def isThresholdContr(N, M, p, c, l, myRho, nu, cm=0):
myShift = (1 - 2 * cm)
xhat = scipy.optimize.minimize(meanShiftOF,
0.01,
args=(c, p, l, myRho),
method='SLSQP',
jac=None)
mu = xhat.x
theta = np.zeros(M)
cgf = np.zeros(M)
qZ = np.zeros([M, N])
G = np.transpose(np.tile(np.random.normal(mu, 1, M), (N, 1)))
e = np.random.normal(0, 1, [M, N])
W = np.random.chisquare(nu, M)
myV = W / myShift
V = np.transpose(np.sqrt(np.tile(myV, (N, 1)) / nu))
num = V * myT.ppf(p, nu) * np.ones((M, 1)) - np.multiply(np.sqrt(myRho), G)
pZ = norm.cdf(np.divide(num, np.sqrt(1 - myRho)))
for n in range(0, M):
theta[n] = vc.getSaddlePoint(pZ[n, :], c, l, 0.0)
qZ[n, :] = getQ(theta[n], c, pZ[n, :])
cgf[n] = vc.computeCGF(theta[n], pZ[n, :], c)
I = np.transpose(1 * np.less(e, norm.ppf(qZ)))
L = np.dot(c, I)
rnChi = np.exp(-cm * myV - (nu / 2) * np.log(myShift))
rnMu = np.exp(-mu * G[:, 0] + 0.5 * mu**2)
rnTwist = computeRND(theta, L, cgf)
rn = rnChi * rnMu * rnTwist
return I, theta, pZ, qZ, cgf, rn
def isThresholdT(N, M, p, c, l, myRho, nu, cm=0):
myShift = (1 - 2 * cm)
mu = getOptimalMeanShift(c, p, l, myRho)
W = np.random.chisquare(nu, M)
myV = W / myShift
theta = np.zeros(M)
cgf = np.zeros(M)
qZ = np.zeros([M, N])
V = np.transpose(np.sqrt(np.tile(myV, (N, 1)) / nu))
e = np.random.normal(0, 1, [M, N])
G = np.transpose(np.tile(np.random.normal(mu, 1, M), (N, 1)))
num = V * (myT.ppf(p, nu) * np.ones((M, 1))) - np.sqrt(myRho) * G
pZ = norm.cdf(np.divide(num, np.sqrt(1 - myRho)))
for n in range(0, M):
theta[n] = vc.getSaddlePoint(pZ[n, :], c, l, 0.0)
qZ[n, :] = getQ(theta[n], c, pZ[n, :])
cgf[n] = vc.computeCGF(theta[n], pZ[n, :], c)
I = np.transpose(1 * np.less(e, norm.ppf(qZ)))
L = np.dot(c, I)
rnChi = np.exp(-cm * myV - (nu / 2) * np.log(myShift))
rnMu = np.exp(-mu * G[:, 0] + 0.5 * (mu**2))
rnTwist = computeRND(theta, L, cgf)
rn = rnChi * rnMu * rnTwist
tailProb = np.mean(np.multiply(L > l, rn))
eShortfall = np.mean(np.multiply(L * (L > l), rn)) / tailProb
return tailProb, eShortfall
def isIndDefault(N, M, p, c, l):
U = np.random.uniform(0, 1, [M, N])
theta = vc.getSaddlePoint(p, c, l, 0.0)
qZ = getQ(theta, c, p)
cgf = vc.computeCGF(theta, p, c)
I = np.transpose(1 * np.less(U, qZ))
L = np.dot(c, I)
rn = computeRND(theta, L, cgf)
tailProb = np.mean(np.multiply(L > l, rn))
eShortfall = np.mean(np.multiply(L * (L > l), rn)) / tailProb
return tailProb, eShortfall
def isMixture(N, M, p, c, l, p1, p2):
theta = np.zeros(M)
cgf = np.zeros(M)
qS = np.zeros([M, N])
S = np.random.gamma(p1, 1 / p1, [M])
wS = np.transpose(np.tile(1 - p2 + p2 * S, [N, 1]))
pS = np.tile(p, [M, 1]) * wS
for n in range(0, M):
theta[n] = vc.getSaddlePoint(pS[n, :], c, l, -0.2)
qS[n, :] = getQ(theta[n], c, pS[n, :])
cgf[n] = vc.computeCGF(theta[n], pS[n, :], c)
I = np.transpose(1 * np.greater_equal(np.random.poisson(qS, [M, N]), 1))
L = np.dot(c, I)
rn = computeRND(theta, L, cgf)
tailProb = np.mean(np.multiply(L > l, rn))
eShortfall = np.mean(np.multiply(L * (L > l), rn)) / tailProb
return tailProb, eShortfall
def isThreshold(N, M, p, c, l, myRho, nu, shiftMean, isT, invVector=0):
mu = 0.0
gamma = 0.0
if shiftMean == 1:
mu = getOptimalMeanShift(c, p, l, myRho)
theta = np.zeros(M)
cgf = np.zeros(M)
qZ = np.zeros([M, N])
G = np.transpose(np.tile(np.random.normal(mu, 1, M), (N, 1)))
e = np.random.normal(0, 1, [M, N])
if isT == 1:
gamma = -2
W = np.random.chisquare(nu, M)
myV = W / (1 - 2 * gamma)
V = np.transpose(np.sqrt(np.tile(myV, (N, 1)) / nu))
num = (1 / V) * myT.ppf(p, nu) * np.ones(
(M, 1)) - np.multiply(np.sqrt(myRho), G)
pZ = norm.cdf(np.divide(num, np.sqrt(1 - myRho)))
elif isT == 2:
V = np.transpose(
np.sqrt(np.tile(np.random.gamma(nu, 1 / nu, M), (N, 1))))
num = (1 / V) * invVector * np.ones(
(M, 1)) - np.multiply(np.sqrt(myRho), G)
pZ = norm.cdf(np.divide(num, np.sqrt(1 - myRho)))
else:
pZ = th.computeP(p, myRho, G)
for n in range(0, M):
theta[n] = vc.getSaddlePoint(pZ[n, :], c, l, 0.0)
qZ[n, :] = getQ(theta[n], c, pZ[n, :])
cgf[n] = vc.computeCGF(theta[n], pZ[n, :], c)
I = np.transpose(1 * np.less(e, norm.ppf(qZ)))
L = np.dot(c, I)
if isT == 1:
rnChi = np.exp(-gamma * myV - (nu / 2) * np.log(1 - 2 * gamma))
else:
rnChi = np.ones(M)
if shiftMean == 1:
rn = computeRND(theta, L, cgf) * np.exp(-mu * G[:, 0] + 0.5 *
(mu**2)) * rnChi
else:
rn = computeRND(theta, L, cgf) * rnChi
tailProb = np.mean(np.multiply(L > l, rn))
eShortfall = np.mean(np.multiply(L * (L > l), rn)) / tailProb
return tailProb, eShortfall
def isThresholdSimple(N, M, p, c, l, myRho):
mu = getOptimalMeanShift(c, p, l, myRho)
theta = np.zeros(M)
cgf = np.zeros(M)
qZ = np.zeros([M, N])
e = np.random.normal(0, 1, [M, N])
G = np.transpose(np.tile(np.random.normal(mu, 1, M), (N, 1)))
num = (norm.ppf(p) * np.ones((M, 1))) - np.sqrt(myRho) * G
pZ = norm.cdf(np.divide(num, np.sqrt(1 - myRho)))
for n in range(0, M):
theta[n] = vc.getSaddlePoint(pZ[n, :], c, l, 0.0)
qZ[n, :] = getQ(theta[n], c, pZ[n, :])
cgf[n] = vc.computeCGF(theta[n], pZ[n, :], c)
I = np.transpose(1 * np.less(e, norm.ppf(qZ)))
L = np.dot(c, I)
rn = np.exp(-mu * G[:, 0] + 0.5 * (mu**2)) * computeRND(theta, L, cgf)
tailProb = np.mean(np.multiply(L > l, rn))
eShortfall = np.mean(np.multiply(L * (L > l), rn)) / tailProb
return tailProb, eShortfall
def meanShiftOF(mu, c, p, l, myRho):
pZ = th.computeP(p, myRho, mu)
theta = vc.getSaddlePoint(pZ, c, l, 0.0)
f_l = -theta * l + vc.computeCGF(theta, pZ, c)
return -(f_l - 0.5 * np.dot(mu, mu))
