def riccati(c_yy, x, y,t):
'''
Description:
Solves the Riccati equation, specifically AOA^T - O + Q = 0
Parameters:
c_yy: a whitened covariance matrix, used in calculation of Q
x,y: matrices that are used to calculate koopman estimators
t: timelaga
Returns:
The matrix, O, solved in the above equation.
'''
# c_xx, c_yy = whitening(x,y)
K_xx, K_xy = koopman_est(x, y ,t)
evals_xx,evect_xx = np.linalg.eigh(K_xx) #hermetian
# evals_xy,evect_xy = np.linalg.eigh(K_xy)
print ('\n', evals_xx, '\n \n', evect_xx)
v_x = evect_xx[:,list(evals_xx).index(1)]
try:
v_y = np.linalg.solve(K_xy, v_x) # I think this v_y is extraneous, there isn't v_y used anywhere
except:
raise Error('cannot find eigenvalue = 1')
chi_bar = np.mean(chi(x).T, axis =1)
gamma_bar = np.mean(gamma(y).T, axis =1)
A = K_xx - np.outer(v_x*chi_bar)
B = K_xy - np.outer(v_x*gamma_bar)
Q = B.dot(np.inverse(c_yy)).dot(B.T)
return solve_discrete_lyapunov(A, Q)
def ComputeG(A0, A1, d, Q0, tau0, beta, mu):
'''
COMPUTEG Compute government income given mu and return tax revenues, and
Policy matrixes for the planner. Here A0,A1,d,Q0,tau0,beta and mu are all
asssumed to be floats
'''
#Create Matrixes
R = array([[0, -A0 / 2, 0, 0], [-A0 / 2, A1 / 2, -mu / 2, 0],
[0, -mu / 2, 0, 0], [0, 0, 0, d / 2]])
A = array([[1, 0, 0, 0], [0, 1, 0, 1], [0, 0, 0, 0],
[-A0 / d, A1 / d, 0, A1 / d + 1 / beta]])
B = array([0, 0, 1, 1 / d]).reshape(-1, 1)
Q = 0
#OLRP to solve the Ramsey Problem.
F, P = olrp(beta, A, B, -R, Q)
#Need y_0 to compute government tax revenue.
P21 = P[3, :3]
P22 = P[3, 3]
z0 = array([1, Q0, tau0]).reshape(-1, 1)
u0 = -P22**(-1) * P21.dot(z0)
y0 = vstack([z0, u0])
#Define A_F and S matricies
AF = A - B.dot(F)
S = array([0, 1, 0, 0]).reshape(-1,
1).dot(array([0, 0, 1, 0]).reshape(1, -1))
#Solves equation (25)
#Omega = solve_sylvester(beta*AF.T,-linalg.inv(AF),-beta*AF.T.dot(S))
Omega = solve_discrete_lyapunov(
sqrt(beta) * AF.T,
beta * AF.T.dot(S).dot(AF))
T0 = y0.T.dot(Omega).dot(y0)
return T0, A, B, F, P
def ComputeG( A0,A1,d,Q0,tau0,beta,mu ):
'''
COMPUTEG Compute government income given mu and return tax revenues, and
Policy matrixes for the planner. Here A0,A1,d,Q0,tau0,beta and mu are all
asssumed to be floats
'''
#Create Matrixes
R = array([[0,-A0/2,0,0],[-A0/2,A1/2, -mu/2, 0],[0,-mu/2,0,0],[0,0,0,d/2]])
A = array([[1,0,0,0],[0,1,0,1],[0,0,0,0],[-A0/d,A1/d,0,A1/d+1/beta]])
B = array([0,0,1,1/d]).reshape(-1,1)
Q = 0;
#OLRP to solve the Ramsey Problem.
F,P = olrp(beta,A,B,-R,Q)
#Need y_0 to compute government tax revenue.
P21 = P[3,:3]; P22 = P[3,3];
z0 = array([1,Q0,tau0]).reshape(-1,1)
u0 = -P22**(-1)*P21.dot(z0)
y0 = vstack([z0,u0])
#Define A_F and S matricies
AF = A-B.dot(F);
S =array([0,1,0,0]).reshape(-1,1).dot(array([0,0,1,0]).reshape(1,-1))
#Solves equation (25)
#Omega = solve_sylvester(beta*AF.T,-linalg.inv(AF),-beta*AF.T.dot(S))
Omega = solve_discrete_lyapunov(sqrt(beta)*AF.T,beta*AF.T.dot(S).dot(AF))
T0 = y0.T.dot(Omega).dot(y0)
return T0,A,B,F,P
def matrin_dist_for_LDS(lds1, lds2):
# According to the formula (9) of the paper, the final matrix A is obtained, and the dimension is (2D, 2D)
A = torch.zeros((lds1[0].shape[0] * 2, lds1[0].shape[1] * 2))
A[:lds1[0].shape[0], :lds1[0].shape[0]] = lds1[0]
A[lds1[0].shape[0]:, lds1[0].shape[0]:] = lds2[0]
# According to the formula (10) of the paper, the final matrix C is obtained, and the dimension is (D, 2D)
C = torch.zeros((lds1[0].shape[0], lds1[0].shape[1] * 2))
C[:, :lds1[0].shape[0]] = lds1[1]
C[:, lds1[0].shape[0]:] = lds2[1]
# According to the paper, find the solution to the Lyapunov equation (A_t P A - P = -Q, where Q is C_t C)
# First find Q in the Lyapunov equation, and then use the scipy package to solve to get P
Q = torch.mm(C.transpose(0, 1), C)
P = solve_discrete_lyapunov(A, Q)
# According to the formula (8) of the paper, the P of the corresponding position is obtained
P11 = P[:lds1[0].shape[0], :lds1[0].shape[0]]
P12 = P[:lds1[0].shape[0], lds1[0].shape[0]:]
P21 = P[lds1[0].shape[0]:, :lds1[0].shape[0]]
P22 = P[lds1[0].shape[0]:, lds1[0].shape[0]:]
# According to the formula (11) of the paper, the corresponding eigenvalues are obtained
eig_P = np.matmul(np.linalg.inv(P11), P12)
eig_P = np.matmul(eig_P, np.linalg.inv(P22))
eig_P = np.matmul(eig_P, P21)
eig_P = np.linalg.eig(eig_P)[0]
# According to the paper formula (12), the final distance is obtained
return np.sqrt(-np.log(np.prod(eig_P**2)))
def check_stationary_initialization_2dim(dtype=float):
endog = np.zeros(10, dtype=dtype)
# 2-dimensional example
mod = MLEModel(endog, k_states=2, k_posdef=2)
mod.ssm.initialize_stationary()
intercept = np.array([2.3, -10.2], dtype=dtype)
phi = np.array([[0.8, 0.1], [-0.2, 0.7]], dtype=dtype)
sigma2 = np.array([[1.4, -0.2], [-0.2, 4.5]], dtype=dtype)
mod['state_intercept'] = intercept
mod['transition'] = phi
mod['selection'] = np.eye(2).astype(dtype)
mod['state_cov'] = sigma2
mod.ssm._initialize_filter()
mod.ssm._initialize_state()
_statespace = mod.ssm._statespace
initial_state = np.array(_statespace.initial_state)
initial_state_cov = np.array(_statespace.initial_state_cov)
desired = np.linalg.solve(np.eye(2).astype(dtype) - phi, intercept)
assert_allclose(initial_state, desired)
desired = solve_discrete_lyapunov(phi, sigma2)
# precision reductions only required for single precision float / complex
assert_allclose(initial_state_cov, desired, atol=1e-5)
def check_stationary_initialization_1dim(dtype=float):
endog = np.zeros(10, dtype=dtype)
# 1-dimensional example
mod = MLEModel(endog, k_states=1, k_posdef=1)
mod.ssm.initialize_stationary()
intercept = np.array([2.3], dtype=dtype)
phi = np.diag([0.9]).astype(dtype)
sigma2 = np.diag([1.3]).astype(dtype)
mod['state_intercept'] = intercept
mod['transition'] = phi
mod['selection'] = np.eye(1).astype(dtype)
mod['state_cov'] = sigma2
mod.ssm._initialize_filter()
mod.ssm._initialize_state()
_statespace = mod.ssm._statespace
initial_state = np.array(_statespace.initial_state)
initial_state_cov = np.array(_statespace.initial_state_cov)
# precision reductions only required for float complex case
# mean = intercept + phi * mean
# intercept = (1 - phi) * mean
# mean = intercept / (1 - phi)
assert_allclose(initial_state, intercept / (1 - phi[0, 0]))
desired = np.linalg.inv(np.eye(1) - phi).dot(intercept)
assert_allclose(initial_state, desired)
# var = phi**2 var + sigma2
# var = sigma2 / (1 - phi**2)
assert_allclose(initial_state_cov, sigma2 / (1 - phi**2))
assert_allclose(initial_state_cov, solve_discrete_lyapunov(phi, sigma2))
def __init__(self,
Y,
A0,
A1,
U0,
U1,
Q,
Phi,
initV='unconditional',
X0=None,
V0=None,
statevar_names=None):
self.Y, self.A0, self.A1, self.U0, self.U1, self.Q, self.Phi, self.initV, self.X0, self.V0, self.statevar_names = Y, A0, A1, U0, U1, \
Q, Phi, initV, X0, V0, statevar_names
n = self.U0.size
if self.X0 is None: # initialize X0 at unconditional mean
self.X0 = np.mat(np.linalg.inv(np.identity(n) - self.U1) * self.U0)
if self.V0 is None:
if initV == 'steady_state': # solve discrete Ricatti equation
self.V0 = np.mat(
solve_discrete_are(np.array(self.U1.T),
np.array(self.A1.T), np.array(self.Q),
np.array(self.Phi.T)))
elif initV == 'unconditional': # solve discrete Lyapunov equation
self.V0 = np.mat(
solve_discrete_lyapunov(np.array(self.U1.T),
np.array(self.Q)))
elif initV == 'identity':
self.V0 = np.mat(np.identity(n))
if statevar_names is None:
self.statevar_names = np.arange(len(self.X0))
Kalman.kalmanCount += 1
def check_stationary_initialization_2dim(dtype=float):
endog = np.zeros(10, dtype=dtype)
# 2-dimensional example
mod = MLEModel(endog, k_states=2, k_posdef=2)
mod.ssm.initialize_stationary()
intercept = np.array([2.3, -10.2], dtype=dtype)
phi = np.array([[0.8, 0.1],
[-0.2, 0.7]], dtype=dtype)
sigma2 = np.array([[1.4, -0.2],
[-0.2, 4.5]], dtype=dtype)
mod['state_intercept'] = intercept
mod['transition'] = phi
mod['selection'] = np.eye(2).astype(dtype)
mod['state_cov'] = sigma2
mod.ssm._initialize_filter()
mod.ssm._initialize_state()
_statespace = mod.ssm._statespace
initial_state = np.array(_statespace.initial_state)
initial_state_cov = np.array(_statespace.initial_state_cov)
desired = np.linalg.solve(np.eye(2).astype(dtype) - phi, intercept)
assert_allclose(initial_state, desired)
desired = solve_discrete_lyapunov(phi, sigma2)
# precision reductions only required for single precision float / complex
assert_allclose(initial_state_cov, desired, atol=1e-5)
def gen_results(i):
"""
This function follows Zha to redraw coefficients from the regression and
re-estimate the model parameters using those coefficients. Each of these
draws has the ability to be weighted in relative importance by the marginal
likelihood of X0 given the drawn VAR coefficients. Note that this function
makes calls to np.block and la.solve_discrete_lyapunov, which as of yet are
not compatible with numba's jit compiler.
Input:
i (int): Keeps track of the iteration number of the specific call. Used
for random number generator seeding
"""
if current_process().pid % cpus == 0:
pbar.update(cpus) # Track progress on one core only
# Get coefficient draws
[G, BB, mx] = MLEVARsim(n, T, b_hat0, Lam, dt, lags, cn, i + current_seed,
noncinds)
# Check if the matrix G is explosive; if so, discard the draw. Otherwise, proceed.
if np.all(np.abs(la.eigvals(G)) <= 1):
Sigma = la.solve_discrete_lyapunov(
G, BB) # Written as Sigma_j in the paper
# Check Sigma_j for invertibility conditions
if np.linalg.cond(Sigma) <= cond_tol:
return process_VAR(G, Sigma, num_vars, uc, mx, BB, X0)
return 0, 0, 0, 0, 0, 0, False
def check_stationary_initialization_1dim(dtype=float):
endog = np.zeros(10, dtype=dtype)
# 1-dimensional example
mod = MLEModel(endog, k_states=1, k_posdef=1)
mod.ssm.initialize_stationary()
intercept = np.array([2.3], dtype=dtype)
phi = np.diag([0.9]).astype(dtype)
sigma2 = np.diag([1.3]).astype(dtype)
mod['state_intercept'] = intercept
mod['transition'] = phi
mod['selection'] = np.eye(1).astype(dtype)
mod['state_cov'] = sigma2
mod.ssm._initialize_filter()
mod.ssm._initialize_state()
_statespace = mod.ssm._statespace
initial_state = np.array(_statespace.initial_state)
initial_state_cov = np.array(_statespace.initial_state_cov)
# precision reductions only required for float complex case
# mean = intercept + phi * mean
# intercept = (1 - phi) * mean
# mean = intercept / (1 - phi)
assert_allclose(initial_state, intercept / (1 - phi[0, 0]))
desired = np.linalg.inv(np.eye(1) - phi).dot(intercept)
assert_allclose(initial_state, desired)
# var = phi**2 var + sigma2
# var = sigma2 / (1 - phi**2)
assert_allclose(initial_state_cov, sigma2 / (1 - phi**2))
assert_allclose(initial_state_cov, solve_discrete_lyapunov(phi, sigma2))
def _H2P(A, B, analog):
"""Computes the positive-definite P matrix for determining the H2-norm."""
if analog:
P = solve_lyapunov(A, -np.dot(B, B.T)) # AP + PA^T = -BB^T
else:
# Note: discretization is not performed for the user
P = solve_discrete_lyapunov(A, np.dot(B, B.T)) # APA^T - P = -BB^T
return P
def steady_state_cost(A, B, Q, R, w_covr, e_covr, K):
AK = A + np.dot(B, K)
QK = Q + mdot(K.T, R, K)
# Steady-state covariance of the state
Xss = sla.solve_discrete_lyapunov(AK, w_covr + mdot(B, e_covr, B.T))
# Steady-state cost
css = np.trace(np.dot(QK, Xss))
return css
def checkdstable(A):
n = len(A)
P = solve_discrete_lyapunov(A.T, np.identity(n))
S = sqrtm(P)
invS = np.linalg.inv(S)
UB = S.dot(A).dot(invS)
[U, B] = polar(UB)
B = projectPSD(B, 0, 1)
return P, S, U, B
def P_and_Pe_associated_to_K(self, K):
if self.is_stable(K):
cl_map = self.a_cl(K)
P = LA.solve_discrete_lyapunov(cl_map.T, self.Q + K.T @ self.R @ K)
distP = LA.norm(P - self.P_opt, 2) / LA.norm(self.P_opt, 2)
else:
P = 100.0 * np.eye(self.n)
distP = float("inf")
return P, distP
def log_lik(self, p0, y=None, return_filtered=False):
yy = np.array(self.yy)
TT, RR, QQ, DD, ZZ, HH = self.system_matrices(p0)
EXPeffect_f = self.get_EXPeffect(p0)
RQR = np.dot(np.dot(RR, QQ), RR.T)
try:
Pt = solve_discrete_lyapunov(TT, RQR)
Pt = TT @ Pt @ TT.T + RQR
except:
return -1000000000.0
initial_distribution = np.array(self.initial_distribution(p0),
dtype=float).squeeze()
strans = self.transition(p0)
sigrq = np.asarray(np.sqrt(self.conditional_variance(p0))).squeeze()
epsr_m = self.conditional_mean(p0).squeeze()
nobs = yy.shape[0]
shocks = np.zeros(nobs)
for t in range(1, nobs):
shocks[t] = self.construct_shock(p0, yy[t], yy[t - 1])
epsr_m = np.atleast_1d(np.array(epsr_m))
try:
res = _markov_switching_learning_lik(
yy, TT, RR, QQ, DD.squeeze(), np.asarray(ZZ, dtype=float), HH,
EXPeffect_f, Pt, self.obs_ind, self.shock_ind,
np.asarray(np.atleast_1d(initial_distribution), dtype=float),
np.asarray(np.atleast_2d(strans),
dtype=float), self.shock_scaling,
np.asarray(epsr_m, dtype=float), sigrq**2, shocks)
except:
res = -100000000.0, 0.0, 0.0
loglh, filtered_mean_mat, filtered_prob_mat = res
if return_filtered:
Ats = p.DataFrame(filtered_mean_mat,
columns=self.state_names,
index=self.yy.index)
xts = p.DataFrame(filtered_prob_mat,
columns=['pj%d' % d for d in range(nj)],
index=self.yy.index)
res = p.concat([Ats, xts], axis=1)
return loglh, res
else:
if np.isnan(loglh): loglh = -10000000000
return loglh
def dlyap(A, Q):
"""
Solve the discrete-time Lyapunov equation.
Wrapper around scipy.linalg.solve_discrete_lyapunov.
Pass a copy of input matrices to protect them from modification.
"""
try:
return solve_discrete_lyapunov(np.copy(A), np.copy(Q))
except ValueError:
return np.full_like(Q, np.inf)
def update_secondary_params(self):
if self.Q is not None: # Given QRS
try:
A_Eigs = linalg.eig(self.A)[0]
except Exception as e:
print('Error in eig ({})... Tying again!'.format(e))
A_Eigs = linalg.eig(self.A)[0] # Try again!
isStable = np.max(np.abs(A_Eigs)) < 1
if isStable:
self.XCov = linalg.solve_discrete_lyapunov(self.A, self.Q)
self.G = self.A @ self.XCov @ self.C.T + self.S
self.YCov = self.C @ self.XCov @ self.C.T + self.R
self.YCov = (self.YCov + self.YCov.T) / 2
else:
self.XCov = np.ones(self.A.shape)
self.XCov[:] = np.nan
self.YCov = np.ones(self.A.shape)
self.XCov[:] = np.nan
try:
self.Pp = linalg.solve_discrete_are(
self.A.T, self.C.T, self.Q, self.R,
s=self.S) # Solves Katayama eq. 5.42a
self.innovCov = self.C @ self.Pp @ self.C.T + self.R
innovCovInv = np.linalg.pinv(self.innovCov)
self.K = (self.A @ self.Pp @ self.C.T + self.S) @ innovCovInv
self.Kf = self.Pp @ self.C.T @ innovCovInv
self.Kv = self.S @ innovCovInv
self.A_KC = self.A - self.K @ self.C
except:
print('Could not solve DARE')
self.Pp = np.empty(self.A.shape)
self.Pp[:] = np.nan
self.K = np.empty((self.A.shape[0], self.R.shape[0]))
self.K[:] = np.nan
self.Kf = np.array(self.K)
self.Kv = np.array(self.K)
self.innovCov = np.empty(self.R.shape)
self.innovCov[:] = np.nan
self.A_KC = np.empty(self.A.shape)
self.A_KC[:] = np.nan
self.P2 = self.XCov - self.Pp # (should give the solvric solution) Proof: Katayama Theorem 5.3 and A.3 in pvo book
elif self.K is not None: # Given K
self.XCov = None
self.G = None
self.YCov = None
self.Pp = None
self.Kf = None
self.Kv = None
self.A_KC = self.A - self.K @ self.C
self.P2 = None
def get_invariant_distribution(self, transition_matrix, noise_covariance):
A = transition_matrix - np.eye(transition_matrix.shape[0])
B = np.zeros(transition_matrix.shape[0], dtype='float')
print("get invariant distribution")
print(transition_matrix)
print(noise_covariance)
s_0 = np.linalg.solve(A, B)
p_0 = scipy.solve_discrete_lyapunov(transition_matrix,
noise_covariance)
return NormalVectorDistribution(s_0, np.matrix(p_0, dtype=float))
def _terminalObj(self):
# estado terminal
XN = self.X_pred[-1] # terminal state
XdN = XN[self.nxs:self.nxs+self.nxd]
Vt = 0
# Adição do custo terminal
# Q terminal
Q_lyap = [email protected]@[email protected]@self.sys.F
Q_bar = solve_discrete_lyapunov(self.sys.F, Q_lyap, method='bilinear')
Vt = csd.dot(XdN**2, csd.diag(Q_bar))
self.Q_bar = Q_bar
return Vt
def test_global_stationary():
# Test for global approximate diffuse initialization
# - 1-dimensional -
endog = np.zeros(10)
mod = sarimax.SARIMAX(endog, order=(1, 0, 0), trend='c')
# no intercept
intercept = 0
phi = 0.5
sigma2 = 2.
mod.update(np.r_[intercept, phi, sigma2])
init = Initialization(mod.k_states, 'stationary')
check_initialization(mod, init, [0], np.diag([0]),
np.eye(1) * sigma2 / (1 - phi**2))
# intercept
intercept = 1.2
phi = 0.5
sigma2 = 2.
mod.update(np.r_[intercept, phi, sigma2])
init = Initialization(mod.k_states, 'stationary')
check_initialization(mod, init, [intercept / (1 - phi)], np.diag([0]),
np.eye(1) * sigma2 / (1 - phi**2))
# - n-dimensional -
endog = np.zeros(10)
mod = sarimax.SARIMAX(endog, order=(2, 0, 0), trend='c')
# no intercept
intercept = 0
phi = [0.5, -0.2]
sigma2 = 2.
mod.update(np.r_[intercept, phi, sigma2])
init = Initialization(mod.k_states, 'stationary')
T = np.array([[0.5, 1],
[-0.2, 0]])
Q = np.diag([sigma2, 0])
desired_cov = solve_discrete_lyapunov(T, Q)
check_initialization(mod, init, [0, 0], np.diag([0, 0]), desired_cov)
# intercept
intercept = 1.2
phi = [0.5, -0.2]
sigma2 = 2.
mod.update(np.r_[intercept, phi, sigma2])
init = Initialization(mod.k_states, 'stationary')
desired_intercept = np.linalg.inv(np.eye(2) - T).dot([intercept, 0])
check_initialization(mod, init, desired_intercept, np.diag([0, 0]),
desired_cov)
def test_univariate(self):
# Real case
a = np.array([[0.5]])
q = np.array([[10.]])
actual = tools.solve_discrete_lyapunov(a, q)
desired = solve_discrete_lyapunov(a, q)
assert_allclose(actual, desired)
# Complex case (where the Lyapunov equation is taken as a complex
# function)
a = np.array([[0.5+1j]])
q = np.array([[10.]])
actual = tools.solve_discrete_lyapunov(a, q)
desired = solve_discrete_lyapunov(a, q)
assert_allclose(actual, desired)
# Complex case (where the Lyapunov equation is taken as a real
# function)
a = np.array([[0.5+1j]])
q = np.array([[10.]])
actual = tools.solve_discrete_lyapunov(a, q, complex_step=True)
desired = self.solve_dicrete_lyapunov_direct(a, q, complex_step=True)
assert_allclose(actual, desired)
def test_univariate(self):
# Real case
a = np.array([[0.5]])
q = np.array([[10.]])
actual = tools.solve_discrete_lyapunov(a, q)
desired = solve_discrete_lyapunov(a, q)
assert_allclose(actual, desired)
# Complex case (where the Lyapunov equation is taken as a complex
# function)
a = np.array([[0.5 + 1j]])
q = np.array([[10.]])
actual = tools.solve_discrete_lyapunov(a, q)
desired = solve_discrete_lyapunov(a, q)
assert_allclose(actual, desired)
# Complex case (where the Lyapunov equation is taken as a real
# function)
a = np.array([[0.5 + 1j]])
q = np.array([[10.]])
actual = tools.solve_discrete_lyapunov(a, q, complex_step=True)
desired = self.solve_dicrete_lyapunov_direct(a, q, complex_step=True)
assert_allclose(actual, desired)
def test_global_stationary():
# Test for global approximate diffuse initialization
# - 1-dimensional -
endog = np.zeros(10)
mod = sarimax.SARIMAX(endog, order=(1, 0, 0), trend='c')
# no intercept
intercept = 0
phi = 0.5
sigma2 = 2.
mod.update(np.r_[intercept, phi, sigma2])
init = Initialization(mod.k_states, 'stationary')
check_initialization(mod, init, [0], np.diag([0]),
np.eye(1) * sigma2 / (1 - phi**2))
# intercept
intercept = 1.2
phi = 0.5
sigma2 = 2.
mod.update(np.r_[intercept, phi, sigma2])
init = Initialization(mod.k_states, 'stationary')
check_initialization(mod, init, [intercept / (1 - phi)], np.diag([0]),
np.eye(1) * sigma2 / (1 - phi**2))
# - n-dimensional -
endog = np.zeros(10)
mod = sarimax.SARIMAX(endog, order=(2, 0, 0), trend='c')
# no intercept
intercept = 0
phi = [0.5, -0.2]
sigma2 = 2.
mod.update(np.r_[intercept, phi, sigma2])
init = Initialization(mod.k_states, 'stationary')
T = np.array([[0.5, 1], [-0.2, 0]])
Q = np.diag([sigma2, 0])
desired_cov = solve_discrete_lyapunov(T, Q)
check_initialization(mod, init, [0, 0], np.diag([0, 0]), desired_cov)
# intercept
intercept = 1.2
phi = [0.5, -0.2]
sigma2 = 2.
mod.update(np.r_[intercept, phi, sigma2])
init = Initialization(mod.k_states, 'stationary')
desired_intercept = np.linalg.inv(np.eye(2) - T).dot([intercept, 0])
check_initialization(mod, init, desired_intercept, np.diag([0, 0]),
desired_cov)
def is_controllable(self):
"""Tests if the disc object is controllable.
Returns:
bool: Boolean, true if the disc configuration is controllable
ndarray: Controllability gramiam from discrete Lyapunov equation
"""
if (self.ready()):
if (self.B is not None):
q = np.matmul(self.B,self.B.conj().T)
x_c = linalg.solve_discrete_lyapunov(self.A.conj().T,q)
controllable = (linalg.eigvals(x_c)>0).sum() == np.shape(self.A)[0]
return [controllable,x_c]
else:
print("Please set B.")
def get_p_init_lyapunov(self, Q):
pmat = self.precalc_mat[0]
qmat = self.precalc_mat[1]
F = np.vstack(
(pmat[1, 0][:, :-self.neps], qmat[1, 0][:-self.neps, :-self.neps]))
E = np.vstack((pmat[1, 0][:, -self.neps:], qmat[1, 0][:-self.neps,
-self.neps:]))
Q = E @ Q @ E.T
p4 = sl.solve_discrete_lyapunov(F[self.dimp:, :], Q[self.dimp:,
self.dimp:])
return F @ p4 @ F.T + Q
def __init__(self, Y, A0, A1, U0, U1, Q, Phi, initV='unconditional', X0 = None, V0 = None, statevar_names = None):
self.Y, self.A0, self.A1, self.U0, self.U1, self.Q, self.Phi, self.initV, self.X0, self.V0, self.statevar_names = Y, A0, A1, U0, U1, \
Q, Phi, initV, X0, V0, statevar_names
n = self.U0.size
if self.X0 is None: # initialize X0 at unconditional mean
self.X0 = np.mat(np.linalg.inv(np.identity(n) - self.U1)*self.U0)
if self.V0 is None:
if initV == 'steady_state': # solve discrete Ricatti equation
self.V0 = np.mat(solve_discrete_are(np.array(self.U1.T), np.array(self.A1.T), np.array(self.Q), np.array(self.Phi.T)))
elif initV == 'unconditional': # solve discrete Lyapunov equation
self.V0 = np.mat(solve_discrete_lyapunov(np.array(self.U1.T), np.array(self.Q)))
elif initV == 'identity':
self.V0 = np.mat(np.identity(n))
if statevar_names is None:
self.statevar_names = np.arange(len(self.X0))
Kalman.kalmanCount += 1
def controllability_gramian(self):
"""Controllability gramian matrix."""
from scipy import linalg
B = self.B.evaluate()
Q = -B @ B.T
# Find Wc given A @ Wc + Wc @ A.T = Q
# Wc > o if (A, B) controllable
Wc = linalg.solve_discrete_lyapunov(self.A.evaluate(), Q)
# Wc should be symmetric positive semi-definite
Wc = (Wc + Wc.T) / 2
return Matrix(Wc)
def observability_gramian(self):
"""Observability gramian matrix."""
from scipy import linalg
C = self.C.evaluate()
Q = -C.T @ C
# Find Wo given A.T @ Wo + Wo @ A = Q
# Wo > o if (C, A) observable
Wo = linalg.solve_discrete_lyapunov(self.A.evaluate().T, Q)
# Wo should be symmetric positive semi-definite
Wo = (Wo + Wo.T) / 2
return Matrix(Wo)
def initial_state(TT, RR, C, QQ):
"""
Compute the initial state x_0 and state covariance matrix P_0
"""
Ns = TT.shape[0]
e = np.linalg.eigvals(TT)
if all(np.abs(e) < 1.0):
x_0 = np.linalg.inv(np.eye(Ns) - TT) @ C
P_0 = solve_discrete_lyapunov(TT, RR @ QQ @ RR.T)
else:
x_0 = C
P_0 = 1e+6 * np.eye(Ns)
return x_0, P_0
def is_observable(self):
"""Tests if the disc object is observable.
Returns:
bool: Boolean, true if the disc configuration is observable
ndarray: Observability gramiam from discrete Lyapunov equation
"""
if (self.ready()):
if (self.C is not None):
q = np.matmul(self.C.conj().T,self.C)
y_o = linalg.solve_discrete_lyapunov(self.A,q)
y_o = y_o.conj().T
observable = (linalg.eigvals(y_o)>0).sum() == np.shape(self.A)[0]
return [observable,y_o]
else:
print("Please set C.")
def cost_inf_K(A,B,Q,R,K):
'''
Arguments:
State transition matrices (A,B)
LQR Costs (Q,R)
Control Gain K
Outputs:
cost: Infinite time horizon LQR cost of static gain K
'''
cl_map = A+B.dot(K)
if np.amax(np.abs(LA.eigvals(cl_map)))<(1.0-1.0e-6):
cost = np.trace(LA.solve_discrete_lyapunov(cl_map.T,Q+np.dot(K.T,R.dot(K))))
else:
cost = float("inf")
return cost
def evaluate_F(self, F):
"""
Given a fixed policy F, with the interpretation u = -F x, this
function computes the matrix P_F and constant d_F associated with
discounted cost J_F(x) = x' P_F x + d_F.
Parameters
==========
F : array_like
A self.k x self.n array
Returns
=======
P_F : array_like, dtype = float
Matrix for discounted cost
d_F : scalar
Constant for discounted cost
K_F : array_like, dtype = float
Worst case policy
O_F : array_like, dtype = float
Matrix for discounted entropy
o_F : scalar
Constant for discounted entropy
"""
# == Simplify names == #
Q, R, A, B, C = self.Q, self.R, self.A, self.B, self.C
beta, theta = self.beta, self.theta
# == Solve for policies and costs using agent 2's problem == #
K_F, neg_P_F = self.F_to_K(F)
P_F = - neg_P_F
I = np.identity(self.j)
H = inv(I - C.T.dot(P_F.dot(C)) / theta)
d_F = log(det(H))
# == Compute O_F and o_F == #
sig = -1.0 / theta
AO = sqrt(beta) * (A - dot(B, F) + dot(C, K_F))
O_F = solve_discrete_lyapunov(AO.T, beta * dot(K_F.T, K_F))
ho = (trace(H - 1) - d_F) / 2.0
tr = trace(dot(O_F, C.dot(H.dot(C.T))))
o_F = (ho + beta * tr) / (1 - beta)
return K_F, P_F, d_F, O_F, o_F
def evaluate_F(self, F):
"""
Given a fixed policy F, with the interpretation u = -F x, this
function computes the matrix P_F and constant d_F associated with
discounted cost J_F(x) = x' P_F x + d_F.
Parameters
==========
F : array_like
A self.k x self.n array
Returns
=======
P_F : array_like, dtype = float
Matrix for discounted cost
d_F : scalar
Constant for discounted cost
K_F : array_like, dtype = float
Worst case policy
O_F : array_like, dtype = float
Matrix for discounted entropy
o_F : scalar
Constant for discounted entropy
"""
# == Simplify names == #
Q, R, A, B, C = self.Q, self.R, self.A, self.B, self.C
beta, theta = self.beta, self.theta
# == Solve for policies and costs using agent 2's problem == #
K_F, neg_P_F = self.F_to_K(F)
P_F = -neg_P_F
I = np.identity(self.j)
H = inv(I - C.T.dot(P_F.dot(C)) / theta)
d_F = log(det(H))
# == Compute O_F and o_F == #
sig = -1.0 / theta
AO = sqrt(beta) * (A - dot(B, F) + dot(C, K_F))
O_F = solve_discrete_lyapunov(AO.T, beta * dot(K_F.T, K_F))
ho = (trace(H - 1) - d_F) / 2.0
tr = trace(dot(O_F, C.dot(H.dot(C.T))))
o_F = (ho + beta * tr) / (1 - beta)
return K_F, P_F, d_F, O_F, o_F
def riccati(self):
# if (1-self.epsilon) <= np.linalg.eigh(self.K_xx)[0][-1] <= 1 + self.epsilon:
# self.v_x = np.linalg.eigh(self.K_xx)[1][-1]
self.v_x = np.linalg.eigh(self.K_xx)[1][-1] # just setting the value of v_x to the largest evals evect
# else:
# raise # Something bad
self.chi_bar = np.vstack(self.x_whitened).mean(0) # are these suppose to uphold K_xx.T(chi_bar) = chi_bar
self.gamma_bar = np.vstack(self.y_whitened).mean(0) # and K_xy.T(chi_bar) = gamma_bar
self.chi_bar = np.insert(self.chi_bar, len(self.chi_bar), 1) # Adding the final element to chi and gamma bar to match shape with Kxx and Kxy
self.gamma_bar = np.insert(self.gamma_bar, len(self.gamma_bar), 1)
self.A = self.K_xx - np.outer(self.v_x, self.chi_bar)
self.B = self.K_xy - np.outer(self.v_x, self.gamma_bar)
Q = np.dot(self.B, self.B.T)
self.O = solve_discrete_lyapunov(self.A,Q)
return self.O
def controllability_gramian_discrete(mat_a, mat_b):
"""
Calculates the controllabilty gramian of a discrete time LTI system.
Parameters
----------
mat_a : ndarray
The state matrix.
mat_b : ndarray
The input matrix.
Returns
-------
mat_wc : ndarray
The controllability gramian.
"""
mat_q = mat_b @ mat_b.T
mat_wc = linalg.solve_discrete_lyapunov(mat_a, mat_q)
return mat_wc
def observability_gramian_discrete(mat_a, mat_c):
"""
Calculates the observability gramian of a discrete time LTI system.
Parameters
----------
mat_a : ndarray
The state matrix.
mat_c : ndarray
The output matrix.
Returns
-------
mat_wo : ndarray
The observability gramian.
"""
mat_q = mat_c.T @ mat_c
mat_wo = linalg.solve_discrete_lyapunov(mat_a.T, mat_q)
return mat_wo
def test_nested():
endog = np.zeros(10)
mod = sarimax.SARIMAX(endog, order=(6, 0, 0))
phi = [0.5, -0.2, 0.1, 0.0, 0.1, 0.0]
sigma2 = 2.
mod.update(np.r_[phi, sigma2])
# Create the initialization object as a series of nested objects
init1_1 = Initialization(3)
init1_1_1 = Initialization(2, 'stationary')
init1_1_2 = Initialization(1, 'approximate_diffuse',
approximate_diffuse_variance=1e9)
init1_1.set((0, 2), init1_1_1)
init1_1.set(2, init1_1_2)
init1_2 = Initialization(3)
init1_2_1 = Initialization(1, 'known', constant=[1], stationary_cov=[[2.]])
init1_2.set(0, init1_2_1)
init1_2_2 = Initialization(1, 'diffuse')
init1_2.set(1, init1_2_2)
init1_2_3 = Initialization(1, 'approximate_diffuse')
init1_2.set(2, init1_2_3)
init = Initialization(6)
init.set((0, 3), init1_1)
init.set((3, 6), init1_2)
# Check the output
desired_cov = np.zeros((6, 6))
T = np.array([[0.5, 1],
[-0.2, 0]])
Q = np.array([[sigma2, 0],
[0, 0]])
desired_cov[:2, :2] = solve_discrete_lyapunov(T, Q)
desired_cov[2, 2] = 1e9
desired_cov[3, 3] = 2.
desired_cov[5, 5] = 1e6
check_initialization(mod, init, [0, 0, 0, 1, 0, 0],
np.diag([0, 0, 0, 0, 1, 0]),
desired_cov)
def _trainVARMA(self, data):
"""
Train correlated ARMA model on white noise ARMA, with Fourier already removed
@ In, data, np.array(np.array(float)), data on which to train with shape (# pivot values, # targets)
@ Out, results, statsmodels.tsa.arima_model.ARMAResults, fitted VARMA
@ Out, stateDist, Distributions.MultivariateNormal, MVN from which VARMA noise is taken
@ Out, initDist, Distributions.MultivariateNormal, MVN from which VARMA initial state is taken
"""
Pmax = self.Pmax
Qmax = self.Qmax
model = sm.tsa.VARMAX(endog=data, order=(Pmax, Qmax))
self.raiseADebug('... fitting VARMA ...')
results = model.fit(disp=False, maxiter=1000)
lenHist, numVars = data.shape
# train multivariate normal distributions using covariances, keep it around so we can control the RNG
## it appears "measurement" always has 0 covariance, and so is all zeros (see _generateVARMASignal)
## all the noise comes from the stateful properties
stateDist = self._trainMultivariateNormal(
numVars, np.zeros(numVars), model.ssm['state_cov'.encode('ascii')])
# train initial state sampler
## Used to pick an initial state for the VARMA by sampling from the multivariate normal noise
# and using the AR and MA initial conditions. Implemented so we can control the RNG internally.
# Implementation taken directly from statsmodels.tsa.statespace.kalman_filter.KalmanFilter.simulate
## get mean
smoother = model.ssm
mean = np.linalg.solve(
np.eye(smoother.k_states) -
smoother['transition'.encode('ascii'), :, :, 0],
smoother['state_intercept'.encode('ascii'), :, 0])
## get covariance
r = smoother['selection'.encode('ascii'), :, :, 0]
q = smoother['state_cov'.encode('ascii'), :, :, 0]
selCov = r.dot(q).dot(r.T)
cov = solve_discrete_lyapunov(
smoother['transition'.encode('ascii'), :, :, 0], selCov)
# FIXME it appears this is always resulting in a lowest-value initial state. Why?
initDist = self._trainMultivariateNormal(len(mean), mean, cov)
# NOTE: uncomment this line to get a printed summary of a lot of information about the fitting.
#self.raiseADebug('VARMA model training summary:\n',results.summary())
return model, stateDist, initDist
def test_nested():
endog = np.zeros(10)
mod = sarimax.SARIMAX(endog, order=(6, 0, 0))
phi = [0.5, -0.2, 0.1, 0.0, 0.1, 0.0]
sigma2 = 2.
mod.update(np.r_[phi, sigma2])
# Create the initialization object as a series of nested objects
init1_1 = Initialization(3)
init1_1_1 = Initialization(2, 'stationary')
init1_1_2 = Initialization(1,
'approximate_diffuse',
approximate_diffuse_variance=1e9)
init1_1.set((0, 2), init1_1_1)
init1_1.set(2, init1_1_2)
init1_2 = Initialization(3)
init1_2_1 = Initialization(1, 'known', constant=[1], stationary_cov=[[2.]])
init1_2.set(0, init1_2_1)
init1_2_2 = Initialization(1, 'diffuse')
init1_2.set(1, init1_2_2)
init1_2_3 = Initialization(1, 'approximate_diffuse')
init1_2.set(2, init1_2_3)
init = Initialization(6)
init.set((0, 3), init1_1)
init.set((3, 6), init1_2)
# Check the output
desired_cov = np.zeros((6, 6))
T = np.array([[0.5, 1], [-0.2, 0]])
Q = np.array([[sigma2, 0], [0, 0]])
desired_cov[:2, :2] = solve_discrete_lyapunov(T, Q)
desired_cov[2, 2] = 1e9
desired_cov[3, 3] = 2.
desired_cov[5, 5] = 1e6
check_initialization(mod, init, [0, 0, 0, 1, 0, 0],
np.diag([0, 0, 0, 0, 1, 0]), desired_cov)
def test_multivariate(self):
# Real case
a = tools.companion_matrix([1, -0.4, 0.5])
q = np.diag([10., 5.])
actual = tools.solve_discrete_lyapunov(a, q)
desired = solve_discrete_lyapunov(a, q)
assert_allclose(actual, desired)
# Complex case (where the Lyapunov equation is taken as a complex
# function)
a = tools.companion_matrix([1, -0.4 + 0.1j, 0.5])
q = np.diag([10., 5.])
actual = tools.solve_discrete_lyapunov(a, q, complex_step=False)
desired = self.solve_dicrete_lyapunov_direct(a, q, complex_step=False)
assert_allclose(actual, desired)
# Complex case (where the Lyapunov equation is taken as a real
# function)
a = tools.companion_matrix([1, -0.4 + 0.1j, 0.5])
q = np.diag([10., 5.])
actual = tools.solve_discrete_lyapunov(a, q, complex_step=True)
desired = self.solve_dicrete_lyapunov_direct(a, q, complex_step=True)
assert_allclose(actual, desired)
def test_multivariate(self):
# Real case
a = tools.companion_matrix([1, -0.4, 0.5])
q = np.diag([10., 5.])
actual = tools.solve_discrete_lyapunov(a, q)
desired = solve_discrete_lyapunov(a, q)
assert_allclose(actual, desired)
# Complex case (where the Lyapunov equation is taken as a complex
# function)
a = tools.companion_matrix([1, -0.4+0.1j, 0.5])
q = np.diag([10., 5.])
actual = tools.solve_discrete_lyapunov(a, q, complex_step=False)
desired = self.solve_dicrete_lyapunov_direct(a, q, complex_step=False)
assert_allclose(actual, desired)
# Complex case (where the Lyapunov equation is taken as a real
# function)
a = tools.companion_matrix([1, -0.4+0.1j, 0.5])
q = np.diag([10., 5.])
actual = tools.solve_discrete_lyapunov(a, q, complex_step=True)
desired = self.solve_dicrete_lyapunov_direct(a, q, complex_step=True)
assert_allclose(actual, desired)
def computeG(A0, A1, d, Q0, tau0, beta, mu):
"""
Compute government income given mu and return tax revenues and
policy matrixes for the planner.
Parameters
----------
A0 : float
A constant parameter for the inverse demand function
A1 : float
A constant parameter for the inverse demand function
d : float
A constant parameter for quadratic adjustment cost of production
Q0 : float
An initial condition for production
tau0 : float
An initial condition for taxes
beta : float
A constant parameter for discounting
mu : float
Lagrange multiplier
Returns
-------
T0 : array(float)
Present discounted value of government spending
A : array(float)
One of the transition matrices for the states
B : array(float)
Another transition matrix for the states
F : array(float)
Policy rule matrix
P : array(float)
Value function matrix
"""
# Create Matrices for solving Ramsey problem
R = np.array([[0, -A0/2, 0, 0],
[-A0/2, A1/2, -mu/2, 0],
[0, -mu/2, 0, 0],
[0, 0, 0, d/2]])
A = np.array([[1, 0, 0, 0],
[0, 1, 0, 1],
[0, 0, 0, 0],
[-A0/d, A1/d, 0, A1/d+1/beta]])
B = np.array([0, 0, 1, 1/d]).reshape(-1, 1)
Q = 0
# Use LQ to solve the Ramsey Problem.
lq = LQ(Q, -R, A, B, beta=beta)
P, F, d = lq.stationary_values()
# Need y_0 to compute government tax revenue.
P21 = P[3, :3]
P22 = P[3, 3]
z0 = np.array([1, Q0, tau0]).reshape(-1, 1)
u0 = -P22**(-1) * P21.dot(z0)
y0 = np.vstack([z0, u0])
# Define A_F and S matricies
AF = A - B.dot(F)
S = np.array([0, 1, 0, 0]).reshape(-1, 1).dot(np.array([[0, 0, 1, 0]]))
# Solves equation (25)
temp = beta * AF.T.dot(S).dot(AF)
Omega = solve_discrete_lyapunov(np.sqrt(beta) * AF.T, temp)
T0 = y0.T.dot(Omega).dot(y0)
return T0, A, B, F, P
Cit = convert_arb_mat(Ct) Dit = convert_arb_mat(Dt) # version numpy.mat+arb Anit = arb(1)*At Bnit = arb(1)*Bt Cnit = arb(1)*Ct Dnit = arb(1)*Dt # Résolution de l'Equation de Lyapunov: W = A W At + B Bt # References Wscipy = solve_discrete_lyapunov( A, B*Bt) #ref = A*Wscipy*At + B*Bt #print( "\"Précision\" résolution scipy: " + str(np.amax( Wscipy - ref)) ) Wschur = dlyap_schur( A, B*Bt) #ref = A*Wschur*At + B*Bt #print( "Précision résolution schur: " + str(np.amax( Wschur - ref)) ) Wslycot = dlyap_slycot( A, B*Bt) #ref = A*Wslycot*At + B*Bt #print( "\"Précision\" résolution slycot: " + str(np.amax( Wslycot - ref)) ) # Méthodes "maison" Wnaiv = lyap_naiv(A, B*Bt) #ref = A*Wnaiv*At + B*Bt
def check_discrete_case(self, a, q):
x = solve_discrete_lyapunov(a, q)
assert_array_almost_equal(np.dot(np.dot(a, x),a.conj().transpose()) - x, -1.0*q)
def _compute_multivariate_acovf_from_coefficients(
coefficients, error_variance, maxlag=None,
forward_autocovariances=False):
r"""
Compute multivariate autocovariances from vector autoregression coefficient
matrices
Parameters
----------
coefficients : array or list
The coefficients matrices. If a list, should be a list of length
`order`, where each element is an array sized `k_endog` x `k_endog`. If
an array, should be the coefficient matrices horizontally concatenated
and sized `k_endog` x `k_endog * order`.
error_variance : array
The variance / covariance matrix of the error term. Should be sized
`k_endog` x `k_endog`.
maxlag : integer, optional
The maximum autocovariance to compute. Default is `order`-1. Can be
zero, in which case it returns the variance.
forward_autocovariances : boolean, optional
Whether or not to compute forward autocovariances
:math:`E(y_t y_{t+j}')`. Default is False, so that backward
autocovariances :math:`E(y_t y_{t-j}')` are returned.
Returns
-------
autocovariances : list
A list of the first `maxlag` autocovariance matrices. Each matrix is
shaped `k_endog` x `k_endog`.
Notes
-----
Computes
..math::
\Gamma(j) = E(y_t y_{t-j}')
for j = 1, ..., `maxlag`, unless `forward_autocovariances` is specified,
in which case it computes:
..math::
E(y_t y_{t+j}') = \Gamma(j)'
Coefficients are assumed to be provided from the VAR model:
.. math::
y_t = A_1 y_{t-1} + \dots + A_p y_{t-p} + \varepsilon_t
Autocovariances are calculated by solving the associated discrete Lyapunov
equation of the state space representation of the VAR process.
"""
from scipy import linalg
# Convert coefficients to a list of matrices, for use in
# `companion_matrix`; get dimensions
if type(coefficients) == list:
order = len(coefficients)
k_endog = coefficients[0].shape[0]
else:
k_endog, order = coefficients.shape
order //= k_endog
coefficients = [
coefficients[:k_endog, i*k_endog:(i+1)*k_endog]
for i in range(order)
]
if maxlag is None:
maxlag = order-1
# Start with VAR(p): w_{t+1} = phi_1 w_t + ... + phi_p w_{t-p+1} + u_{t+1}
# Then stack the VAR(p) into a VAR(1) in companion matrix form:
# z_{t+1} = F z_t + v_t
companion = companion_matrix(
[1] + [-coefficients[i] for i in range(order)]
).T
# Compute the error variance matrix for the stacked form: E v_t v_t'
selected_variance = np.zeros(companion.shape)
selected_variance[:k_endog, :k_endog] = error_variance
# Compute the unconditional variance of z_t: E z_t z_t'
stacked_cov = linalg.solve_discrete_lyapunov(companion, selected_variance)
# The first (block) row of the variance of z_t gives the first p-1
# autocovariances of w_t: \Gamma_i = E w_t w_t+i with \Gamma_0 = Var(w_t)
# Note: these are okay, checked against ArmaProcess
autocovariances = [
stacked_cov[:k_endog, i*k_endog:(i+1)*k_endog]
for i in range(min(order, maxlag+1))
]
for i in range(maxlag - (order-1)):
stacked_cov = np.dot(companion, stacked_cov)
autocovariances += [
stacked_cov[:k_endog, -k_endog:]
]
if forward_autocovariances:
for i in range(len(autocovariances)):
autocovariances[i] = autocovariances[i].T
return autocovariances
def kalman_filter(model, return_loglike=False):
# Parameters
dtype = model.dtype
# Kalman filter properties
filter_method = model.filter_method
inversion_method = model.inversion_method
stability_method = model.stability_method
conserve_memory = model.conserve_memory
tolerance = model.tolerance
loglikelihood_burn = model.loglikelihood_burn
# Check for acceptable values
if not filter_method == FILTER_CONVENTIONAL:
warn('The pure Python version of the kalman filter only supports the'
' conventional Kalman filter')
implemented_inv_methods = INVERT_NUMPY | INVERT_UNIVARIATE | SOLVE_CHOLESKY
if not inversion_method & implemented_inv_methods:
warn('The pure Python version of the kalman filter only performs'
' inversion using `numpy.linalg.inv`.')
if not tolerance == 0:
warn('The pure Python version of the kalman filter does not check'
' for convergence.')
# Convergence (this implementation does not consider convergence)
converged = False
period_converged = 0
# Dimensions
nobs = model.nobs
k_endog = model.k_endog
k_states = model.k_states
k_posdef = model.k_posdef
# Allocate memory for variables
filtered_state = np.zeros((k_states, nobs), dtype=dtype)
filtered_state_cov = np.zeros((k_states, k_states, nobs), dtype=dtype)
predicted_state = np.zeros((k_states, nobs+1), dtype=dtype)
predicted_state_cov = np.zeros((k_states, k_states, nobs+1), dtype=dtype)
forecast = np.zeros((k_endog, nobs), dtype=dtype)
forecast_error = np.zeros((k_endog, nobs), dtype=dtype)
forecast_error_cov = np.zeros((k_endog, k_endog, nobs), dtype=dtype)
loglikelihood = np.zeros((nobs+1,), dtype=dtype)
# Selected state covariance matrix
selected_state_cov = (
np.dot(
np.dot(model.selection[:, :, 0],
model.state_cov[:, :, 0]),
model.selection[:, :, 0].T
)
)
# Initial values
if model.initialization == 'known':
initial_state = model._initial_state.astype(dtype)
initial_state_cov = model._initial_state_cov.astype(dtype)
elif model.initialization == 'approximate_diffuse':
initial_state = np.zeros((k_states,), dtype=dtype)
initial_state_cov = (
np.eye(k_states).astype(dtype) * model._initial_variance
)
elif model.initialization == 'stationary':
initial_state = np.zeros((k_states,), dtype=dtype)
initial_state_cov = solve_discrete_lyapunov(
np.array(model.transition[:, :, 0], dtype=dtype),
np.array(selected_state_cov[:, :], dtype=dtype),
)
else:
raise RuntimeError('Statespace model not initialized.')
# Copy initial values to predicted
predicted_state[:, 0] = initial_state
predicted_state_cov[:, :, 0] = initial_state_cov
# print(predicted_state_cov[:, :, 0])
# Setup indices for possibly time-varying matrices
design_t = 0
obs_intercept_t = 0
obs_cov_t = 0
transition_t = 0
state_intercept_t = 0
selection_t = 0
state_cov_t = 0
# Iterate forwards
time_invariant = model.time_invariant
for t in range(nobs):
# Get indices for possibly time-varying arrays
if not time_invariant:
if model.design.shape[2] > 1: design_t = t
if model.obs_intercept.shape[1] > 1: obs_intercept_t = t
if model.obs_cov.shape[2] > 1: obs_cov_t = t
if model.transition.shape[2] > 1: transition_t = t
if model.state_intercept.shape[1] > 1: state_intercept_t = t
if model.selection.shape[2] > 1: selection_t = t
if model.state_cov.shape[2] > 1: state_cov_t = t
# Selected state covariance matrix
if model.selection.shape[2] > 1 or model.state_cov.shape[2] > 1:
selected_state_cov = (
np.dot(
np.dot(model.selection[:, :, selection_t],
model.state_cov[:, :, state_cov_t]),
model.selection[:, :, selection_t].T
)
)
# #### Forecast for time t
# `forecast` $= Z_t a_t + d_t$
#
# *Note*: $a_t$ is given from the initialization (for $t = 0$) or
# from the previous iteration of the filter (for $t > 0$).
forecast[:, t] = (
np.dot(model.design[:, :, design_t], predicted_state[:, t]) +
model.obs_intercept[:, obs_intercept_t]
)
# *Intermediate calculation* (used just below and then once more)
# `tmp1` array used here, dimension $(m \times p)$
# $\\#_1 = P_t Z_t'$
# $(m \times p) = (m \times m) (p \times m)'$
tmp1 = np.dot(predicted_state_cov[:, :, t],
model.design[:, :, design_t].T)
# #### Forecast error for time t
# `forecast_error` $\equiv v_t = y_t -$ `forecast`
forecast_error[:, t] = model.obs[:, t] - forecast[:, t]
# #### Forecast error covariance matrix for time t
# $F_t \equiv Z_t P_t Z_t' + H_t$
forecast_error_cov[:, :, t] = (
np.dot(model.design[:, :, design_t], tmp1) +
model.obs_cov[:, :, obs_cov_t]
)
# Store the inverse
if k_endog == 1 and inversion_method & INVERT_UNIVARIATE:
forecast_error_cov_inv = 1.0 / forecast_error_cov[0, 0, t]
determinant = forecast_error_cov[0, 0, t]
tmp2 = forecast_error_cov_inv * forecast_error[:, t]
tmp3 = forecast_error_cov_inv * model.design[:, :, design_t]
elif inversion_method & SOLVE_CHOLESKY:
U, info = lapack.dpotrf(forecast_error_cov[:, :, t])
determinant = np.product(U.diagonal())**2
tmp2, info = lapack.dpotrs(U, forecast_error[:, t])
tmp3, info = lapack.dpotrs(U, model.design[:, :, design_t])
else:
forecast_error_cov_inv = np.linalg.inv(forecast_error_cov[:, :, t])
determinant = np.linalg.det(forecast_error_cov[:, :, t])
tmp2 = np.dot(forecast_error_cov_inv, forecast_error[:, t])
tmp3 = np.dot(forecast_error_cov_inv, model.design[:, :, design_t])
# #### Filtered state for time t
# $a_{t|t} = a_t + P_t Z_t' F_t^{-1} v_t$
# $a_{t|t} = 1.0 * \\#_1 \\#_2 + 1.0 a_t$
filtered_state[:, t] = (
predicted_state[:, t] + np.dot(tmp1, tmp2)
)
# #### Filtered state covariance for time t
# $P_{t|t} = P_t - P_t Z_t' F_t^{-1} Z_t P_t$
# $P_{t|t} = P_t - \\#_1 \\#_3 P_t$
filtered_state_cov[:, :, t] = (
predicted_state_cov[:, :, t] -
np.dot(
np.dot(tmp1, tmp3),
predicted_state_cov[:, :, t]
)
)
# #### Loglikelihood
loglikelihood[t] = -0.5 * (
np.log((2*np.pi)**k_endog * determinant) +
np.dot(forecast_error[:, t], tmp2)
)
# #### Predicted state for time t+1
# $a_{t+1} = T_t a_{t|t} + c_t$
predicted_state[:, t+1] = (
np.dot(model.transition[:, :, transition_t],
filtered_state[:, t]) +
model.state_intercept[:, state_intercept_t]
)
# #### Predicted state covariance matrix for time t+1
# $P_{t+1} = T_t P_{t|t} T_t' + Q_t^*$
predicted_state_cov[:, :, t+1] = (
np.dot(
np.dot(model.transition[:, :, transition_t],
filtered_state_cov[:, :, t]),
model.transition[:, :, transition_t].T
) + selected_state_cov
)
# Enforce symmetry of predicted covariance matrix
predicted_state_cov[:, :, t+1] = (
predicted_state_cov[:, :, t+1] + predicted_state_cov[:, :, t+1].T
) / 2
if return_loglike:
return np.array(loglikelihood)
else:
kwargs = dict(
(k, v) for k, v in locals().items()
if k in _kalman_filter._fields
)
kwargs['model'] = _statespace(
initial_state=initial_state, initial_state_cov=initial_state_cov
)
kfilter = _kalman_filter(**kwargs)
return FilterResults(model, kfilter)
def test_mixed_stationary():
# More specific tests when one or more blocks are initialized as stationary
endog = np.zeros(10)
mod = sarimax.SARIMAX(endog, order=(2, 1, 0))
phi = [0.5, -0.2]
sigma2 = 2.
mod.update(np.r_[phi, sigma2])
init = Initialization(mod.k_states)
init.set(0, 'diffuse')
init.set((1, 3), 'stationary')
desired_cov = np.zeros((3, 3))
T = np.array([[0.5, 1],
[-0.2, 0]])
Q = np.diag([sigma2, 0])
desired_cov[1:, 1:] = solve_discrete_lyapunov(T, Q)
check_initialization(mod, init, [0, 0, 0], np.diag([1, 0, 0]), desired_cov)
init.clear()
init.set(0, 'diffuse')
init.set(1, 'stationary')
init.set(2, 'approximate_diffuse')
T = np.array([[0.5]])
Q = np.diag([sigma2])
desired_cov = np.diag([0, np.squeeze(solve_discrete_lyapunov(T, Q)), 1e6])
check_initialization(mod, init, [0, 0, 0], np.diag([1, 0, 0]), desired_cov)
init.clear()
init.set(0, 'diffuse')
init.set(1, 'stationary')
init.set(2, 'stationary')
desired_cov[2, 2] = 0
check_initialization(mod, init, [0, 0, 0], np.diag([1, 0, 0]), desired_cov)
# Test with a VAR model
endog = np.zeros((10, 2))
mod = varmax.VARMAX(endog, order=(1, 0), )
intercept = [1.5, -0.1]
transition = np.array([[0.5, -0.2],
[0.1, 0.8]])
cov = np.array([[1.2, -0.4],
[-0.4, 0.4]])
tril = np.tril_indices(2)
params = np.r_[intercept, transition.ravel(),
np.linalg.cholesky(cov)[tril]]
mod.update(params)
# > stationary, global
init = Initialization(mod.k_states, 'stationary')
desired_intercept = np.linalg.solve(np.eye(2) - transition, intercept)
desired_cov = solve_discrete_lyapunov(transition, cov)
check_initialization(mod, init, desired_intercept, np.diag([0, 0]),
desired_cov)
# > diffuse, global
init.set(None, 'diffuse')
check_initialization(mod, init, [0, 0], np.eye(2), np.diag([0, 0]))
# > stationary, individually
init.unset(None)
init.set(0, 'stationary')
init.set(1, 'stationary')
a, Pinf, Pstar = init(model=mod)
desired_intercept = [intercept[0] / (1 - transition[0, 0]),
intercept[1] / (1 - transition[1, 1])]
desired_cov = np.diag([cov[0, 0] / (1 - transition[0, 0]**2),
cov[1, 1] / (1 - transition[1, 1]**2)])
check_initialization(mod, init, desired_intercept, np.diag([0, 0]),
desired_cov)
