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

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STOCHASTIC HYDROLOGY Bivariate Simulation. Bivariate normal Bivariate exponential Bivariate gamma. Professor Ke-Sheng Cheng Department of Bioenvironmental Systems Engineering National Taiwan University. - PowerPoint PPT Presentation

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Page 1: STOCHASTIC HYDROLOGY Bivariate Simulation

STOCHASTIC HYDROLOGYBivariate Simulation

Professor Ke-Sheng ChengDepartment of Bioenvironmental Systems Engineering

National Taiwan University

Bivariate normalBivariate exponential

Bivariate gamma

Page 2: STOCHASTIC HYDROLOGY Bivariate Simulation

• Unlike the univariate stochastic simulation, bivariate simulation not only needs to consider the marginal densities but also the covariation of the two random variables.

04/22/23 2Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

Page 3: STOCHASTIC HYDROLOGY Bivariate Simulation

Bivariate normal simulation I. Using conditional density

• Joint density

where and and are respectively the mean vector and covariance matrix of X1 and X2.

1

2

1

21

2

1)(

XXeX

)( 21 XXX T

04/22/23 3Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

Page 4: STOCHASTIC HYDROLOGY Bivariate Simulation

• 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.

2

22

111

222

22122

1

)()(

2

1exp

)1(2

1)|(

xx

xxXf

04/22/23 4Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

Page 5: STOCHASTIC HYDROLOGY Bivariate Simulation

• 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.

111

22

x 2

2 1

),,,( 11211 nxxx

),,,( 22221 nxxx

04/22/23 5Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

Page 6: STOCHASTIC HYDROLOGY Bivariate Simulation

Bivariate normal simulation II. Using the PC Transformation

04/22/23 6Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

Page 7: STOCHASTIC HYDROLOGY Bivariate Simulation

04/22/23 7Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

Page 8: STOCHASTIC HYDROLOGY Bivariate Simulation

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.

04/22/23 8Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

Page 9: STOCHASTIC HYDROLOGY Bivariate Simulation

• 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).

04/22/23 9Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

Page 10: STOCHASTIC HYDROLOGY Bivariate Simulation

• 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.

04/22/23 10Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

Page 11: STOCHASTIC HYDROLOGY Bivariate Simulation

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

543

2

232

63

1

661

66

3

1

61

XXX

XXT

zz

zzzzK

[A]

Standard normal deviate

04/22/23 11Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

Page 12: STOCHASTIC HYDROLOGY Bivariate Simulation

04/22/23 12Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

Page 13: STOCHASTIC HYDROLOGY Bivariate Simulation

• General equation for hydrological frequency analysis

XTXT KX

04/22/23 13Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

Page 14: STOCHASTIC HYDROLOGY Bivariate Simulation

• 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.

543

2

232

63

1

661

66

3

1

61

XXX

XXT

zz

zzzzK

[A]

04/22/23 14Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

Page 15: STOCHASTIC HYDROLOGY Bivariate Simulation

Gamma density

xex

xf xX 0,

)(

1),;( /

1

0

02

2

0

2

04/22/23 15Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

Page 16: STOCHASTIC HYDROLOGY Bivariate Simulation

543

22

32

63

1

661

66

3

1

61

XXXXX

T zzzzzzK

• 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.

04/22/23 16Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

Page 17: STOCHASTIC HYDROLOGY Bivariate Simulation

543

22

32

63

1

661

66

3

1

61

XXXXX

T zzzzzzK

04/22/23 17Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

Page 18: STOCHASTIC HYDROLOGY Bivariate Simulation

• 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.

04/22/23 18Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

Page 19: STOCHASTIC HYDROLOGY Bivariate Simulation

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 .

)|(| uUvUV

2

22 12

1exp

)1(2

1

UV

UV

UV

uv

uUV 21 UV

04/22/23 19Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

Page 20: STOCHASTIC HYDROLOGY Bivariate Simulation

)|(| uUvUV

2

22 12

1exp

)1(2

1

UV

UV

UV

uv

04/22/23 20Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

Page 21: STOCHASTIC HYDROLOGY Bivariate Simulation

XX KX

04/22/23 21Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

Page 22: STOCHASTIC HYDROLOGY Bivariate Simulation

Flowchart of BVG simulation (1/2)

04/22/23 22Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

Page 23: STOCHASTIC HYDROLOGY Bivariate Simulation

Flowchart of BVG simulation (2/2)

04/22/23 23Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

Page 24: STOCHASTIC HYDROLOGY Bivariate Simulation

04/22/23 24Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

Page 25: STOCHASTIC HYDROLOGY Bivariate Simulation

Conversion ~ UVXY

32 62

933

UVYXUVYX

UVYXYXYXYXXY

CCBB

CCACCAAA

4

61

X

XA 3

66

XX

XB 2

63

1

X

XC

4

61

Y

YA 3

66

YY

YB 2

63

1

Y

YC

[B]

04/22/23 25Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

Page 26: STOCHASTIC HYDROLOGY Bivariate Simulation

04/22/23 26Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

Page 27: STOCHASTIC HYDROLOGY Bivariate Simulation

04/22/23 27Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

Page 28: STOCHASTIC HYDROLOGY Bivariate Simulation

• 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 .

543

22

32

63

1

661

66

3

1

61

XXXXX

X UUUUUUK

543

22

32

63

1

661

66

3

1

61

YYYYY

Y VVVVVVK

UV

04/22/23 28Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

Page 29: STOCHASTIC HYDROLOGY Bivariate Simulation

• Correlation coefficient of KX and KY can be derived as follows:

YXYXKK KKEKKCovYX

),(

543

22

32

5432

232

63

1

661

66

3

1

61

63

1

661

66

3

1

61

YYYYY

XXXXX

VVVVVV

UUUUUU

E

04/22/23 29Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

Page 30: STOCHASTIC HYDROLOGY Bivariate Simulation

52

33

24

63

1

66

3

1

661

61

XXXXX

X UUUUK

XXXX DUUCUBUA 61 32

52

33

24

63

1

66

3

1

661

61

YYYYY

Y VVVVK

YYYY DVVCVBVA 61 32

5234

63

1,

63

1,

66,

61

X

XX

XXX

XX

X DCBA

5234

63

1,

63

1,

66,

61

Y

YY

YYY

YY

Y DCBA

04/22/23 30Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

Page 31: STOCHASTIC HYDROLOGY Bivariate Simulation

YXYXYXYX

YXYX

YXYX

YXYX

YXYXYX

YXYXYX

YX

DDVVCDVBDVAD

UUDCVVUUCC

VUUBCVUUAC

UDBVVUCB

VUBBUVABUDA

VVUCAVUBAUVAA

E

KKE

61

666

166

161

111

61

32

333

233

232

222

32

04/22/23 31Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

Page 32: STOCHASTIC HYDROLOGY Bivariate Simulation

Since KX and KY are distributed with zero means, it follows that

YXKKE

YXYXYX

YXYXYX

DDVVUUCCVUUAC

VUBBVVUCAUVAAE

666

116333

223

XXXXXX DDUUCUBUAEKE 61][ 32

0 YX DD

04/22/23 32Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

Page 33: STOCHASTIC HYDROLOGY Bivariate Simulation

VVUUCCVUUAC

VUBBVVUCAUVAAE

KKE

YXYX

YXYXYX

YXKK YX

666

116

333

223

16 223 VUEBBUVECAAA YXUVYXUVYX

UVYX

UVYX

VUEUVEVUECC

VUEAC

3666

63333

3

04/22/23 33Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

Page 34: STOCHASTIC HYDROLOGY Bivariate Simulation

• It can also be shown that

Thus,

12 222 UVVUE UVUVVUE 96 333

UVUVEVUE 333

32 62

933

UVYXUVYX

UVYXYXYXYXKKXY

CCBB

CCACCAAAYX

04/22/23 34Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

Page 35: STOCHASTIC HYDROLOGY Bivariate Simulation

04/22/23 35Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

Page 36: STOCHASTIC HYDROLOGY Bivariate Simulation

ipRelationsh Value-Single ~ UVXY

We have also proved that Eq. (B) is indeed a single-value function.

04/22/23 36Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

Page 37: STOCHASTIC HYDROLOGY Bivariate Simulation

Proof of Eq. (B) as a single-value function

04/22/23 37Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

Page 38: STOCHASTIC HYDROLOGY Bivariate Simulation

• Therefore,

YXYXYXYX CCACCAAA 933

222222

61

661

6YYXX

04/22/23 38Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

Page 39: STOCHASTIC HYDROLOGY Bivariate Simulation

04/22/23 39Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

Page 40: STOCHASTIC HYDROLOGY Bivariate Simulation

04/22/23 40Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

Page 41: STOCHASTIC HYDROLOGY Bivariate Simulation

04/22/23 41Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

Page 42: STOCHASTIC HYDROLOGY Bivariate Simulation

04/22/23 42Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

Page 43: STOCHASTIC HYDROLOGY Bivariate Simulation

• The above equation indicates increases with increasing , and thus Eq. (B) is a single-value function.

XYUV

04/22/23 43Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

Page 44: STOCHASTIC HYDROLOGY Bivariate Simulation

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.

04/22/23 44Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

Page 45: STOCHASTIC HYDROLOGY Bivariate Simulation

Statistical properties of typhoon events at two raingauge stations

04/22/23 45Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

Page 46: STOCHASTIC HYDROLOGY Bivariate Simulation

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

04/22/23 46Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

Page 47: STOCHASTIC HYDROLOGY Bivariate Simulation

Variation of the sample means with respect to sample size n.

10,000 samples

04/22/23 47Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

Page 48: STOCHASTIC HYDROLOGY Bivariate Simulation

Variation of the sample skewness with respect to sample size n.

10,000 samples

04/22/23 48Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

Page 49: STOCHASTIC HYDROLOGY Bivariate Simulation

Variation of the sample correlation coefficient with respect to sample size n.

10,000 samples

04/22/23 49Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

Page 50: STOCHASTIC HYDROLOGY Bivariate Simulation

Comparing CDF and ECDF

04/22/23 50Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

Page 51: STOCHASTIC HYDROLOGY Bivariate Simulation

A scatter plot of simulated random samples with inappropriate pattern (adapted from Schmeiser and Lal,

1982).

04/22/23 51Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

Page 52: STOCHASTIC HYDROLOGY Bivariate Simulation

Scattering of random samples

04/22/23 52Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

Page 53: STOCHASTIC HYDROLOGY Bivariate Simulation

Feasible region of XY

04/22/23 53Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

Page 54: STOCHASTIC HYDROLOGY Bivariate Simulation

04/22/23 54Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

Page 55: STOCHASTIC HYDROLOGY Bivariate Simulation

04/22/23 55Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

Page 56: STOCHASTIC HYDROLOGY Bivariate Simulation

04/22/23 56Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

Page 57: STOCHASTIC HYDROLOGY Bivariate Simulation

Joint BVG Density • Random samples generated by the

proposed approach are distributed with the following joint PDF of the Moran bivariate gamma model:

)1(2

)(2)(exp

),;(),;(1

1),(

2

22

2

UV

UVUVUV

yyYxxX

UV

XY

vuvu

yfxfyxf

)],;([1xxX xFu )],;([1

yyY yFv

04/22/23 57Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

Page 58: STOCHASTIC HYDROLOGY Bivariate Simulation

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. 1967. A Generalized Bivariate Exponential Distribution. Journal of Applied Probability, Vol. 4, 291-302.

04/22/23 58Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

Page 59: STOCHASTIC HYDROLOGY Bivariate Simulation

where 1, 2 and 12 are parameters. The expected values of X

and Y and the correlation coefficient (X,Y) are expressed by

0 , )(exp

0 , )(exp),(

2121

1221,

xyyx

yxyxyxF YX

121

1

X

122

1

Y

1221

12),(

YX

04/22/23 59Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

Page 60: STOCHASTIC HYDROLOGY Bivariate Simulation

Simulation of the bivariate exponential distribution of Equation (1) is achieved by independently generating random numbers of three univariate exponential densities (Z1, Z2, and Z12) with

parameters 1, 2 and 12, respectively. Then a pair of

random number of (X,Y) is obtained by setting

x=min(z1, z12) and y=min(z2, z12).

04/22/23 60Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

Page 61: STOCHASTIC HYDROLOGY Bivariate Simulation

61

Another example – target cancer risk

Page 62: STOCHASTIC HYDROLOGY Bivariate Simulation

62

Modeling MCSinorg – Log-normal

Page 63: STOCHASTIC HYDROLOGY Bivariate Simulation

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.