Stochastic Hydrology Stochastic Simulation of Bivariate Gamma Distribution Professor Ke-sheng Cheng Department of Bioenvironmental Systems Engineering National Taiwan University
Unlike the univariate stochastic simulation, bivariate simulation not only needs to consider the marginal densities but also the covariation of the two random variables. 7/25/2019 Dept. of Bioenvironmental Systems Engineering, National Taiwan University
Bivariate normal simulation I. Using conditional density Joint density where and and are respectively the mean vector and covariance matrix of X1 and X2. 7/25/2019 Dept. of Bioenvironmental Systems Engineering, National Taiwan University
Conditional density where i and i (i = 1, 2) are respectively the mean and standard deviation of Xi, and is the correlation coefficient between X1 and X2. 7/25/2019 Dept. of Bioenvironmental Systems Engineering, National Taiwan University
Random number generation of a BVN distribution can be done by The conditional distribution of X2 given X1=x1 is also a normal distribution with mean and standard deviation respectively equal to and . Random number generation of a BVN distribution can be done by Generating a random sample of X1, say . Generating corresponding random sample of X2| x1, i.e. , using the conditional density. 7/25/2019 Dept. of Bioenvironmental Systems Engineering, National Taiwan University
Bivariate normal simulation II. Using the PC Transformation 7/25/2019 Dept. of Bioenvironmental Systems Engineering, National Taiwan University
7/25/2019 Dept. of Bioenvironmental Systems Engineering, National Taiwan University
Stochastic simulation of bivariate gamma distribution Importance of the bivariate gamma distribution Many environmental variables are non-negative and asymmetric. The gamma distribution is a special case of the more general Pearson type III distribution. Total depth and storm duration have been found to be jointly distributed with gamma marginal densities. 7/25/2019 Dept. of Bioenvironmental Systems Engineering, National Taiwan University
Many bivariate gamma distribution models are difficult to be implemented to solve practical problems, and seldom succeeded in gaining popularity among practitioners in the field of hydrological frequency analysis (Yue et al., 2001). Additionally, there is no agreement about what the multivariate gamma distribution should be and in practical applications we often only need to specify the marginal gamma distributions and the correlations between the component random variables (Law, 2007). 7/25/2019 Dept. of Bioenvironmental Systems Engineering, National Taiwan University
Simulation of bivariate gamma distribution based on the frequency factor which is well-known to scientists and engineers in water resources field. The proposed approach aims to yield random vectors which have not only the desired marginal distributions but also a pre-specified correlation coefficient between component variates. 7/25/2019 Dept. of Bioenvironmental Systems Engineering, National Taiwan University
Rationale of BVG simulation using frequency factor From the view point of random number generation, the frequency factor can be considered as a random variable K, and KT is a value of K with exceedence probability 1/T. Frequency factor of the Pearson type III distribution can be approximated by Standard normal deviate [A] 7/25/2019 Dept. of Bioenvironmental Systems Engineering, National Taiwan University
7/25/2019 Dept. of Bioenvironmental Systems Engineering, National Taiwan University
General equation for hydrological frequency analysis 7/25/2019 Dept. of Bioenvironmental Systems Engineering, National Taiwan University
The gamma distribution is a special case of the Pearson type III distribution with a zero location parameter. Therefore, it seems plausible to generate random samples of a bivariate gamma distribution based on two jointly distributed frequency factors. [A] 7/25/2019 Dept. of Bioenvironmental Systems Engineering, National Taiwan University
Gamma density 7/25/2019 Dept. of Bioenvironmental Systems Engineering, National Taiwan University
Assume two gamma random variables X and Y are jointly distributed. The two random variables are respectively associated with their frequency factors KX and KY . Equation (A) indicates that the frequency factor KX of a random variable X with gamma density is approximated by a function of the standard normal deviate and the coefficient of skewness of the gamma density. 7/25/2019 Dept. of Bioenvironmental Systems Engineering, National Taiwan University
7/25/2019 Dept. of Bioenvironmental Systems Engineering, National Taiwan University
Thus, random number generation of the second frequency factor KY must take into consideration the correlation between KX and KY which stems from the correlation between U and V. 7/25/2019 Dept. of Bioenvironmental Systems Engineering, National Taiwan University
Conditional normal density Given a random number of U, say u, the conditional density of V is expressed by the following conditional normal density with mean and variance . 7/25/2019 Dept. of Bioenvironmental Systems Engineering, National Taiwan University
7/25/2019 Dept. of Bioenvironmental Systems Engineering, National Taiwan University
7/25/2019 Dept. of Bioenvironmental Systems Engineering, National Taiwan University
Flowchart of BVG simulation (1/2) 7/25/2019 Dept. of Bioenvironmental Systems Engineering, National Taiwan University
Flowchart of BVG simulation (2/2) 7/25/2019 Dept. of Bioenvironmental Systems Engineering, National Taiwan University
7/25/2019 Dept. of Bioenvironmental Systems Engineering, National Taiwan University
[B] 7/25/2019 Dept. of Bioenvironmental Systems Engineering, National Taiwan University
7/25/2019 Dept. of Bioenvironmental Systems Engineering, National Taiwan University
7/25/2019 Dept. of Bioenvironmental Systems Engineering, National Taiwan University
Frequency factors KX and KY can be respectively approximated by where U and V both are random variables with standard normal density and are correlated with correlation coefficient . 7/25/2019 Dept. of Bioenvironmental Systems Engineering, National Taiwan University
Correlation coefficient of KX and KY can be derived as follows: 7/25/2019 Dept. of Bioenvironmental Systems Engineering, National Taiwan University
7/25/2019 Dept. of Bioenvironmental Systems Engineering, National Taiwan University
7/25/2019 Dept. of Bioenvironmental Systems Engineering, National Taiwan University
Since KX and KY are distributed with zero means, it follows that 7/25/2019 Dept. of Bioenvironmental Systems Engineering, National Taiwan University
7/25/2019 Dept. of Bioenvironmental Systems Engineering, National Taiwan University
It can also be shown that Thus, 7/25/2019 Dept. of Bioenvironmental Systems Engineering, National Taiwan University
7/25/2019 Dept. of Bioenvironmental Systems Engineering, National Taiwan University
We have also proved that Eq. (B) is indeed a single-value function. 7/25/2019 Dept. of Bioenvironmental Systems Engineering, National Taiwan University
Proof of Eq. (B) as a single-value function 7/25/2019 Dept. of Bioenvironmental Systems Engineering, National Taiwan University
Therefore, 7/25/2019 Dept. of Bioenvironmental Systems Engineering, National Taiwan University
7/25/2019 Dept. of Bioenvironmental Systems Engineering, National Taiwan University
7/25/2019 Dept. of Bioenvironmental Systems Engineering, National Taiwan University
7/25/2019 Dept. of Bioenvironmental Systems Engineering, National Taiwan University
7/25/2019 Dept. of Bioenvironmental Systems Engineering, National Taiwan University
Simulation and validation We chose to base our simulation on real rainfall data observed at two raingauge stations (C1I020 and C1G690) in central Taiwan. Results of a previous study show that total rainfall depth (in mm) and duration (in hours) of typhoon events can be modeled as a joint gamma distribution. 7/25/2019 Dept. of Bioenvironmental Systems Engineering, National Taiwan University
Statistical properties of typhoon events at two raingauge stations 7/25/2019 Dept. of Bioenvironmental Systems Engineering, National Taiwan University
Assessing simulation results Variation of the sample means with respect to sample size n. Variation of the sample skewness with respect to sample size n. Variation of the sample correlation coefficient with respect to sample size n. Comparing CDF and ECDF Scattering pattern of random samples 7/25/2019 Dept. of Bioenvironmental Systems Engineering, National Taiwan University
Variation of the sample means with respect to sample size n. 7/25/2019 Dept. of Bioenvironmental Systems Engineering, National Taiwan University
Variation of the sample skewness with respect to sample size n. 7/25/2019 Dept. of Bioenvironmental Systems Engineering, National Taiwan University
Variation of the sample correlation coefficient with respect to sample size n. 7/25/2019 Dept. of Bioenvironmental Systems Engineering, National Taiwan University
Comparing CDF and ECDF 7/25/2019 Dept. of Bioenvironmental Systems Engineering, National Taiwan University
A scatter plot of simulated random samples with inappropriate pattern (adapted from Schmeiser and Lal, 1982). 7/25/2019 Dept. of Bioenvironmental Systems Engineering, National Taiwan University
Scattering of random samples 7/25/2019 Dept. of Bioenvironmental Systems Engineering, National Taiwan University
Feasible region of 7/25/2019 Dept. of Bioenvironmental Systems Engineering, National Taiwan University
7/25/2019 Dept. of Bioenvironmental Systems Engineering, National Taiwan University
7/25/2019 Dept. of Bioenvironmental Systems Engineering, National Taiwan University
7/25/2019 Dept. of Bioenvironmental Systems Engineering, National Taiwan University
Joint BVG Density Random samples generated by the proposed approach are distributed with the following joint PDF of the Moran bivariate gamma model: 7/25/2019 Dept. of Bioenvironmental Systems Engineering, National Taiwan University