def idistinv_jacobi(u, n, alph, bet):
r"""
[x] = idistinv_jacobi(u, n, alph, bet)
Computes the inverse of the order-n induced primitive for the Jacobi
distribution with parameters alph and bet. Uses a bisection method in
conjunction with forward evaluation given by idist_jacobi.
"""
if np.isscalar(n):
n = np.array([n])
assert((np.all(u >= 0)) and (np.all(u <= 1)))
assert((alph > -1) and (bet > -1))
assert(np.all(n >= 0))
x = np.zeros(u.shape)
supp = [-1., 1.]
if n.shape[0] == 1:
primitive = partial(idist_jacobi, n=n, alph=alph, bet=bet)
# Need 2*n + K coefficients, where K is the size of the
# Markov-Stiltjies binning procedure
recursion_coeffs = jacobi_recurrence(2*n + 400, alph, bet)
# All functions that accept b assume they are receiving b
# but recusion_coeffs:,1=np.sqrt(b)
a = recursion_coeffs[:, 0:1]
b = recursion_coeffs[:, 1:]**2
x = idist_inverse(u, n, primitive, a, b, supp)
else:
nmax = n.max()
# [nn, ~, bin] = histcounts(n, -0.5:(nmax+0.5))
edges = np.arange(-0.5, nmax+0.5)
nn, bins = histcounts(n, edges)
recursion_coeffs = jacobi_recurrence(2*nmax + 400, alph, bet)
# All functions that accept b assume they are receiving b
# but recusion_coeffs:,1=np.sqrt(b)
a = recursion_coeffs[:, 0:1]
b = recursion_coeffs[:, 1:]**2
# need to use u_flat code to be be consistent with akils code
# inside idist_inverse. but if that code is correct final result
# will be the same
#u_flat = u.flatten(order='F')
for qq in range(0, nmax+1):
flags = bins == (qq+1)
#flat_flags = flags.flatten(order='F')
primitive = partial(idist_jacobi, n=qq, alph=alph, bet=bet)
# xtemp = idist_inverse(
# u_flat[flat_flags], qq, primitive, a, b, supp)
xtemp = idist_inverse(
u[flags], qq, primitive, a, b, supp)
x[flags] = xtemp
return x
def demo_idist_jacobi():
alph = -0.8
bet = np.sqrt(101)
#n = 8; M = 1001
n = 4
M = 5
x = np.cos(np.linspace(0, np.pi, M + 2))[:, np.newaxis]
x = x[1:-1]
F = idist_jacobi(x, n, alph, bet)
recursion_coeffs = jacobi_recurrence(n + 1, alph, bet, True)
#recursion_coeffs[:,1]=np.sqrt(recursion_coeffs[:,1])
polys = evaluate_orthonormal_polynomial_1d(x[:, 0], n, recursion_coeffs)
wt_function = (1 - x)**alph * (1 + x)**bet / (2**(alph + bet + 1) *
betafn(bet + 1, alph + 1))
f = polys[:, -1:]**2 * wt_function
fig, ax = plt.subplots(1, 1)
plt.plot(x, f)
ax.set_xlabel('$x$')
ax.set_ylabel('$p_n^2(x) \mathrm{d}\mu(x)$')
ax.set_xlim(-1, 1)
ax.set_ylim(0, 4)
fig, ax = plt.subplots(1, 1)
plt.plot(x, F)
ax.set_xlabel('$x$')
ax.set_ylabel('$F_n(x)$')
ax.set_xlim(-1, 1)
plt.show()
def gauss_jacobi_pts_wts_1D(num_samples, alpha_poly, beta_poly):
"""
Return Gauss Jacobi quadrature rule that exactly integrates polynomials
of num_samples 2*num_samples-1 with respect to the probabilty density
function of Beta random variables on [-1,1]
C*(1+x)^(beta_poly)*(1-x)^alpha_poly
where
C = 1/(2**(alpha_poly+beta_poly)*beta_fn(beta_poly+1,alpha_poly+1))
or equivalently
C*(1+x)**(alpha_stat-1)*(1-x)**(beta_stat-1)
where
C = 1/(2**(alpha_stat+beta_stat-2)*beta_fn(alpha_stat,beta_stat))
Parameters
----------
num_samples : integer
The number of samples in the quadrature rule
alpha_poly : float
The Jaocbi parameter alpha = beta_stat-1
beta_poly : float
The Jacobi parameter beta = alpha_stat-1
Returns
-------
x : np.ndarray(num_samples)
Quadrature samples
w : np.ndarray(num_samples)
Quadrature weights
"""
ab = jacobi_recurrence(num_samples,
alpha=alpha_poly,
beta=beta_poly,
probability=True)
return gauss_quadrature(ab, num_samples)
def get_jacobi_recursion_coefficients(poly_type, opts, num_coefs):
"""
Get the recursion coefficients of a Jacobi polynomial.
Parameters
----------
opts : dictionary
Dictionary with the following attributes
alpha_poly : float
The alpha parameter of the jacobi polynomial. Only used and required
if poly_type is not None
beta_poly : float
The beta parameter of the jacobi polynomial. Only used and required
if poly_type is not None
shapes : dictionary
Shape parameters of the Beta distribution. shapes['a'] is the
a parameter of the Beta distribution and shapes['a'] is the
b parameter of the Beta distribution.
The parameter of the Jacobi polynomial are determined using the
following relationships: alpha_poly = b-1, beta_poly = a-1.
This option is not required or ignored when poly_type is not None
Returns
-------
recursion_coeffs : np.ndarray (num_coefs, 2)
"""
if poly_type is not None:
alpha_poly, beta_poly = opts['alpha_poly'], opts['beta_poly']
else:
alpha_poly, beta_poly = opts['shapes']['b'] - \
1, opts['shapes']['a']-1
return jacobi_recurrence(
num_coefs, alpha=alpha_poly, beta=beta_poly, probability=True)
def idist_jacobi(x, n, alph, bet, M=10):
r"""
idist_jacobi -- Evaluation of induced distribution
F = idist_jacobi(x, n, alph, bet, {M = 10})
Evaluates the integral
F = \int_{-1}**x p_n**2(x) \dx{\mu(x)},
where mu is the (a,b) Jacobi polynomial measure, scaled to be a
probability distribution on [-1,1], and p_n is the corresponding
degree-n orthonormal polynomial.
This function evaluates this via a transformation, measure modification,
and Gauss quadrature, the ending Gauss quadrature has M points.
"""
assert ((alph > -1) and (bet > -1))
assert (np.all(np.abs(x) <= 1))
assert np.all(n >= 0)
if x.ndim == 2:
assert x.shape[1] == 1
x = x[:, 0]
A = int(np.floor(abs(alph))) # is an integer
Aa = alph - A
F = np.zeros((x.shape[0], 1))
mrs_centroid = medapprox_jacobi(alph, bet, n)
xreflect = x > mrs_centroid
if x[xreflect].shape[0] > 0:
F[xreflect] = 1 - idist_jacobi(-x[xreflect], n, bet, alph, M)
recursion_coeffs = jacobi_recurrence(n + 1, alph, bet, True)
# All functions that accept b assume they are receiving b
# but recusion_coeffs:,1=np.sqrt(b)
a = recursion_coeffs[:, 0]
b = recursion_coeffs[:, 1]**2
assert b[0] == 1 # To make it a probability measure
if n > 0:
# Zeros of p_n
xn = gauss_quadrature(recursion_coeffs, n)[0]
# This is the (inverse) n'th root of the leading coefficient square
# of p_n. We'll use it for scaling later
kn_factor = np.exp(-1. / n * np.sum(np.log(b)))
for xq in range(x.shape[0]):
if x[xq] == -1:
F[xq] = 0
continue
if xreflect[xq]:
continue
# Recurrence coefficients for quadrature rule
recursion_coeffs = jacobi_recurrence(2 * n + A + M + 1, 0, bet, True)
# All functions that accept b assume they are receiving b
# but recusion_coeffs:,1=np.sqrt(b)
a = recursion_coeffs[:, 0:1]
b = recursion_coeffs[:, 1:]**2
assert b[0] == 1 # To make it a probability measure
if n > 0:
# Transformed
un = (2. / (x[xq] + 1.)) * (xn + 1) - 1
# Keep this so that bet(1) always equals what it did before
logfactor = 0
# Successive quadratic measure modifications
for j in range(n):
a, b = quadratic_modification_C(a, b, un[j])
logfactor += np.log(b[0] * ((x[xq] + 1) / 2)**2 * kn_factor)
b[0] = 1
# Linear modification by factors (2 - 1/2*(u+1)*(x+1)),
# having root u = (3-x)/(1+x)
root = (3. - x[xq]) / (1. + x[xq])
for aq in range(A):
[a, b] = linear_modification(a, b, root)
logfactor += np.log(b[0] * 1 / 2 * (x[xq] + 1))
b[0] = 1
# M-point Gauss quadrature for evaluation of auxilliary integral I
# gauss quadrature requires np.sqrt(b)
u, w = gauss_quadrature(np.hstack((a, np.sqrt(b))), M)
I = np.dot(w.T, (2. - 1. / 2. * (u + 1.) * (x[xq] + 1))**Aa)
F[xq] = np.exp(logfactor - alph * np.log(2) -
betaln(bet + 1, alph + 1) - np.log(bet + 1) +
(bet + 1) * np.log((x[xq] + 1) / 2)) * I
return F
def test_random_function_train(self):
np.random.seed(5)
num_vars = 2
degree = 5
rank = 2
sparsity_ratio = 0.2
sample_ratio = .9
ranks = rank*np.ones(num_vars+1,dtype=np.uint64)
ranks[0]=1; ranks[-1]=1
alpha=0; beta=0;
recursion_coeffs = jacobi_recurrence(
degree+1, alpha=alpha,beta=beta,probability=True)
ft_data = generate_random_sparse_function_train(
num_vars,rank,degree+1,sparsity_ratio)
true_sol = ft_data[1]
num_ft_params = true_sol.shape[0]
num_samples = int(sample_ratio*num_ft_params)
samples = np.random.uniform(-1.,1.,(num_vars,num_samples))
function = lambda samples: evaluate_function_train(
samples,ft_data,recursion_coeffs)
values = function(samples)
assert np.linalg.norm(values)>0, (np.linalg.norm(values))
num_validation_samples=100
validation_samples = np.random.uniform(
-1.,1.,(num_vars,num_validation_samples))
validation_values = function(validation_samples)
zero_ft_data = copy.deepcopy(ft_data)
zero_ft_data[1]=np.zeros_like(zero_ft_data[1])
# DO NOT use ft_data in following two functions.
# These function only overwrites parameters associated with the
# active indices the rest of the parameters are taken from ft_data.
# If ft_data is used some of the true data will be kept and give
# an unrealisticaly accurate answer
approx_eval = partial(modify_and_evaluate_function_train,samples,
zero_ft_data,recursion_coeffs,None)
apply_approx_adjoint_jacobian = partial(
apply_function_train_adjoint_jacobian,samples,zero_ft_data,
recursion_coeffs,1e-3)
sparsity = np.where(true_sol!=0)[0].shape[0]
print(('sparsity',sparsity,'num_samples',num_samples))
# sparse
project = partial(s_sparse_projection,sparsity=sparsity)
# non-linear least squres
#project = partial(s_sparse_projection,sparsity=num_ft_params)
# use uninormative initial guess
#initial_guess = np.zeros_like(true_sol)
# use linear approximation as initial guess
linear_ft_data = ft_linear_least_squares_regression(
samples,values,degree,perturb=None)
initial_guess = linear_ft_data[1]
# use initial guess that is close to true solution
# num_samples required to obtain accruate answer decreases signficantly
# over linear or uniformative guesses. As size of perturbation from
# truth increases num_samples must increase
initial_guess = true_sol.copy()+np.random.normal(0.,.1,(num_ft_params))
tol = 5e-3
max_iter=1000
result = iterative_hard_thresholding(
approx_eval, apply_approx_adjoint_jacobian, project,
values[:,0], initial_guess, tol, max_iter, verbosity=1)
sol = result[0]
residnorm = result[1]
recovered_ft_data=copy.deepcopy(ft_data)
recovered_ft_data[1]=sol
ft_validation_values = evaluate_function_train(
validation_samples,recovered_ft_data,recursion_coeffs)
validation_error = np.linalg.norm(
validation_values-ft_validation_values)
rel_validation_error=validation_error/np.linalg.norm(validation_values)
# compare relative error because exit condition is based upon
# relative residual
assert rel_validation_error<10*tol, rel_validation_error
def test_sparse_function_train(self):
np.random.seed(5)
num_vars = 2
degree = 5
rank = 2
tol = 1e-5
sparsity_ratio = 0.2
sample_ratio = 0.6
ranks = rank*np.ones(num_vars+1,dtype=np.uint64)
ranks[0]=1; ranks[-1]=1
alpha=0; beta=0;
recursion_coeffs = jacobi_recurrence(
degree+1, alpha=alpha,beta=beta,probability=True)
ft_data = generate_random_sparse_function_train(
num_vars,rank,degree+1,sparsity_ratio)
true_sol = ft_data[1]
num_ft_params = true_sol.shape[0]
num_samples = int(sample_ratio*num_ft_params)
samples = np.random.uniform(-1.,1.,(num_vars,num_samples))
function = lambda samples: evaluate_function_train(
samples,ft_data,recursion_coeffs)
#function = lambda samples: np.cos(samples.sum(axis=0))[:,np.newaxis]
values = function(samples)
print(values.shape)
assert np.linalg.norm(values)>0, (np.linalg.norm(values))
num_validation_samples=100
validation_samples = np.random.uniform(
-1.,1.,(num_vars,num_validation_samples))
validation_values = function(validation_samples)
zero_ft_data = copy.deepcopy(ft_data)
zero_ft_data[1]=np.zeros_like(zero_ft_data[1])
# DO NOT use ft_data in following two functions.
# These function only overwrites parameters associated with the
# active indices the rest of the parameters are taken from ft_data.
# If ft_data is used some of the true data will be kept and give
# an unrealisticaly accurate answer
approx_eval = partial(modify_and_evaluate_function_train,samples,
zero_ft_data,recursion_coeffs,None)
apply_approx_adjoint_jacobian = partial(
apply_function_train_adjoint_jacobian,samples,zero_ft_data,
recursion_coeffs,1e-3)
def least_squares_regression(indices,initial_guess):
#if initial_guess is None:
st0 = np.random.get_state()
np.random.seed(1)
initial_guess = np.random.normal(0.,.01,indices.shape[0])
np.random.set_state(st0)
result = ft_non_linear_least_squares_regression(
samples, values, ft_data, recursion_coeffs, initial_guess,
indices,{'gtol':tol,'ftol':tol,'xtol':tol,'verbosity':0})
return result[indices]
sparsity = np.where(true_sol!=0)[0].shape[0]
print(('sparsity',sparsity,'num_samples',num_samples,
'num_ft_params',num_ft_params))
print(true_sol)
active_indices = None
use_omp = True
#use_omp = False
if not use_omp:
sol = least_squares_regression(np.arange(num_ft_params),None)
else:
result = orthogonal_matching_pursuit(
approx_eval, apply_approx_adjoint_jacobian,
least_squares_regression, values[:,0], active_indices,
num_ft_params, tol, min(num_samples,num_ft_params), verbosity=1)
sol = result[0]
residnorm = result[1]
recovered_ft_data=copy.deepcopy(ft_data)
recovered_ft_data[1]=sol
ft_validation_values = evaluate_function_train(
validation_samples,recovered_ft_data,recursion_coeffs)
validation_error = np.linalg.norm(
validation_values-ft_validation_values)
rel_validation_error=validation_error/np.linalg.norm(validation_values)
# compare relative error because exit condition is based upon
# relative residual
print(rel_validation_error)
assert rel_validation_error<100*tol, rel_validation_error
def get_recursion_coefficients(self, opts, num_coefs):
poly_type = opts.get('poly_type', None)
var_type = None
if poly_type is None:
var_type = opts['rv_type']
if poly_type == 'legendre' or var_type == 'uniform':
recursion_coeffs = jacobi_recurrence(num_coefs,
alpha=0,
beta=0,
probability=True)
elif poly_type == 'jacobi' or var_type == 'beta':
if poly_type is not None:
alpha_poly, beta_poly = opts['alpha_poly'], opts['beta_poly']
else:
alpha_poly, beta_poly = opts['shapes']['b'] - 1, opts[
'shapes']['a'] - 1
recursion_coeffs = jacobi_recurrence(num_coefs,
alpha=alpha_poly,
beta=beta_poly,
probability=True)
elif poly_type == 'hermite' or var_type == 'norm':
recursion_coeffs = hermite_recurrence(num_coefs,
rho=0.,
probability=True)
elif poly_type == 'krawtchouk' or var_type == 'binom':
if poly_type is None:
opts = opts['shapes']
n, p = opts['n'], opts['p']
num_coefs = min(num_coefs, n)
recursion_coeffs = krawtchouk_recurrence(num_coefs, n, p)
elif poly_type == 'hahn' or var_type == 'hypergeom':
if poly_type is not None:
apoly, bpoly = opts['alpha_poly'], opts['beta_poly']
N = opts['N']
else:
M, n, N = [opts['shapes'][key] for key in ['M', 'n', 'N']]
apoly, bpoly = -(n + 1), -M - 1 + n
num_coefs = min(num_coefs, N)
recursion_coeffs = hahn_recurrence(num_coefs, N, apoly, bpoly)
elif poly_type == 'discrete_chebyshev' or var_type == 'discrete_chebyshev':
if poly_type is not None:
N = opts['N']
else:
N = opts['shapes']['xk'].shape[0]
assert np.allclose(opts['shapes']['xk'], np.arange(N))
assert np.allclose(opts['shapes']['pk'], np.ones(N) / N)
num_coefs = min(num_coefs, N)
recursion_coeffs = discrete_chebyshev_recurrence(num_coefs, N)
elif poly_type == 'discrete_numeric' or var_type == 'float_rv_discrete':
if poly_type is None:
opts = opts['shapes']
xk, pk = opts['xk'], opts['pk']
#shapes['xk'] will be in [0,1] but canonical domain is [-1,1]
xk = xk * 2 - 1
assert xk.min() >= -1 and xk.max() <= 1
if num_coefs > xk.shape[0]:
msg = 'Number of coefs requested is larger than number of '
msg += 'probability masses'
raise Exception(msg)
recursion_coeffs = modified_chebyshev_orthonormal(
num_coefs, [xk, pk])
p = evaluate_orthonormal_polynomial_1d(np.asarray(xk, dtype=float),
num_coefs - 1,
recursion_coeffs)
error = np.absolute((p.T * pk).dot(p) - np.eye(num_coefs)).max()
if error > self.numerically_generated_poly_accuracy_tolerance:
msg = f'basis created is ill conditioned. '
msg += f'Max error: {error}. Max terms: {xk.shape[0]}, '
msg += f'Terms requested: {num_coefs}'
raise Exception(msg)
elif poly_type == 'monomial':
recursion_coeffs = None
else:
if poly_type is not None:
raise Exception('poly_type (%s) not supported' % poly_type)
else:
raise Exception('var_type (%s) not supported' % var_type)
return recursion_coeffs
def test_evaluate_multivariate_orthonormal_polynomial(self):
num_vars = 2
alpha = 0.
beta = 0.
degree = 2
deriv_order = 1
probability_measure = True
ab = jacobi_recurrence(degree + 1,
alpha=alpha,
beta=beta,
probability=probability_measure)
x, w = np.polynomial.legendre.leggauss(degree)
samples = cartesian_product([x] * num_vars, 1)
weights = outer_product([w] * num_vars)
indices = compute_hyperbolic_indices(num_vars, degree, 1.0)
# sort lexographically to make testing easier
I = np.lexsort((indices[0, :], indices[1, :], indices.sum(axis=0)))
indices = indices[:, I]
basis_matrix = evaluate_multivariate_orthonormal_polynomial(
samples, indices, ab, deriv_order)
exact_basis_vals_1d = []
exact_basis_derivs_1d = []
for dd in range(num_vars):
x = samples[dd, :]
exact_basis_vals_1d.append(
np.asarray([1 + 0. * x, x, 0.5 * (3. * x**2 - 1)]).T)
exact_basis_derivs_1d.append(
np.asarray([0. * x, 1.0 + 0. * x, 3. * x]).T)
exact_basis_vals_1d[-1] /= np.sqrt(1. /
(2 * np.arange(degree + 1) + 1))
exact_basis_derivs_1d[-1] /= np.sqrt(
1. / (2 * np.arange(degree + 1) + 1))
exact_basis_matrix = np.asarray([
exact_basis_vals_1d[0][:, 0], exact_basis_vals_1d[0][:, 1],
exact_basis_vals_1d[1][:, 1], exact_basis_vals_1d[0][:, 2],
exact_basis_vals_1d[0][:, 1] * exact_basis_vals_1d[1][:, 1],
exact_basis_vals_1d[1][:, 2]
]).T
# x1 derivative
exact_basis_matrix = np.vstack(
(exact_basis_matrix,
np.asarray([
0. * x, exact_basis_derivs_1d[0][:, 1], 0. * x,
exact_basis_derivs_1d[0][:, 2],
exact_basis_derivs_1d[0][:, 1] * exact_basis_vals_1d[1][:, 1],
0. * x
]).T))
# x2 derivative
exact_basis_matrix = np.vstack(
(exact_basis_matrix,
np.asarray([
0. * x, 0. * x, exact_basis_derivs_1d[1][:, 1], 0. * x,
exact_basis_vals_1d[0][:, 1] * exact_basis_derivs_1d[1][:, 1],
exact_basis_derivs_1d[1][:, 2]
]).T))
def func(x):
return evaluate_multivariate_orthonormal_polynomial(
x, indices, ab, 0)
basis_matrix_derivs = basis_matrix[samples.shape[1]:]
basis_matrix_derivs_fd = np.empty_like(basis_matrix_derivs)
for ii in range(samples.shape[1]):
basis_matrix_derivs_fd[ii::samples.shape[1], :] = approx_fprime(
samples[:, ii:ii + 1], func, 1e-7)
assert np.allclose(exact_basis_matrix[samples.shape[1]:],
basis_matrix_derivs_fd)
assert np.allclose(exact_basis_matrix, basis_matrix)
def get_recursion_coefficients(
opts,
num_coefs,
numerically_generated_poly_accuracy_tolerance=1e-12):
"""
Parameters
----------
num_coefs : interger
The number of recursion coefficients desired
numerically_generated_poly_accuracy_tolerance : float
Tolerance used to construct any numerically generated polynomial
basis functions.
opts : dictionary
Dictionary with the following attributes
rv_type : string
The type of variable associated with the polynomial. If poly_type
is not provided then the recursion coefficients chosen is selected
using the Askey scheme. E.g. uniform -> legendre, norm -> hermite.
rv_type is assumed to be the name of the distribution of scipy.stats
variables, e.g. for gaussian rv_type = norm(0, 1).dist
poly_type : string
The type of polynomial which overides rv_type. Supported types
['legendre', 'hermite', 'jacobi', 'krawtchouk', 'hahn',
'discrete_chebyshev', 'discrete_numeric', 'continuous_numeric',
'function_indpnt_vars', 'product_indpnt_vars', 'monomial']
Note 'monomial' does not produce an orthogonal basis
The remaining options are specific to rv_type and poly_type. See
- :func:`pyapprox.univariate_quadrature.get_jacobi_recursion_coefficients`
- :func:`pyapprox.univariate_quadrature.get_function_independent_vars_recursion_coefficients`
- :func:`pyapprox.univariate_quadrature.get_product_independent_vars_recursion_coefficients`
Note Legendre is just a special instance of a Jacobi polynomial with
alpha_poly, beta_poly = 0, 0 and alpha_stat, beta_stat = 1, 1
Returns
-------
recursion_coeffs : np.ndarray (num_coefs, 2)
"""
# variables that require numerically generated polynomials with
# predictor corrector method
from scipy import stats
from scipy.stats import _continuous_distns
poly_type = opts.get('poly_type', None)
var_type = None
if poly_type is None:
var_type = opts['rv_type']
if poly_type == 'legendre' or var_type == 'uniform':
recursion_coeffs = jacobi_recurrence(
num_coefs, alpha=0, beta=0, probability=True)
elif poly_type == 'jacobi' or var_type == 'beta':
recursion_coeffs = get_jacobi_recursion_coefficients(
poly_type, opts, num_coefs)
elif poly_type == 'hermite' or var_type == 'norm':
recursion_coeffs = hermite_recurrence(
num_coefs, rho=0., probability=True)
elif poly_type == 'krawtchouk' or var_type == 'binom':
# although bounded the krwatchouk polynomials are not defined
# on the canonical domain [-1,1] but rather the user and
# canconical domain are the same
if poly_type is None:
opts = opts['shapes']
n, p = opts['n'], opts['p']
num_coefs = min(num_coefs, n)
recursion_coeffs = krawtchouk_recurrence(
num_coefs, n, p)
elif poly_type == 'hahn' or var_type == 'hypergeom':
# although bounded the hahn polynomials are not defined
# on the canonical domain [-1,1] but rather the user and
# canconical domain are the same
if poly_type is not None:
apoly, bpoly = opts['alpha_poly'], opts['beta_poly']
N = opts['N']
else:
M, n, N = [opts['shapes'][key] for key in ['M', 'n', 'N']]
apoly, bpoly = -(n+1), -M-1+n
num_coefs = min(num_coefs, N)
recursion_coeffs = hahn_recurrence(num_coefs, N, apoly, bpoly)
xk = np.arange(max(0, N-M+n), min(n, N)+1, dtype=float)
elif poly_type == 'discrete_chebyshev' or var_type == 'discrete_chebyshev':
# although bounded the discrete_chebyshev polynomials are not defined
# on the canonical domain [-1,1] but rather the user and
# canconical domain are the same
if poly_type is not None:
N = opts['N']
else:
N = opts['shapes']['xk'].shape[0]
assert np.allclose(opts['shapes']['xk'], np.arange(N))
assert np.allclose(opts['shapes']['pk'], np.ones(N)/N)
num_coefs = min(num_coefs, N)
recursion_coeffs = discrete_chebyshev_recurrence(num_coefs, N)
elif poly_type == 'discrete_numeric' or var_type == 'float_rv_discrete':
if poly_type is None:
opts = opts['shapes']
xk, pk = opts['xk'], opts['pk']
# shapes['xk'] will be in [0, 1] but canonical domain is [-1, 1]
xk = xk*2-1
assert xk.min() >= -1 and xk.max() <= 1
if num_coefs > xk.shape[0]:
msg = 'Number of coefs requested is larger than number of '
msg += 'probability masses'
raise Exception(msg)
#recursion_coeffs = modified_chebyshev_orthonormal(num_coefs, [xk, pk])
recursion_coeffs = lanczos(xk, pk, num_coefs)
p = evaluate_orthonormal_polynomial_1d(
np.asarray(xk, dtype=float), num_coefs-1, recursion_coeffs)
error = np.absolute((p.T*pk).dot(p)-np.eye(num_coefs)).max()
if error > numerically_generated_poly_accuracy_tolerance:
msg = f'basis created is ill conditioned. '
msg += f'Max error: {error}. Max terms: {xk.shape[0]}, '
msg += f'Terms requested: {num_coefs}'
raise Exception(msg)
elif (poly_type == 'continuous_numeric' or
var_type == 'continuous_rv_sample'):
if poly_type is None:
opts = opts['shapes']
xk, pk = opts['xk'], opts['pk']
if num_coefs > xk.shape[0]:
msg = 'Number of coefs requested is larger than number of '
msg += 'samples'
raise Exception(msg)
#print(num_coefs)
#recursion_coeffs = modified_chebyshev_orthonormal(num_coefs, [xk, pk])
#recursion_coeffs = lanczos(xk, pk, num_coefs)
recursion_coeffs = predictor_corrector(
num_coefs, (xk, pk), xk.min(), xk.max(),
interval_size=xk.max()-xk.min())
p = evaluate_orthonormal_polynomial_1d(
np.asarray(xk, dtype=float), num_coefs-1, recursion_coeffs)
error = np.absolute((p.T*pk).dot(p)-np.eye(num_coefs)).max()
if error > numerically_generated_poly_accuracy_tolerance:
msg = f'basis created is ill conditioned. '
msg += f'Max error: {error}. Max terms: {xk.shape[0]}, '
msg += f'Terms requested: {num_coefs}'
raise Exception(msg)
elif poly_type == 'monomial':
recursion_coeffs = None
elif var_type in _continuous_distns._distn_names:
quad_options = {
'nquad_samples': 10,
'atol': numerically_generated_poly_accuracy_tolerance,
'rtol': numerically_generated_poly_accuracy_tolerance,
'max_steps': 10000, 'verbose': 0}
rv = getattr(stats, var_type)(**opts['shapes'])
recursion_coeffs = predictor_corrector_known_scipy_pdf(
num_coefs, rv, quad_options)
elif poly_type == 'function_indpnt_vars':
recursion_coeffs = get_function_independent_vars_recursion_coefficients(
opts, num_coefs)
elif poly_type == 'product_indpnt_vars':
recursion_coeffs = get_product_independent_vars_recursion_coefficients(
opts, num_coefs)
else:
if poly_type is not None:
raise Exception('poly_type (%s) not supported' % poly_type)
else:
raise Exception('var_type (%s) not supported' % var_type)
return recursion_coeffs