std_X2 = np.sqrt(mu_X2 - np.log(mod_log_X2)) # std of the associated Gaussian
# %% definition of the random variables
n = 2 # number of random variables (dimensions)
# assign data: 1st variable is Lognormal
dist_x1 = ERADist('lognormal', 'PAR', [mu_X1, std_X1])
# assign data: 2nd variable is Lognormal
dist_x2 = ERADist('lognormal', 'PAR', [mu_X2, std_X2])
# distributions
dist_X = [dist_x1, dist_x2]
# correlation matrix
R = np.eye(n) # independent case
# object with distribution information
T_nataf = ERANataf(dist_X, R)
# %% likelihood function
lam = np.array([1, 1]) # means of the prediction error
i = 9 # simulation level
var_eps = 0.5**(i - 1) # variance of the prediction error
f_tilde = np.array([3.13, 9.83]) # measured eigenfrequencies [Hz]
# shear building model
f = lambda x: shear_building_2DOF(m1, m2, kn * x[0], kn * x[1])
# modal measure-of-fit function
J = lambda x: np.sum((lam**2) * (((f(x)**2) / f_tilde**2) - 1)**2)
# likelihood function
likelihood = lambda x: np.exp(-J(x) / (2 * var_eps))
Papaioannou et al.
Probabilistic Engineering Mechanics 41 (2015) 83-103.
---------------------------------------------------------------------------
"""
# %% definition of the random variables
d = 100 # number of dimensions
pi_pdf = list()
for i in range(d):
pi_pdf.append(ERADist('exponential', 'PAR', 1)) # n independent rv
# correlation matrix
R = np.eye(d) # independent case
# object with distribution information
pi_pdf = ERANataf(pi_pdf, R) # if you want to include dependence
# %% limit state function and its gradient in the original space
Ca = 140
g = lambda x: Ca - np.sum(x)
dg = lambda x: np.tile([-1], [d, 1])
# %% Solve the optimization problem of the First Order Reliability Method
# OPC 1. FORM using Hasofer-Lind-Rackwitz-Fiessler algorithm HLRF (Ref.1 Pag.128)
[u_star_hlrf, x_star_hlrf, beta_hlrf, Pf_hlrf] = FORM_HLRF(g, dg, pi_pdf)
# OPC 2. FORM using Python scipy.optimize.minimize()
[u_star_fmc, x_star_fmc, beta_fmc, Pf_fmc] = FORM_fmincon(g, pi_pdf)
# exact solution
"""
# %% definition of the random variables
n = 12 # number of random variables (dimensions)
# assign data: 1st variable is Lognormal
dist_x = ERADist('standardnormal', 'PAR', [0, 1])
# distributions
dist_X = list()
for i in range(n):
dist_X.append(dist_x)
# correlation matrix
R = np.eye(n) # independent case
# object with distribution information
T_nataf = ERANataf(dist_X, R)
# %% likelihood function
mu_l = 0.462
sigma_l = 0.6
likelihood = lambda u: np.prod(sp.stats.norm.pdf(
(u - mu_l) / sigma_l), axis=0) / sigma_l
realmin = np.finfo(np.double).tiny # realmin to avoid Inf values in log(0)
log_likelihood = lambda u: np.log(likelihood(u) + realmin)
# %% aBUS-SuS
N = 2000 # number of samples per level
p0 = 0.1 # probability of each subset
# run the BUS_SuS.m function
[h, samplesU, samplesX, cE, c, lam_new] = aBUS_SuS(N, p0, log_likelihood,
Rho = np.array([[1.0, 0.5, 0.5], [0.5, 1.0, 0.5], [0.5, 0.5, 1.0]])
elif (example == 2):
M.append(ERADist('rayleigh', 'PAR', 1))
M.append(ERADist('gumbel', 'MOM', [0, 1]))
Rho = np.array([[1.0, 0.6], [0.6, 1.0]])
elif (example == 3):
M.append(ERADist('gamma', 'PAR', [2, 1]))
M.append(ERADist('chisquare', 'PAR', 5))
Rho = np.array([[1.0, 0.5], [0.5, 1.0]])
elif (example == 4):
M.append(ERADist('gamma', 'PAR', [1, 2]))
M.append(ERADist('weibull', 'PAR', [4, 5]))
Rho = np.array([[1.0, 0.1], [0.1, 1.0]])
# applying Nataf transformation
T_Nataf = ERANataf(M, Rho)
# generation of of random samples from the joint distribution
N = 1000
X = T_Nataf.random(N)
# samples in standard space
U = T_Nataf.X2U(X)
# samples in physical space
X = T_Nataf.U2X(U)
n = len(M)
xx = np.zeros(shape=[n, N])
f_X = np.zeros(shape=[n, N])
F_X = np.zeros(shape=[n, N])