/usr/share/openturns/validation/ValidClaytonCopula.txt is in openturns-validation 1.5-7build2.
This file is owned by root:root, with mode 0o644.
The actual contents of the file can be viewed below.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 | > restart:
> F:=exp(-log(u1^(-theta)/(u1*u2)^(theta/(theta+1))+1-u1^(-theta))/theta
> );
(-theta)
u1 (-theta)
ln(------------------ + 1 - u1 )
/ theta \
|---------|
\theta + 1/
(u1 u2)
F := exp(- ---------------------------------------)
theta
> simplify(F);
/ 1 \
|- -----|
\ theta/
/ / theta \ \
| |- ---------| |
| (-theta) \ theta + 1/ (-theta)|
\u1 (u1 u2) + 1 - u1 /
> CDF_clayton:=(u^(-theta)+v^(-theta)-1)^(-1/theta);
/ 1 \
|- -----|
\ theta/
(-theta) (-theta)
CDF_clayton := (u + v - 1)
> PDF_clayton:=factor(diff(diff(CDF_clayton,u),v));
> du:=factor(diff(PDF_clayton, u));
> dv:=factor(diff(PDF_clayton, v));
> with(codegen,optimize,makeproc,cost,prep2trans):
> optimize(du);
> optimize(dv);
/ 1 \
|- -----|
\ theta/
(-theta) (-theta) (-theta)
PDF_clayton := (u + v - 1) v
(-theta) / (-theta) (-theta) 2
u (theta + 1) / (v (u + v - 1) u)
/
/ 1 \
|- -----|
\ theta/
(-theta) (-theta) (-theta) (-theta)
du := (u + v - 1) v u
(theta + 1)
(-theta) (-theta) (-theta)
(u theta - theta v + theta - v + 1)
/ 2 (-theta) (-theta) 3
/ (u (u + v - 1) v)
/
/ 1 \
|- -----|
\ theta/
(-theta) (-theta) (-theta) (-theta)
dv := - (u + v - 1) v u
(theta + 1)
(-theta) (-theta) (-theta)
(u theta - theta v - theta + u - 1)
/ 2 (-theta) (-theta) 3
/ (v (u + v - 1) u)
/
/ 1 \
|- -----|
(-theta) (-theta) \ theta/
t1 = u , t2 = v , t3 = t1 + t2 - 1, t5 = t3 ,
2 2
t13 = u , t16 = t3 , t22 =
t5 t2 t1 (theta + 1) (t1 theta - theta t2 + theta - t2 + 1)
-----------------------------------------------------------
t13 t16 t3 v
/ 1 \
|- -----|
(-theta) (-theta) \ theta/
t1 = u , t2 = v , t3 = t1 + t2 - 1, t5 = t3 ,
2 2
t13 = v , t16 = t3 , t23 =
t5 t2 t1 (theta + 1) (t1 theta - theta t2 - theta + t1 - 1)
- -----------------------------------------------------------
t13 t16 t3 u
> optimize(factor(diff(PDF_clayton, theta)));
/ 1 \
|- -----|
(-theta) (-theta) \ theta/
t1 = u , t2 = v , t3 = t1 + t2 - 1, t5 = t3 ,
2
t7 = ln(v), t8 = theta , t9 = t8 theta, t10 = t7 t9,
t13 = ln(u), t15 = t7 t8, t16 = t13 t8, t19 = ln(t3),
t20 = t19 t1, t21 = t19 t2, t22 = t8 t1, t25 = t8 t2, t29 =
-t10 t1 + theta t1 t13 + t15 + t16 + theta t2 t7 - t8 + t20
+ t21 + 2 t22 t13 + 2 t25 t7 + t20 theta, t33 = t13 t9, t38
= t21 theta + t10 t2 - t15 t1 + t33 t1 - t33 t2 - t16 t2
2
- t19 - t19 theta + t10 + t33 + t22 + t25, t43 = t3 ,
t5 t2 t1 (t29 + t38)
t51 = --------------------
t8 t43 t3 v u
> theta:=2.5;evalf(Int(Int((u-0.5)*(v-0.5)*PDF_clayton,u=0..1),v=0..1));
theta := 2.5
0.06183689859
> evalf(subs(theta=2.5,Int(Int((u-1/2)*(v-1/2)*PDF_clayton,u=0..1),v=0..
> 1)));
0.06183689859
> subs(u=0.2,v=0.2,PDF_clayton);
> subs(u=0.2,v=0.2,CDF_clayton);
3.387900732
0.1521173556
>
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