Risk Calc code

FROM SCOTT, FOUND IN JACK'S VERSION OF MANUSCRIPT

Y = X + e

X ^ e

Y ~ N(20, 7)

e ~ N(0, 5) _________ _____

X ~ N(E(Y) - E(e), Ö V(Y) -V(e)) ~ N(20 - 0, Ö7² - 5²) » N(20, 4.9)

/////////////////////////////////////////////////////////////////////////////////////////////////////////////

func E() return mean($1)

func V() return variance($1)

// independence, normal

Y ~ N(20, 7)

e ~ N(0, 5)

clear; show Y; show e in darkgray

X ~ N(E(Y) - E(e), sqrt(V(Y) - V(e)))

clear; show X; show Y in gray

Xindependent = X

// perfect dependence, normal

Y ~ N(20, 7)

e ~ N(0, 5)

clear; show Y; show e in darkgray

X ~ N(E(Y) - E(e), sqrt(V(Y)) - sqrt(V(e)))

clear; show X; show Y in gray

a = convolve(X,e,perfect)

clear; show a in red; show Y; show X;

Y

~normal(range=[1.96919,38.0308], mean=20, var=49)

a

~(range=[1.96919,38.0308], mean=20, var=[44.7,49])

Xperfect = X

// specified correlation, normal

Y ~ N(20, 7)

e ~ N(0, 5)

r = 0.5

clear; show Y; show e in darkgray

X ~ N(E(Y) - E(e), sqrt(V(e)*(r^2-1)+V(Y)) - r*sqrt(V(e)))

clear; show X; show Y in gray

r = 1; X ~ N(E(Y) - E(e), sqrt(V(e)*(r^2-1)+V(Y)) - r*sqrt(V(e)))

r = .9;X ~ N(E(Y) - E(e), sqrt(V(e)*(r^2-1)+V(Y)) - r*sqrt(V(e)))

r = .8;X ~ N(E(Y) - E(e), sqrt(V(e)*(r^2-1)+V(Y)) - r*sqrt(V(e)))

r = .7;X ~ N(E(Y) - E(e), sqrt(V(e)*(r^2-1)+V(Y)) - r*sqrt(V(e)))

r = .6;X ~ N(E(Y) - E(e), sqrt(V(e)*(r^2-1)+V(Y)) - r*sqrt(V(e)))

r = .5;X ~ N(E(Y) - E(e), sqrt(V(e)*(r^2-1)+V(Y)) - r*sqrt(V(e)))

r = .4;X ~ N(E(Y) - E(e), sqrt(V(e)*(r^2-1)+V(Y)) - r*sqrt(V(e)))

r = .3;X ~ N(E(Y) - E(e), sqrt(V(e)*(r^2-1)+V(Y)) - r*sqrt(V(e)))

r = .2;X ~ N(E(Y) - E(e), sqrt(V(e)*(r^2-1)+V(Y)) - r*sqrt(V(e)))

r = .1;X ~ N(E(Y) - E(e), sqrt(V(e)*(r^2-1)+V(Y)) - r*sqrt(V(e)))

r = 0;X ~ N(E(Y) - E(e), sqrt(V(e)*(r^2-1)+V(Y)) - r*sqrt(V(e)))

r = -.1;X ~ N(E(Y) - E(e), sqrt(V(e)*(r^2-1)+V(Y)) - r*sqrt(V(e)))

r = -.2;X ~ N(E(Y) - E(e), sqrt(V(e)*(r^2-1)+V(Y)) - r*sqrt(V(e)))

r = -.3;X ~ N(E(Y) - E(e), sqrt(V(e)*(r^2-1)+V(Y)) - r*sqrt(V(e)))

r = -.4;X ~ N(E(Y) - E(e), sqrt(V(e)*(r^2-1)+V(Y)) - r*sqrt(V(e)))

r = -.5;X ~ N(E(Y) - E(e), sqrt(V(e)*(r^2-1)+V(Y)) - r*sqrt(V(e)))

r = -.6;X ~ N(E(Y) - E(e), sqrt(V(e)*(r^2-1)+V(Y)) - r*sqrt(V(e)))

r = -.7;X ~ N(E(Y) - E(e), sqrt(V(e)*(r^2-1)+V(Y)) - r*sqrt(V(e)))

r = -.8;X ~ N(E(Y) - E(e), sqrt(V(e)*(r^2-1)+V(Y)) - r*sqrt(V(e)))

r = -.9;X ~ N(E(Y) - E(e), sqrt(V(e)*(r^2-1)+V(Y)) - r*sqrt(V(e)))

r = -1;X ~ N(E(Y) - E(e), sqrt(V(e)*(r^2-1)+V(Y)) - r*sqrt(V(e)))

r = 0;X ~ N(E(Y) - E(e), sqrt(V(e)*(r^2-1)+V(Y)) - r*sqrt(V(e)))

clear; show X; show Xindependent in red

r = 1; X ~ N(E(Y) - E(e), sqrt(V(e)*(r^2-1)+V(Y)) - r*sqrt(V(e)))

clear; show X; show Xperfect in red

// unknown correlation, normal

Y ~ N(20, 7)

e ~ N(0, 5)

X ~ N(E(Y) - E(e), [sd(Y) - sd(e),sd(Y) + sd(e)])

clear; show X; show Y in gray

r = -1;Xopposite ~ N(E(Y) - E(e), sqrt(V(e)*(r^2-1)+V(Y)) - r*sqrt(V(e)))

show Xperfect,Xopposite in red

// Frechet, normal marginals

Y ~ N(20, 7)

e ~ N(0, 5)

X ~ Y - e

clear; show X; show Y in gray

r = -1;Xopposite ~ N(E(Y) - E(e), sqrt(V(e)*(r^2-1)+V(Y)) - r*sqrt(V(e)))

show Xperfect,Xopposite in red

/////////////////////////////////////////////////////////////////////////////////////////////////////////////////

// distributional p-boxes (i.e., normals with interval parameters)

// these formulations may not be quite correct (they may need backcalc formulas)

Y = N( [19,21], [6.5,7.5])

e = N( [-0.1,0.1], [4.8,5.2])

clear; show Y; show e in darkgray

// independence, normal

X ~ N(E(Y) - E(e), sqrt(V(Y) - V(e)))

clear; show X; show Y in gray

Xindependent = X

// perfect dependence, normal

X ~ N(E(Y) - E(e), sqrt(V(Y)) - sqrt(V(e)))

a = convolve(X,e,perfect)

clear; show a in red; show Y; show X;

Y

~normal(range=[-0.31872,40.3187], mean=[19,21], var=[42.2,56.3])

a

~(range=[-1.54905,41.5491], mean=[18.79,21.2], var=[24,63])

Xperfect = X

// specified correlation, normal

r = 0.5

X ~ N(E(Y) - E(e), sqrt(V(e)*(r^2-1)+V(Y)) - r*sqrt(V(e)))

clear; show X; show Y in gray

r = 1; X ~ N(E(Y) - E(e), sqrt(V(e)*(r^2-1)+V(Y)) - r*sqrt(V(e)))

r = .9;X ~ N(E(Y) - E(e), sqrt(V(e)*(r^2-1)+V(Y)) - r*sqrt(V(e)))

r = .8;X ~ N(E(Y) - E(e), sqrt(V(e)*(r^2-1)+V(Y)) - r*sqrt(V(e)))

r = .7;X ~ N(E(Y) - E(e), sqrt(V(e)*(r^2-1)+V(Y)) - r*sqrt(V(e)))

r = .6;X ~ N(E(Y) - E(e), sqrt(V(e)*(r^2-1)+V(Y)) - r*sqrt(V(e)))

r = .5;X ~ N(E(Y) - E(e), sqrt(V(e)*(r^2-1)+V(Y)) - r*sqrt(V(e)))

r = .4;X ~ N(E(Y) - E(e), sqrt(V(e)*(r^2-1)+V(Y)) - r*sqrt(V(e)))

r = .3;X ~ N(E(Y) - E(e), sqrt(V(e)*(r^2-1)+V(Y)) - r*sqrt(V(e)))

r = .2;X ~ N(E(Y) - E(e), sqrt(V(e)*(r^2-1)+V(Y)) - r*sqrt(V(e)))

r = .1;X ~ N(E(Y) - E(e), sqrt(V(e)*(r^2-1)+V(Y)) - r*sqrt(V(e)))

r = 0;X ~ N(E(Y) - E(e), sqrt(V(e)*(r^2-1)+V(Y)) - r*sqrt(V(e)))

r = -.1;X ~ N(E(Y) - E(e), sqrt(V(e)*(r^2-1)+V(Y)) - r*sqrt(V(e)))

r = -.2;X ~ N(E(Y) - E(e), sqrt(V(e)*(r^2-1)+V(Y)) - r*sqrt(V(e)))

r = -.3;X ~ N(E(Y) - E(e), sqrt(V(e)*(r^2-1)+V(Y)) - r*sqrt(V(e)))

r = -.4;X ~ N(E(Y) - E(e), sqrt(V(e)*(r^2-1)+V(Y)) - r*sqrt(V(e)))

r = -.5;X ~ N(E(Y) - E(e), sqrt(V(e)*(r^2-1)+V(Y)) - r*sqrt(V(e)))

r = -.6;X ~ N(E(Y) - E(e), sqrt(V(e)*(r^2-1)+V(Y)) - r*sqrt(V(e)))

r = -.7;X ~ N(E(Y) - E(e), sqrt(V(e)*(r^2-1)+V(Y)) - r*sqrt(V(e)))

r = -.8;X ~ N(E(Y) - E(e), sqrt(V(e)*(r^2-1)+V(Y)) - r*sqrt(V(e)))

r = -.9;X ~ N(E(Y) - E(e), sqrt(V(e)*(r^2-1)+V(Y)) - r*sqrt(V(e)))

r = -1;X ~ N(E(Y) - E(e), sqrt(V(e)*(r^2-1)+V(Y)) - r*sqrt(V(e)))

r = 0;X ~ N(E(Y) - E(e), sqrt(V(e)*(r^2-1)+V(Y)) - r*sqrt(V(e)))

clear; show X; show Xindependent in red

r = 1; X ~ N(E(Y) - E(e), sqrt(V(e)*(r^2-1)+V(Y)) - r*sqrt(V(e)))

clear; show X; show Xperfect in red

// unknown correlation, normal

X ~ N(E(Y) - E(e), [sd(Y) - sd(e),sd(Y) + sd(e)])

clear; show X; show Y in gray

r = -1;Xopposite ~ N(E(Y) - E(e), sqrt(V(e)*(r^2-1)+V(Y)) - r*sqrt(V(e)))

show Xperfect,Xopposite in red

// Frechet, normal marginals

X ~ Y - e

clear; show X; show Y in gray

r = -1;Xopposite ~ N(E(Y) - E(e), sqrt(V(e)*(r^2-1)+V(Y)) - r*sqrt(V(e)))

show Xperfect,Xopposite in red

/////////////////////////////////////////////////////////////////////////////////////////////////////////////////

// general p-boxes (i.e., don’t have to be normal)

// p-boxes

y = N( [18,22], [6,9])

e = N( [-1,1], [4,5])

x = N( decon(mean(e), mean(y)), sqrt( decon(var(e),var(y))))

clear; show y; show x in blue

Y = x |+| e

clear; show y; show Y in red

x = N( mean(y) - mean(e), sqrt( var(y) - var(e)))

clear; show y; show x in blue

Y = x |+| e

clear; show y; show Y in red

70/4

17.5

// big epistemic uncertainty

clear; show Y; show e in darkgray

Y = N( 20, 10)

e = N( [-3,3], [4,6])

Y = N( [15,25], [7,11])

e = N( 0, 5)

Y = N( [15,25], [7,11])

e = N( [-3,3], [4,6])

Y = N( [15,25], [7,11])

e = N( [-5,5], [ 0, 5] )

e = [-5,5]

// independence, normal

X ~ N(E(Y) - E(e), sqrt(V(Y) - V(e)))

X ~ N(decon(E(e), E(Y)), sqrt(decon(V(e),V(Y))))

clear; show X; show Y in gray

a = X |+| e

clear; show a in red; show Y; show X;

Xindependent = X

// perfect dependence, normal

X ~ N(E(Y) - E(e), sqrt(V(Y)) - sqrt(V(e)))

X ~ N(decon(E(e),E(Y)), decon(sqrt(V(e)),sqrt(V(Y))))

a = convolve(X,e,perfect)

clear; show a in red; show Y; show X;

Y

~normal(range=[-0.31872,40.3187], mean=[19,21], var=[42.2,56.3])

a

~(range=[-0.31872,40.3187], mean=[19,21], var=[29.5,56.3])

Xperfect = X