A new procedure with a Box-Cox transformation (Box&Cox,
1964) have been introduced when calculation log likelihood from validation data.
A Technical module switch /Box-Cox
transformation/ is used to turn the
transformation on. The Box-Cox transformation is using two parameters (/Box Cox power/ and /Box
Cox offset/) that are part of the parameter
table “Scaling of Observed Variables” to filter the observed and simulated data:
Newvalue=((value+offset)**power-1)/power when power>0.00001
and
Newvalue=log(value+offset) when power<0.00001.
The likelihood of the untransformed values are equal to the likelihood
based on the transformed values multiplied with the derivative of the
transformation. Thus, the log likelihood is calculated as the sum of the
loglikelihood of the transformed values plus the log of the transform derivative
calculated for the measured value:
The values are calculated as=
((value+offset)**(power-1)))
This procedure was used by Yang et al
(2007) for a hydrological application.
References:
Box, G. E. P. and
Cox, D. R. (1964): An Analysis of Transformations. Journal of the Royal
Statistical Society. Series B (Methodological), Vol. 26, No. 2 (1964), pp.
211-252.
Yang, J., Reichert, P., Abbaspour, K. C., and Yang, H. Hydrological
modelling of the Chaohe
Basin in China: Statistical model formulation and
Bayesian inference. Journal of
Hydrology 340, 167– 182,
2007.