def gen_LDS(self):
self.log.info('forming LDS model')
# form the state space representation
A = np.hstack([
self.Phi_inv * outer(self.phi, self.phi, F) for F in self.F
])
# this is [A_1 A_2 .. A_p; I 0]
I = np.eye(len(self.phi)*(self.p-1))
O = np.zeros((len(self.phi)*(self.p-1), len(self.phi)))
A = np.vstack([A, np.hstack([I, O])])
if self.G:
B = np.hstack([
self.Phi_inv * outer(self.phi, self.phi, G)
for G in self.G
])
# this is [B_1 B_2 .. B_q; I 0]
O = np.zeros((len(self.phi)*(self.p-1), self.q*len(self.phi)))
B = np.vstack([B, O])
else:
B = None
# C = <H,phi>(s1 ... sn)
# <H,phi> = a^T <psi, phi>
Cop = np.array([inner(self.H, phi_i) for phi_i in self.phi])
# So we need to go through each basis function in Cop (C-operator) and
# evalutate it at each observation location. This begs the question of
# where the observation locations should come into the program. For
# the moment, though, we'll assume them fixed and store them in the
# HIDEX object
C = np.zeros((len(self.obs_locns),len(self.phi)*(self.p)))
for i,s in enumerate(self.obs_locns):
for j,basis_function in enumerate(Cop):
C[i,j] = basis_function(s)
# make the covariance matrices
Sw = np.zeros(
(len(self.phi)*(self.p), len(self.phi)*(self.p))
,dtype=float
)
for i, basis_i in enumerate(self.phi):
for j, basis_j in enumerate(self.phi):
Sw[i,j] = inner(basis_i, basis_j, self.Q)
# form the initial state
x0 = np.hstack([self.f0.weights,np.zeros((len(self.phi)*(self.p-1)))])
# turn everything into matrices for the LDS model
A = np.matrix(A)
B = np.matrix(B)
C = np.matrix(C)
Sw = np.matrix(Sw)
Sv = np.matrix(self.R)
x0 = np.matrix(x0).T
Pi0 = np.matrix(self.Pi0)
# make a state space model
return LDS.LDS(A, B, C, Sw, Sv, x0, Pi0)
def __init__(self, F, G, H, f0, Q, R, Pi0, obs_locns):
"""
Parameters
==========
F : list
list of dynamic kernels
G : list
list of input kernels
H : Kernel object
observation kernel
f0 : Field object
initial field
Q : CovarianceFunction object
covariance of the field
R : matrix
covariance matrix of the observation noise
Pi0 : matrix
initial covariance matrix of the field weights
obs_locns : list
list of observation locations
"""
# process and store arguments
self.p = len(F)
self.q = len(G)
self.psi = F[0].bases
self.phi = f0.bases
self.F = F
self.G = G
self.H = H
self.Q = Q
self.R = R
self.f0 = f0
self.Pi0 = Pi0
self.obs_locns = obs_locns
# form Phi = \int phi(s) phi^T(s) ds
self.Phi = outer(self.phi, self.phi)
# store the inversion
self.Phi_inv = np.linalg.inv(self.Phi)
# logging
self.log = logging.getLogger('Field')
self.log.info('Initialised new HIDEX model')
# genreate an LDS representation
self.LDS = self.gen_LDS()