def E_cond(poly, freeze, dist, **kws):
assert not dist.dependent()
if poly.dim<len(dist):
poly = po.setdim(poly, len(dist))
freeze = po.Poly(freeze)
freeze = po.setdim(freeze, len(dist))
keys = freeze.A.keys()
if len(keys)==1 and keys[0]==(0,)*len(dist):
freeze = freeze.A.values()[0]
else:
freeze = np.array(keys)
freeze = freeze.reshape(freeze.size/len(dist), len(dist))
shape = poly.shape
poly = po.flatten(poly)
kmax = np.max(poly.keys, 0)+1
keys = [i for i in np.ndindex(*kmax)]
vals = dist.mom(np.array(keys).T, **kws).T
mom = dict(zip(keys, vals))
A = poly.A.copy()
keys = A.keys()
out = {}
zeros = [0]*poly.dim
for i in xrange(len(keys)):
key = list(keys[i])
a = A[tuple(key)]
for d in xrange(poly.dim):
for j in xrange(len(freeze)):
if freeze[j,d]:
key[d], zeros[d] = zeros[d], key[d]
break
tmp = a*mom[tuple(key)]
if tuple(zeros) in out:
out[tuple(zeros)] = out[tuple(zeros)] + tmp
else:
out[tuple(zeros)] = tmp
for d in xrange(poly.dim):
for j in xrange(len(freeze)):
if freeze[j,d]:
key[d], zeros[d] = zeros[d], key[d]
break
out = po.Poly(out, poly.dim, poly.shape, float)
out = po.reshape(out, shape)
return out
def fit_regression(P, x, u, rule="LS", retall=False, **kws):
"""
Fit a polynomial chaos expansion using linear regression.
Parameters
----------
P : Poly
Polynomial chaos expansion with `P.shape=(M,)` and `P.dim=D`.
x : array_like
Collocation nodes with `x.shape=(D,K)`.
u : array_like
Model evaluations with `len(u)=K`.
retall : bool
If True return uhat in addition to R
rule : str
Regression method used.
The follwong methods uses scikits-learn as backend.
See `sklearn.linear_model` for more details.
Key Scikit-learn Description
--- ------------ -----------
Parameters Description
---------- -----------
"BARD" ARDRegression Bayesian ARD Regression
n_iter=300 Maximum iterations
tol=1e-3 Optimization tolerance
alpha_1=1e-6 Gamma scale parameter
alpha_2=1e-6 Gamma inverse scale parameter
lambda_1=1e-6 Gamma shape parameter
lambda_2=1e-6 Gamma inverse scale parameter
threshold_lambda=1e-4 Upper pruning threshold
"BR" BayesianRidge Bayesian Ridge Regression
n_iter=300 Maximum iterations
tol=1e-3 Optimization tolerance
alpha_1=1e-6 Gamma scale parameter
alpha_2=1e-6 Gamma inverse scale parameter
lambda_1=1e-6 Gamma shape parameter
lambda_2=1e-6 Gamma inverse scale parameter
"EN" ElastiNet Elastic Net
alpha=1.0 Dampening parameter
rho Mixing parameter in [0,1]
max_iter=300 Maximum iterations
tol Optimization tolerance
"ENC" ElasticNetCV EN w/Cross Validation
rho Dampening parameter(s)
eps=1e-3 min(alpha)/max(alpha)
n_alphas Number of alphas
alphas List of alphas
max_iter Maximum iterations
tol Optimization tolerance
cv=3 Cross validation folds
"LA" Lars Least Angle Regression
n_nonzero_coefs Number of non-zero coefficients
eps Cholesky regularization
"LAC" LarsCV LAR w/Cross Validation
max_iter Maximum iterations
cv=5 Cross validation folds
max_n_alphas Max points for residuals in cv
"LAS" Lasso Least Absolute Shrinkage and
Selection Operator
alpha=1.0 Dampening parameter
max_iter Maximum iterations
tol Optimization tolerance
"LASC" LassoCV LAS w/Cross Validation
eps=1e-3 min(alpha)/max(alpha)
n_alphas Number of alphas
alphas List of alphas
max_iter Maximum iterations
tol Optimization tolerance
cv=3 Cross validation folds
"LL" LassoLars Lasso and Lars model
max_iter Maximum iterations
eps Cholesky regularization
"LLC" LassoLarsCV LL w/Cross Validation
max_iter Maximum iterations
cv=5 Cross validation folds
max_n_alphas Max points for residuals in cv
eps Cholesky regularization
"LLIC" LassoLarsIC LL w/AIC or BIC
criterion "AIC" or "BIC" criterion
max_iter Maximum iterations
eps Cholesky regularization
"OMP" OrthogonalMatchingPursuit
n_nonzero_coefs Number of non-zero coefficients
tol Max residual norm (instead of non-zero coef)
Local methods
Key Description
--- -----------
"LS" Ordenary Least Squares
"T" Ridge Regression/Tikhonov Regularization
order Order of regularization (or custom matrix)
alpha Dampning parameter (else estimated from gcv)
"TC" T w/Cross Validation
order Order of regularization (or custom matrix)
alpha Dampning parameter (else estimated from gcv)
Returns
-------
R[, uhat]
R : Poly
Fitted polynomial with `R.shape=u.shape[1:]` and `R.dim=D`.
uhat : np.ndarray
The Fourier coefficients in the estimation.
Examples
--------
>>> P = cp.Poly([1, x, y])
>>> s = [[-1,-1,1,1], [-1,1,-1,1]]
>>> u = [0,1,1,2]
>>> print fit_regression(P, s, u)
0.5q1+0.5q0+1.0
"""
x = np.array(x)
if len(x.shape) == 1:
x = x.reshape(1, *x.shape)
u = np.array(u)
Q = P(*x).T
shape = u.shape[1:]
u = u.reshape(u.shape[0], int(np.prod(u.shape[1:])))
rule = rule.upper()
# Local rules
if rule == "LS":
uhat = la.lstsq(Q, u)[0].T
elif rule == "T":
uhat, alphas = rlstsq(Q, u, kws.get("order", 0),
kws.get("alpha", None), False, True)
uhat = uhat.T
elif rule == "TC":
uhat = rlstsq(Q, u, kws.get("order", 0), kws.get("alpha", None), True)
uhat = uhat.T
else:
# Scikit-learn wrapper
try:
_ = lm
except:
raise NotImplementedError("sklearn not installed")
if rule == "BARD":
solver = lm.ARDRegression(fit_intercept=False, copy_X=False, **kws)
elif rule == "BR":
kws["fit_intercept"] = kws.get("fit_intercept", False)
solver = lm.BayesianRidge(**kws)
elif rule == "EN":
kws["fit_intercept"] = kws.get("fit_intercept", False)
solver = lm.ElasticNet(**kws)
elif rule == "ENC":
kws["fit_intercept"] = kws.get("fit_intercept", False)
solver = lm.ElasticNetCV(**kws)
elif rule == "LA": # success
kws["fit_intercept"] = kws.get("fit_intercept", False)
solver = lm.Lars(**kws)
elif rule == "LAC":
kws["fit_intercept"] = kws.get("fit_intercept", False)
solver = lm.LarsCV(**kws)
elif rule == "LAS":
kws["fit_intercept"] = kws.get("fit_intercept", False)
solver = lm.Lasso(**kws)
elif rule == "LASC":
kws["fit_intercept"] = kws.get("fit_intercept", False)
solver = lm.LassoCV(**kws)
elif rule == "LL":
kws["fit_intercept"] = kws.get("fit_intercept", False)
solver = lm.LassoLars(**kws)
elif rule == "LLC":
kws["fit_intercept"] = kws.get("fit_intercept", False)
solver = lm.LassoLarsCV(**kws)
elif rule == "LLIC":
kws["fit_intercept"] = kws.get("fit_intercept", False)
solver = lm.LassoLarsIC(**kws)
elif rule == "OMP":
solver = lm.OrthogonalMatchingPursuit(**kws)
uhat = solver.fit(Q, u).coef_
u = u.reshape(u.shape[0], *shape)
R = po.sum((P * uhat), -1)
R = po.reshape(R, shape)
if retall == 1:
return R, uhat
elif retall == 2:
if rule == "T":
return R, uhat, Q, alphas
return R, uhat, Q
return R
def pcm_lr(func,
order,
dist_out,
sample=None,
dist_in=None,
rule="H",
orth=3,
regression="LS",
retall=False):
"""
Probabilistic Collocation Method using Linear Least Squares fit
Parameters
----------
Required arguemnts
func : callable
The model to be approximated. Must accept arguments on the
form `z` is an 1-dimensional array with `len(z)==len(dist)`.
order : int
The order of chaos expansion.
dist_out : Dist
Distributions for models parameter.
Optional arguments
sample : int
The order of the sample scheme to be used.
If omited it defaults to 2*len(orth).
dist_in : Dist
If included, space will be mapped using a Rosenblatt
transformation from dist_out to dist_in before creating an
expansin in terms of dist_in
rule:
rule for generating samples, where d is the number of
dimensions.
Key Name Nested
---- ---------------- ------
"K" Korobov no
"R" (Pseudo-)Random no
"L" Latin hypercube no
"S" Sobol yes
"H" Halton yes
"M" Hammersley yes
orth : int, str, callable, Poly
Orthogonal polynomial generation.
int, str :
orth will be passed to orth_select
for selection of orthogonalization.
See orth_select doc for more details.
callable :
the return of orth(M, dist) will be used.
Poly :
it will be used directly.
It must be of length N+1=comb(M+D, M)
regression : str
Linear regression method used.
See fit_regression for more details.
retall : bool
If True, return extra values.
# Examples
# --------
#
# Define function:
# >>> func = lambda z: -z[1]**2 + 0.1*z[0]
#
# Define distribution:
# >>> dist = cp.J(cp.Normal(), cp.Normal())
#
# Perform pcm:
# >>> q, x, y = cp.pcm_lr(func, 2, dist, retall=True)
# >>> print cp.around(q, 10)
# -q1^2+0.1q0
# >>> print len(x.T)
# 12
"""
if dist_in is None:
dist = dist_out
else:
dist = dist_in
# orthogonalization
if orth is None:
if dist.dependent():
orth = "chol"
else:
orth = "ttr"
if isinstance(orth, (str, int, long)):
orth = orth_select(orth)
if not isinstance(orth, po.Poly):
orth = orth(order, dist)
# sampling
if sample is None:
sample = 2 * len(orth)
x = samplegen(sample, dist, rule)
# Rosenblatt
if not (dist_in is None):
x = dist_out.ppf(dist_in.cdf(x))
# evals
y = np.array(map(func, x.T))
shape = y.shape[1:]
y = y.reshape(len(y), y.size / len(y))
if sample == 0:
y_ = y[:]
R = orth * y
else:
R, y_ = fit_regression(orth, x, y, regression, retall=1)
R = po.reshape(R, shape)
if retall:
return R, x, y
return R
def fit_regression(P, x, u, rule="LS", retall=False, **kws):
"""
Fit a polynomial chaos expansion using linear regression.
Parameters
----------
P : Poly
Polynomial chaos expansion with `P.shape=(M,)` and `P.dim=D`.
x : array_like
Collocation nodes with `x.shape=(D,K)`.
u : array_like
Model evaluations with `len(u)=K`.
retall : bool
If True return uhat in addition to R
rule : str
Regression method used.
The follwong methods uses scikits-learn as backend.
See `sklearn.linear_model` for more details.
Key Scikit-learn Description
--- ------------ -----------
Parameters Description
---------- -----------
"BARD" ARDRegression Bayesian ARD Regression
n_iter=300 Maximum iterations
tol=1e-3 Optimization tolerance
alpha_1=1e-6 Gamma scale parameter
alpha_2=1e-6 Gamma inverse scale parameter
lambda_1=1e-6 Gamma shape parameter
lambda_2=1e-6 Gamma inverse scale parameter
threshold_lambda=1e-4 Upper pruning threshold
"BR" BayesianRidge Bayesian Ridge Regression
n_iter=300 Maximum iterations
tol=1e-3 Optimization tolerance
alpha_1=1e-6 Gamma scale parameter
alpha_2=1e-6 Gamma inverse scale parameter
lambda_1=1e-6 Gamma shape parameter
lambda_2=1e-6 Gamma inverse scale parameter
"EN" ElastiNet Elastic Net
alpha=1.0 Dampening parameter
rho Mixing parameter in [0,1]
max_iter=300 Maximum iterations
tol Optimization tolerance
"ENC" ElasticNetCV EN w/Cross Validation
rho Dampening parameter(s)
eps=1e-3 min(alpha)/max(alpha)
n_alphas Number of alphas
alphas List of alphas
max_iter Maximum iterations
tol Optimization tolerance
cv=3 Cross validation folds
"LA" Lars Least Angle Regression
n_nonzero_coefs Number of non-zero coefficients
eps Cholesky regularization
"LAC" LarsCV LAR w/Cross Validation
max_iter Maximum iterations
cv=5 Cross validation folds
max_n_alphas Max points for residuals in cv
"LAS" Lasso Least Absolute Shrinkage and
Selection Operator
alpha=1.0 Dampening parameter
max_iter Maximum iterations
tol Optimization tolerance
"LASC" LassoCV LAS w/Cross Validation
eps=1e-3 min(alpha)/max(alpha)
n_alphas Number of alphas
alphas List of alphas
max_iter Maximum iterations
tol Optimization tolerance
cv=3 Cross validation folds
"LL" LassoLars Lasso and Lars model
max_iter Maximum iterations
eps Cholesky regularization
"LLC" LassoLarsCV LL w/Cross Validation
max_iter Maximum iterations
cv=5 Cross validation folds
max_n_alphas Max points for residuals in cv
eps Cholesky regularization
"LLIC" LassoLarsIC LL w/AIC or BIC
criterion "AIC" or "BIC" criterion
max_iter Maximum iterations
eps Cholesky regularization
"OMP" OrthogonalMatchingPursuit
n_nonzero_coefs Number of non-zero coefficients
tol Max residual norm (instead of non-zero coef)
Local methods
Key Description
--- -----------
"LS" Ordenary Least Squares
"T" Ridge Regression/Tikhonov Regularization
order Order of regularization (or custom matrix)
alpha Dampning parameter (else estimated from gcv)
"TC" T w/Cross Validation
order Order of regularization (or custom matrix)
alpha Dampning parameter (else estimated from gcv)
Returns
-------
R[, uhat]
R : Poly
Fitted polynomial with `R.shape=u.shape[1:]` and `R.dim=D`.
uhat : np.ndarray
The Fourier coefficients in the estimation.
Examples
--------
>>> P = cp.Poly([1, x, y])
>>> x = [[-1,-1,1,1], [-1,1,-1,1]]
>>> u = [0,1,1,2]
>>> print fit_regression(P, x, u)
0.5q1+0.5q0+1.0
"""
x = np.array(x)
if len(x.shape)==1:
x = x.reshape(1, *x.shape)
u = np.array(u)
Q = P(*x).T
shape = u.shape[1:]
u = u.reshape(u.shape[0], np.prod(u.shape[1:]))
rule = rule.upper()
# Local rules
if rule=="LS":
uhat = la.lstsq(Q, u)[0]
elif rule=="T":
uhat = rlstsq(Q, u, kws.get("order",0),
kws.get("alpha", None), False)
elif rule=="TC":
uhat = rlstsq(Q, u, kws.get("order",0),
kws.get("alpha", None), True)
else:
# Scikit-learn wrapper
try:
_ = lm
except:
raise NotImplementedError(
"sklearn not installed")
if rule=="BARD":
solver = lm.ARDRegression(fit_intercept=False,
copy_X=False, **kws)
elif rule=="BR":
kws["fit_intercept"] = kws.get("fit_intercept", False)
solver = lm.BayesianRidge(**kws)
elif rule=="EN":
kws["fit_intercept"] = kws.get("fit_intercept", False)
solver = lm.ElasticNet(**kws)
elif rule=="ENC":
kws["fit_intercept"] = kws.get("fit_intercept", False)
solver = lm.ElasticNetCV(**kws)
elif rule=="LA":
kws["fit_intercept"] = kws.get("fit_intercept", False)
solver = lm.Lars(**kws)
elif rule=="LAC":
kws["fit_intercept"] = kws.get("fit_intercept", False)
solver = lm.LarsCV(**kws)
elif rule=="LAS":
kws["fit_intercept"] = kws.get("fit_intercept", False)
solver = lm.Lasso(**kws)
elif rule=="LASC":
kws["fit_intercept"] = kws.get("fit_intercept", False)
solver = lm.LassoCV(**kws)
elif rule=="LL":
kws["fit_intercept"] = kws.get("fit_intercept", False)
solver = lm.LassoLars(**kws)
elif rule=="LLC":
kws["fit_intercept"] = kws.get("fit_intercept", False)
solver = lm.LassoLarsCV(**kws)
elif rule=="LLIC":
kws["fit_intercept"] = kws.get("fit_intercept", False)
solver = lm.LassoLarsIC(**kws)
elif rule=="OMP":
solver = lm.OrthogonalMatchingPursuit(**kws)
uhat = solver.fit(Q, u).coef_
u = u.reshape(u.shape[0], *shape)
R = po.sum((P*uhat.T), -1)
R = po.reshape(R, shape)
if retall==1:
return R, uhat
elif retall==2:
return R, uhat, Q
return R
def pcm_lr(func, order, dist_out, sample=None,
dist_in=None, rule="H",
orth=3, regression="LS", retall=False):
"""
Probabilistic Collocation Method using Linear Least Squares fit
Parameters
----------
Required arguemnts
func : callable
The model to be approximated. Must accept arguments on the
form `z` is an 1-dimensional array with `len(z)==len(dist)`.
order : int
The order of chaos expansion.
dist_out : Dist
Distributions for models parameter.
Optional arguments
sample : int
The order of the sample scheme to be used.
If omited it defaults to 2*len(orth).
dist_in : Dist
If included, space will be mapped using a Rosenblatt
transformation from dist_out to dist_in before creating an
expansin in terms of dist_in
rule:
rule for generating samples, where d is the number of
dimensions.
Key Name Nested
---- ---------------- ------
"K" Korobov no
"R" (Pseudo-)Random no
"L" Latin hypercube no
"S" Sobol yes
"H" Halton yes
"M" Hammersley yes
orth : int, str, callable, Poly
Orthogonal polynomial generation.
int, str :
orth will be passed to orth_select
for selection of orthogonalization.
See orth_select doc for more details.
callable :
the return of orth(M, dist) will be used.
Poly :
it will be used directly.
It must be of length N+1=comb(M+D, M)
regression : str
Linear regression method used.
See fit_regression for more details.
retall : bool
If True, return extra values.
# Examples
# --------
#
# Define function:
# >>> func = lambda z: -z[1]**2 + 0.1*z[0]
#
# Define distribution:
# >>> dist = cp.J(cp.Normal(), cp.Normal())
#
# Perform pcm:
# >>> q, x, y = cp.pcm_lr(func, 2, dist, retall=True)
# >>> print cp.around(q, 10)
# -q1^2+0.1q0
# >>> print len(x.T)
# 12
"""
if dist_in is None:
dist = dist_out
else:
dist = dist_in
# orthogonalization
if orth is None:
if dist.dependent():
orth = "chol"
else:
orth = "ttr"
if isinstance(orth, (str, int, long)):
orth = orth_select(orth)
if not isinstance(orth, po.Poly):
orth = orth(order, dist)
# sampling
if sample is None:
sample = 2*len(orth)
x = samplegen(sample, dist, rule)
# Rosenblatt
if not (dist_in is None):
x = dist_out.ppf(dist_in.cdf(x))
# evals
y = np.array(map(func, x.T))
shape = y.shape[1:]
y = y.reshape(len(y), y.size/len(y))
if sample==0:
y_ = y[:]
R = orth * y
else:
R, y_ = fit_regression(orth, x, y, regression, retall=1)
R = po.reshape(R, shape)
if retall:
return R, x, y
return R