def test_G_explosive_root(f_arma32):
"""
Test how to handle explosive root in Smoother
"""
# Initialize the model
x = 1 # used to calculate stationary mean
model = BCM()
model.set_f(f_arma32)
theta_intend = np.array([0.1, 0.2, 0.25])
theta_test = np.array([0.59358299, 0.91708496, 0.54634604])
T = 250
my_ft = lambda theta, T, **kwargs: ft(theta, f_arma32, T, **kwargs)
x_col = ['const']
Xt = pd.DataFrame({x_col[0]: x * np.ones(T)})
# Build simulated data
df, y_col, xi_col = model.simulated_data(input_theta=theta_intend, Xt=Xt)
Xt = df_to_tensor(df, x_col)
Yt = df_to_tensor(df, y_col)
kf = Filter(my_ft, Yt, Xt, for_smoother=True)
kf.fit(theta_test)
ks = Smoother()
ks.fit(kf)
result = ks.G(theta_test)
assert not np.isnan(result)
def test_get_smoothed_val_not_smoothed(ft_ll_mvar_1d, Yt_1d):
"""
Test error message when fit is not run
"""
kf = Filter(ft_ll_mvar_1d, Yt_1d, for_smoother=True)
ks = Smoother()
with pytest.raises(TypeError) as error:
ks.get_smoothed_val()
expected_result = 'The Kalman smoother object is not fitted yet'
result = str(error.value)
assert result == expected_result
def test_init_attr_smoother_not_fitted(ft_rw_1, theta_rw, Yt_1d, Xt_1d):
"""
Test error message Kalman filter not fitted
"""
kf = Filter(ft_rw_1, Yt_1d, Xt_1d, for_smoother=True)
ks = Smoother()
with pytest.raises(TypeError) as error:
ks.init_attr_smoother(kf)
msg = str(error.value)
e_msg = 'The Kalman filter object is not fitted yet'
assert msg == e_msg
def test_get_smoothed_val_all_xi(ft_ll_mvar_diffuse, Yt_mvar_diffuse_missing,
theta_ll_mvar_diffuse):
"""
Test df with all xi
"""
kf = Filter(ft_ll_mvar_diffuse, Yt_mvar_diffuse_missing, for_smoother=True)
kf.fit(theta_ll_mvar_diffuse)
ks = Smoother()
ks.fit(kf)
y_t_T, yP_t_T, xi_T, P_T = ks.get_smoothed_val()
np.testing.assert_array_equal(xi_T[2], ks.xi_t_T[2])
np.testing.assert_array_equal(P_T[2], ks.P_t_T[2])
np.testing.assert_array_equal(P_T[3], ks.P_t_T[3])
Mt = ft_ll_mvar_diffuse(theta_ll_mvar_diffuse, 4)
Rt = Mt['Rt'][0]
Dt = Mt['Dt'][0]
Ht = Mt['Ht'][0]
# Test smoothed y
R2_0 = np.array([[0.5 - 0.4 / 0.6 * 0.4]])
B0 = 0.4 / 0.6
delta_H0 = Ht[0:1] - B0 * Ht[1:]
eps0 = B0 * (Yt_mvar_diffuse_missing[0][1] - Dt[1:].dot(ks.Xt[0]) -
Ht[1:].dot(ks.xi_t_T[0]))
y0 = Ht[0:1].dot(ks.xi_t_T[0]) + Dt[0:1].dot(ks.Xt[0]) + eps0
yP_0 = (delta_H0.dot(ks.P_t_T[0]).dot(delta_H0.T) + R2_0).item()
R2_2 = np.array([[0.6 - 0.4 / 0.5 * 0.4]])
B2 = 0.4 / 0.5
delta_H2 = Ht[1] - B2 * Ht[0]
eps2 = B2 * (Yt_mvar_diffuse_missing[2][0] - Dt[:1].dot(ks.Xt[2]) -
Ht[:1].dot(ks.xi_t_T[2]))
y2 = Ht[1:].dot(ks.xi_t_T[2]) + Dt[1:].dot(ks.Xt[2]) + eps2
yP_2 = (delta_H2.dot(ks.P_t_T[2]).dot(delta_H2.T) + R2_2).item()
expected_y_t_T = [
np.array([[y0], [2]]),
Ht.dot(ks.xi_t_T[1] + Dt.dot(ks.Xt[1])),
np.array([[2.5], [y2]]),
np.array([[3], [5]])
]
expected_Pcov_T = [
np.array([[yP_0, 0], [0, 0]]),
ks.Ht[1].dot(ks.P_t_T[1]).dot(ks.Ht[1].T) + ks.Rt[1],
np.array([[0, 0], [0, yP_2]]),
np.zeros([2, 2])
]
for t in range(4):
np.testing.assert_array_almost_equal(yP_t_T[t], expected_Pcov_T[t])
np.testing.assert_array_almost_equal(y_t_T[t], expected_y_t_T[t])
def test_init_attr_smoother_diffuse(ft_ll_mvar_diffuse, theta_ll_mvar_diffuse,
Yt_mvar_diffuse_missing):
"""
Test initialization for diffuse smoother
"""
kf = Filter(ft_ll_mvar_diffuse, Yt_mvar_diffuse_missing, for_smoother=True)
kf.fit(theta_ll_mvar_diffuse)
ks = Smoother()
ks.init_attr_smoother(kf)
e_r1 = np.zeros([2, 1])
e_N1 = np.zeros([2, 2])
np.testing.assert_array_equal(e_r1, ks.r1_t[ks.t_q - 1])
np.testing.assert_array_equal(e_N1, ks.N1_t[ks.t_q - 1])
assert ks.t_q == 3
def test_get_smoothed_y_selected_xi(ft_ll_mvar_diffuse,
Yt_mvar_diffuse_missing,
theta_ll_mvar_diffuse):
"""
Test df with selected xi
"""
kf = Filter(ft_ll_mvar_diffuse, Yt_mvar_diffuse_missing, for_smoother=True)
kf.fit(theta_ll_mvar_diffuse)
ks = Smoother()
ks.fit(kf)
_, _, xi_T, P_T = ks.get_smoothed_val(xi_col=[1])
np.testing.assert_array_equal(xi_T[2], ks.xi_t_T[2][[1]])
np.testing.assert_array_equal(P_T[2], ks.P_t_T[2][[1]][:, [1]])
def test_init_attr_smoother(ft_rw_1, theta_rw, Yt_1d, Xt_1d):
"""
Test normal run
"""
kf = Filter(ft_rw_1, Yt_1d, Xt_1d, for_smoother=True)
kf.fit(theta_rw)
ks = Smoother()
ks.init_attr_smoother(kf)
e_r0 = np.zeros([1, 1])
e_N0 = np.zeros([1, 1])
r0 = ks.r0_t[kf.T - 1]
N0 = ks.N0_t[kf.T - 1]
np.testing.assert_array_equal(e_r0, r0)
np.testing.assert_array_equal(e_N0, N0)
assert ks.r1_t.shape[0] == 0
def test_E_chi2(ft_ll_mvar_diffuse, theta_ll_mvar_diffuse,
Yt_mvar_diffuse_smooth):
"""
Test normal run
"""
kf = Filter(ft_ll_mvar_diffuse, Yt_mvar_diffuse_smooth, for_smoother=True)
kf.fit(theta_ll_mvar_diffuse)
ks = Smoother()
ks.fit(kf)
theta_test = theta_ll_mvar_diffuse + 1
Mt = ft_ll_mvar_diffuse(theta_test, ks.T)
# Test first missing
t = 2
chi2 = ks._E_chi2(Mt, t)
Ht = Mt['Ht'][t][1]
Dt = Mt['Dt'][t][1]
e_chi = ks.Yt[t][0] - Ht.dot(ks.xi_t_T[t]) - \
Dt.dot(ks.Xt[t])
e_chi2 = e_chi.dot(e_chi.T) + Ht.dot(ks.P_t_T[t]).dot(Ht.T)
np.testing.assert_array_equal(e_chi2, chi2)
# # Test all present
t = 3
chi2 = ks._E_chi2(Mt, t)
Ht = Mt['Ht'][t]
Dt = Mt['Dt'][t]
e_chi = ks.Yt[t] - Ht.dot(ks.xi_t_T[t]) - \
Dt.dot(ks.Xt[t])
e_chi2 = e_chi.dot(e_chi.T) + Ht.dot(ks.P_t_T[t]).dot(Ht.T)
np.testing.assert_array_equal(e_chi2, chi2)
def test_E_delta2(ft_ll_mvar_diffuse, theta_ll_mvar_diffuse,
Yt_mvar_diffuse_smooth):
"""
Test normal run
"""
kf = Filter(ft_ll_mvar_diffuse, Yt_mvar_diffuse_smooth, for_smoother=True)
kf.fit(theta_ll_mvar_diffuse)
ks = Smoother()
ks.fit(kf)
t = 2
theta_test = [i + 0.1 for i in theta_ll_mvar_diffuse]
Mt = ft_ll_mvar_diffuse(theta_test, ks.T)
delta2 = ks._E_delta2(Mt, t)
nt = ks.xi_t_T[t] - Mt['Ft'][t-1].dot(ks.xi_t_T[t-1]) - \
Mt['Bt'][t-1].dot(ks.Xt[t-1])
e_delta2 = nt.dot(nt.T) + ks.P_t_T[t] + Mt['Ft'][t - 1].dot(
ks.P_t_T[t - 1]).dot(Mt['Ft'][t - 1].T) - Mt['Ft'][t - 1].dot(
ks.Pcov_t_t1[t - 1]) - (ks.Pcov_t_t1[t - 1].T).dot(
Mt['Ft'][t - 1].T)
np.testing.assert_array_almost_equal(e_delta2, delta2)
def test_sequential_smooth(ft_ar2_mvar, theta_ar2_mvar, Yt_ar2_mvar,
Xt_ar2_mvar):
"""
Test whether it gives the same result as using direct approach
"""
kf = Filter(ft_ar2_mvar, Yt_ar2_mvar, Xt_ar2_mvar, for_smoother=True)
kf.fit(theta_ar2_mvar)
ks = Smoother()
ks.fit(kf)
# Test period 1
t = 0
Ft = ks.Ft[t]
Ht = ks.Ht[t]
xi_t = ks.xi_t[t][0]
P_t = ks.P_star_t[t][0]
Rt = ks.Rt[t]
Dt = ks.Dt[t]
Upsilon_t = Ht.dot(P_t).dot(Ht.T) + Rt
Upsilon_t_inv = linalg.pinv(Upsilon_t)
Kt = P_t.dot(Ht.T).dot(Upsilon_t_inv)
vt = ks.Yt[t] - Ht.dot(xi_t) - Dt.dot(ks.Xt[t])
Lt = Ft.dot(ks.I - Kt.dot(Ht))
rt = ks.r0_t[t + 1]
Nt = ks.N0_t[t + 1]
r1t = (Ht.T).dot(Upsilon_t_inv).dot(vt) + (Lt.T).dot(rt)
N1t = (Ht.T).dot(Upsilon_t_inv).dot(Ht) + \
(Lt.T).dot(Nt).dot(Lt)
e_xi_t_T = xi_t + P_t.dot(r1t)
e_P_t_T = P_t - P_t.dot(N1t).dot(P_t)
e_P_cov_t_t1 = P_t.dot(Lt.T).dot(ks.I - Nt.dot(ks.P_star_t[t + 1][0]))
e_PL = ks.P_star_t[t][ks.n_t[t]].dot(Ft.T)
np.testing.assert_array_almost_equal(e_xi_t_T, ks.xi_t_T[t])
np.testing.assert_array_almost_equal(e_P_t_T, ks.P_t_T[t])
np.testing.assert_array_almost_equal(e_PL, P_t.dot(Lt.T))
np.testing.assert_array_almost_equal(e_P_cov_t_t1, ks.Pcov_t_t1[t])
np.testing.assert_array_almost_equal(r1t, ks.r0_t[t])
np.testing.assert_array_almost_equal(N1t, ks.N0_t[t])
# Test period 2
t = 1
n_t = ks.n_t[t]
Ft = ks.Ft[t]
Ht = ks.Ht[t][:n_t]
xi_t = ks.xi_t[t][0]
P_t = ks.P_star_t[t][0]
Rt = ks.Rt[t][:n_t][:, :n_t]
Dt = ks.Dt[t][:n_t]
Yt = ks.Yt[t][:n_t]
Upsilon_t = Ht.dot(P_t).dot(Ht.T) + Rt
Upsilon_t_inv = linalg.pinv(Upsilon_t)
Kt = P_t.dot(Ht.T).dot(Upsilon_t_inv)
vt = Yt - Ht.dot(xi_t) - Dt.dot(ks.Xt[t])
Lt = Ft.dot(ks.I - Kt.dot(Ht))
rt = ks.r0_t[t + 1]
Nt = ks.N0_t[t + 1]
r1t = (Ht.T).dot(Upsilon_t_inv).dot(vt) + (Lt.T).dot(rt)
N1t = (Ht.T).dot(Upsilon_t_inv).dot(Ht) + \
(Lt.T).dot(Nt).dot(Lt)
e_xi_t_T = xi_t + P_t.dot(r1t)
e_P_t_T = P_t - P_t.dot(N1t).dot(P_t)
e_P_cov_t_t1 = P_t.dot(Lt.T).dot(ks.I - Nt.dot(ks.P_star_t[t + 1][0]))
np.testing.assert_array_almost_equal(e_xi_t_T, ks.xi_t_T[t])
np.testing.assert_array_almost_equal(e_P_t_T, ks.P_t_T[t])
np.testing.assert_array_almost_equal(e_P_cov_t_t1, ks.Pcov_t_t1[t])
np.testing.assert_array_almost_equal(r1t, ks.r0_t[t])
np.testing.assert_array_almost_equal(N1t, ks.N0_t[t])
# Test period 3
t = 2
n_t = ks.n_t[t]
Ft = ks.Ft[t]
xi_t = ks.xi_t[t][0]
P_t = ks.P_star_t[t][0]
rt = ks.r0_t[t + 1]
Nt = ks.N0_t[t + 1]
r1t = (Ft.T).dot(rt)
N1t = (Ft.T).dot(Nt).dot(Ft)
e_xi_t_T = xi_t + P_t.dot(r1t)
e_P_t_T = P_t - P_t.dot(N1t).dot(P_t)
e_P_cov_t_t1 = P_t.dot(Ft.T).dot(ks.I - Nt.dot(ks.P_star_t[t + 1][0]))
np.testing.assert_array_almost_equal(e_xi_t_T, ks.xi_t_T[t])
np.testing.assert_array_almost_equal(e_P_t_T, ks.P_t_T[t])
np.testing.assert_array_almost_equal(e_P_cov_t_t1, ks.Pcov_t_t1[t])
np.testing.assert_array_almost_equal(r1t, ks.r0_t[t])
np.testing.assert_array_almost_equal(N1t, ks.N0_t[t])
# Test period 4
t = 3
n_t = ks.n_t[t]
Ft = ks.Ft[t]
Ht = ks.Ht[t][:n_t]
xi_t = ks.xi_t[t][0]
P_t = ks.P_star_t[t][0]
Rt = ks.Rt[t][:n_t][:, :n_t]
Dt = ks.Dt[t][:n_t]
Yt = ks.Yt[t][:n_t]
Upsilon_t = Ht.dot(P_t).dot(Ht.T) + Rt
Upsilon_t_inv = linalg.pinv(Upsilon_t)
Kt = P_t.dot(Ht.T).dot(Upsilon_t_inv)
vt = Yt - Ht.dot(xi_t) - Dt.dot(ks.Xt[t])
Lt = Ft.dot(ks.I - Kt.dot(Ht))
rt = np.zeros([2, 1])
Nt = np.zeros([2, 2])
r1t = (Ht.T).dot(Upsilon_t_inv).dot(vt) + (Lt.T).dot(rt)
N1t = (Ht.T).dot(Upsilon_t_inv).dot(Ht) + \
(Lt.T).dot(Nt).dot(Lt)
e_xi_t_T = xi_t + P_t.dot(r1t)
e_P_t_T = P_t - P_t.dot(N1t).dot(P_t)
np.testing.assert_array_almost_equal(e_xi_t_T, ks.xi_t_T[t])
np.testing.assert_array_almost_equal(e_P_t_T, ks.P_t_T[t])
np.testing.assert_array_almost_equal(r1t, ks.r0_t[t])
np.testing.assert_array_almost_equal(N1t, ks.N0_t[t])
def test_sequential_smooth_diffuse_vec(ft_ll_mvar_diffuse,
theta_ll_mvar_diffuse,
Yt_mvar_diffuse_smooth_vec):
"""
Test cases with diffuse initial conditions. I test the case
where there are multiple observations at time t, and only the
first measurement matters (in local linear models).
"""
kf = Filter(ft_ll_mvar_diffuse,
Yt_mvar_diffuse_smooth_vec,
for_smoother=True)
kf.fit(theta_ll_mvar_diffuse)
ks = Smoother()
ks.fit(kf)
# Test period 3
t = 2
n_t = ks.n_t[t]
Ft = ks.Ft[t]
Ht = ks.Ht[t][:n_t]
xi_t = ks.xi_t[t][0]
P_t = ks.P_star_t[t][0]
Rt = ks.Rt[t][:n_t][:, :n_t]
Dt = ks.Dt[t][:n_t]
Yt = ks.Yt[t][:n_t]
Upsilon_t = Ht.dot(P_t).dot(Ht.T) + Rt
Upsilon_t_inv = linalg.pinv(Upsilon_t)
Kt = P_t.dot(Ht.T).dot(Upsilon_t_inv)
vt = Yt - Ht.dot(xi_t) - Dt.dot(ks.Xt[t])
Lt = Ft.dot(ks.I - Kt.dot(Ht))
rt = np.zeros([2, 1])
Nt = np.zeros([2, 2])
r1t = (Ht.T).dot(Upsilon_t_inv).dot(vt) + (Lt.T).dot(rt)
N1t = (Ht.T).dot(Upsilon_t_inv).dot(Ht) + \
(Lt.T).dot(Nt).dot(Lt)
e_xi_t_T = xi_t + P_t.dot(r1t)
e_P_t_T = P_t - P_t.dot(N1t).dot(P_t)
np.testing.assert_array_almost_equal(e_xi_t_T, ks.xi_t_T[t])
np.testing.assert_array_almost_equal(e_P_t_T, ks.P_t_T[t])
np.testing.assert_array_almost_equal(r1t, ks.r0_t[t])
np.testing.assert_array_almost_equal(N1t, ks.N0_t[t])
# Test period 2 (refer to Koopman 1997)
t = 1
Ft = ks.Ft[t]
n_t = ks.n_t[t]
xi_t = ks.xi_t[t][0]
Rt = ks.Rt[t][:n_t][:, :n_t]
Ht = ks.Ht[t][:n_t]
Dt = ks.Dt[t][:n_t]
Yt = ks.Yt[t][:n_t]
vt = Yt - Ht.dot(xi_t) - Dt.dot(ks.Xt[t])
P_inf_t = ks.P_inf_t[t][0]
P_star_t = ks.P_star_t[t][0]
r_star_t = ks.r0_t[t + 1]
N_star_t = ks.N0_t[t + 1]
Upsilon_inf_t = Ht.dot(P_inf_t).dot(Ht.T)
Upsilon_star_t = Ht.dot(P_star_t).dot(Ht.T) + Rt
K_ = np.array([[0, 0.5], [0.5, -0.25]])
star = K_.dot(Upsilon_star_t).dot(K_)
L_ = np.eye(2)
L_[0][1] = -star[1][0] / star[1][1]
L_star_L = L_.dot(star).dot(L_.T)
L_[1][1] = 1 / np.sqrt(L_star_L[1][1])
J = (L_.dot(K_)).T
J1 = J[:, 0:1]
J2 = J[:, 1:2]
F_star_neg = J2.dot(J2.T)
F_inf_neg = J1.dot(J1.T)
F2 = -F_inf_neg.dot(Upsilon_star_t).dot(F_inf_neg)
M_star = Ft.dot(P_star_t).dot(Ht.T)
M_inf = Ft.dot(P_inf_t).dot(Ht.T)
K_star = M_star.dot(F_star_neg) + M_inf.dot(F_inf_neg)
K_inf = M_star.dot(F_inf_neg) - M_inf.dot(F_inf_neg).dot(
Upsilon_star_t).dot(F_inf_neg)
L_star = Ft - K_star.dot(Ht)
L_inf = -K_inf.dot(Ht)
r_inf_1t = (Ht.T).dot(F_inf_neg).dot(vt) + (L_inf.T).dot(r_star_t)
r_star_1t = (Ht.T).dot(F_star_neg).dot(vt) + (L_star.T).dot(r_star_t)
N_star_1t = (Ht.T).dot(F_star_neg).dot(Ht) + (
L_star.T).dot(N_star_t).dot(L_star)
N_1_1t = (Ht.T).dot(F_inf_neg).dot(Ht) + (L_inf.T).dot(N_star_t).dot(
L_star) + (L_star.T).dot(N_star_t).dot(L_inf)
N_2_1t = (Ht.T).dot(F2).dot(Ht) + (L_inf.T).dot(N_star_t).dot(L_inf)
e_xi_t_T = xi_t + P_star_t.dot(r_star_1t) + P_inf_t.dot(r_inf_1t)
e_P_t_T = P_star_t - P_star_t.dot(N_star_1t).dot(P_star_t) - \
P_star_t.dot(N_1_1t).dot(P_inf_t) - P_inf_t.dot(N_1_1t).dot(
P_star_t) - P_inf_t.dot(N_2_1t).dot(P_inf_t)
e_Pcov = (P_star_t.dot(L_star.T) + P_inf_t.dot(L_inf.T)).dot(ks.I - \
N_star_t.dot(ks.P_star_t[t+1][0]))
np.testing.assert_array_almost_equal(e_xi_t_T, ks.xi_t_T[t])
np.testing.assert_array_almost_equal(e_P_t_T, ks.P_t_T[t])
np.testing.assert_array_almost_equal(e_Pcov, ks.Pcov_t_t1[t])
def test_sequential_smooth_diffuse_missing(ft_ll_mvar_diffuse,
theta_ll_mvar_diffuse,
Yt_mvar_diffuse_smooth):
"""
Test cases with diffuse initial conditions. I test period 1
in a seperate test, as the sequential smoother uses a different
system of r and N.
"""
kf = Filter(ft_ll_mvar_diffuse, Yt_mvar_diffuse_smooth, for_smoother=True)
kf.fit(theta_ll_mvar_diffuse)
ks = Smoother()
ks.fit(kf)
# Test period 2
t = 1
Ft = ks.Ft[t]
n_t = ks.n_t[t]
xi_t = ks.xi_t[t][0]
P_inf_t = ks.P_inf_t[t][0]
P_star_t = ks.P_star_t[t][0]
r0_t = ks.r0_t[t + 1]
r1_t = ks.r1_t[t + 1]
N0_t = ks.N0_t[t + 1]
N1_t = ks.N1_t[t + 1]
N2_t = ks.N2_t[t + 1]
r0_1t = (Ft.T).dot(r0_t)
r1_1t = (Ft.T).dot(r1_t)
e_xi_T = xi_t + P_star_t.dot(r0_1t) + P_inf_t.dot(r1_1t)
N0_1t = (Ft.T).dot(N0_t).dot(Ft)
N1_1t = (Ft.T).dot(N1_t).dot(Ft)
N2_1t = (Ft.T).dot(N2_t).dot(Ft)
e_P_T = P_star_t - P_star_t.dot(N0_1t).dot(P_star_t) - P_star_t.dot(
N1_1t).dot(P_inf_t) - P_inf_t.dot(N1_1t).dot(P_star_t) - \
P_inf_t.dot(N2_1t).dot(P_inf_t)
e_P_cov_t_t1 = (P_star_t.dot(Ft.T) + P_inf_t.dot(Ft.T)).dot(
ks.I - N1_t.dot(ks.P_inf_t[t + 1][0]) -
N0_t.dot(ks.P_star_t[t + 1][0])) - P_inf_t.dot(Ft.T).dot(
N2_t.dot(ks.P_inf_t[t + 1][0]) + N1_t.dot(ks.P_star_t[t + 1][0]))
np.testing.assert_array_almost_equal(e_xi_T, ks.xi_t_T[t])
np.testing.assert_array_almost_equal(e_P_T, ks.P_t_T[t])
np.testing.assert_array_almost_equal(e_P_cov_t_t1, ks.Pcov_t_t1[t])
np.testing.assert_array_almost_equal(r0_1t, ks.r0_t[t])
np.testing.assert_array_almost_equal(r1_1t, ks.r1_t[t])
np.testing.assert_array_almost_equal(N0_1t, ks.N0_t[t])
np.testing.assert_array_almost_equal(N1_1t, ks.N1_t[t])
np.testing.assert_array_almost_equal(N2_1t, ks.N2_t[t])
# Test period 3
t = 2
Ft = ks.Ft[t]
n_t = ks.n_t[t]
xi_t = ks.xi_t[t][0]
Rt = ks.Rt[t][:n_t][:, :n_t]
Ht = ks.Ht[t][:n_t]
Dt = ks.Dt[t][:n_t]
Yt = ks.Yt[t][:n_t]
vt = Yt - Ht.dot(xi_t) - Dt.dot(ks.Xt[t])
P_inf_t = ks.P_inf_t[t][0]
P_star_t = ks.P_star_t[t][0]
Upsilon_inf_t = Ht.dot(P_inf_t).dot(Ht.T)
Upsilon_star_t = Ht.dot(P_star_t).dot(Ht.T) + Rt
Upsilon_1 = linalg.pinv(Upsilon_inf_t)
Upsilon_2 = -Upsilon_1.dot(Upsilon_star_t).dot(Upsilon_1)
K0 = P_inf_t.dot(Ht.T).dot(Upsilon_1)
K1 = P_star_t.dot(Ht.T).dot(Upsilon_1) + \
P_inf_t.dot(Ht.T).dot(Upsilon_2)
L0 = Ft.dot(ks.I - K0.dot(Ht))
L1 = -Ft.dot(K1.dot(Ht))
r0_t = ks.r0_t[t + 1]
r1_t = np.zeros([2, 1])
N0_t = ks.N0_t[t + 1]
N1_t = np.zeros([2, 2])
N2_t = np.zeros([2, 2])
r0_1t = (L0.T).dot(r0_t)
r1_1t = (Ht.T).dot(Upsilon_1).dot(vt) + \
(L0.T).dot(r1_t) + (L1.T).dot(r0_t)
e_xi_T = xi_t + P_star_t.dot(r0_1t) + P_inf_t.dot(r1_1t)
N0_1t = (L0.T).dot(N0_t).dot(L0)
N1_1t = (Ht.T).dot(Upsilon_1).dot(Ht) + (L0.T).dot(N1_t).dot(L0.T) + \
(L1.T).dot(N0_t).dot(L0) + (L0.T).dot(N0_t).dot(L1)
N2_1t = (Ht.T).dot(Upsilon_2).dot(Ht) + (L0.T).dot(N2_t).dot(L0) + \
(L0.T).dot(N1_t).dot(L1) + (L1.T).dot(N1_t.T).dot(L0) + \
(L1.T).dot(N0_t).dot(L1)
e_P_T = P_star_t - P_star_t.dot(N0_1t).dot(P_star_t) - P_star_t.dot(
N1_1t).dot(P_inf_t) - P_inf_t.dot(N1_1t).dot(P_star_t) - \
P_inf_t.dot(N2_1t).dot(P_inf_t)
e_P_cov_t_t1 = (P_star_t.dot(L0.T) + P_inf_t.dot(L1.T)).dot(
ks.I - N1_t.dot(ks.P_inf_t[t + 1][0]) -
N0_t.dot(ks.P_star_t[t + 1][0])) - P_inf_t.dot(L0.T).dot(
N2_t.dot(ks.P_inf_t[t + 1][0]) + N1_t.dot(ks.P_star_t[t + 1][0]))
np.testing.assert_array_almost_equal(e_xi_T, ks.xi_t_T[t])
np.testing.assert_array_almost_equal(e_P_T, ks.P_t_T[t])
np.testing.assert_array_almost_equal(e_P_cov_t_t1, ks.Pcov_t_t1[t])
np.testing.assert_array_almost_equal(r0_1t, ks.r0_t[t])
np.testing.assert_array_almost_equal(r1_1t, ks.r1_t[t])
np.testing.assert_array_almost_equal(N0_1t, ks.N0_t[t])
np.testing.assert_array_almost_equal(N1_1t, ks.N1_t[t])
np.testing.assert_array_almost_equal(N2_1t, ks.N2_t[t])
# Test period 4
t = 3
n_t = ks.n_t[t]
Ft = ks.Ft[t]
Ht = ks.Ht[t][:n_t]
xi_t = ks.xi_t[t][0]
P_t = ks.P_star_t[t][0]
Rt = ks.Rt[t][:n_t][:, :n_t]
Dt = ks.Dt[t][:n_t]
Yt = ks.Yt[t][:n_t]
Upsilon_t = Ht.dot(P_t).dot(Ht.T) + Rt
Upsilon_t_inv = linalg.pinv(Upsilon_t)
Kt = P_t.dot(Ht.T).dot(Upsilon_t_inv)
vt = Yt - Ht.dot(xi_t) - Dt.dot(ks.Xt[t])
Lt = Ft.dot(ks.I - Kt.dot(Ht))
rt = np.zeros([2, 1])
Nt = np.zeros([2, 2])
r1t = (Ht.T).dot(Upsilon_t_inv).dot(vt) + (Lt.T).dot(rt)
N1t = (Ht.T).dot(Upsilon_t_inv).dot(Ht) + \
(Lt.T).dot(Nt).dot(Lt)
e_xi_t_T = xi_t + P_t.dot(r1t)
e_P_t_T = P_t - P_t.dot(N1t).dot(P_t)
np.testing.assert_array_almost_equal(e_xi_t_T, ks.xi_t_T[t])
np.testing.assert_array_almost_equal(e_P_t_T, ks.P_t_T[t])
np.testing.assert_array_almost_equal(r1t, ks.r0_t[t])
np.testing.assert_array_almost_equal(N1t, ks.N0_t[t])