From power law to linear fit

Linear fits are not only used to infer the parameters of a linear model, but also of other models that are linearized via a suitable transformation of the variables. The best known cases are the exponential law, linearized taking the log of the ordinate, and the power low, linearized taking the log of both coordinates. Linearizion is particularly important to provide a visual evidence in support of the claimed model. However, quantitative inference based on the transformed variable is not so obvious, if high accuracy in the determination of the model parameters is desired. Let us make some comments on the power law, in which both variables are log-transformed and therefore more general.

We start hypothesizing a model

$$B = \kappa \,A^\gamma\,,$$ (69)

that is linearized as

$$\log B = \gamma\, \log A + \log \kappa\,.$$ (70)

We identify then $\log B$ with $\mu_y$ of the linear case, $\log A$ with $\mu_x$, $\gamma$ with $m$ and $\log \kappa$ with $c$. But this identification does not allows us yet to use tout court the formulas derived above, because each of them depends on a well defined model. Let us see where are the possible problems.

• In the simplest model $a_i$ is normally distributed around $A_i$ and $b_i$ around $B_i$ (we indicate by ${\mbox{\boldmath$a$}}$ and ${\mbox{\boldmath$b$}}$ the set of observations in the original variables). But, in general, $x_i\equiv \log a_i$ and $y_i\equiv \log b_i$ are not normally distributed around $\mu_{x_i}\equiv \log A$ and $\mu_{y_i}\equiv \log B$, respectively. They are only when the measurements are very precise, i.e. $\sigma_{a_i}/ a_i \ll 1$ and $\sigma_{b_i}/b_i \ll 1$. This the case in which standard `error propagation', based on the well known formulas base on linearization, holds.
• If the precision is not very high, i.e. $\sigma_{a_i}/ a_i$ and $\sigma_{b_i}/b_i$ are not very small, non-linear effects in the transformations could be important (see e.g. Ref. [15]).
• When some of $\sigma_{a_i}/ a_i$ and $\sigma_{b_i}/b_i$ approach unity it becomes important to consider the error functions and the priors about $A$ and $B$ with the due care. For example, very often the quantities $A$ and $B$ are defined positive - and if we take their logarithms, they have to be positive. This requires the model to be correctly set up in order to prevent negative values of $A$ and $B$.

Further considerations would require a good knowledge of the the experimental apparatus and of the physics under study. Therefore I refrain from indicating a toy model, that could be used acritically in serious applications. Instead I encourage to draw a graphical representation of the model, as done in Figs. 1 and 2 and to make the inventory of the ingredients. Sometimes the representation in terms of Bayesian network is almost equivalent to solve the problem, thanks also to the methods developed in the past decades to calculate the relevant integrals, using e.g. Markov Chain Monte Carlo (MCMC), see e.g. Ref. [13] and references therein. In case of simple models one can even use free available software, like BUGS [11].