac{\partial S}{\partial A} = 0[/tex], [tex]\frac{\partial S}{\partial B} = 0[/tex], and [tex]\frac{\partial S}{\partial C} = 0[/tex], the only requirements we need to 'place' the line as near to the dots as possible (if possible?) ?
     
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