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 _obs_reg(self, M, dev_new, m_old):
# reg_mat = chol(C(obs_mesh_*, obs_mesh_*)).T.I * M.dev
reg_mat_new = -1. * dot(
self.Uo[:m_old, m_old:].T,
trisolve(self.Uo[:m_old, :m_old], M.dev, uplo='U', transa='T')).T
trisolve(self.Uo[m_old:, m_old:].T, reg_mat_new, 'L', inplace=True)
reg_mat_new += asmatrix(
trisolve(self.Uo[m_old:, m_old:], dev_new.T, uplo='U',
transa='T')).T
return asmatrix(vstack((M.reg_mat, reg_mat_new)))
def observe(self, obs_mesh, obs_V, output_type='o'):
__doc__ = Covariance.observe.__doc__
ndim = obs_mesh.shape[1]
nobs = obs_mesh.shape[0]
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
# bases_o is the basis evaluated on the observation mesh.
basis_o = asmatrix(self.eval_basis(obs_mesh, regularize=False))
# chol(basis_o.T * coef_cov * basis_o)
chol_inner = self.coef_U * basis_o
if output_type=='s':
C_eval = dot(chol_inner.T, chol_inner).copy('F')
U_eval = linalg.cholesky(C_eval).T.copy('F')
U = U_eval
m = U_eval.shape[0]
piv = arange(m)
else:
# chol(basis_o.T * coef_cov * basis_o + diag(obs_V)). Really should do this as low-rank update of covariance
# of V.
U, piv, m = ichol_basis(basis=chol_inner, nug=obs_V, reltol=self.relative_precision)
U = asmatrix(U)
piv_new = piv[:m]
self.obs_piv = piv_new
obs_mesh_new = obs_mesh[piv_new,:]
self.Uo = U[:m,:m]
# chol(basis_o.T * coef_cov * basis_o + diag(obs_V)) ^ -T * basis_o.T
self.Uo_cov = trisolve(self.Uo, basis_o[:,piv[:m]].T, uplo='U', transa='T')
# chol(basis_o.T * coef_cov * basis_o + diag(obs_V)).T.I * basis_o.T * coef_cov
self.Uo_cov = self.Uo_cov * self.coef_cov
# coef_cov = coef_cov - coef_cov * basis_o * (basis_o.T * coef_cov * basis_o + diag(obs_V)).I * basis_o.T * coef_cov
self.coef_cov = self.coef_cov - self.Uo_cov.T * self.Uo_cov
# coef_U = chol(coef_cov)
U, m, piv = ichol_full(c=self.coef_cov, reltol=self.relative_precision)
U = asmatrix(U)
self.coef_U = U[:m,argsort(piv)]
self.m = m
if output_type=='o':
return piv_new, obs_mesh_new
if output_type=='s':
return U_eval, C_eval, basis_o
raise ValueError, 'Output type not recognized.'
def _obs_eval(self, M, M_out, x, Uo_Cxo=None):
if Uo_Cxo is None:
Uo_Cxo = trisolve(M.Uo,
self(M.obs_mesh, x, observed=False),
uplo='U',
transa='T')
M_out += dot(asarray(M.reg_mat).squeeze(), asarray(Uo_Cxo)).squeeze()
return M_out
def observe(self, obs_mesh, obs_V):
__doc__ = Covariance.observe.__doc__
ndim = obs_mesh.shape[1]
nobs = obs_mesh.shape[0]
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
# bases_o is the basis evaluated on the observation mesh.
basis_o = asmatrix(self.eval_basis(obs_mesh, regularize=False))
# chol(basis_o.T * coef_cov * basis_o)
chol_inner = self.coef_U * basis_o
# chol(basis_o.T * coef_cov * basis_o + diag(obs_V)). Really should do this as low-rank update of covariance
# of V.
U, piv, m = ichol_basis(basis=chol_inner, nug=obs_V, reltol=self.relative_precision)
U = asmatrix(U)
piv_new = piv[:m]
self.obs_piv = piv_new
obs_mesh_new = obs_mesh[piv_new,:]
self.Uo = U[:m,:m]
# chol(basis_o.T * coef_cov * basis_o + diag(obs_V)) ^ -T * basis_o.T
self.Uo_cov = trisolve(self.Uo, basis_o[:,piv[:m]].T, uplo='U', transa='T')
# chol(basis_o.T * coef_cov * basis_o + diag(obs_V)).T.I * basis_o.T * coef_cov
self.Uo_cov = self.Uo_cov * self.coef_cov
# coef_cov -= coef_cov * basis_o * (basis_o.T * coef_cov * basis_o + diag(obs_V)).I * basis_o.T * coef_cov
self.coef_cov -= self.Uo_cov.T * self.Uo_cov
# coef_U = chol(coef_cov)
U, m, piv = ichol_full(c=self.coef_cov, reltol=self.relative_precision)
U = asmatrix(U)
self.coef_U = U[:m,argsort(piv)]
self.m = m
return piv_new, obs_mesh_new, None
def _unobs_reg(self, M):
# reg_mat = chol(C(obs_mesh_*, obs_mesh_*)).T.I * M.dev
return asmatrix(trisolve(self.Uo, M.dev.T, uplo='U', transa='T')).T
def __call__(self,
x,
y=None,
observed=True,
regularize=True,
return_Uo_Cxo=False):
if y is x:
symm = True
else:
symm = False
# Remember shape of x, and then 'regularize' it.
orig_shape = shape(x)
if len(orig_shape) > 1:
orig_shape = orig_shape[:-1]
if regularize:
x = regularize_array(x)
ndimx = x.shape[-1]
lenx = x.shape[0]
if return_Uo_Cxo:
Uo_Cxo = None
# Safety
if self.ndim is not None:
if not self.ndim == ndimx:
raise ValueError, "The number of spatial dimensions of x, "+\
ndimx.__str__()+\
", does not match the number of spatial dimensions of the Covariance instance's base mesh, "+\
self.ndim.__str__()+"."
# If there are observation points, prepare self(obs_mesh, x)
# and chol(self(obs_mesh, obs_mesh)).T.I * self(obs_mesh, x)
# ==========================================================
# = If only one argument is provided, return the diagonal. =
# ==========================================================
if y is None:
# Special fast-path for functions that have an 'amp' parameter
if hasattr(self.eval_fun, 'diag_call'):
V = self.eval_fun.diag_call(x, **self.params)
# Otherwise, evaluate the diagonal in a loop.
else:
V = empty(lenx, dtype=float)
for i in xrange(lenx):
this_x = x[i].reshape((1, -1))
V[i] = self.eval_fun(this_x, this_x, **self.params)
if self.observed and observed:
sqpart = empty(lenx, dtype=float)
Cxo = self.eval_fun(self.obs_mesh, x, **self.params)
Uo_Cxo = trisolve(self.Uo, Cxo, uplo='U', transa='T')
square_and_sum(Uo_Cxo, sqpart)
V -= sqpart
if return_Uo_Cxo:
return V.reshape(orig_shape), Uo_Cxo
else:
return V.reshape(orig_shape)
else:
# ====================================================
# = # If x and y are the same array, save some work: =
# ====================================================
if symm:
C = self.eval_fun(x, x, symm=True, **self.params)
# Update return value using observations.
if self.observed and observed:
Cxo = self.eval_fun(self.obs_mesh, x, **self.params)
Uo_Cxo = trisolve(self.Uo, Cxo, uplo='U', transa='T')
C -= Uo_Cxo.T * Uo_Cxo
if return_Uo_Cxo:
return C, Uo_Cxo
else:
return C
# ======================================
# = # If x and y are different arrays: =
# ======================================
else:
if regularize:
y = regularize_array(y)
ndimy = y.shape[-1]
leny = y.shape[0]
if not ndimx == ndimy:
raise ValueError, 'The last dimension of x and y (the number of spatial dimensions) must be the same.'
C = self.eval_fun(x, y, **self.params)
# Update return value using observations.
if self.observed and observed:
# If there are observation points, prepare self(obs_mesh, y)
# and chol(self(obs_mesh, obs_mesh)).T.I * self(obs_mesh, y)
Cyo = self.eval_fun(self.obs_mesh, y, **self.params)
Uo_Cyo = trisolve(self.Uo, Cyo, uplo='U', transa='T')
C -= Uo_Cxo.T * Uo_Cyo
return C
def continue_cholesky(self,
x,
x_old,
chol_dict_old,
apply_pivot=True,
observed=True,
nugget=None,
regularize=True,
assume_full_rank=False,
rank_limit=0):
"""
U = C.continue_cholesky(x, x_old, chol_dict_old[, observed=True, nugget=None,
rank_limit=0])
returns {'pivots': piv, 'U': U}
Computes incomplete Cholesky factorization of self(z,z), without
actually evaluating the matrix first. Here z is the concatenation of x
and x_old. 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.
- `chol_dict_old`: A dictionary with kbasis_ys ['pivots', 'U']. Would be the
output of either this method or C.cholesky().
- `apply_pivot`: A flag. If it's set to 'True', it returns a
matrix U (not necessarily triangular) such that U.T*U=C(x,x).
If it's set to 'False', the return value is a dictionary.
Item 'pivots' is a vector of pivots, and item 'U' is an
upper-triangular matrix (not necessarily square) such that
U[:,argsort(piv)].T * U[:,argsort(piv)] = C(x,x).
- `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.
- `rank_limit`: If rank_limit > 0, the factor will have at most
rank_limit rows.
"""
if regularize:
x = regularize_array(x)
# Concatenation of the old points and new points.
xtot = vstack((x_old, x))
# Extract information from chol_dict_old.
U_old = chol_dict_old['U']
m_old = U_old.shape[0]
piv_old = chol_dict_old['pivots']
# Number of old points.
N_old = x_old.shape[0]
# Number of new points.
N_new = x.shape[0]
if rank_limit == 0:
m_new_max = N_new
else:
m_new_max = min(N_new, max(0, rank_limit - m_old))
# get-row function
def rowfun(i, xpiv, rowvec):
"""
A function that can be used to overwrite an input array with superdiagonal rows.
"""
rowvec[i:] = self.__call__(x=xpiv[i - 1, :].reshape(1, -1),
y=xpiv[i:, :],
regularize=False,
observed=observed)
# diagonal
diag = self.__call__(x, y=None, regularize=False, observed=observed)
# not really implemented yet.
if nugget is not None:
diag += nugget.ravel()
# Arrange U for input to ichol. See documentation.
U = asmatrix(
zeros((m_new_max + m_old, N_old + N_new), dtype=float, order='F'))
U[:m_old, :m_old] = U_old[:, :m_old]
U[:m_old, N_new + m_old:] = U_old[:, m_old:]
offdiag = self.__call__(x=x_old[piv_old[:m_old], :],
y=x,
observed=observed,
regularize=False)
trisolve(U_old[:, :m_old], offdiag, uplo='U', transa='T', inplace=True)
U[:m_old, m_old:N_new + m_old] = offdiag
# Initialize pivot vector:
# [old_posdef_pivots new_pivots old_singular_pivots]
# - old_posdef_pivots are the indices of the rows that made it into the Cholesky factor so far.
# - old_singular_pivots are the indices of the rows that haven't made it into the Cholesky factor so far.
# - new_pivots are the indices of the rows that are going to be incorporated now.
piv = zeros(N_new + N_old, dtype=int)
piv[:m_old] = piv_old[:m_old]
piv[N_new + m_old:] = piv_old[m_old:]
piv[m_old:N_new + m_old] = arange(N_new) + N_old
# ============================================
# = Call to Fortran function ichol_continue. =
# ============================================
# Early return if rank is all used up.
if m_new_max > 0:
# ============================================
# = Call to Fortran function ichol_continue. =
# ============================================
if not assume_full_rank:
m, piv = ichol_continue(U,
diag=diag,
reltol=self.relative_precision,
rowfun=rowfun,
piv=piv,
x=xtot[piv, :],
mold=m_old)
else:
m = m_old + N_new
C_eval = self.__call__(x, x, observed=True, regularize=False)
U2 = cholesky(C_eval).T
U[m_old:, m_old:N_new + m_old] = U2
if m_old < N_old:
offdiag2 = self.__call__(x=x,
y=x_old[piv_old[m_old:]],
observed=observed,
regularize=False)
trisolve(U2, offdiag2, uplo='U', transa='T', inplace=True)
U[m_old:, N_new + m_old:] = offdiag2
else:
m = m_old
# Arrange output matrix and return.
if m < 0:
raise ValueError, 'Matrix does not appear positive semidefinite.'
if not apply_pivot:
# Useful for self.observe. U is upper triangular.
U = U[:m, :]
if assume_full_rank:
return {'pivots': piv, 'U': U, 'C_eval': C_eval, 'U_new': U2}
else:
return {'pivots': piv, 'U': U}
else:
# Useful for the user. U.T * U = C(x,x).
return U[:m, argsort(piv)]
def _obs_reg(self, M, dev_new, m_old):
# reg_mat = chol(self.basis_o.T * self.coef_cov * self.basis_o + diag(obs_V)).T.I * self.basis_o.T * self.coef_cov *
# chol(self(obs_mesh_*, obs_mesh_*)).T.I * M.dev
M.reg_mat += self.Uo_cov.T * asmatrix(trisolve(self.Uo, dev_new, uplo='U', transa='T')).T
return M.reg_mat
def _unobs_reg(self, M):
# reg_mat = chol(self.basis_o.T * self.coef_cov * self.basis_o + diag(obs_V)).T.I * self.basis_o.T * self.coef_cov *
# chol(self(obs_mesh_*, obs_mesh_*)).T.I * M.dev
return self.Uo_cov.T * asmatrix(trisolve(self.Uo, M.dev, uplo='U', transa='T')).T
def continue_cholesky(self, x, x_old, chol_dict_old, apply_pivot = True, observed=True, nugget=None, regularize=True, assume_full_rank = False, rank_limit=0):
"""
U = C.continue_cholesky(x, x_old, chol_dict_old[, observed=True, nugget=None,
rank_limit=0])
returns {'pivots': piv, 'U': U}
Computes incomplete Cholesky factorization of self(z,z), without
actually evaluating the matrix first. Here z is the concatenation of x
and x_old. 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.
- `chol_dict_old`: A dictionary with kbasis_ys ['pivots', 'U']. Would be the
output of either this method or C.cholesky().
- `apply_pivot`: A flag. If it's set to 'True', it returns a
matrix U (not necessarily triangular) such that U.T*U=C(x,x).
If it's set to 'False', the return value is a dictionary.
Item 'pivots' is a vector of pivots, and item 'U' is an
upper-triangular matrix (not necessarily square) such that
U[:,argsort(piv)].T * U[:,argsort(piv)] = C(x,x).
- `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.
- `rank_limit`: If rank_limit > 0, the factor will have at most
rank_limit rows.
"""
if regularize:
x=regularize_array(x)
# Concatenation of the old points and new points.
xtot = vstack((x_old,x))
# Extract information from chol_dict_old.
U_old = chol_dict_old['U']
m_old = U_old.shape[0]
piv_old = chol_dict_old['pivots']
# Number of old points.
N_old = x_old.shape[0]
# Number of new points.
N_new = x.shape[0]
if rank_limit == 0:
m_new_max = N_new
else:
m_new_max = min(N_new,max(0,rank_limit-m_old))
# get-row function
def rowfun(i,xpiv,rowvec):
"""
A function that can be used to overwrite an input array with superdiagonal rows.
"""
rowvec[i:] = self.__call__(x=xpiv[i-1,:].reshape(1,-1), y=xpiv[i:,:], regularize=False, observed = observed)
# diagonal
diag = self.__call__(x, y=None, regularize=False, observed = observed)
# not really implemented yet.
if nugget is not None:
diag += nugget.ravel()
# Arrange U for input to ichol. See documentation.
U = asmatrix(zeros((m_new_max + m_old, N_old + N_new), dtype=float, order='F'))
U[:m_old, :m_old] = U_old[:,:m_old]
U[:m_old,N_new+m_old:] = U_old[:,m_old:]
offdiag = self.__call__(x=x_old[piv_old[:m_old],:], y=x, observed=observed, regularize=False)
trisolve(U_old[:,:m_old],offdiag,uplo='U',transa='T', inplace=True)
U[:m_old, m_old:N_new+m_old] = offdiag
# Initialize pivot vector:
# [old_posdef_pivots new_pivots old_singular_pivots]
# - old_posdef_pivots are the indices of the rows that made it into the Cholesky factor so far.
# - old_singular_pivots are the indices of the rows that haven't made it into the Cholesky factor so far.
# - new_pivots are the indices of the rows that are going to be incorporated now.
piv = zeros(N_new + N_old, dtype=int)
piv[:m_old] = piv_old[:m_old]
piv[N_new + m_old:] = piv_old[m_old:]
piv[m_old:N_new + m_old] = arange(N_new)+N_old
# ============================================
# = Call to Fortran function ichol_continue. =
# ============================================
# Early return if rank is all used up.
if m_new_max > 0:
# ============================================
# = Call to Fortran function ichol_continue. =
# ============================================
if not assume_full_rank:
m, piv = ichol_continue(U, diag = diag, reltol = self.relative_precision, rowfun = rowfun, piv=piv, x=xtot[piv,:], mold=m_old)
else:
m = m_old + N_new
C_eval = self.__call__(x,x,observed=True,regularize=False)
U2 = cholesky(C_eval).T
U[m_old:,m_old:N_new+m_old] = U2
if m_old < N_old:
offdiag2 = self.__call__(x=x, y=x_old[piv_old[m_old:]], observed=observed, regularize=False)
trisolve(U2,offdiag2,uplo='U',transa='T',inplace=True)
U[m_old:,N_new+m_old:] = offdiag2
else:
m = m_old
# Arrange output matrix and return.
if m<0:
raise ValueError, 'Matrix does not appear positive semidefinite.'
if not apply_pivot:
# Useful for self.observe. U is upper triangular.
U = U[:m,:]
if assume_full_rank:
return {'pivots': piv, 'U': U, 'C_eval':C_eval, 'U_new': U2}
else:
return {'pivots': piv, 'U': U}
else:
# Useful for the user. U.T * U = C(x,x).
return U[:m,argsort(piv)]
def _obs_eval(self, M, M_out, x):
Cxo = self(M.obs_mesh, x, observed = False)
Cxo_Uo_inv = trisolve(M.Uo, Cxo, uplo='U', transa='T')
M_out += dot(asarray(M.reg_mat).T,asarray(Cxo_Uo_inv)).squeeze()
return M_out
def continue_cholesky(self, x, x_old, chol_dict_old, apply_pivot = True, observed=True, nugget=None, regularize=True, assume_full_rank=False):
"""
U = C.continue_cholesky(x, x_old, chol_dict_old[, observed=True, nugget=None])
{'pivots': piv, 'U': U} = \
C.cholesky(x, x_old, chol_dict_old, apply_pivot = False[, observed=True, nugget=None])
Computes incomplete Cholesky factorization of self(z,z). Here z is the
concatenation of x and x_old. 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.
- `chol_dict_old`: A dictionary with kbasis_ys ['pivots', 'U']. Would be the
output of either this method or C.cholesky().
- `apply_pivot`: A flag. If it's set to 'True', it returns a
matrix U (not necessarily triangular) such that U.T*U=C(x,x).
If it's set to 'False', the return value is a dictionary.
Item 'pivots' is a vector of pivots, and item 'U' is an
upper-triangular matrix (not necessarily square) such that
U[:,argsort(piv)].T * U[:,argsort(piv)] = C(x,x).
- `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.
"""
if regularize:
x=regularize_array(x)
# Concatenation of the old points and new points.
xtot = vstack((x_old,x))
# Extract information from chol_dict_old.
U_old = chol_dict_old['U']
m_old = U_old.shape[0]
piv_old = chol_dict_old['pivots']
# Number of old points.
N_old = x_old.shape[0]
# Number of new points.
N_new = x.shape[0]
# Compute off-diagonal part of Cholesky factor
offdiag = self.__call__(x=x_old[piv_old[:m_old],:], y=x, observed=observed, regularize=False)
trisolve(U_old[:,:m_old],offdiag,uplo='U',transa='T', inplace=True)
# Compute new diagonal part of Cholesky factor
C_new = self.__call__(x=x, y=x, observed=observed, regularize=False)
if nugget is not None:
for i in xrange(N_new):
C_new[i,i] += nugget[i]
C_new -= offdiag.T*offdiag
if not assume_full_rank:
U_new, m_new, piv_new = ichol_full(c=C_new, reltol=self.relative_precision)
else:
U_new = cholesky(C_new).T
m_new = U_new.shape[0]
piv_new = arange(m_new)
U_new = asmatrix(U_new[:m_new,:])
U = asmatrix(zeros((m_new + m_old, N_old + N_new), dtype=float, order='F'))
# Top portion of U
U[:m_old, :m_old] = U_old[:,:m_old]
U[:m_old,N_new+m_old:] = U_old[:,m_old:]
offdiag=offdiag[:,piv_new]
U[:m_old, m_old:N_new+m_old] = offdiag
# Lower portion of U
U[m_old:,m_old:m_old+N_new] = U_new
if m_old < N_old and m_new > 0:
offdiag_lower = self.__call__( x=x[piv_new[:m_new],:],
y=x_old[piv_old[m_old:],:], observed=observed, regularize=False)
offdiag_lower -= offdiag[:,:m_new].T*U[:m_old,m_old+N_new:]
trisolve(U_new[:,:m_new],offdiag_lower,uplo='U',transa='T', inplace=True)
U[m_old:,m_old+N_new:] = offdiag_lower
# Rank and pivots
m=m_old+m_new
piv=hstack((piv_old[:m_old],piv_new+N_old,piv_old[m_old:]))
# Arrange output matrix and return.
if m<0:
raise ValueError, 'Matrix does not appear positive semidefinite.'
if not apply_pivot:
# Useful for self.observe. U is upper triangular.
return {'pivots': piv, 'U': U}
else:
# Useful for the user. U.T * U = C(x,x).
return U[:,argsort(piv)]
def _obs_eval(self, M, M_out, x, Uo_Cxo=None):
if Uo_Cxo is None:
Uo_Cxo = trisolve(M.Uo, self(M.obs_mesh, x, observed = False), uplo='U', transa='T')
M_out += dot(asarray(M.reg_mat).squeeze(),asarray(Uo_Cxo)).squeeze()
return M_out
def _obs_reg(self, M, dev_new, m_old):
# reg_mat = chol(C(obs_mesh_*, obs_mesh_*)).T.I * M.dev
reg_mat_new = -1.*dot(self.Uo[:m_old,m_old:].T , trisolve(self.Uo[:m_old,:m_old], M.dev, uplo='U', transa='T')).T
trisolve(self.Uo[m_old:,m_old:].T, reg_mat_new, 'L', inplace=True)
reg_mat_new += asmatrix(trisolve(self.Uo[m_old:,m_old:], dev_new.T, uplo='U', transa='T')).T
return asmatrix(vstack((M.reg_mat,reg_mat_new)))
def _unobs_reg(self, M):
# reg_mat = chol(C(obs_mesh_*, obs_mesh_*)).T.I * M.dev
return asmatrix(trisolve(self.Uo, M.dev.T, uplo='U',transa='T')).T
def __call__(self, x, y=None, observed=True, regularize=True, return_Uo_Cxo=False):
if y is x:
symm=True
else:
symm=False
# Remember shape of x, and then 'regularize' it.
orig_shape = shape(x)
if len(orig_shape)>1:
orig_shape = orig_shape[:-1]
if regularize:
x=regularize_array(x)
ndimx = x.shape[-1]
lenx = x.shape[0]
if return_Uo_Cxo:
Uo_Cxo = None
# Safety
if self.ndim is not None:
if not self.ndim == ndimx:
raise ValueError, "The number of spatial dimensions of x, "+\
ndimx.__str__()+\
", does not match the number of spatial dimensions of the Covariance instance's base mesh, "+\
self.ndim.__str__()+"."
# If there are observation points, prepare self(obs_mesh, x)
# and chol(self(obs_mesh, obs_mesh)).T.I * self(obs_mesh, x)
# ==========================================================
# = If only one argument is provided, return the diagonal. =
# ==========================================================
if y is None:
# Special fast-path for functions that have an 'amp' parameter
if hasattr(self.eval_fun, 'diag_call'):
V = self.eval_fun.diag_call(x, **self.params)
# Otherwise, evaluate the diagonal in a loop.
else:
V=empty(lenx,dtype=float)
for i in xrange(lenx):
this_x = x[i].reshape((1,-1))
V[i] = self.eval_fun(this_x, this_x, **self.params)
if self.observed and observed:
sqpart = empty(lenx,dtype=float)
Cxo = self.eval_fun(self.obs_mesh, x, **self.params)
Uo_Cxo = trisolve(self.Uo, Cxo, uplo='U', transa='T')
square_and_sum(Uo_Cxo, sqpart)
V -= sqpart
if return_Uo_Cxo:
return V.reshape(orig_shape), Uo_Cxo
else:
return V.reshape(orig_shape)
else:
# ====================================================
# = # If x and y are the same array, save some work: =
# ====================================================
if symm:
C=self.eval_fun(x,x,symm=True,**self.params)
# Update return value using observations.
if self.observed and observed:
Cxo = self.eval_fun(self.obs_mesh, x, **self.params)
Uo_Cxo = trisolve(self.Uo, Cxo, uplo='U', transa='T')
C -= Uo_Cxo.T * Uo_Cxo
if return_Uo_Cxo:
return C, Uo_Cxo
else:
return C
# ======================================
# = # If x and y are different arrays: =
# ======================================
else:
if regularize:
y=regularize_array(y)
ndimy = y.shape[-1]
leny = y.shape[0]
if not ndimx==ndimy:
raise ValueError, 'The last dimension of x and y (the number of spatial dimensions) must be the same.'
C = self.eval_fun(x,y,**self.params)
# Update return value using observations.
if self.observed and observed:
# If there are observation points, prepare self(obs_mesh, y)
# and chol(self(obs_mesh, obs_mesh)).T.I * self(obs_mesh, y)
Cyo = self.eval_fun(self.obs_mesh, y, **self.params)
Uo_Cyo = trisolve(self.Uo, Cyo,uplo='U', transa='T')
C -= Uo_Cxo.T * Uo_Cyo
return C
def observe(self, obs_mesh, obs_V, output_type='o'):
__doc__ = Covariance.observe.__doc__
ndim = obs_mesh.shape[1]
nobs = obs_mesh.shape[0]
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
# bases_o is the basis evaluated on the observation mesh.
basis_o = asmatrix(self.eval_basis(obs_mesh, regularize=False))
# chol(basis_o.T * coef_cov * basis_o)
chol_inner = self.coef_U * basis_o
if output_type == 's':
C_eval = dot(chol_inner.T, chol_inner).copy('F')
U_eval = linalg.cholesky(C_eval).T.copy('F')
U = U_eval
m = U_eval.shape[0]
piv = arange(m)
else:
# chol(basis_o.T * coef_cov * basis_o + diag(obs_V)). Really should do this as low-rank update of covariance
# of V.
U, piv, m = ichol_basis(basis=chol_inner,
nug=obs_V,
reltol=self.relative_precision)
U = asmatrix(U)
piv_new = piv[:m]
self.obs_piv = piv_new
obs_mesh_new = obs_mesh[piv_new, :]
self.Uo = U[:m, :m]
# chol(basis_o.T * coef_cov * basis_o + diag(obs_V)) ^ -T * basis_o.T
self.Uo_cov = trisolve(self.Uo,
basis_o[:, piv[:m]].T,
uplo='U',
transa='T')
# chol(basis_o.T * coef_cov * basis_o + diag(obs_V)).T.I * basis_o.T * coef_cov
self.Uo_cov = self.Uo_cov * self.coef_cov
# coef_cov = coef_cov - coef_cov * basis_o * (basis_o.T * coef_cov * basis_o + diag(obs_V)).I * basis_o.T * coef_cov
self.coef_cov = self.coef_cov - self.Uo_cov.T * self.Uo_cov
# coef_U = chol(coef_cov)
U, m, piv = ichol_full(c=self.coef_cov, reltol=self.relative_precision)
U = asmatrix(U)
self.coef_U = U[:m, argsort(piv)]
self.m = m
if output_type == 'o':
return piv_new, obs_mesh_new
if output_type == 's':
return U_eval, C_eval, basis_o
raise ValueError, 'Output type not recognized.'
def _unobs_reg(self, M):
# reg_mat = chol(self.basis_o.T * self.coef_cov * self.basis_o + diag(obs_V)).T.I * self.basis_o.T * self.coef_cov *
# chol(self(obs_mesh_*, obs_mesh_*)).T.I * M.dev
return self.Uo_cov.T * asmatrix(
trisolve(self.Uo, M.dev, uplo='U', transa='T')).T
def _obs_reg(self, M, dev_new, m_old):
# reg_mat = chol(self.basis_o.T * self.coef_cov * self.basis_o + diag(obs_V)).T.I * self.basis_o.T * self.coef_cov *
# chol(self(obs_mesh_*, obs_mesh_*)).T.I * M.dev
M.reg_mat = M.reg_mat + self.Uo_cov.T * asmatrix(
trisolve(self.Uo, dev_new, uplo='U', transa='T')).T
return M.reg_mat
def continue_cholesky(self, x, x_old, chol_dict_old, apply_pivot = True, observed=True, nugget=None, regularize=True, assume_full_rank=False, rank_limit=0):
"""
U = C.continue_cholesky(x, x_old, chol_dict_old[, observed=True, nugget=None,
rank_limit=0])
Returns {'pivots': piv, 'U': U}
Computes incomplete Cholesky factorization of self(z,z). Here z is the
concatenation of x and x_old. 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.
- `chol_dict_old`: A dictionary with kbasis_ys ['pivots', 'U']. Would be the
output of either this method or C.cholesky().
- `apply_pivot`: A flag. If it's set to 'True', it returns a
matrix U (not necessarily triangular) such that U.T*U=C(x,x).
If it's set to 'False', the return value is a dictionary.
Item 'pivots' is a vector of pivots, and item 'U' is an
upper-triangular matrix (not necessarily square) such that
U[:,argsort(piv)].T * U[:,argsort(piv)] = C(x,x).
- `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.
- `rank_limit`: If rank_limit > 0, the factor will have at most
rank_limit rows.
"""
if regularize:
x=regularize_array(x)
if rank_limit > 0:
raise ValueError, 'NearlyFullRankCovariance does not accept a rank_limit argument. Use Covariance instead.'
if rank_limit > 0:
raise ValueError, 'NearlyFullRankCovariance does not accept a rank_limit argument. Use Covariance instead.'
# Concatenation of the old points and new points.
xtot = vstack((x_old,x))
# Extract information from chol_dict_old.
U_old = chol_dict_old['U']
m_old = U_old.shape[0]
piv_old = chol_dict_old['pivots']
# Number of old points.
N_old = x_old.shape[0]
# Number of new points.
N_new = x.shape[0]
# Compute off-diagonal part of Cholesky factor
offdiag = self.__call__(x=x_old[piv_old[:m_old],:], y=x, observed=observed, regularize=False)
trisolve(U_old[:,:m_old],offdiag,uplo='U',transa='T', inplace=True)
# Compute new diagonal part of Cholesky factor
C_new = self.__call__(x=x, y=x, observed=observed, regularize=False)
if nugget is not None:
for i in xrange(N_new):
C_new[i,i] += nugget[i]
C_new -= offdiag.T*offdiag
if not assume_full_rank:
U_new, m_new, piv_new = ichol_full(c=C_new, reltol=self.relative_precision)
else:
U_new = cholesky(C_new).T
m_new = U_new.shape[0]
piv_new = arange(m_new)
U_new = asmatrix(U_new[:m_new,:])
U = asmatrix(zeros((m_new + m_old, N_old + N_new), dtype=float, order='F'))
# Top portion of U
U[:m_old, :m_old] = U_old[:,:m_old]
U[:m_old,N_new+m_old:] = U_old[:,m_old:]
offdiag=offdiag[:,piv_new]
U[:m_old, m_old:N_new+m_old] = offdiag
# Lower portion of U
U[m_old:,m_old:m_old+N_new] = U_new
if m_old < N_old and m_new > 0:
offdiag_lower = self.__call__( x=x[piv_new[:m_new],:],
y=x_old[piv_old[m_old:],:], observed=observed, regularize=False)
offdiag_lower -= offdiag[:,:m_new].T*U[:m_old,m_old+N_new:]
trisolve(U_new[:,:m_new],offdiag_lower,uplo='U',transa='T', inplace=True)
U[m_old:,m_old+N_new:] = offdiag_lower
# Rank and pivots
m=m_old+m_new
piv=hstack((piv_old[:m_old],piv_new+N_old,piv_old[m_old:]))
# Arrange output matrix and return.
if m<0:
raise ValueError, 'Matrix does not appear positive semidefinite.'
if not apply_pivot:
# Useful for self.observe. U is upper triangular.
if assume_full_rank:
return {'pivots':piv,'U':U,'C_eval':C_new,'U_new':U_new}
else:
return {'pivots': piv, 'U': U}
else:
# Useful for the user. U.T * U = C(x,x).
return U[:,argsort(piv)]
def __call__(self, x, y=None, observed=True, regularize=True):
if y is x:
symm=True
else:
symm=False
# Remember shape of x, and then 'regularize' it.
orig_shape = shape(x)
if len(orig_shape)>1:
orig_shape = orig_shape[:-1]
if regularize:
x=regularize_array(x)
ndimx = x.shape[-1]
lenx = x.shape[0]
# Safety
if self.ndim is not None:
if not self.ndim == ndimx:
raise ValueError, "The number of spatial dimensions of x, "+\
ndimx.__str__()+\
", does not match the number of spatial dimensions of the Covariance instance's base mesh, "+\
self.ndim.__str__()+"."
# If there are observation points, prepare self(obs_mesh, x)
# and chol(self(obs_mesh, obs_mesh)).T.I * self(obs_mesh, x)
if self.observed and observed:
Cxo = self.eval_fun(self.obs_mesh, x, **self.params)
Uo_Cxo = trisolve(self.Uo, Cxo, uplo='U', transa='T')
# ==========================================================
# = If only one argument is provided, return the diagonal. =
# ==========================================================
# See documentation.
if y is None:
# V = diag_call(x=x, cov_fun = self.diag_cov_fun)
# =============================================================================
# = Come up with a better solution, the diagonal is not necessarily constant. =
# =============================================================================
V=empty(lenx,dtype=float)
# V.fill(self.params['amp']**2)
for i in xrange(lenx):
this_x = x[i].reshape((1,-1))
V[i] = self.eval_fun(this_x, this_x,**self.params)
if self.observed and observed:
for i in range(lenx):
this_Uo_Cxo = Uo_Cxo[:,i]
V[i] -= this_Uo_Cxo.T*this_Uo_Cxo
return V.reshape(orig_shape)
else:
# ====================================================
# = # If x and y are the same array, save some work: =
# ====================================================
if symm:
C=self.eval_fun(x,x,symm=True,**self.params)
# Update return value using observations.
if self.observed and observed:
C -= Uo_Cxo.T * Uo_Cxo
return C
# ======================================
# = # If x and y are different arrays: =
# ======================================
else:
if regularize:
y=regularize_array(y)
ndimy = y.shape[-1]
leny = y.shape[0]
if not ndimx==ndimy:
raise ValueError, 'The last dimension of x and y (the number of spatial dimensions) must be the same.'
C = self.eval_fun(x,y,**self.params)
# Update return value using observations.
if self.observed and observed:
# If there are observation points, prepare self(obs_mesh, y)
# and chol(self(obs_mesh, obs_mesh)).T.I * self(obs_mesh, y)
Cyo = self.eval_fun(self.obs_mesh, y, **self.params)
Uo_Cyo = trisolve(self.Uo, Cyo,uplo='U', transa='T')
C -= Uo_Cxo.T * Uo_Cyo
return C
