SPEI
====
Obtaining location (:math:`\mu`), scale (:math:`\sigma`), and shape (:math:`\kappa`) from L-moments (see J. R. M. Hosking and
James R. Wallis (): Regional Frequency Analysis: An Approach Based on L-Moments):
.. math::
k = -\tau_3
.. math::
\sigma = \frac{\lambda_2 \sin(\kappa \pi)}{\kappa \pi}
.. math::
\mu = \lambda_1 - \frac{\sigma}{\kappa} \left[ 1 - \frac{\kappa \pi}{\sin(\kappa \pi)} \right]
The `shifted log-logistic distribution `_
has a three parameters cumulative distribution function defined as:
.. math::
F(x; \mu, \sigma, \kappa) = \frac{1}{1 + \left[ 1 + \kappa \frac{x - \mu}{\sigma} \right]^{-\frac{1}{\kappa}} }
The `logistic distribution `_
has a three parameters cumulative distribution function defined as:
.. math::
F(x; \mu, \sigma) = \frac{1}{1 + e^{-\frac{x - \mu}{\sigma}}}