STOCHASTIC HYDROLOGY Stochastic Simulation of Bivariate Distributions Professor Ke-Sheng Cheng Department of Bioenvironmental Systems Engineering National Taiwan University Bivariate normal Bivariate exponential Bivariate gamma

Unlike the univariate stochastic simulation, bivariate simulation not only needs to consider the marginal densities but also the covariation of the two random variables. 1/15/ Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

Bivariate normal simulation I. Using conditional density Joint density where and  and  are respectively the mean vector and covariance matrix of X 1 and X 2. 1/15/ Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

Conditional density where  i and  i (i = 1, 2) are respectively the mean and standard deviation of X i, and  is the correlation coefficient between X 1 and X 2. 1/15/ Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

The conditional distribution of X 2 given X 1 =x 1 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 X 1, say. – Generating corresponding random sample of X 2 | x 1, i.e., using the conditional density. 1/15/ Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

Bivariate normal simulation II. Using the PC Transformation 1/15/ Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

1/15/ Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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. 1/15/ Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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). 1/15/ Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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. 1/15/ Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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 K T is a value of K with exceedence probability 1/T. Frequency factor of the Pearson type III distribution can be approximated by [A] Standard normal deviate 1/15/ Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

1/15/ Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

General equation for hydrological frequency analysis 1/15/ Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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] 1/15/ Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

Gamma density 1/15/ Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

Assume two gamma random variables X and Y are jointly distributed. The two random variables are respectively associated with their frequency factors K X and K Y. Equation (A) indicates that the frequency factor K X 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. 1/15/ Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

1/15/ Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

Thus, random number generation of the second frequency factor K Y must take into consideration the correlation between K X and K Y which stems from the correlation between U and V. 1/15/ Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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. 1/15/ Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

1/15/ Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

1/15/ Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

Flowchart of BVG simulation (1/2) 1/15/ Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

Flowchart of BVG simulation (2/2) 1/15/ Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

1/15/ Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

[B] 1/15/ Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

1/15/ Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

1/15/ Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

Frequency factors K X and K Y can be respectively approximated by where U and V both are random variables with standard normal density and are correlated with correlation coefficient. 1/15/ Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

Correlation coefficient of K X and K Y can be derived as follows: 1/15/ Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

1/15/ Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

1/15/ Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

Since K X and K Y are distributed with zero means, it follows that 1/15/ Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

1/15/ Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

It can also be shown that Thus, 1/15/ Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

1/15/ Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

We have also proved that Eq. (B) is indeed a single-value function. 1/15/ Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

Proof of Eq. (B) as a single-value function 1/15/ Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

Therefore, 1/15/ Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

1/15/ Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

1/15/ Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

1/15/ Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

1/15/ Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

The above equation indicates increases with increasing, and thus Eq. (B) is a single-value function. 1/15/ Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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. 1/15/ Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

Statistical properties of typhoon events at two raingauge stations 1/15/ Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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 1/15/ Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

Variation of the sample means with respect to sample size n. 10,000 samples 1/15/ Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

Variation of the sample skewness with respect to sample size n. 10,000 samples 1/15/ Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

Variation of the sample correlation coefficient with respect to sample size n. 10,000 samples 1/15/ Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

Comparing CDF and ECDF 1/15/ Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

A scatter plot of simulated random samples with inappropriate pattern (adapted from Schmeiser and Lal, 1982). 1/15/ Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

Scattering of random samples 1/15/ Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

Feasible region of 1/15/ Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

1/15/ Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

1/15/ Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

1/15/ Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

Joint BVG Density Random samples generated by the proposed approach are distributed with the following joint PDF of the Moran bivariate gamma model: 1/15/ Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

Stochastic Simulation of Bivariate Exponential Distribution A bivariate exponential distribution simulation algorithm was proposed by Marshall and Olkin (1967). Let X and Y be two jointly distributed exponenttial random variables. The joint exponential distribution function of Marshall and Olkin model (MOBED) has the following form: Marshall, A.W. & Olkin, I A Generalized Bivariate Exponential Distribution. Journal of Applied Probability, Vol. 4, /15/ Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

where 1, 2 and 12 are parameters. The expected values of X and Y and the correlation coefficient  (X,Y) are expressed by 1/15/ Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

Simulation of the bivariate exponential distribution of Equation (1) is achieved by independently generating random numbers of three univariate exponential densities (Z 1, Z 2, and Z 12 ) with parameters 1, 2 and 12, respectively. Then a pair of random number of (X,Y) is obtained by setting x=min(z 1, z 12 ) and y=min(z 2, z 12 ). 1/15/ Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

61 Another example – target cancer risk

62 Modeling MCS inorg – Log-normal

63 Cumulative distribution of the target cancer risk There is no need for stochastic simulation since the risk is completely dependent on only one random variable (MCS). Once the parameters of MCS are determined, the distribution of TR is completely specified.