def stat_function(x):
assert x.ndim == 1
#vals = sd_opt_problem.smoother1(
#x[np.newaxis,:]-ygrid[:,np.newaxis]).mean(axis=1)
vals = EmpiricalCDF(x)(ygrid)
return vals
def plot_truncated_lognormal_example_exact_quantities(num_samples=int(1e5),
plot=False,
mu=0,
sigma=1):
if plot:
assert num_samples <= 1e5
num_vars = 1.
lb, ub = -1, 3
f, f_cdf, f_pdf, VaR, CVaR, ssd, ssd_disutil = \
get_truncated_lognormal_example_exact_quantities(lb, ub, mu, sigma)
# lb,ub passed to truncnorm_rv are defined for standard normal.
# Adjust for mu and sigma using
alpha, beta = (lb - mu) / sigma, (ub - mu) / sigma
samples = truncnorm_rv.rvs(alpha, beta, mu, sigma,
size=num_samples)[np.newaxis, :]
values = f(samples)[:, 0]
fig, axs = plt.subplots(1, 6, sharey=False, figsize=(16, 6))
from pyapprox.density import EmpiricalCDF
ygrid = np.linspace(np.exp(lb) - 1, np.exp(ub) * 1.1, 100)
ecdf = EmpiricalCDF(values)
if plot:
axs[0].plot(ygrid, ecdf(ygrid), '-')
axs[0].plot(ygrid, f_cdf(ygrid), '--')
axs[0].set_xlim(ygrid.min(), ygrid.max())
axs[0].set_title('CDF')
if plot:
ygrid = np.linspace(np.exp(lb) - 1, np.exp(ub) * 1.1, 100)
axs[1].hist(values, bins='auto', density=True)
axs[1].plot(ygrid, f_pdf(ygrid), '--')
axs[1].set_xlim(ygrid.min(), ygrid.max())
axs[1].set_title('PDF')
pgrid = np.linspace(0.01, 1 - 1e-2, 10)
evar = np.array([value_at_risk(values, p)[0] for p in pgrid]).squeeze()
if plot:
axs[2].plot(pgrid, evar, '-')
axs[2].plot(pgrid, VaR(pgrid), '--')
axs[2].set_title('VaR')
else:
assert np.allclose(evar, VaR(pgrid), rtol=2e-2)
pgrid = np.linspace(0, 1 - 1e-2, 100)
ecvar = np.array([conditional_value_at_risk(values, p) for p in pgrid])
# CVaR for alpha=0 should be the mean
assert np.allclose(ecvar[0], values.mean())
if plot:
axs[3].plot(pgrid, ecvar, '-')
axs[3].plot(pgrid, CVaR(pgrid), '--')
axs[3].set_xlim(pgrid.min(), pgrid.max())
axs[3].set_title('CVaR')
else:
assert np.allclose(ecvar.squeeze(), CVaR(pgrid).squeeze(), rtol=1e-2)
ygrid = np.linspace(np.exp(lb) - 10, np.exp(ub) + 1, 100)
essd = compute_conditional_expectations(ygrid, values, False)
if plot:
axs[4].plot(ygrid, essd, '-')
axs[4].plot(ygrid, ssd(ygrid), '--')
axs[4].set_xlim(ygrid.min(), ygrid.max())
axs[4].set_title(r'$E[(\eta-Y)^+]$')
axs[5].set_xlabel(r'$\eta$')
else:
assert np.allclose(essd.squeeze(), ssd(ygrid), rtol=2e-2)
# zoom into ygrid over high probability region of -Y
ygrid = -ygrid[::-1]
disutil_essd = compute_conditional_expectations(ygrid, values, True)
assert np.allclose(disutil_essd,
compute_conditional_expectations(ygrid, -values, False))
# print(np.linalg.norm(disutil_essd.squeeze()-ssd_disutil(ygrid),ord=np.inf))
if plot:
axs[5].plot(ygrid, disutil_essd, '-', label='Empirical')
axs[5].plot(ygrid, ssd_disutil(ygrid), '--', label='Exact')
axs[5].set_xlim(ygrid.min(), ygrid.max())
axs[5].set_title(r'$E[(\eta-(-Y))^+]$')
axs[5].set_xlabel(r'$\eta$')
axs[5].legend()
plt.show()
def help_test_stochastic_dominance(self,
solver,
nsamples,
degree,
disutility=None,
plot=False):
"""
disutilty is none plot emprical CDF
disutility is True plot disutility SSD
disutility is False plot standard SSD
"""
from pyapprox.multivariate_polynomials import PolynomialChaosExpansion
from pyapprox.variable_transformations import \
define_iid_random_variable_transformation
from pyapprox.indexing import compute_hyperbolic_indices
num_vars = 1
mu, sigma = 0, 1
f, f_cdf, f_pdf, VaR, CVaR, ssd, ssd_disutil = \
get_lognormal_example_exact_quantities(mu,sigma)
samples = np.random.normal(0, 1, (1, nsamples))
samples = np.sort(samples)
values = f(samples[0, :])[:, np.newaxis]
pce = PolynomialChaosExpansion()
var_trans = define_iid_random_variable_transformation(
normal_rv(mu, sigma), num_vars)
pce.configure({'poly_type': 'hermite', 'var_trans': var_trans})
indices = compute_hyperbolic_indices(1, degree, 1.)
pce.set_indices(indices)
eta_indices = None
#eta_indices=np.argsort(values[:,0])[nsamples//2:]
coef, sd_opt_problem = solver(samples,
values,
pce.basis_matrix,
eta_indices=eta_indices)
pce.set_coefficients(coef[:, np.newaxis])
pce_values = pce(samples)[:, 0]
ygrid = pce_values.copy()
if disutility is not None:
if disutility:
ygrid = -ygrid[::-1]
stat_function = partial(compute_conditional_expectations,
ygrid,
disutility_formulation=disutility)
if disutility:
# Disutility SSD
eps = 1e-14
assert np.all(
stat_function(values[:, 0]) <= stat_function(pce_values) +
eps)
else:
# SSD
assert np.all(
stat_function(pce_values) <= stat_function(values[:, 0]))
else:
# FSD
from pyapprox.density import EmpiricalCDF
stat_function = lambda x: EmpiricalCDF(x)(ygrid)
assert np.all(
stat_function(pce_values) <= stat_function(values[:, 0]))
if plot:
lstsq_pce = PolynomialChaosExpansion()
lstsq_pce.configure({
'poly_type': 'hermite',
'var_trans': var_trans
})
lstsq_pce.set_indices(indices)
lstsq_coef = solve_least_squares_regression(
samples, values, lstsq_pce.basis_matrix)
lstsq_pce.set_coefficients(lstsq_coef)
#axs[1].plot(ygrid,stat_function(values[:,0]),'ko',ms=12)
#axs[1].plot(ygrid,stat_function(pce_values),'rs')
#axs[1].plot(ygrid,stat_function(lstsq_pce(samples)[:,0]),'b*')
ylb,yub = values.min()-abs(values.max())*.1,\
values.max()+abs(values.max())*.1
ygrid = np.linspace(ylb, yub, 101)
ygrid = np.sort(np.concatenate([ygrid, pce_values]))
if disutility is not None:
if disutility:
ygrid = -ygrid[::-1]
stat_function = partial(compute_conditional_expectations,
ygrid,
disutility_formulation=disutility)
else:
print('here')
print(ygrid)
def stat_function(x):
assert x.ndim == 1
#vals = sd_opt_problem.smoother1(
#x[np.newaxis,:]-ygrid[:,np.newaxis]).mean(axis=1)
vals = EmpiricalCDF(x)(ygrid)
return vals
fig, axs = plot_1d_functions_and_statistics(
[f, pce, lstsq_pce], ['Exact', 'SSD', 'Lstsq'], samples,
values, stat_function, ygrid)
plt.show()