def av_integrate(avfun, avmap, N):
"""Approximate the integral of a function of active variables.
Parameters
----------
avfun : function
a function of the active variables
avmap : ActiveVariableMap
a domains.ActiveVariableMap
N : int
the number of points in the quadrature rule
Returns
-------
mu : float
an estimate of the integral
Notes
-----
This function is usually used when one has already constructed a response
surface on the active variables and wants to estimate its integral.
"""
if not isinstance(avmap, ActiveVariableMap):
raise TypeError('avmap should be an ActiveVariableMap.')
if not isinstance(N, Integral):
raise TypeError('N should be an integer.')
Yp, Yw = av_quadrature_rule(avmap, N)
if isinstance(avfun, ActiveSubspaceResponseSurface):
avf = avfun.predict_av(Yp)[0]
else:
avf = SimulationRunner(avfun).run(Yp)
mu = np.dot(Yw.T, avf)[0, 0]
return mu
def build_from_interface(self, m, fun, dfun=None):
self.m = m
# number of gradient samples
M = 6 * (m + 1) * np.log(m)
# sample points
if self.bflag:
X = np.random.uniform(-1.0, 1.0, size=(M, m))
else:
X = np.random.normal(size=(M, m))
self.X = X
# sample the simulation's outputs
sr = SimulationRunner(fun)
self.f_runner = sr
# sample the simulation's gradients
if dfun == None:
df = finite_difference_gradients(X, sr)
else:
sgr = SimulationGradientRunner(dfun)
df = sgr.run(X)
self.df_runner = sgr
self.df = df
# compute the active subspaces
self.subspaces = build_subspaces(df)
def finite_difference_gradients(X, fun, h=1e-6):
"""Compute finite difference gradients with a given interface.
Parameters
----------
X : ndarray
M-by-m matrix that contains the points to estimate the gradients with
finite differences
fun : function
function that returns the simulation's quantity of interest given inputs
h : float, optional
the finite difference step size (default 1e-6)
Returns
-------
df : ndarray
M-by-m matrix that contains estimated partial derivatives approximated
by finite differences
"""
X, M, m = process_inputs(X)
# points to run simulations including the perturbed inputs
XX = np.kron(np.ones((m+1, 1)),X) + \
h*np.kron(np.vstack((np.zeros((1, m)), np.eye(m))), np.ones((M, 1)))
# run the simulation
if isinstance(fun, SimulationRunner):
F = fun.run(XX)
else:
F = SimulationRunner(fun).run(XX)
df = (F[M:].reshape((m, M)).transpose() - F[:M]) / h
return df.reshape((M,m))
def train_with_interface(self, fun, N, NMC=10):
Y, X, ind = as_design(self.avmap, N, NMC=NMC)
if isinstance(self.avmap.domain, BoundedActiveVariableDomain):
X = np.vstack((X, self.avmap.domain.vertX))
Y = np.vstack((Y, self.avmap.domain.vertY))
il = np.amax(ind) + 1
iu = np.amax(ind) + self.avmap.domain.vertX.shape[0] + 1
iind = np.arange(il, iu)
ind = np.vstack(( ind, iind.reshape((iind.size,1)) ))
# run simulation interface at all design points
sr = SimulationRunner(fun)
f = sr.run(X)
Ef, Vf = conditional_expectations(f, ind)
self.train(Y, Ef, v=Vf)
def build_from_interface(self, m, fun, dfun=None, avdim=None):
self.m = m
# number of gradient samples
M = int(np.floor(6*(m+1)*np.log(m)))
# sample points for gradients
if self.bndflag:
X = np.random.uniform(-1.0, 1.0, size=(M, m))
else:
X = np.random.normal(size=(M, m))
funr = SimulationRunner(fun)
f = funr.run(X)
self.X, self.f, self.funr = X, f, funr
# sample the simulation's gradients
if dfun == None:
df = finite_difference_gradients(X, fun)
else:
dfunr = SimulationGradientRunner(dfun)
df = dfunr.run(X)
self.dfunr = dfunr
# compute the active subspace
ss = Subspaces()
ss.compute(df)
if avdim is not None:
ss.partition(avdim)
self.n = ss.W1.shape[1]
print 'Dimension of subspace is {:d}'.format(self.n)
# set up the active variable domain and map
if self.bndflag:
avdom = BoundedActiveVariableDomain(ss)
avmap = BoundedActiveVariableMap(avdom)
else:
avdom = UnboundedActiveVariableDomain(ss)
avmap = UnboundedActiveVariableMap(avdom)
# build the response surface
avrs = ActiveSubspaceResponseSurface(avmap)
avrs.train_with_interface(fun, int(np.power(5,self.n)))
self.av_respsurf = avrs
def train_with_interface(self, fun, N, NMC=10):
"""Train the response surface with input/output pairs.
Parameters
----------
fun : function
a function that returns the simulation quantity of interest given a
point in the input space as an 1-by-m ndarray
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
Notes
-----
The training methods exploit the eigenvalues from the active subspace
analysis to determine length scales for each variable when tuning
the parameters of the radial bases.
The method sets attributes of the object for further use.
The method uses the response_surfaces.av_design function to get the
design for the appropriate `avmap`.
"""
Y, X, ind = av_design(self.avmap, N, NMC=NMC)
if isinstance(self.avmap.domain, BoundedActiveVariableDomain):
X = np.vstack((X, self.avmap.domain.vertX))
Y = np.vstack((Y, self.avmap.domain.vertY))
il = np.amax(ind) + 1
iu = np.amax(ind) + self.avmap.domain.vertX.shape[0] + 1
iind = np.arange(il, iu)
ind = np.vstack((ind, iind.reshape((iind.size, 1))))
# run simulation interface at all design points
if isinstance(fun, SimulationRunner):
f = fun.run(X)
else:
f = SimulationRunner(fun).run(X)
Ef, Vf = conditional_expectations(f, ind)
self._train(Y, Ef, v=Vf)
def as_integrate(fun, domain, subspace, N, NMC=10):
"""
Description of as_integrate
Arguments:
fun:
domain:
subspace:
N:
NMC: (default=10)
Outputs:
mu:
lb:
ub:
"""
w, x, ind = as_quadrature_rule(domain, subspace, N, NMC)
f = SimulationRunner(fun).run(x)
return compute_integral(w, f, ind)
def build_from_interface(self, fun, dfun=None, avdim=None):
"""
Build the active subspace-enabled model with interfaces to the
simulation.
:param function fun: A function that interfaces with the simulation.
It should take an ndarray of shape 1-by-m (e.g., a row of `X`), and
it should return a scalar. That scalar is the quantity of interest from the simulation.
:param function dfun: A function that interfaces with the simulation.
It should take an ndarray of shape 1-by-m (e.g., a row of `X`), and it
should return the gradient of the quantity of interest as an ndarray of shape 1-by-m.
:param int avdim: The dimension of the active subspace. If `avdim` is not
present, it is chosen after computing the active subspaces using
the given interfaces.
**Notes**
This method follows these steps:
#. Draw random points according to the weight function on the space\
of simulation inputs.
#. Compute the quantity of interest and its gradient at the sampled\
inputs. If `dfun` is None, use finite differences.
#. Use the collection of gradients to estimate the eigenvectors and\
eigenvalues that determine and define the active subspace.
#. Train a response surface using the interface, which uses a careful\
design of experiments on the space of active variables. This design\
uses about 5 points per dimension of the active subspace.
"""
if not hasattr(fun, '__call__'):
raise TypeError('fun should be a callable function.')
if dfun is not None:
if not hasattr(dfun, '__call__'):
raise TypeError('dfun should be a callable function.')
if avdim is not None:
if not isinstance(avdim, int):
raise TypeError('avdim should be an integer')
m = self.m
# number of gradient samples
M = int(np.floor(6*(m+1)*np.log(m)))
# sample points for gradients
if self.bounded_inputs:
X = np.random.uniform(-1.0, 1.0, size=(M, m))
else:
X = np.random.normal(size=(M, m))
fun = SimulationRunner(fun)
f = fun.run(X)
self.X, self.f, self.fun = X, f, fun
# sample the simulation's gradients
if dfun == None:
df = finite_difference_gradients(X, fun)
else:
dfun = SimulationGradientRunner(dfun)
df = dfun.run(X)
self.dfun = dfun
# compute the active subspace
ss = Subspaces()
ss.compute(df)
if avdim is not None:
ss.partition(avdim)
self.n = ss.W1.shape[1]
print 'The dimension of the active subspace is {:d}.'.format(self.n)
# set up the active variable domain and map
if self.bounded_inputs:
avdom = BoundedActiveVariableDomain(ss)
avmap = BoundedActiveVariableMap(avdom)
else:
avdom = UnboundedActiveVariableDomain(ss)
avmap = UnboundedActiveVariableMap(avdom)
# build the response surface
asrs = ActiveSubspaceResponseSurface(avmap)
asrs.train_with_interface(fun, int(np.power(5,self.n)))
# compute testing error as an R-squared
self.Rsqr = 1.0 - ( np.linalg.norm(asrs.predict(X)[0] - f)**2 \
/ np.var(f) )
self.as_respsurf = asrs
def integrate(fun, avmap, N, NMC=10):
"""Approximate the integral of a function of m variables.
Parameters
----------
fun : function
an interface to the simulation that returns the quantity of interest
given inputs as an 1-by-m ndarray
avmap : ActiveVariableMap
a domains.ActiveVariableMap
N : int
the number of points in the quadrature rule
NMC : int, optional
the number of points in the Monte Carlo estimates of the conditional
expectation and conditional variance (default 10)
Returns
-------
mu : float
an estimate of the integral of the function computed against the weight
function on the simulation inputs
lb : float
a central-limit-theorem 95% lower confidence from the Monte Carlo part
of the integration
ub : float
a central-limit-theorem 95% upper confidence from the Monte Carlo part
of the integration
See Also
--------
integrals.quadrature_rule
Notes
-----
The CLT-based bounds `lb` and `ub` are likely poor estimators of the error.
They only account for the variance from the Monte Carlo portion. They do
not include any error from the integration rule on the active variables.
"""
if not isinstance(avmap, ActiveVariableMap):
raise TypeError('avmap should be an ActiveVariableMap.')
if not isinstance(N, Integral):
raise TypeError('N should be an integer')
# get the quadrature rule
Xp, Xw, ind = quadrature_rule(avmap, N, NMC=NMC)
# compute the simulation output at each quadrature node
if isinstance(fun, SimulationRunner):
f = fun.run(Xp)
else:
f = SimulationRunner(fun).run(Xp)
# estimate conditional expectations and variances
Ef, Vf = conditional_expectations(f, ind)
# get weights for the conditional expectations
w = conditional_expectations(Xw * NMC, ind)[0]
# estimate the average
mu = np.dot(Ef.T, w)
# estimate the variance due to Monte Carlo
sig2 = np.dot(Vf.T, w * w) / NMC
# compute 95% confidence bounds from the Monte Carlo
lb, ub = mu - 1.96 * np.sqrt(sig2), mu + 1.96 * np.sqrt(sig2)
return mu[0, 0], lb[0, 0], ub[0, 0]