These functions will fit distributions to a set of values using
maximum-likelihood estimation. In the context of the 'spatialwarnings'
package, they are most-often used to fit parametric distributions on patch
size distributions. As a result, these functions assume that the data
contains only integer, strictly positive values. The type of distribution
depends on the prefix of the function: 'pl' for power-law, 'tpl' for
truncated power-law, 'lnorm' for lognormal and 'exp' for an exponential
distribution.
In the context of distribution-fitting, 'xmin' represents the minimum value
that a distribution can take. It is often used to represent the minimum
scale at which a power-law model is appropriate (Clauset et al. 2009), and
can be estimated on an empirical distribution using
xmin_estim
. Again, please note that the fitting procedure
assumes here that xmin is equal or grater than one.
Please note that a best effort is made to have the fit converge, but
it may sometimes fail when the parameters are far from their usual
range. It is good practice to make sure the fits are sensible when
convergence warnings are reported.
For reference, the shape of the distributions is as follow:
power-law \(x^{-a}\) where a is the power-law exponent
exponential \(exp(-bx)\) where b is the truncation rate
of the exponential
truncated power-law \(x^{-a}exp(-bx)\) where a
and b are the exponent of the power law and the rate of truncation
The lognormal form follows the standard definition.