Power law model fitting

L. El Ghaoui

Power law model fitting

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

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

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

$\tilde{y}_i = a^T \tilde{x}^{(i)} +b, \quad i=1,\cdots, m.$

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 reality. Finding the best fit can be addressed via the optimization problem

$\min\limits_z ||X^Tz-y||_2,$

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

See also: Power laws.

Power law model fitting

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