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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

that is linearized as

We identify then with of the linear case,
with , with and with .
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 is normally distributed around
and around
(we indicate by
and
the set of observations
in the original variables).
But, in general,
and
are not normally distributed
around
and
, respectively.
They are only when the measurements are very precise, i.e.
and
. 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.
and
are not very small,
non-linear effects in the transformations could be important
(see e.g. Ref. [15]).
- When some of
and
approach
unity it becomes important to consider
the error functions and the priors about and
with the due care. For example,
very often the quantities and 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 and .

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].

** Next:** Systematic errors
** Up:** Fits, and especially linear
** Previous:** Computational issues: normalization, fit
Giulio D'Agostini
2005-11-21