def fit_quadrature(orth, nodes, weights, solves, retall=False,
norms=None, **kws):
"""
Using spectral projection to create a polynomial approximation over
distribution space.
Parameters
----------
orth : Poly
Orthogonal polynomial expansion. Must be orthogonal for the
approximation to be accurate.
nodes : array_like
Where to evaluate the polynomial expansion and model to
approximate.
nodes.shape==(D,K) where D is the number of dimensions and K is
the number of nodes.
weights : array_like
Weights when doing numerical integration.
weights.shape==(K,)
solves : array_like, callable
The model to approximate.
If array_like is provided, it must have len(solves)==K.
If callable, it must take a single argument X with len(X)==D,
and return a consistent numpy compatible shape.
norms : array_like
In the of TTR using coefficients to estimate the polynomial
norm is more stable than manual calculation.
Calculated using quadrature if no provided.
norms.shape==(len(orth),)
"""
orth = po.Poly(orth)
nodes = np.asfarray(nodes)
weights = np.asfarray(weights)
if hasattr(solves, "__call__"):
solves = [solves(q) for q in nodes.T]
solves = np.asfarray(solves)
shape = solves.shape
solves = solves.reshape(weights.size, solves.size/weights.size)
ovals = orth(*nodes)
vals1 = [(val*solves.T*weights).T for val in ovals]
if norms is None:
vals2 = [(val**2*weights).T for val in ovals]
norms = np.sum(vals2, 1)
else:
norms = np.array(norms).flatten()
assert len(norms)==len(orth)
coefs = (np.sum(vals1, 1).T/norms).T
coefs = coefs.reshape(len(coefs), *shape[1:])
Q = po.transpose(po.sum(orth*coefs.T, -1))
if retall:
return Q, coefs
return Q
def pcm_gq(func,
order,
dist_out,
dist_in=None,
acc=None,
orth=None,
retall=False,
sparse=False):
"""
Probabilistic Collocation Method using optimal Gaussian quadrature
Parameters
----------
Required arguments
func : callable
The model to be approximated.
Must accept arguments on the form `func(z, *args, **kws)`
where `z` is an 1-dimensional array with `len(z)==len(dist)`.
order : int
The order of the polynomial approximation
dist_out : Dist
Distributions for models parameter
Optional arguments
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
acc : float
The order of the sample scheme used
If omitted order+1 will be used
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.
All polynomials must be orthogonal for method to work
properly.
args : itterable
Extra positional arguments passed to `func`.
kws : dict
Extra keyword arguments passed to `func`.
retall : bool
If True, return also number of evaluations
sparse : bool
If True, Smolyak sparsegrid will be used instead of full
tensorgrid
Returns
-------
Q[, X]
Q : Poly
Polynomial estimate of a given a model.
X : np.ndarray
Values used in evaluation
# Examples
# --------
# Define function:
# >>> func = lambda z: z[1]*z[0]
#
# Define distribution:
# >>> dist = cp.J(cp.Normal(), cp.Normal())
#
# Perform pcm:
# >>> p, x, w, y = cp.pcm_gq(func, 2, dist, acc=3, retall=True)
# >>> print cp.around(p, 10)
# q0q1
# >>> print len(w)
# 16
"""
if acc is None:
acc = order + 1
if dist_in is None:
z, w = qu.generate_quadrature(acc,
dist_out,
100,
sparse=sparse,
rule="G")
x = z
dist = dist_out
else:
z, w = qu.generate_quadrature(acc,
dist_in,
100,
sparse=sparse,
rule="G")
x = dist_out.ppf(dist_in.cdf(z))
dist = dist_in
y = np.array(map(func, x.T))
shape = y.shape
y = y.reshape(w.size, y.size / w.size)
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)
ovals = orth(*z)
vals1 = [(val * y.T * w).T for val in ovals]
vals2 = [(val**2 * w).T for val in ovals]
coef = (np.sum(vals1, 1).T / np.sum(vals2, 1)).T
coef = coef.reshape(len(coef), *shape[1:])
Q = po.transpose(po.sum(orth * coef.T, -1))
if retall:
return Q, x, w, y
return Q
def fit_quadrature(orth,
nodes,
weights,
solves,
retall=False,
norms=None,
**kws):
"""
Using spectral projection to create a polynomial approximation over
distribution space.
Parameters
----------
orth : Poly
Orthogonal polynomial expansion. Must be orthogonal for the
approximation to be accurate.
nodes : array_like
Where to evaluate the polynomial expansion and model to
approximate.
nodes.shape==(D,K) where D is the number of dimensions and K is
the number of nodes.
weights : array_like
Weights when doing numerical integration.
weights.shape==(K,)
solves : array_like, callable
The model to approximate.
If array_like is provided, it must have len(solves)==K.
If callable, it must take a single argument X with len(X)==D,
and return a consistent numpy compatible shape.
norms : array_like
In the of TTR using coefficients to estimate the polynomial
norm is more stable than manual calculation.
Calculated using quadrature if no provided.
norms.shape==(len(orth),)
"""
orth = po.Poly(orth)
nodes = np.asfarray(nodes)
weights = np.asfarray(weights)
if hasattr(solves, "__call__"):
solves = [solves(q) for q in nodes.T]
solves = np.asfarray(solves)
shape = solves.shape
solves = solves.reshape(weights.size, solves.size / weights.size)
ovals = orth(*nodes)
vals1 = [(val * solves.T * weights).T for val in ovals]
if norms is None:
vals2 = [(val**2 * weights).T for val in ovals]
norms = np.sum(vals2, 1)
else:
norms = np.array(norms).flatten()
assert len(norms) == len(orth)
coefs = (np.sum(vals1, 1).T / norms).T
coefs = coefs.reshape(len(coefs), *shape[1:])
Q = po.transpose(po.sum(orth * coefs.T, -1))
if retall:
return Q, coefs
return Q
def fit_adaptive(func,
poly,
dist,
abserr=1.e-8,
relerr=1.e-8,
budget=0,
norm=0,
bufname="",
retall=False):
"""Adaptive estimation of Fourier coefficients.
Parameters
----------
func : callable
Should take a single argument `q` which is 1D array
`len(q)=len(dist)`.
Must return something compatible with np.ndarray.
poly : Poly
Polynomial vector for which to create Fourier coefficients for.
dist : Dist
A distribution to optimize the Fourier coefficients to.
abserr : float
Absolute error tolerance.
relerr : float
Relative error tolerance.
budget : int
Soft maximum number of function evaluations.
0 means unlimited.
norm : int
Specifies the norm that is used to measure the error and
determine convergence properties (irrelevant for single-valued
functions). The `norm` argument takes one of the values:
0 : L0-norm
1 : L0-norm on top of paired the L2-norm. Good for complex
numbers where each conseqtive pair of the solution is real
and imaginery.
2 : L2-norm
3 : L1-norm
4 : L_infinity-norm
bufname : str, optional
Buffer evaluations to file such that the fit_adaptive can be
run again without redooing all evaluations.
retall : bool
If true, returns extra values.
Returns
-------
estimate[, coeffs, norms, coeff_error, norm_error]
estimate : Poly
The polynomial chaos expansion representation of func.
coeffs : np.ndarray
The Fourier coefficients.
norms : np.ndarray
The norm of the orthogonal polynomial squared.
coeff_error : np.ndarray
Estimated integration error of the coeffs.
norm_error : np.ndarray
Estimated integration error of the norms.
Examples
--------
>>> func = lambda q: q[0]*q[1]
>>> poly = cp.basis(0,2,2)
>>> dist = cp.J(cp.Uniform(0,1), cp.Uniform(0,1))
>>> res = cp.fit_adaptive(func, poly, dist, budget=100)
>>> print res
"""
if bufname:
func = lazy_eval(func, load=bufname)
dim = len(dist)
n = [0, 0]
dummy_x = dist.inv(.5 * np.ones(dim, dtype=np.float64))
val = np.array(func(dummy_x), np.float64)
xmin = np.zeros(dim, np.float64)
xmax = np.ones(dim, np.float64)
def f1(u, ns, *args):
qs = dist.inv(u.reshape(ns, dim))
out = (poly(*qs.T)**2).T.flatten()
return out
dim1 = len(poly)
val1 = np.empty(dim1, dtype=np.float64)
err1 = np.empty(dim1, dtype=np.float64)
_cubature(f1, dim1, xmin, xmax, (), "h", abserr, relerr, norm, budget,
True, val1, err1)
val1 = np.tile(val1, val.size)
dim2 = np.prod(val.shape) * dim1
val2 = np.empty(dim2, dtype=np.float64)
err2 = np.empty(dim2, dtype=np.float64)
def f2(u, ns, *args):
n[0] += ns
n[1] += 1
qs = dist.inv(u.reshape(ns, dim))
Y = np.array([func(q) for q in qs])
Q = poly(*qs.T)
out = np.array([Y.T * q1 for q1 in Q]).T.flatten()
out = out / np.tile(val1, ns)
return out
try:
_ = _cubature
except:
raise NotImplementedError("cubature not install properly")
_cubature(f2, dim2, xmin, xmax, (), "h", abserr, relerr, norm, budget,
True, val2, err2)
shape = (dim1, ) + val.shape
val2 = val2.reshape(shape[::-1]).T
out = po.transpose(po.sum(poly * val2.T, -1))
if retall:
return out, val2, val1, err2, err1
return val2
def fit_adaptive(func, poly, dist, abserr=1.e-8, relerr=1.e-8,
budget=0, norm=0, bufname="", retall=False):
"""Adaptive estimation of Fourier coefficients.
Parameters
----------
func : callable
Should take a single argument `q` which is 1D array
`len(q)=len(dist)`.
Must return something compatible with np.ndarray.
poly : Poly
Polynomial vector for which to create Fourier coefficients for.
dist : Dist
A distribution to optimize the Fourier coefficients to.
abserr : float
Absolute error tolerance.
relerr : float
Relative error tolerance.
budget : int
Soft maximum number of function evaluations.
0 means unlimited.
norm : int
Specifies the norm that is used to measure the error and
determine convergence properties (irrelevant for single-valued
functions). The `norm` argument takes one of the values:
0 : L0-norm
1 : L0-norm on top of paired the L2-norm. Good for complex
numbers where each conseqtive pair of the solution is real
and imaginery.
2 : L2-norm
3 : L1-norm
4 : L_infinity-norm
bufname : str, optional
Buffer evaluations to file such that the fit_adaptive can be
run again without redooing all evaluations.
retall : bool
If true, returns extra values.
Returns
-------
estimate[, coeffs, norms, coeff_error, norm_error]
estimate : Poly
The polynomial chaos expansion representation of func.
coeffs : np.ndarray
The Fourier coefficients.
norms : np.ndarray
The norm of the orthogonal polynomial squared.
coeff_error : np.ndarray
Estimated integration error of the coeffs.
norm_error : np.ndarray
Estimated integration error of the norms.
Examples
--------
>>> func = lambda q: q[0]*q[1]
>>> poly = cp.basis(0,2,2)
>>> dist = cp.J(cp.Uniform(0,1), cp.Uniform(0,1))
>>> res = cp.fit_adaptive(func, poly, dist, budget=100)
>>> print res
"""
if bufname:
func = lazy_eval(func, load=bufname)
dim = len(dist)
n = [0,0]
dummy_x = dist.inv(.5*np.ones(dim, dtype=np.float64))
val = np.array(func(dummy_x), np.float64)
xmin = np.zeros(dim, np.float64)
xmax = np.ones(dim, np.float64)
def f1(u, ns, *args):
qs = dist.inv(u.reshape(ns, dim))
out = (poly(*qs.T)**2).T.flatten()
return out
dim1 = len(poly)
val1 = np.empty(dim1, dtype=np.float64)
err1 = np.empty(dim1, dtype=np.float64)
_cubature(f1, dim1, xmin, xmax, (), "h", abserr, relerr, norm,
budget, True, val1, err1)
val1 = np.tile(val1, val.size)
dim2 = np.prod(val.shape)*dim1
val2 = np.empty(dim2, dtype=np.float64)
err2 = np.empty(dim2, dtype=np.float64)
def f2(u, ns, *args):
n[0] += ns
n[1] += 1
qs = dist.inv(u.reshape(ns, dim))
Y = np.array([func(q) for q in qs])
Q = poly(*qs.T)
out = np.array([Y.T*q1 for q1 in Q]).T.flatten()
out = out/np.tile(val1, ns)
return out
try:
_ = _cubature
except:
raise NotImplementedError(
"cubature not install properly")
_cubature(f2, dim2, xmin, xmax, (), "h", abserr, relerr, norm,
budget, True, val2, err2)
shape = (dim1,)+val.shape
val2 = val2.reshape(shape[::-1]).T
out = po.transpose(po.sum(poly*val2.T, -1))
if retall:
return out, val2, val1, err2, err1
return val2
def pcm_gq(func, order, dist_out, dist_in=None, acc=None,
orth=None, retall=False, sparse=False):
"""
Probabilistic Collocation Method using optimal Gaussian quadrature
Parameters
----------
Required arguments
func : callable
The model to be approximated.
Must accept arguments on the form `func(z, *args, **kws)`
where `z` is an 1-dimensional array with `len(z)==len(dist)`.
order : int
The order of the polynomial approximation
dist_out : Dist
Distributions for models parameter
Optional arguments
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
acc : float
The order of the sample scheme used
If omitted order+1 will be used
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.
All polynomials must be orthogonal for method to work
properly.
args : itterable
Extra positional arguments passed to `func`.
kws : dict
Extra keyword arguments passed to `func`.
retall : bool
If True, return also number of evaluations
sparse : bool
If True, Smolyak sparsegrid will be used instead of full
tensorgrid
Returns
-------
Q[, X]
Q : Poly
Polynomial estimate of a given a model.
X : np.ndarray
Values used in evaluation
# Examples
# --------
# Define function:
# >>> func = lambda z: z[1]*z[0]
#
# Define distribution:
# >>> dist = cp.J(cp.Normal(), cp.Normal())
#
# Perform pcm:
# >>> p, x, w, y = cp.pcm_gq(func, 2, dist, acc=3, retall=True)
# >>> print cp.around(p, 10)
# q0q1
# >>> print len(w)
# 16
"""
if acc is None:
acc = order+1
if dist_in is None:
z,w = qu.generate_quadrature(acc, dist_out, 100, sparse=sparse,
rule="G")
x = z
dist = dist_out
else:
z,w = qu.generate_quadrature(acc, dist_in, 100, sparse=sparse,
rule="G")
x = dist_out.ppf(dist_in.cdf(z))
dist = dist_in
y = np.array(map(func, x.T))
shape = y.shape
y = y.reshape(w.size, y.size/w.size)
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)
ovals = orth(*z)
vals1 = [(val*y.T*w).T for val in ovals]
vals2 = [(val**2*w).T for val in ovals]
coef = (np.sum(vals1, 1).T/np.sum(vals2, 1)).T
coef = coef.reshape(len(coef), *shape[1:])
Q = po.transpose(po.sum(orth*coef.T, -1))
if retall:
return Q, x, w, y
return Q