Modeling and simulation of bivariate distribution of shear strength parameters using copulas - Risk and Reliability in Geotechnical Engineering - page 80

Table 2.1 Summary of the adopted bivariate copula functions and their parameter domains

Generator
function, φ θ (t) Copula Copula function, C (u 1 , u 2 ; θ ) Copula density function, D (u 1 , u 2 ; θ ) Range of θ

Gaussian

θ (

1

( u ),

1

( ) u - ΦΦ Φ

− −

1

1 22

2
21

2 22

ς
ς

=
=

−

1

() u

(()

[ − 1, 1]

ςθ θςςςθ
θ

− + Φ

Φ

1 2 1 1

12

exp , −

(

2

) −

1

u 2

1

2

−

− θ

2

[( )( 1 1 uu 2 uu )]

+− +−
+− +

θθ
θ

-

Plackett (0, ∞ )\{1}

SS uu

2

4 ( 1 ) −− −

−

θθ

1 2 12

12

,

{[ 1 ( 1 )( uu )]

2

4 uu ( 1 )} (/

32 )

− θ θ −

21 ( ) θ 1 2 12

S 1 ( 1 )( u u ) =+ −+ θ

1 2

( e 1 ) e

( uu )

ln e
e

θ
θ

1
1

1 ( e

u

1 )( e

u

1 )

− −
−+ −

θ

− θ − θ + −

−

−
−

− θ

−

− θ

−

Frank ( −∞ , ∞ )\{0}

12

1 2

ln 1

−

− +

2

θ

e 1

− θ

−

( e 1 ) ( e

u

1 )( e

u

1 )

− θ − θ − θ

−

1 2

1
2

2

θ
t

‡
‰
‰

Ž
‰
‰

1
2

11 1

No.16 S ++

2

4 ,

θ θ

θ

1( t ) [0, ∞ ) + −

( )

1 1 S

(1/2)

S

1

uu 1 1, + +

−

−

−

+ −− +− θ +

1 2

uu

u

1 2

u

2

1 2

2

11 1

Suu 1 =+−− +− θ

1 2

uu

2

11 1

1 2

Suu 1 4 =+−− +− θ + θ

1 2

uu

1 2

Note: The symbol “—” denotes that the generator function is not available; Φ − 1 is the inverse standard normal distribution function; and Φ θ is the bivariate standard normal distribution
function with Pearson linear correlation coefficient θ .