1. Stochastic Hydrology Stochastic Simulation of Bivariate Gamma Distribution
Professor Ke-sheng Cheng
Department of Bioenvironmental Systems Engineering
National Taiwan University

2. 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

3. 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

4. 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

5. 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.
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Dept. of Bioenvironmental Systems Engineering, National Taiwan University

6. Bivariate normal simulation II. Using the PC Transformation
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Dept. of Bioenvironmental Systems Engineering, National Taiwan University

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Dept. of Bioenvironmental Systems Engineering, National Taiwan University

8. 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

9. 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

10. 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

11. 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]
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Dept. of Bioenvironmental Systems Engineering, National Taiwan University

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Dept. of Bioenvironmental Systems Engineering, National Taiwan University

13. General equation for hydrological frequency analysis
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Dept. of Bioenvironmental Systems Engineering, National Taiwan University

14. 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]
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Dept. of Bioenvironmental Systems Engineering, National Taiwan University

15. Gamma density
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Dept. of Bioenvironmental Systems Engineering, National Taiwan University

16. 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

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Dept. of Bioenvironmental Systems Engineering, National Taiwan University

18. 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

19. 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
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Dept. of Bioenvironmental Systems Engineering, National Taiwan University

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Dept. of Bioenvironmental Systems Engineering, National Taiwan University

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Dept. of Bioenvironmental Systems Engineering, National Taiwan University

22. Flowchart of BVG simulation (1/2)
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Dept. of Bioenvironmental Systems Engineering, National Taiwan University

23. Flowchart of BVG simulation (2/2)
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Dept. of Bioenvironmental Systems Engineering, National Taiwan University

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Dept. of Bioenvironmental Systems Engineering, National Taiwan University

25. [B]
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Dept. of Bioenvironmental Systems Engineering, National Taiwan University

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Dept. of Bioenvironmental Systems Engineering, National Taiwan University

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Dept. of Bioenvironmental Systems Engineering, National Taiwan University

28. 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
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Dept. of Bioenvironmental Systems Engineering, National Taiwan University

29. Correlation coefficient of KX and KY can be derived as follows:
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Dept. of Bioenvironmental Systems Engineering, National Taiwan University

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Dept. of Bioenvironmental Systems Engineering, National Taiwan University

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Dept. of Bioenvironmental Systems Engineering, National Taiwan University

32. Since KX and KY are distributed with zero means, it follows that
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Dept. of Bioenvironmental Systems Engineering, National Taiwan University

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Dept. of Bioenvironmental Systems Engineering, National Taiwan University

34. It can also be shown that
Thus,
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Dept. of Bioenvironmental Systems Engineering, National Taiwan University

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Dept. of Bioenvironmental Systems Engineering, National Taiwan University

36. We have also proved that Eq. (B) is indeed a single-value function.
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Dept. of Bioenvironmental Systems Engineering, National Taiwan University

37. Proof of Eq. (B) as a single-value function
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Dept. of Bioenvironmental Systems Engineering, National Taiwan University

38. Therefore,
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Dept. of Bioenvironmental Systems Engineering, National Taiwan University

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Dept. of Bioenvironmental Systems Engineering, National Taiwan University

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Dept. of Bioenvironmental Systems Engineering, National Taiwan University

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Dept. of Bioenvironmental Systems Engineering, National Taiwan University

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Dept. of Bioenvironmental Systems Engineering, National Taiwan University

43. 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

44. Statistical properties of typhoon events at two raingauge stations
7/25/2019
Dept. of Bioenvironmental Systems Engineering, National Taiwan University

45. 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

46. Variation of the sample means with respect to sample size n.
7/25/2019
Dept. of Bioenvironmental Systems Engineering, National Taiwan University

47. Variation of the sample skewness with respect to sample size n.
7/25/2019
Dept. of Bioenvironmental Systems Engineering, National Taiwan University

48. Variation of the sample correlation coefficient with respect to sample size n.
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Dept. of Bioenvironmental Systems Engineering, National Taiwan University

49. Comparing CDF and ECDF 7/25/2019
Dept. of Bioenvironmental Systems Engineering, National Taiwan University

50. A scatter plot of simulated random samples with inappropriate pattern (adapted from Schmeiser and Lal, 1982).
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Dept. of Bioenvironmental Systems Engineering, National Taiwan University

51. Scattering of random samples
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Dept. of Bioenvironmental Systems Engineering, National Taiwan University

52. Feasible region of
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Dept. of Bioenvironmental Systems Engineering, National Taiwan University

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Dept. of Bioenvironmental Systems Engineering, National Taiwan University

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Dept. of Bioenvironmental Systems Engineering, National Taiwan University

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Dept. of Bioenvironmental Systems Engineering, National Taiwan University

56. 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