def momgen(order, domain, acc=100, sparse=False, rule="C",
composite=1, part=None, trans=lambda x:x, **kws):
if isinstance(domain, di.Dist):
dim = len(domain)
else:
dim = np.array(domain[0]).size
x0 = trans(np.zeros(dim))
if np.array(x0).shape==():
func = trans
trans = lambda x: [func(x)]
if part is None:
X,W = generate_quadrature(order, domain=domain, acc=acc, sparse=sparse,
rule=rule, composite=composite, part=part, **kws)
Y = np.array(trans(X))
def _mom(k):
out = np.sum(np.prod(Y.T**k, -1)*W, 0)
return out
else:
isdist = not isinstance(domain, (tuple, list, np.ndarray))
if isdist:
lo,up = domain.range()
else:
lo,up = np.array(domain)
X,W,Y = [], [], []
for I in np.ndindex(*part):
x,w = clenshaw_curtis(order, lo, up, part=(I, part))
y = np.array(trans(x))
if isdist:
w *= domain.pdf(x).flatten()
if np.any(w):
X.append(x); W.append(w); Y.append(y)
def _mom(k):
out = 0.
for i in xrange(len(X)):
out += np.sum(np.prod(Y[i].T**k, -1)*W[i], 0)
return out
_mom = lazy_eval(_mom, tuple)
def mom(K, **kws):
out = np.array([_mom(k) for k in K.T])
return out
return mom
def momgen(order, domain, acc=100, sparse=False, rule="C",
composite=1, part=None, trans=lambda x:x, **kws):
if isinstance(domain, di.Dist):
dim = len(domain)
else:
dim = np.array(domain[0]).size
x0 = trans(np.zeros(dim))
if np.array(x0).shape==():
func = trans
trans = lambda x: [func(x)]
if part is None:
X,W = generate_quadrature(order, domain=domain, acc=acc, sparse=sparse,
rule=rule, composite=composite, part=part, **kws)
Y = np.array(trans(X))
def _mom(k):
out = np.sum(np.prod(Y.T**k, -1)*W, 0)
return out
else:
isdist = not isinstance(domain, (tuple, list, np.ndarray))
if isdist:
lo,up = domain.range()
else:
lo,up = np.array(domain)
X,W,Y = [], [], []
for I in np.ndindex(*part):
x,w = clenshaw_curtis(order, lo, up, part=(I, part))
y = np.array(trans(x))
if isdist:
w *= domain.pdf(x).flatten()
if np.any(w):
X.append(x); W.append(w); Y.append(y)
def _mom(k):
out = 0.
for i in xrange(len(X)):
out += np.sum(np.prod(Y[i].T**k, -1)*W[i], 0)
return out
_mom = lazy_eval(_mom, tuple)
def mom(K, **kws):
out = np.array([_mom(k) for k in K.T])
return out
return mom
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
if ax is None:
ax = dim - np.argmin(1 * (np.array(I)[::-1] == 0)) - 1
if not i: return i, ax
if I[ax] == 0:
j = parent(parent(i, dim)[0], dim)[0]
while child(j + 1, dim, ax) < i:
j += 1
return j, ax
out = np.array(I) - 1 * (np.eye(dim)[ax])
return single_index(out), ax
parent = lazy_eval(parent, int)
def child(i, dim, ax):
"""
Child node according to Bertran's notation.
Parameters
----------
i : int
Index of the parent node.
dim : int
Dimensionality of the problem.
ax : int
Dimension direction to define a child.
Must have `0<=ax<dim`
def rule_generator(*funcs):
"""
Constructor for creating quadrature generator
Parameters
----------
*funcs : callable
One dimensional integration rule where
for func in funcs:
nodes, weights = func(order)
order : int
Order of integration rule
nodes : array_like
Where to evaluate.
weights : array_like
Weights corresponding to each node.
Returns
-------
mv_rule : callable
Multidimensional integration rule
nodes, weights = rule_gen(order)
order : int, array_like
Order of integration rule. If array_like, order along each
axis.
nodes : np.ndarray
Where to evaluate with nodes.shape==(D,K), where
D=len(funcs) and K is the number of points to evaluate.
weights : np.ndarray
Weights to go with the nodes with weights.shape=(K,).
"""
dim = len(funcs)
def tensprod_rule(N, part=None):
N = N * np.ones(dim, int)
q = [np.array(funcs[i](N[i])) \
for i in xrange(dim)]
x = [_[0] for _ in q]
x = combine(x, part=part).T
w = [_[1] for _ in q]
w = np.prod(combine(w, part=part), -1)
return x, w
tensprod_rule = lazy_eval(tensprod_rule)
def mv_rule(order, sparse=False, growth=None, part=None):
"""
Multidimensional integration rule
Parameters
----------
order : int, array_like
Order of integration rule. If array_like, order along each
axis.
Returns
-------
nodes, weights
nodes : np.ndarray
Where to evaluate with nodes.shape==(D,K), where
D=len(funcs) and K is the number of points to evaluate.
weights : np.ndarray
Weights to go with the nodes with weights.shape=(K,).
"""
if growth:
def foo(N):
N = N * np.ones(dim, int)
return tensprod_rule([growth(n) for n in N], part=part)
else:
def foo(N):
return tensprod_rule(N, part=part)
if sparse:
order = np.ones(dim, dtype=int) * order
m = np.min(order)
skew = [o - m for o in order]
return sparse_grid(foo, m, dim, skew=skew)
return foo(order)
return mv_rule
def rule_generator(*funcs):
"""
Constructor for creating quadrature generator
Parameters
----------
*funcs : callable
One dimensional integration rule where
for func in funcs:
nodes, weights = func(order)
order : int
Order of integration rule
nodes : array_like
Where to evaluate.
weights : array_like
Weights corresponding to each node.
Returns
-------
mv_rule : callable
Multidimensional integration rule
nodes, weights = rule_gen(order)
order : int, array_like
Order of integration rule. If array_like, order along each
axis.
nodes : np.ndarray
Where to evaluate with nodes.shape==(D,K), where
D=len(funcs) and K is the number of points to evaluate.
weights : np.ndarray
Weights to go with the nodes with weights.shape=(K,).
"""
dim = len(funcs)
def tensprod_rule(N, part=None):
N = N*np.ones(dim, int)
q = [np.array(funcs[i](N[i])) \
for i in xrange(dim)]
x = [_[0] for _ in q]
x = combine(x, part=part).T
w = [_[1] for _ in q]
w = np.prod(combine(w, part=part), -1)
return x, w
tensprod_rule = lazy_eval(tensprod_rule)
def mv_rule(order, sparse=False, growth=None, part=None):
"""
Multidimensional integration rule
Parameters
----------
order : int, array_like
Order of integration rule. If array_like, order along each
axis.
Returns
-------
nodes, weights
nodes : np.ndarray
Where to evaluate with nodes.shape==(D,K), where
D=len(funcs) and K is the number of points to evaluate.
weights : np.ndarray
Weights to go with the nodes with weights.shape=(K,).
"""
if growth:
def foo(N):
N = N*np.ones(dim, int)
return tensprod_rule([growth(n) for n in N], part=part)
else:
def foo(N):
return tensprod_rule(N, part=part)
if sparse:
order = np.ones(dim, dtype=int)*order
m = np.min(order)
skew = [o-m for o in order]
return sparse_grid(foo, m, dim, skew=skew)
return foo(order)
return mv_rule
I = multi_index(i, dim)
if ax is None:
ax = dim - np.argmin(1*(np.array(I)[::-1]==0))-1
if not i: return i, ax
if I[ax]==0:
j = parent(parent(i, dim)[0], dim)[0]
while child(j+1, dim, ax)<i: j += 1
return j, ax
out = np.array(I) - 1*(np.eye(dim)[ax])
return single_index(out), ax
parent = lazy_eval(parent, int)
def child(i, dim, ax):
"""
Child node according to Bertran's notation.
Parameters
----------
i : int
Index of the parent node.
dim : int
Dimensionality of the problem.
ax : int
Dimension direction to define a child.
Must have `0<=ax<dim`