How do you model a linear equation when there's no apparent pattern between x and y? For example x=30, y=134, x=31, y=155, x=33, y=165, x=35,...

Sunday, January 12, 2014

How do you model a linear equation when there's no apparent pattern between x and y? For example x=30, y=134, x=31, y=155, x=33, y=165, x=35,...

Hello!

It is obvious that there is no linear formula exactly connecting these x's and y's, if we consider the slopes between neighbor points:

$\displaystyle\frac{{{155}-{134}}}{{{31}-{30}}}={21},$ $\displaystyle\frac{{{165}-{155}}}{{{33}-{31}}}={5},$ $\displaystyle\frac{{{167}-{165}}}{{{35}-{33}}}={1}.$

For a single line, all these slopes must be the same.

But this isn't the whole story. We may seek such a line $\displaystyle{y}={a}{x}+{b}$ that would be the closest to all these points. The simplest criteria of such a proximity is the least squares one, which means we try to minimize

$\displaystyle{\sum_{{{k}={1}}}^{{n}}}{{\left({y}_{{n}}-{\left({a}{x}_{{n}}+{b}\right)}\right)}}^{{2}}.$

This problem has the exact unique answer (see for example the link attached). We have to compute the numbers

$\displaystyle{p}={\sum_{{{k}={1}}}^{{n}}}{{x}_{{k}}^{{2}}},$ $\displaystyle{q}={\sum_{{{k}={1}}}^{{n}}}{x}_{{k}},$ $\displaystyle{r}={\sum_{{{k}={1}}}^{{n}}}{x}_{{k}}{y}_{{k}}$ and $\displaystyle{s}={\sum_{{{k}={1}}}^{{n}}}{y}_{{k}}.$

In our case $\displaystyle{p}={4175},$ $\displaystyle{q}={129},$ $\displaystyle{r}={20115}$ and $\displaystyle{s}={621}.$ Then we solve the linear system for the unknowns a and b,

$\displaystyle{p}{a}+{q}{b}={r},$ $\displaystyle{q}{a}+{n}{b}={s}$ (here n=4).

I hope you know how to solve such systems, the solution for this is $\displaystyle{a}=\frac{{351}}{{59}},$ $\displaystyle{b}=-\frac{{2160}}{{59}}.$

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Sunday, January 12, 2014