Wm. Snyder et al., NORMALIZATION OF A HYDROLOGIC SAMPLE PROBABILITY DENSITY-FUNCTION BY TRANSFORM OPTIMIZATION, Journal of hydrology, 149(1-4), 1993, pp. 97-110
Transformation of variates is the conventional procedure for deriving
a probability density function of a variate y when a probability densi
ty function of a variate x and a function y = f(x) are known. In this
study the probability density functions p(x) and p(y) are assumed know
n and the transform is treated as a differential equation which is sol
ved to yield an optimal variate transform function x = f(y). Specifica
lly, p(x) is considered a sample probability density function and p(y)
is the normal probability density function. The solution for x = f(y)
thus provides an optimal normalization of the sample. Properties of t
he normal distribution can then be used for estimating confidence inte
rvals of the mean and tolerance limits of outer values. Such estimates
of risk and uncertainty, when de-transformed back to the sampled vari
ate of interest, x, are valuable tools in hydrology and water resource
and environmental protection analysis and planning. The numerical met
hodology for solving the differential equation is not specific to the
particular problem, and can be extended to other situations and other
probability density functions.