dim_m = 4 m = np.linspace(1, dim_m, dim_m).reshape((dim_m, 1)) D = np.random.randn(dim_m**2).reshape((dim_m, dim_m)) U,S,VT = np.linalg.svd(D) C = U.dot(np.diag(S**2)).dot(U.T) invC = U.dot(np.diag(1/S**2)).dot(U.T) C_cholesky = U.dot(np.diag(S)) checkvalue = np.linalg.norm(C - C_cholesky.dot(C_cholesky.T))/np.linalg.norm(C) assert checkvalue < 2e-16, checkvalue checkvalue2 = np.linalg.norm(C.dot(invC) - np.eye(dim_m)) assert checkvalue2 < 1e-12, checkvalue2 ####################### Multivariate Gaussian ################## sample_size = 100000 samples = SampleMultivariateNormal(m, C_cholesky, sample_size) m_emp = samples.sum(axis=1).reshape((dim_m,1))/sample_size m_one = m.dot(np.ones((1,sample_size))) C_emp = ((samples-m_one).dot((samples-m_one).T))/sample_size print "Test Multivariate Gaussian:" print "m, m_emp, m-m_emp: " print np.concatenate((m, m_emp, m-m_emp), axis=1) print np.sqrt(np.sum((m_emp-m)**2))/np.sqrt(np.sum(m**2)) print "C, C_emp, C-C_emp: " print C print C_emp print C-C_emp print np.linalg.norm(C-C_emp)/np.linalg.norm(C) ########################## Wishart ############################# sample_size = 10000
def BackwardSampling_SVD(Gt, svd_invW, m0, m_all, svd_C0, C_all, a_all, R_all):
""" Sample from the joint distribution (theta_{0:T} | y_{1:T})
Inputs:
Gt = list of matrices containing definition of DLM
svd_invW = [U, S] such that U.S.U^T = W^{-1}
(W = covariance matrix for evolution equation)
m0, C0 = parameters of initial distribution
m_all, C_all, a_all, R_all = output from KalmanFilter_SVD
Outputs:
thetas = joint draw from (theta_{0:T} | y_{1:T})
maxVarB = maximum variance in B at each time step (diagonal only) """
timesteps, parameters = m_all.shape
thetas = np.zeros((timesteps+1, parameters))
invGam = svd_invW[0].dot(np.diag(np.sqrt(svd_invW[1])))
# Sample at time t=T
h = m_all[timesteps-1,:].reshape((parameters,1))
C = C_all[-1]
B_ch = C[0].dot(np.diag(C[1]))
theta = SampleMultivariateNormal(h, B_ch)
thetas[-1,:] = theta.reshape((1, parameters))
maxVarB = []
# To check, compute sT:
# s_all, S_all = np.zeros(m_all.shape), []
# s = m_all[timesteps-1,:].reshape((parameters,1))
# s_all[timesteps-1,:] = s.T
# Sfull = C[0].dot(np.diag(C[1]**2)).dot(C[0].T)
# S_all.append(Sfull)
# Iterate starting from t=T down to t=1:
ii = timesteps - 2
for G, Rsvd in zip(reversed(Gt), reversed(R_all)):
a = a_all[ii+1,:].reshape((parameters,1))
if ii < 0:
m = m0.reshape((parameters,1))
C = svd_C0[0].dot(np.diag(svd_C0[1])).dot(svd_C0[0].T)
invR = Rsvd[0].dot(np.diag(1/Rsvd[1]**2)).dot(Rsvd[0].T)
CGinvR = C.dot(G.T).dot(invR)
h = m + CGinvR.dot(theta - a)
tmp = (invGam.T).dot(G).dot(svd_C0[0])
A, invD, ET = svd(np.concatenate((tmp, \
np.diag(1/np.sqrt(svd_C0[1]))), axis=0))
U = svd_C0[0].dot(ET.T)
B_ch = U.dot(np.diag(1/invD))
theta = SampleMultivariateNormal(h, B_ch)
thetas[ii+1,:] = theta.reshape((1, parameters))
maxVarB.append(np.abs(np.diag(B_ch.dot(B_ch.T))).max())
break
m = m_all[ii,:].reshape((parameters,1))
Csvd = C_all[ii]
C = Csvd[0].dot(np.diag(Csvd[1]**2)).dot(Csvd[0].T)
invR = Rsvd[0].dot(np.diag(1/Rsvd[1]**2)).dot(Rsvd[0].T)
CGinvR = C.dot(G.T).dot(invR)
h = m + CGinvR.dot(theta - a)
tmp = (invGam.T).dot(G).dot(Csvd[0])
A, invD, ET = svd(np.concatenate((tmp, np.diag(1/Csvd[1])), axis=0))
U = Csvd[0].dot(ET.T)
B_ch = U.dot(np.diag(1/invD))
theta = SampleMultivariateNormal(h, B_ch)
thetas[ii+1,:] = theta.reshape((1, parameters))
maxVarB.append(np.abs(np.diag(B_ch.dot(B_ch.T))).max())
# To check, compute st
# s = m + CGinvR.dot(s - a)
# s_all[ii,:] = s.T
# Bfull = C - CGinvR.dot(G).dot(C)
# Bfull = C - C.dot(G.T).dot(invR).dot(G).dot(C)
# invBfullbis = np.linalg.inv(C) + \
# (G.T).dot(np.linalg.inv(W)).dot(G)
# Bfullbis = np.linalg.inv(invBfullbis)
# Rfull = Rsvd[0].dot(np.diag(Rsvd[1]**2)).dot(Rsvd[0].T)
# Sfull = C - CGinvR.dot(Rfull-Sfull).dot(CGinvR.T)
# S_all.append(Sfull)
ii -= 1
return thetas, maxVarB
def BackwardSampling_SVD(Gt, svd_invW, m0, m_all, svd_C0, C_all, a_all, R_all):
""" Sample from the joint distribution (theta_{0:T} | y_{1:T})
Inputs:
Gt = list of matrices containing definition of DLM
svd_invW = [U, S] such that U.S.U^T = W^{-1}
(W = covariance matrix for evolution equation)
m0, C0 = parameters of initial distribution
m_all, C_all, a_all, R_all = output from KalmanFilter_SVD
Outputs:
thetas = joint draw from (theta_{0:T} | y_{1:T})
maxVarB = maximum variance in B at each time step (diagonal only) """
timesteps, parameters = m_all.shape
thetas = np.zeros((timesteps + 1, parameters))
invGam = svd_invW[0].dot(np.diag(np.sqrt(svd_invW[1])))
# Sample at time t=T
h = m_all[timesteps - 1, :].reshape((parameters, 1))
C = C_all[-1]
B_ch = C[0].dot(np.diag(C[1]))
theta = SampleMultivariateNormal(h, B_ch)
thetas[-1, :] = theta.reshape((1, parameters))
maxVarB = []
# To check, compute sT:
# s_all, S_all = np.zeros(m_all.shape), []
# s = m_all[timesteps-1,:].reshape((parameters,1))
# s_all[timesteps-1,:] = s.T
# Sfull = C[0].dot(np.diag(C[1]**2)).dot(C[0].T)
# S_all.append(Sfull)
# Iterate starting from t=T down to t=1:
ii = timesteps - 2
for G, Rsvd in zip(reversed(Gt), reversed(R_all)):
a = a_all[ii + 1, :].reshape((parameters, 1))
if ii < 0:
m = m0.reshape((parameters, 1))
C = svd_C0[0].dot(np.diag(svd_C0[1])).dot(svd_C0[0].T)
invR = Rsvd[0].dot(np.diag(1 / Rsvd[1]**2)).dot(Rsvd[0].T)
CGinvR = C.dot(G.T).dot(invR)
h = m + CGinvR.dot(theta - a)
tmp = (invGam.T).dot(G).dot(svd_C0[0])
A, invD, ET = svd(np.concatenate((tmp, \
np.diag(1/np.sqrt(svd_C0[1]))), axis=0))
U = svd_C0[0].dot(ET.T)
B_ch = U.dot(np.diag(1 / invD))
theta = SampleMultivariateNormal(h, B_ch)
thetas[ii + 1, :] = theta.reshape((1, parameters))
maxVarB.append(np.abs(np.diag(B_ch.dot(B_ch.T))).max())
break
m = m_all[ii, :].reshape((parameters, 1))
Csvd = C_all[ii]
C = Csvd[0].dot(np.diag(Csvd[1]**2)).dot(Csvd[0].T)
invR = Rsvd[0].dot(np.diag(1 / Rsvd[1]**2)).dot(Rsvd[0].T)
CGinvR = C.dot(G.T).dot(invR)
h = m + CGinvR.dot(theta - a)
tmp = (invGam.T).dot(G).dot(Csvd[0])
A, invD, ET = svd(np.concatenate((tmp, np.diag(1 / Csvd[1])), axis=0))
U = Csvd[0].dot(ET.T)
B_ch = U.dot(np.diag(1 / invD))
theta = SampleMultivariateNormal(h, B_ch)
thetas[ii + 1, :] = theta.reshape((1, parameters))
maxVarB.append(np.abs(np.diag(B_ch.dot(B_ch.T))).max())
# To check, compute st
# s = m + CGinvR.dot(s - a)
# s_all[ii,:] = s.T
# Bfull = C - CGinvR.dot(G).dot(C)
# Bfull = C - C.dot(G.T).dot(invR).dot(G).dot(C)
# invBfullbis = np.linalg.inv(C) + \
# (G.T).dot(np.linalg.inv(W)).dot(G)
# Bfullbis = np.linalg.inv(invBfullbis)
# Rfull = Rsvd[0].dot(np.diag(Rsvd[1]**2)).dot(Rsvd[0].T)
# Sfull = C - CGinvR.dot(Rfull-Sfull).dot(CGinvR.T)
# S_all.append(Sfull)
ii -= 1
return thetas, maxVarB