def test_discrete_chebyshev(self):
num_coef = 5
nmasses = 10
xk = np.array(range(nmasses), dtype='float')
pk = np.ones(nmasses) / nmasses
ab_lanczos = lanczos(xk, pk, num_coef)
ab_stieltjes = stieltjes(xk, pk, num_coef)
ab_exact = discrete_chebyshev_recurrence(num_coef, nmasses)
# ab_lanczos[-1, 0] is a dummy entry so set to exact so
# comparison will pass if all other entries are correct
ab_lanczos[-1, 0] = ab_exact[-1, 0]
assert np.allclose(ab_lanczos, ab_exact)
assert np.allclose(ab_stieltjes, ab_exact)
from pyapprox.univariate_quadrature import gauss_quadrature
x, w = gauss_quadrature(ab_lanczos, num_coef)
moments = np.array([(x**ii).dot(w) for ii in range(num_coef)])
true_moments = np.array([(xk**ii).dot(pk) for ii in range(num_coef)])
assert np.allclose(moments, true_moments)
p = evaluate_orthonormal_polynomial_1d(x, num_coef - 1, ab_lanczos)
assert np.allclose((p.T * w).dot(p), np.eye(num_coef))
p = evaluate_orthonormal_polynomial_1d(xk, num_coef - 1, ab_lanczos)
assert np.allclose((p.T * pk).dot(p), np.eye(num_coef))
def test_predictor_corrector_known_scipy_pdf(self):
nterms = 5
quad_options = {
'nquad_samples': 10,
'atol': 1e-8,
'rtol': 1e-8,
'max_steps': 10000,
'verbose': 1
}
rv = stats.beta(1, 1, -1, 2)
ab = predictor_corrector_known_scipy_pdf(nterms, rv, quad_options)
true_ab = jacobi_recurrence(nterms, 0, 0)
assert np.allclose(ab, true_ab)
rv = stats.norm()
ab = predictor_corrector_known_scipy_pdf(nterms, rv, quad_options)
true_ab = hermite_recurrence(nterms)
assert np.allclose(ab, true_ab)
# lognormal is a very hard test
rv = stats.lognorm(1)
# mean, std = 1e4, 7.5e3
# beta = std*np.sqrt(6)/np.pi
# mu = mean - beta*np.euler_gamma
# rv = stats.gumbel_r(loc=mu, scale=beta)
ab = predictor_corrector_known_scipy_pdf(nterms, rv, quad_options)
def integrand(x):
p = evaluate_orthonormal_polynomial_1d(x, nterms - 1, ab)
G = np.empty((x.shape[0], nterms**2))
kk = 0
for ii in range(nterms):
for jj in range(nterms):
G[:, kk] = p[:, ii] * p[:, jj]
kk += 1
return G * rv.pdf(x)[:, None]
lb, ub = rv.interval(1)
xx, __ = gauss_quadrature(ab, nterms)
interval_size = xx.max() - xx.min()
quad_opts = quad_options.copy()
del quad_opts['nquad_samples']
res = integrate_using_univariate_gauss_legendre_quadrature_unbounded(
integrand,
lb,
ub,
quad_options['nquad_samples'],
interval_size=interval_size,
**quad_opts)
res = np.reshape(res, (nterms, nterms), order='C')
print(np.absolute(res - np.eye(nterms)).max())
assert np.absolute(res - np.eye(nterms)).max() < 2e-4
def test_float_rv_discrete(self):
num_coef,nmasses = 5,10
#works for both lanczos and chebyshev algorithms
#xk = np.geomspace(1,512,num=nmasses)
#pk = np.ones(nmasses)/nmasses
#works only for chebyshev algorithms
pk = np.geomspace(1,512,num=nmasses)
pk /= pk.sum()
xk = np.arange(0,nmasses)
#ab = lanczos(xk,pk,num_coef)
ab = modified_chebyshev_orthonormal(
num_coef,[xk,pk],probability=True)
from pyapprox.univariate_quadrature import gauss_quadrature
x,w = gauss_quadrature(ab,num_coef)
moments = np.array([(x**ii).dot(w) for ii in range(num_coef)])
true_moments = np.array([(xk**ii).dot(pk)for ii in range(num_coef)])
assert np.allclose(moments,true_moments),(moments,true_moments)
p = evaluate_orthonormal_polynomial_1d(x, num_coef-1, ab)
assert np.allclose((p.T*w).dot(p),np.eye(num_coef))
p = evaluate_orthonormal_polynomial_1d(xk, num_coef-1, ab)
assert np.allclose((p.T*pk).dot(p),np.eye(num_coef))
def test_krawtchouk(self):
num_coef=6
ntrials = 10
p = 0.5
xk = np.array(range(ntrials+1),dtype='float')
pk = binom.pmf(xk, ntrials, p)
ab_lanzcos = lanczos(xk,pk,num_coef)
ab_stieltjes = stieltjes(xk,pk,num_coef)
ab_exact = krawtchouk_recurrence(num_coef, ntrials, p)
assert np.allclose(ab_lanzcos,ab_exact)
assert np.allclose(ab_stieltjes,ab_exact)
from pyapprox.univariate_quadrature import gauss_quadrature
x,w = gauss_quadrature(ab_lanzcos,num_coef)
moments = np.array([(x**ii).dot(w) for ii in range(num_coef)])
true_moments = np.array([(xk**ii).dot(pk)for ii in range(num_coef)])
assert np.allclose(moments,true_moments)
p = evaluate_orthonormal_polynomial_1d(x, num_coef-1, ab_lanzcos)
assert np.allclose((p.T*w).dot(p),np.eye(num_coef))
p = evaluate_orthonormal_polynomial_1d(xk, num_coef-1, ab_lanzcos)
assert np.allclose((p.T*pk).dot(p),np.eye(num_coef))