def continue_cholesky(self, x, x_old, U_old, observed=True, nugget=None):
"""
U = C.continue_cholesky(x, x_old, U_old[, observed=True, nugget=None])
Computes Cholesky factorization of self(z,z). Assumes the Cholesky
factorization of self(x_old, x_old) has already been computed.
:Arguments:
- `x`: The input array on which to evaluate the Cholesky factorization.
- `x_old`: The input array on which the Cholesky factorization has been
computed.
- `U_old`: The Cholesky factorization of C(x_old, x_old).
- `observed`: If 'True', any observations are taken into account
when computing the Cholesky factor. If not, the unobserved
version of self is used.
- `nugget`: The 'nugget' parameter, which will essentially be
added to the diagonal of C(x,x) before Cholesky factorizing.
"""
# Concatenation of the old points and new points.
xtot = vstack((x_old, x))
# Number of old points.
N_old = x_old.shape[0]
# Number of new points.
N_new = x.shape[0]
U_new = self.__call__(x, x, regularize=False, observed=observed)
# not really implemented yet.
if nugget is not None:
for i in xrange(N_new):
U_new[i, i] += nugget[i]
U = asmatrix(
zeros((N_new + N_old, N_old + N_new), dtype=float, order='F'))
U[:N_old, :N_old] = U_old
offdiag = self.__call__(x=x_old,
y=x,
observed=observed,
regularize=False)
trisolve(U_old, offdiag, uplo='U', transa='T', inplace=True)
U[:N_old, N_old:] = offdiag
U_new -= offdiag.T * offdiag
info = dpotrf_wrap(U_new)
if info > 0:
raise LinAlgError, "Matrix does not appear to be positive definite by row %i. Consider another Covariance subclass, such as NearlyFullRankCovariance." % info
U[N_old:, N_old:] = U_new
return U
def continue_cholesky(self, x, x_old, U_old, observed=True, nugget=None):
"""
U = C.continue_cholesky(x, x_old, U_old[, observed=True, nugget=None])
Computes Cholesky factorization of self(z,z). Assumes the Cholesky
factorization of self(x_old, x_old) has already been computed.
:Arguments:
- `x`: The input array on which to evaluate the Cholesky factorization.
- `x_old`: The input array on which the Cholesky factorization has been
computed.
- `U_old`: The Cholesky factorization of C(x_old, x_old).
- `observed`: If 'True', any observations are taken into account
when computing the Cholesky factor. If not, the unobserved
version of self is used.
- `nugget`: The 'nugget' parameter, which will essentially be
added to the diagonal of C(x,x) before Cholesky factorizing.
"""
# Concatenation of the old points and new points.
xtot = vstack((x_old,x))
# Number of old points.
N_old = x_old.shape[0]
# Number of new points.
N_new = x.shape[0]
U_new = self.__call__(x, x, regularize=False, observed=observed)
# not really implemented yet.
if nugget is not None:
for i in xrange(N_new):
U_new[i,i] += nugget[i]
U = asmatrix(zeros((N_new + N_old, N_old + N_new), dtype=float, order='F'))
U[:N_old, :N_old] = U_old
offdiag = self.__call__(x=x_old, y=x, observed=observed, regularize=False)
trisolve(U_old,offdiag,uplo='U',transa='T', inplace=True)
U[:N_old, N_old:] = offdiag
U_new -= offdiag.T*offdiag
info = dpotrf_wrap(U_new)
if info>0:
raise LinAlgError, "Matrix does not appear to be positive definite by row %i. Consider another Covariance subclass, such as NearlyFullRankCovariance." %info
U[N_old:,N_old:] = U_new
return U
def cholesky(self, x, observed=True, nugget=None, return_eval_also=False):
"""
U = C.cholesky(x[, observed=True, nugget=None])
Computes Cholesky factorization of self(x,x).
:Arguments:
- `x`: The input array on which to evaluate the covariance.
- `observed`: If 'True', any observations are taken into account
when computing the Cholesky factor. If not, the unobserved
version of self is used.
- `nugget`: The 'nugget' parameter, which will essentially be
added to the diagonal of C(x,x) before Cholesky factorizing.
"""
# Number of points in x.
N_new = x.shape[0]
U=self.__call__(x, x, regularize = False, observed = observed)
if return_eval_also:
C_eval = U.copy('F')
if nugget is not None:
for i in xrange(N_new):
U[i,i] += nugget[i]
# print self.params, x.shape, observed, nugget
info = dpotrf_wrap(U)
if info>0:
raise LinAlgError, "Matrix does not appear to be positive definite by row %i. Consider another Covariance subclass, such as NearlyFullRankCovariance." % info
if return_eval_also:
return U, C_eval
else:
return U
def observe(self, obs_mesh, obs_V, output_type='r'):
"""
Observes self on obs_mesh with observation variance obs_V.
Output_type controls the information returned:
'r' : returns information needed by Realization objects.
'o' : returns information needed by function observe.
's' : returns information needed by the Gaussian process
submodel.
"""
# print 'C.observe called'
# Number of spatial dimensions.
ndim = obs_mesh.shape[1]
if self.ndim is not None:
if not ndim == self.ndim:
raise ValueError, "Dimension of observation mesh is not equal to dimension of base mesh."
else:
self.ndim = ndim
# print ndim
# =====================================
# = If self hasn't been observed yet: =
# =====================================
if not self.observed:
# If self has not been observed, get the Cholesky factor of self(obs_mesh, obs_mesh)
# and the side information and store it.
# Rank so far is 0.
m_old = 0
# Number of observation points so far is 0.
N_old = 0
if output_type != 's':
obs_dict = self.cholesky(obs_mesh,
apply_pivot=False,
nugget=obs_V,
regularize=False,
rank_limit=self.rank_limit)
else:
C_eval = self.__call__(obs_mesh, obs_mesh, regularize=False)
U = C_eval.copy('F')
for i in xrange(U.shape[0]):
U[i, i] += obs_V[i]
info = dpotrf_wrap(U)
if info > 0:
raise LinAlgError, "Matrix does not appear to be positive definite by row %i. Could not observe with assume_full_rank=True." % info
obs_dict = {
'U': U,
'pivots': arange(U.shape[0]),
'U_new': U,
'C_eval': C_eval
}
obs_dict_new = obs_dict
# Rank of self(obs_mesh, obs_mesh)
m_new = obs_dict['U'].shape[0]
# Upper-triangular Cholesky factor of self(obs_mesh, obs_mesh)
self.full_Uo = obs_dict['U']
# print (self.full_Uo[:,argsort(obs_dict['pivots'])].T*self.full_Uo[:,argsort(obs_dict['pivots'])] - self(obs_mesh,obs_mesh)).max()
# Upper-triangular square Cholesky factor of self(obs_mesh_*, obs_mesh_*). See documentation.
self.Uo = obs_dict['U'][:, :m_new]
# Pivots.
piv_new = obs_dict['pivots']
self.full_piv = piv_new
self.obs_piv = piv_new[:m_new]
# Remember full observation mesh.
self.full_obs_mesh = obs_mesh
# relevant slice is the positive-definite indices, which get into obs_mesh_*. See documentation.
relevant_slice = self.obs_piv
# obs_mesh_new is obs_mesh_* from documentation.
obs_mesh_new = obs_mesh[relevant_slice, :]
self.obs_mesh = obs_mesh_new
self.obs_V = obs_V[piv_new]
self.obs_len = m_new
# =======================================
# = If self has been observed already: =
# =======================================
else:
# If self has been observed, get the Cholesky factor of the _full_ observation mesh (new
# and old observations) using continue_cholesky, along with side information, and store it.
# Extract information from self's existing attributes related to the observation mesh..
obs_old, piv_old = self.Uo, self.obs_piv
# Rank of self's evaluation on the observation mesh so far.
m_old = len(self.obs_piv)
# Number of observations so far.
N_old = self.full_obs_mesh.shape[0]
# Number of new observations.
N_new = obs_mesh.shape[0]
# Call to self.continue_cholesky.
obs_dict_new = self.continue_cholesky(
x=obs_mesh,
x_old=self.full_obs_mesh,
chol_dict_old={
'U': self.full_Uo,
'pivots': self.full_piv
},
apply_pivot=False,
observed=False,
regularize=False,
nugget=obs_V,
assume_full_rank=output_type == 's',
rank_limit=self.rank_limit)
if output_type == 's':
C_eval = obs_dict_new['C_eval']
# Full Cholesky factor of self(obs_mesh, obs_mesh), where obs_mesh is the combined observation mesh.
self.full_Uo = obs_dict_new['U']
# Rank of self(obs_mesh, obs_mesh)
m_new = self.full_Uo.shape[0]
# Square upper-triangular Cholesky factor of self(obs_mesh_*, obs_mesh_*). See documentation.
self.Uo = self.full_Uo[:, :m_new]
# Pivots.
piv_new = obs_dict_new['pivots']
self.obs_piv = piv_new[:m_new]
self.full_piv = piv_new
# Concatenate old and new observation meshes.
self.full_obs_mesh = vstack((self.full_obs_mesh, obs_mesh))
relevant_slice = piv_new[m_old:m_new] - N_old
obs_mesh_new = obs_mesh[relevant_slice, :]
# Remember obs_mesh_* and corresponding observation variances.
self.obs_mesh = vstack(
(self.obs_mesh, obs_mesh[relevant_slice, :]))
self.obs_V = hstack((self.obs_V, obs_V[relevant_slice]))
# Length of obs_mesh_*.
self.obs_len = m_new
self.observed = True
# Output expected by Realization
if output_type == 'r':
return relevant_slice, obs_mesh_new, self.full_Uo[
m_old:,
argsort(piv_new)[N_old:]], self.full_Uo[:m_old,
argsort(piv_new
)[N_old:]]
# Ouptut expected by observe
if output_type == 'o':
return relevant_slice, obs_mesh_new
# Output expected by the GP submodel
if output_type == 's':
return obs_dict_new['U_new'], obs_dict_new[
'C_eval'], self.full_Uo[:m_old,
argsort(piv_new)[N_old:]]
def observe(self, obs_mesh, obs_V, assume_full_rank=False):
"""
Observes self at obs_mesh with variance given by obs_V.
Returns the following components of the Cholesky factor:
- `relevant_slice`: The indices included in the incomplete Cholesky factorization.
These correspond to the values of obs_mesh that determine the other values,
but not one another.
- `obs_mesh_new`: obs_mesh sliced according to relevant_slice.
- `U_for_draw`: An upper-triangular Cholesky factor of self's evaluation on obs_mesh
conditional on all previous observations.
The first and second are useful to Mean when it observes itself,
the third is useful to Realization when it draws new values.
"""
# print 'C.observe called'
# Number of spatial dimensions.
ndim = obs_mesh.shape[1]
if self.ndim is not None:
if not ndim==self.ndim:
raise ValueError, "Dimension of observation mesh is not equal to dimension of base mesh."
else:
self.ndim = ndim
# print ndim
# =====================================
# = If self hasn't been observed yet: =
# =====================================
if not self.observed:
# If self has not been observed, get the Cholesky factor of self(obs_mesh, obs_mesh)
# and the side information and store it.
# Rank so far is 0.
m_old = 0
# Number of observation points so far is 0.
N_old = 0
if not assume_full_rank:
obs_dict = self.cholesky(obs_mesh, apply_pivot = False, nugget = obs_V, regularize=False, rank_limit = self.rank_limit)
else:
C = self.__call__(obs_mesh,obs_mesh,regularize=False)
for i in xrange(C.shape[0]):
C[i,i] += obs_V[i]
info = dpotrf_wrap(C)
if info>0:
raise LinAlgError, "Matrix does not appear to be positive definite by row %i. Could not observe with assume_full_rank=True." %info
obs_dict = {'U': C,'pivots': arange(C.shape[0])}
# Rank of self(obs_mesh, obs_mesh)
m_new = obs_dict['U'].shape[0]
# Upper-triangular Cholesky factor of self(obs_mesh, obs_mesh)
self.full_Uo = obs_dict['U']
# print (self.full_Uo[:,argsort(obs_dict['pivots'])].T*self.full_Uo[:,argsort(obs_dict['pivots'])] - self(obs_mesh,obs_mesh)).max()
# Upper-triangular square Cholesky factor of self(obs_mesh_*, obs_mesh_*). See documentation.
self.Uo = obs_dict['U'][:,:m_new]
# Pivots.
piv_new = obs_dict['pivots']
self.full_piv = piv_new
self.obs_piv = piv_new[:m_new]
# Remember full observation mesh.
self.full_obs_mesh = obs_mesh
# relevant slice is the positive-definite indices, which get into obs_mesh_*. See documentation.
relevant_slice = self.obs_piv
# obs_mesh_new is obs_mesh_* from documentation.
obs_mesh_new = obs_mesh[relevant_slice,:]
self.obs_mesh = obs_mesh_new
self.obs_V = obs_V[piv_new]
self.obs_len = m_new
# =======================================
# = If self has been observed already: =
# =======================================
else:
# If self has been observed, get the Cholesky factor of the _full_ observation mesh (new
# and old observations) using continue_cholesky, along with side information, and store it.
# Extract information from self's existing attributes related to the observation mesh..
obs_old, piv_old = self.Uo, self.obs_piv
# Rank of self's evaluation on the observation mesh so far.
m_old = len(self.obs_piv)
# Number of observations so far.
N_old = self.full_obs_mesh.shape[0]
# Number of new observations.
N_new = obs_mesh.shape[0]
# Call to self.continue_cholesky.
obs_dict_new = self.continue_cholesky(x=obs_mesh,
x_old = self.full_obs_mesh,
chol_dict_old = {'U': self.full_Uo, 'pivots': self.full_piv},
apply_pivot = False,
observed = False,
regularize=False,
nugget = obs_V,
assume_full_rank = assume_full_rank,
rank_limit = self.rank_limit)
# Full Cholesky factor of self(obs_mesh, obs_mesh), where obs_mesh is the combined observation mesh.
self.full_Uo = obs_dict_new['U']
# Rank of self(obs_mesh, obs_mesh)
m_new = self.full_Uo.shape[0]
# Square upper-triangular Cholesky factor of self(obs_mesh_*, obs_mesh_*). See documentation.
self.Uo=self.full_Uo[:,:m_new]
# Pivots.
piv_new = obs_dict_new['pivots']
self.obs_piv = piv_new[:m_new]
self.full_piv = piv_new
# Concatenate old and new observation meshes.
self.full_obs_mesh = vstack((self.full_obs_mesh, obs_mesh))
relevant_slice = piv_new[m_old:m_new] - N_old
obs_mesh_new = obs_mesh[relevant_slice,:]
# Remember obs_mesh_* and corresponding observation variances.
self.obs_mesh = vstack((self.obs_mesh, obs_mesh[relevant_slice,:]))
self.obs_V = hstack((self.obs_V, obs_V[relevant_slice]))
# Length of obs_mesh_*.
self.obs_len = m_new
self.observed = True
return relevant_slice, obs_mesh_new, self.full_Uo[m_old:,argsort(piv_new)[N_old:]]
def observe(self, obs_mesh, obs_V, output_type='r'):
"""
Observes self on obs_mesh with observation variance obs_V.
Output_type controls the information returned:
'r' : returns information needed by Realization objects.
'o' : returns information needed by function observe.
's' : returns information needed by the Gaussian process
submodel.
"""
# print 'C.observe called'
# Number of spatial dimensions.
ndim = obs_mesh.shape[1]
if self.ndim is not None:
if not ndim==self.ndim:
raise ValueError, "Dimension of observation mesh is not equal to dimension of base mesh."
else:
self.ndim = ndim
# print ndim
# =====================================
# = If self hasn't been observed yet: =
# =====================================
if not self.observed:
# If self has not been observed, get the Cholesky factor of self(obs_mesh, obs_mesh)
# and the side information and store it.
# Rank so far is 0.
m_old = 0
# Number of observation points so far is 0.
N_old = 0
if output_type != 's':
obs_dict = self.cholesky(obs_mesh, apply_pivot = False, nugget = obs_V, regularize=False, rank_limit = self.rank_limit)
else:
C_eval = self.__call__(obs_mesh,obs_mesh,regularize=False)
U = C_eval.copy('F')
for i in xrange(U.shape[0]):
U[i,i] += obs_V[i]
info = dpotrf_wrap(U)
if info>0:
raise LinAlgError, "Matrix does not appear to be positive definite by row %i. Could not observe with assume_full_rank=True." %info
obs_dict = {'U': U,'pivots': arange(U.shape[0]),'U_new':U,'C_eval':C_eval}
obs_dict_new = obs_dict
# Rank of self(obs_mesh, obs_mesh)
m_new = obs_dict['U'].shape[0]
# Upper-triangular Cholesky factor of self(obs_mesh, obs_mesh)
self.full_Uo = obs_dict['U']
# print (self.full_Uo[:,argsort(obs_dict['pivots'])].T*self.full_Uo[:,argsort(obs_dict['pivots'])] - self(obs_mesh,obs_mesh)).max()
# Upper-triangular square Cholesky factor of self(obs_mesh_*, obs_mesh_*). See documentation.
self.Uo = obs_dict['U'][:,:m_new]
# Pivots.
piv_new = obs_dict['pivots']
self.full_piv = piv_new
self.obs_piv = piv_new[:m_new]
# Remember full observation mesh.
self.full_obs_mesh = obs_mesh
# relevant slice is the positive-definite indices, which get into obs_mesh_*. See documentation.
relevant_slice = self.obs_piv
# obs_mesh_new is obs_mesh_* from documentation.
obs_mesh_new = obs_mesh[relevant_slice,:]
self.obs_mesh = obs_mesh_new
self.obs_V = obs_V[piv_new]
self.obs_len = m_new
# =======================================
# = If self has been observed already: =
# =======================================
else:
# If self has been observed, get the Cholesky factor of the _full_ observation mesh (new
# and old observations) using continue_cholesky, along with side information, and store it.
# Extract information from self's existing attributes related to the observation mesh..
obs_old, piv_old = self.Uo, self.obs_piv
# Rank of self's evaluation on the observation mesh so far.
m_old = len(self.obs_piv)
# Number of observations so far.
N_old = self.full_obs_mesh.shape[0]
# Number of new observations.
N_new = obs_mesh.shape[0]
# Call to self.continue_cholesky.
obs_dict_new = self.continue_cholesky(x=obs_mesh,
x_old = self.full_obs_mesh,
chol_dict_old = {'U': self.full_Uo, 'pivots': self.full_piv},
apply_pivot = False,
observed = False,
regularize=False,
nugget = obs_V,
assume_full_rank = output_type=='s',
rank_limit = self.rank_limit)
if output_type=='s':
C_eval = obs_dict_new['C_eval']
# Full Cholesky factor of self(obs_mesh, obs_mesh), where obs_mesh is the combined observation mesh.
self.full_Uo = obs_dict_new['U']
# Rank of self(obs_mesh, obs_mesh)
m_new = self.full_Uo.shape[0]
# Square upper-triangular Cholesky factor of self(obs_mesh_*, obs_mesh_*). See documentation.
self.Uo=self.full_Uo[:,:m_new]
# Pivots.
piv_new = obs_dict_new['pivots']
self.obs_piv = piv_new[:m_new]
self.full_piv = piv_new
# Concatenate old and new observation meshes.
self.full_obs_mesh = vstack((self.full_obs_mesh, obs_mesh))
relevant_slice = piv_new[m_old:m_new] - N_old
obs_mesh_new = obs_mesh[relevant_slice,:]
# Remember obs_mesh_* and corresponding observation variances.
self.obs_mesh = vstack((self.obs_mesh, obs_mesh[relevant_slice,:]))
self.obs_V = hstack((self.obs_V, obs_V[relevant_slice]))
# Length of obs_mesh_*.
self.obs_len = m_new
self.observed = True
# Output expected by Realization
if output_type == 'r':
return relevant_slice, obs_mesh_new, self.full_Uo[m_old:,argsort(piv_new)[N_old:]], self.full_Uo[:m_old, argsort(piv_new)[N_old:]]
# Ouptut expected by observe
if output_type == 'o':
return relevant_slice, obs_mesh_new
# Output expected by the GP submodel
if output_type=='s':
return obs_dict_new['U_new'], obs_dict_new['C_eval'], self.full_Uo[:m_old, argsort(piv_new)[N_old:]]