Power law model fitting

Returning to the example involving power laws, we ask the question of finding the ‘‘best’’ model of the form

$y = C x_1^{a_1} ldots x_n^{a_n},$

given experiments with several input vectors $x^{(i)}$ and associated outputs y_i , i=1,ldots,m . Here the variables of our problem are , and the vector $a in mathbf{R}^n$ . Taking logarithms, we obtain

$tilde{y}_i = a^Ttilde{x}^{(i)} + b, ;; i=1,ldots,m.$

We can write the above linear equations compactly as

$left(begin{array}{c} y_1 vdots y_m end{array}right) = left(begin{array}{cc} tilde{x}_1^T & 1 vdots & vdots tilde{x}_m^T & 1 end{array}right) left(begin{array}{c} a b end{array}right).$

In practice, the power law model is only an approximate model of the reality. Finding the best fit can be addressed via the optimization problem

$min_{z} : |X^Tz-y|_2,$

where $z = (a,b) in mathbf{R}^{n+1}$ , $X in mathbf{R}^{(n+1) times m}$ , with -th column given by $(tilde{x}_1, 1)$ .

See also: Power laws.