def zonotope_quadrature_rule(vert, W1, N, NX=10000):
"""
Description of zonotope_quadrature_rule
Arguments:
vert:
W1:
N:
NX: (default=10000)
Outputs:
points:
weights:
"""
# number of dimensions
m, n = W1.shape
# points
y = np.vstack((vert, maximin_design(vert, N)))
T = Delaunay(y)
c = []
for t in T.simplices:
c.append(np.mean(T.points[t], axis=0))
points = np.array(c)
# approximate weights
Ysamp = np.dot(np.random.uniform(-1.0, 1.0, size=(NX, m)), W1)
I = T.find_simplex(Ysamp)
weights = np.zeros((T.nsimplex, 1))
for i in range(T.nsimplex):
weights[i] = np.sum(I == i) / float(NX)
return points.reshape((T.nsimplex, n)), weights.reshape((T.nsimplex, 1))
def as_design(avmap, N, NMC=10):
# interpret N as total number of points in the design
if type(N) is not int:
raise Exception('N should be an integer.')
m, n = avmap.domain.subspaces.W1.shape
if isinstance(avmap.domain, UnboundedActiveVariableDomain):
NN = [int(np.floor(np.power(N, 1.0/n))) for i in range(n)]
Y = dn.gauss_hermite_design(NN)
elif isinstance(avmap.domain, BoundedActiveVariableDomain):
if n==1:
a, b = avmap.domain.vertY[0,0], avmap.domain.vertY[1,0]
Y = dn.interval_design(a, b, N)
else:
vertices = avmap.domain.vertY
Y = dn.maximin_design(vertices, N)
else:
raise Exception('There is a problem with the avmap.domain.')
X, ind = avmap.inverse(Y, NMC)
return Y, X, ind
def zonotope_quadrature_rule(avmap, N, NX=10000):
"""
Description of zonotope_quadrature_rule
Arguments:
vert:
W1:
N:
NX: (default=10000)
Outputs:
points:
weights:
"""
vert = avmap.domain.vertY
W1 = avmap.domain.subspaces.W1
# number of dimensions
m, n = W1.shape
# points
y = np.vstack((vert, maximin_design(vert, N)))
T = Delaunay(y)
c = []
for t in T.simplices:
c.append(np.mean(T.points[t], axis=0))
points = np.array(c)
# approximate weights
Y_samples = np.dot(np.random.uniform(-1.0, 1.0, size=(NX,m)), W1)
I = T.find_simplex(Y_samples)
weights = np.zeros((T.nsimplex, 1))
for i in range(T.nsimplex):
weights[i] = np.sum(I==i) / float(NX)
return points.reshape((T.nsimplex,n)), weights.reshape((T.nsimplex,1))
def zonotope_quadrature_rule(avmap, N, NX=10000):
"""Quadrature rule on a zonotope.
Quadrature when the dimension of the active subspace is greater than 1 and
the simulation parameter space is bounded.
Parameters
----------
avmap : ActiveVariableMap
a domains.ActiveVariableMap
N : int
the number of quadrature nodes in the active variables
NX : int, optional
the number of samples to use to estimate the quadrature weights (default
10000)
Returns
-------
Yp : ndarray
quadrature nodes on the active variables
Yw : ndarray
quadrature weights on the active variables
See Also
--------
integrals.quadrature_rule
"""
vert = avmap.domain.vertY
W1 = avmap.domain.subspaces.W1
# number of dimensions
m, n = W1.shape
# points
y = np.vstack((vert, maximin_design(vert, N)))
T = Delaunay(y)
c = []
for t in T.simplices:
c.append(np.mean(T.points[t], axis=0))
points = np.array(c)
# approximate weights
Y_samples = np.dot(np.random.uniform(-1.0, 1.0, size=(NX, m)), W1)
I = T.find_simplex(Y_samples)
weights = np.zeros((T.nsimplex, 1))
for i in range(T.nsimplex):
weights[i] = np.sum(I == i) / float(NX)
Yp, Yw = points.reshape((T.nsimplex, n)), weights.reshape((T.nsimplex, 1))
return Yp, Yw
def zonotope_quadrature_rule(avmap, N, NX=10000):
"""Quadrature rule on a zonotope.
Quadrature when the dimension of the active subspace is greater than 1 and
the simulation parameter space is bounded.
Parameters
----------
avmap : ActiveVariableMap
a domains.ActiveVariableMap
N : int
the number of quadrature nodes in the active variables
NX : int, optional
the number of samples to use to estimate the quadrature weights (default
10000)
Returns
-------
Yp : ndarray
quadrature nodes on the active variables
Yw : ndarray
quadrature weights on the active variables
See Also
--------
integrals.quadrature_rule
"""
vert = avmap.domain.vertY
W1 = avmap.domain.subspaces.W1
# number of dimensions
m, n = W1.shape
# points
y = np.vstack((vert, maximin_design(vert, N)))
T = Delaunay(y)
c = []
for t in T.simplices:
c.append(np.mean(T.points[t], axis=0))
points = np.array(c)
# approximate weights
Y_samples = np.dot(np.random.uniform(-1.0, 1.0, size=(NX,m)), W1)
I = T.find_simplex(Y_samples)
weights = np.zeros((T.nsimplex, 1))
for i in range(T.nsimplex):
weights[i] = np.sum(I==i) / float(NX)
Yp, Yw = points.reshape((T.nsimplex,n)), weights.reshape((T.nsimplex,1))
return Yp, Yw
def av_design(avmap, N, NMC=10):
"""
A wrapper that returns the design for the response surface in the space of
the active variables.
:param ActiveVariableMap avmap: A domains.ActiveVariable map that includes
the active variable domain, which includes the active and inactive
subspaces.
:param int N: The number of points used in the design-of-experiments for
constructing the response surface.
:param int NMC: The number of points used to estimate the conditional
expectation and conditional variance of the function given a value
of the active variables. (Default is 10)
:return: Y, N-by-n matrix that contains the design points in the space of
active variables.
:rtype: ndarray
:return: X, (N*NMC)-by-m matrix that contains points in the simulation input
space to run the simulation.
:rtype: ndarray
:return: ind, Indices that map points in `X` to points in `Y`.
:rtype: ndarray
**See Also**
utils.designs.gauss_hermite_design
utils.designs.interval_design
utils.designs.maximin_design
"""
if not isinstance(avmap, ActiveVariableMap):
raise TypeError('avmap should be an ActiveVariableMap.')
# interpret N as total number of points in the design
if not isinstance(N, int):
raise Exception('N should be an integer.')
if not isinstance(NMC, int):
raise Exception('NMC should be an integer.')
m, n = avmap.domain.subspaces.W1.shape
if isinstance(avmap.domain, UnboundedActiveVariableDomain):
NN = [int(np.floor(np.power(N, 1.0/n))) for i in range(n)]
Y = dn.gauss_hermite_design(NN)
elif isinstance(avmap.domain, BoundedActiveVariableDomain):
if n==1:
a, b = avmap.domain.vertY[0,0], avmap.domain.vertY[1,0]
Y = dn.interval_design(a, b, N)
else:
vertices = avmap.domain.vertY
Y = dn.maximin_design(vertices, N)
else:
raise Exception('There is a problem with the avmap.domain.')
X, ind = avmap.inverse(Y, NMC)
return Y, X, ind
def av_design(avmap, N, NMC=10):
"""Design on active variable space.
A wrapper that returns the design for the response surface in the space of
the active variables.
Parameters
----------
avmap : ActiveVariableMap
a domains.ActiveVariable map that includes the active variable domain,
which includes the active and inactive subspaces
N : int
the number of points used in the design-of-experiments for constructing
the response surface
NMC : int, optional
the number of points used to estimate the conditional expectation and
conditional variance of the function given a value of the active
variables (Default is 10)
Returns
-------
Y : ndarray
N-by-n matrix that contains the design points in the space of active
variables
X : ndarray
(N*NMC)-by-m matrix that contains points in the simulation input space
to run the simulation
ind : ndarray
indices that map points in `X` to points in `Y`
See Also
--------
utils.designs.gauss_hermite_design
utils.designs.interval_design
utils.designs.maximin_design
"""
if not isinstance(avmap, ActiveVariableMap):
raise TypeError('avmap should be an ActiveVariableMap.')
# interpret N as total number of points in the design
if not isinstance(N, int):
raise Exception('N should be an integer.')
if not isinstance(NMC, int):
raise Exception('NMC should be an integer.')
m, n = avmap.domain.subspaces.W1.shape
if isinstance(avmap.domain, UnboundedActiveVariableDomain):
NN = [int(np.floor(np.power(N, 1.0 / n))) for i in range(n)]
Y = dn.gauss_hermite_design(NN)
elif isinstance(avmap.domain, BoundedActiveVariableDomain):
if n == 1:
a, b = avmap.domain.vertY[0, 0], avmap.domain.vertY[1, 0]
Y = dn.interval_design(a, b, N)
else:
vertices = avmap.domain.vertY
Y = dn.maximin_design(vertices, N)
else:
raise Exception('There is a problem with the avmap.domain.')
X, ind = avmap.inverse(Y, NMC)
return Y, X, ind