Power laws

Consider a physical process that has inputs x_j, j= 1,\cdots, n, and a scalar output y. Inputs and output are physical, positive quantities, such as volume, height, or temperature. In many cases, we can (at least empirically) describe such physical processes by power laws, which are non-linear models of the form

    \begin{align*} y &= \alpha x_1^{a_1} \cdots x_n^{a_n}, \end{align*}

where \alpha>0, and the coefficients a_j, j= 1,\cdots, n are real numbers. For example, the relationship between area, volume, and size of basic geometric objects; the Coulomb law in electrostatics; birth and survival rates of (say) bacteria as functions of concentrations of chemicals; heat flows and losses in pipes, as functions of the pipe geometry; analog circuit properties as functions of circuit parameters; etc.

The relationship x \rightarrow y is not linear nor affine, but if we take the logarithm of both sides and introduce the new variables

    \begin{align*} \tilde{y} := \log(y), \quad \tilde{x_j} := \log(x_j), \quad j= 1,\cdots, n. \end{align*}

then the above equation becomes an affine one:

    \begin{align*} \tilde{y} &= \log(\alpha) + \sum\limits_{j=1}^{n} a_j \log(x_j) = \log(\alpha) + \sum\limits_{j=1}^{n} a_j \tilde{x}_j = a^T\tilde{x} + b. \end{align*}

where b:=\log(\alpha).

See also: Fitting power laws to data.

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