Accounting for high-order correlations in probabilistic characterization of environmental variables, and evaluation

Abstract

Probabilistic characterization of environmental variables or data typically involves distributional fitting. Correlations, when present in variables or data, can considerably complicate the fitting process. In this work, effects of high-order correlations on distributional fitting were examined, and how they are technically accounted for was described using two multi-dimensional formulation methods: maximum entropy (ME) and Koehler–Symanowski (KS). The ME method formulates a least-biased distribution by maximizing its entropy, and the KS method uses a formulation that conserves specified marginal distributions. Two bivariate environmental data sets, ambient particulate matter and water quality, were chosen for illustration and discussion. Three metrics (log-likelihood function, root-mean-square error, and bivariate Kolmogorov–Smirnov statistic) were used to evaluate distributional fit. Bootstrap confidence intervals were also employed to help inspect the degree of agreement between distributional and sample moments. It is shown that both methods are capable of fitting the data well and have the potential for practical use. The KS distributions were found to be of good quality, and using the maximum likelihood method for the parameter estimation of a KS distribution is computationally efficient.

Appendix: Univariate distributions

The forms of the univariate distributions used in this work are summarized below:

$$\hbox{Lognormal:}\quad f_X (x)=\frac{1}{\sqrt {2\pi } b x}\exp \, \left[ {\frac{-(\ln x-a)^2}{2\,b^2}} \right]\; \hbox{and}\; F_X (x)=\frac{1}{2}\left[ {1+\hbox{erf}\,\left( {\frac{\ln x-a}{\sqrt 2 b}} \right)} \right] $$

for x ∈ (0, ∞), a ∈(−∞, ∞), and b ∈ (0, ∞), where a and b are the scale and shape parameters, respectively.

$$ \hbox{Gamma:}\quad f_X (x)=\frac{1}{{{\Upgamma}}(a)\,{b^a}}x^{a-1}\exp \,\left( {\frac{-x}{b}} \right)\; \hbox{and}\; F_X (x)=P\left( {a,\frac{x}{b}} \right) $$

for x ∈ [0, ∞), a ∈ (0, ∞), and b ∈ (0, ∞), where a and b are the scale and shape parameters, respectively. P(.) is an incomplete gamma function (Weisstein 2003).

$$ \hbox{Weibull:}\quad f_X (x)=b\,a^{-b}x^{b-1}\exp \,\left[ {-\left( {\frac{x}{a}} \right)^b} \right]\; \hbox{and}\; F_X (x)=1-\exp \left[ {-\left( {\frac{x}{a}} \right)^b} \right] $$

for x ∈[0, ∞), a ∈(0, ∞), and b ∈(0, ∞), where a and b are the scale and shape parameters, respectively.