class TestLinearStateSpace(unittest.TestCase):
def setUp(self):
# Initial Values
A = .95
C = .05
G = 1.
mu_0 = .75
self.ss = LinearStateSpace(A, C, G, mu_0=mu_0)
def tearDown(self):
del self.ss
def test_stationarity(self):
vals = self.ss.stationary_distributions(max_iter=1000, tol=1e-9)
ssmux, ssmuy, sssigx, sssigy = vals
self.assertTrue(abs(ssmux - ssmuy) < 2e-8)
self.assertTrue(abs(sssigx - sssigy) < 2e-8)
self.assertTrue(abs(ssmux) < 2e-8)
self.assertTrue(abs(sssigx - self.ss.C**2/(1 - self.ss.A**2)) < 2e-8)
def test_replicate(self):
xval, yval = self.ss.replicate(T=100, num_reps=5000)
assert_allclose(xval, yval)
self.assertEqual(xval.size, 5000)
self.assertLessEqual(abs(np.mean(xval)), .05)
def setUp(self):
# Initial Values
A = .95
C = .05
G = 1.
mu_0 = .75
self.ss = LinearStateSpace(A, C, G, mu_0=mu_0)
class TestLinearStateSpace(unittest.TestCase):
def setUp(self):
# Initial Values
A = .95
C = .05
G = 1.
mu_0 = .75
self.ss = LinearStateSpace(A, C, G, mu_0=mu_0)
def tearDown(self):
del self.ss
def test_stationarity(self):
vals = self.ss.stationary_distributions(max_iter=1000, tol=1e-9)
ssmux, ssmuy, sssigx, sssigy = vals
self.assertTrue(abs(ssmux - ssmuy) < 2e-8)
self.assertTrue(abs(sssigx - sssigy) < 2e-8)
self.assertTrue(abs(ssmux) < 2e-8)
self.assertTrue(abs(sssigx - self.ss.C**2/(1 - self.ss.A**2)) < 2e-8)
def test_simulate(self):
ss = self.ss
sim = ss.simulate(ts_length=250)
for arr in sim:
self.assertTrue(len(arr[0])==250)
def test_simulate_with_seed(self):
ss = self.ss
xval, yval = ss.simulate(ts_length=5, random_state=5)
expected_output = np.array([0.75 , 0.73456137, 0.6812898, 0.76876387,
.71772107])
assert_allclose(xval[0], expected_output)
assert_allclose(yval[0], expected_output)
def test_replicate(self):
xval, yval = self.ss.replicate(T=100, num_reps=5000)
assert_allclose(xval, yval)
self.assertEqual(xval.size, 5000)
self.assertLessEqual(abs(np.mean(xval)), .05)
def test_replicate_with_seed(self):
xval, yval = self.ss.replicate(T=100, num_reps=5, random_state=5)
expected_output = np.array([0.06871204, 0.06937119, -0.1478022,
0.23841252, -0.06823762])
assert_allclose(xval[0], expected_output)
assert_allclose(yval[0], expected_output)
class TestLinearStateSpace(unittest.TestCase):
def setUp(self):
# Initial Values
A = .95
C = .05
G = 1.
mu_0 = .75
self.ss = LinearStateSpace(A, C, G, mu_0=mu_0)
def tearDown(self):
del self.ss
def test_stationarity(self):
vals = self.ss.stationary_distributions(max_iter=1000, tol=1e-9)
ssmux, ssmuy, sssigx, sssigy = vals
self.assertTrue(abs(ssmux - ssmuy) < 2e-8)
self.assertTrue(abs(sssigx - sssigy) < 2e-8)
self.assertTrue(abs(ssmux) < 2e-8)
self.assertTrue(abs(sssigx - self.ss.C**2/(1 - self.ss.A**2)) < 2e-8)
def test_simulate(self):
ss = self.ss
sim = ss.simulate(ts_length=250)
for arr in sim:
self.assertTrue(len(arr[0])==250)
def test_simulate_with_seed(self):
ss = self.ss
xval, yval = ss.simulate(ts_length=5, random_state=5)
expected_output = np.array([0.75 , 0.73456137, 0.6812898, 0.76876387,
.71772107])
assert_allclose(xval[0], expected_output)
assert_allclose(yval[0], expected_output)
def test_replicate(self):
xval, yval = self.ss.replicate(T=100, num_reps=5000)
assert_allclose(xval, yval)
self.assertEqual(xval.size, 5000)
self.assertLessEqual(abs(np.mean(xval)), .05)
def test_replicate_with_seed(self):
xval, yval = self.ss.replicate(T=100, num_reps=5, random_state=5)
expected_output = np.array([0.06871204, 0.06937119, -0.1478022,
0.23841252, -0.06823762])
assert_allclose(xval[0], expected_output)
assert_allclose(yval[0], expected_output)
def setUp(self):
# Initial Values
A = .95
C = .05
G = 1.
mu_0 = .75
self.ss = LinearStateSpace(A, C, G, mu_0=mu_0)
def setUp(self):
# Example 1
A = .95
C = .05
G = 1.
mu_0 = .75
self.ss1 = LinearStateSpace(A, C, G, mu_0=mu_0)
# Example 2
ρ1 = 0.5
ρ2 = 0.3
α = 0.5
A = np.array([[ρ1, ρ2, α], [1, 0, 0], [0, 0, 1]])
C = np.array([[1], [0], [0]])
G = np.array([[1, 0, 0]])
mu_0 = [0.5, 0.5, 1]
self.ss2 = LinearStateSpace(A, C, G, mu_0=mu_0)
def setUp(self):
# Initial Values
self.A = np.array([[.95, 0], [0., .95]])
self.C = np.eye(2) * np.sqrt(0.5)
self.G = np.eye(2) * .5
self.H = np.eye(2) * np.sqrt(0.2)
self.Q = np.dot(self.C, self.C.T)
self.R = np.dot(self.H, self.H.T)
ss = LinearStateSpace(self.A, self.C, self.G, self.H)
self.kf = Kalman(ss)
def test_non_square_A():
A = np.zeros((1, 2))
C = np.zeros((1, 1))
G = np.zeros((1, 1))
LinearStateSpace(A, C, G)
class TestLinearStateSpace(unittest.TestCase):
def setUp(self):
# Example 1
A = .95
C = .05
G = 1.
mu_0 = .75
self.ss1 = LinearStateSpace(A, C, G, mu_0=mu_0)
# Example 2
ρ1 = 0.5
ρ2 = 0.3
α = 0.5
A = np.array([[ρ1, ρ2, α], [1, 0, 0], [0, 0, 1]])
C = np.array([[1], [0], [0]])
G = np.array([[1, 0, 0]])
mu_0 = [0.5, 0.5, 1]
self.ss2 = LinearStateSpace(A, C, G, mu_0=mu_0)
def tearDown(self):
del self.ss1
del self.ss2
def test_stationarity(self):
vals = self.ss1.stationary_distributions()
ssmux, ssmuy, sssigx, sssigy, sssigyx = vals
self.assertTrue(abs(ssmux - ssmuy) < 2e-8)
self.assertTrue(abs(sssigx - sssigy) < 2e-8)
self.assertTrue(abs(ssmux) < 2e-8)
self.assertTrue(abs(sssigx - self.ss1.C**2/(1 - self.ss1.A**2)) < 2e-8)
self.assertTrue(abs(sssigyx - self.ss1.G @ sssigx) < 2e-8)
vals = self.ss2.stationary_distributions()
ssmux, ssmuy, sssigx, sssigy, sssigyx = vals
assert_allclose(ssmux.flatten(), np.array([2.5, 2.5, 1]))
assert_allclose(ssmuy.flatten(), np.array([2.5]))
assert_allclose(sssigx, self.ss2.A @ sssigx @ self.ss2.A.T + self.ss2.C @ self.ss2.C.T)
assert_allclose(sssigy, self.ss2.G @ sssigx @ self.ss2.G.T)
assert_allclose(sssigyx, self.ss2.G @ sssigx)
def test_simulate(self):
ss = self.ss1
sim = ss.simulate(ts_length=250)
for arr in sim:
self.assertTrue(len(arr[0])==250)
def test_simulate_with_seed(self):
ss = self.ss1
xval, yval = ss.simulate(ts_length=5, random_state=5)
expected_output = np.array([0.75 , 0.73456137, 0.6812898, 0.76876387,
.71772107])
assert_allclose(xval[0], expected_output)
assert_allclose(yval[0], expected_output)
def test_replicate(self):
xval, yval = self.ss1.replicate(T=100, num_reps=5000)
assert_allclose(xval, yval)
self.assertEqual(xval.size, 5000)
self.assertLessEqual(abs(np.mean(xval)), .05)
def test_replicate_with_seed(self):
xval, yval = self.ss1.replicate(T=100, num_reps=5, random_state=5)
expected_output = np.array([0.06871204, 0.06937119, -0.1478022,
0.23841252, -0.06823762])
assert_allclose(xval[0], expected_output)
assert_allclose(yval[0], expected_output)
def whitener_lss(self):
r"""
This function takes the linear state space system
that is an input to the Kalman class and it converts
that system to the time-invariant whitener represenation
given by
.. math::
\tilde{x}_{t+1}^* = \tilde{A} \tilde{x} + \tilde{C} v
a = \tilde{G} \tilde{x}
where
.. math::
\tilde{x}_t = [x+{t}, \hat{x}_{t}, v_{t}]
and
.. math::
\tilde{A} =
\begin{bmatrix}
A & 0 & 0 \\
KG & A-KG & KH \\
0 & 0 & 0 \\
\end{bmatrix}
.. math::
\tilde{C} =
\begin{bmatrix}
C & 0 \\
0 & 0 \\
0 & I \\
\end{bmatrix}
.. math::
\tilde{G} =
\begin{bmatrix}
G & -G & H \\
\end{bmatrix}
with :math:`A, C, G, H` coming from the linear state space system
that defines the Kalman instance
Returns
-------
whitened_lss : LinearStateSpace
This is the linear state space system that represents
the whitened system
"""
K = self.K_infinity
# Get the matrix sizes
n, k, m, l = self.ss.n, self.ss.k, self.ss.m, self.ss.l
A, C, G, H = self.ss.A, self.ss.C, self.ss.G, self.ss.H
Atil = np.vstack([np.hstack([A, np.zeros((n, n)), np.zeros((n, l))]),
np.hstack([np.dot(K, G),
A-np.dot(K, G),
np.dot(K, H)]),
np.zeros((l, 2*n + l))])
Ctil = np.vstack([np.hstack([C, np.zeros((n, l))]),
np.zeros((n, m+l)),
np.hstack([np.zeros((l, m)), np.eye(l)])])
Gtil = np.hstack([G, -G, H])
whitened_lss = LinearStateSpace(Atil, Ctil, Gtil)
self.whitened_lss = whitened_lss
return whitened_lss
class TestLinearStateSpace:
def setup(self):
# Example 1
A = .95
C = .05
G = 1.
mu_0 = .75
self.ss1 = LinearStateSpace(A, C, G, mu_0=mu_0)
# Example 2
ρ1 = 0.5
ρ2 = 0.3
α = 0.5
A = np.array([[ρ1, ρ2, α], [1, 0, 0], [0, 0, 1]])
C = np.array([[1], [0], [0]])
G = np.array([[1, 0, 0]])
mu_0 = [0.5, 0.5, 1]
self.ss2 = LinearStateSpace(A, C, G, mu_0=mu_0)
def tearDown(self):
del self.ss1
del self.ss2
def test_stationarity(self):
vals = self.ss1.stationary_distributions()
ssmux, ssmuy, sssigx, sssigy, sssigyx = vals
assert_(abs(ssmux - ssmuy) < 2e-8)
assert_(abs(sssigx - sssigy) < 2e-8)
assert_(abs(ssmux) < 2e-8)
assert_(abs(sssigx - self.ss1.C**2/(1 - self.ss1.A**2)) < 2e-8)
assert_(abs(sssigyx - self.ss1.G @ sssigx) < 2e-8)
vals = self.ss2.stationary_distributions()
ssmux, ssmuy, sssigx, sssigy, sssigyx = vals
assert_allclose(ssmux.flatten(), np.array([2.5, 2.5, 1]))
assert_allclose(ssmuy.flatten(), np.array([2.5]))
assert_allclose(
sssigx,
self.ss2.A @ sssigx @ self.ss2.A.T + self.ss2.C @ self.ss2.C.T
)
assert_allclose(sssigy, self.ss2.G @ sssigx @ self.ss2.G.T)
assert_allclose(sssigyx, self.ss2.G @ sssigx)
def test_simulate(self):
ss = self.ss1
sim = ss.simulate(ts_length=250)
for arr in sim:
assert_(len(arr[0]) == 250)
def test_simulate_with_seed(self):
ss = self.ss1
xval, yval = ss.simulate(ts_length=5, random_state=5)
expected_output = np.array([0.75, 0.69595649, 0.78269723, 0.73095776,
0.69989036])
assert_allclose(xval[0], expected_output)
assert_allclose(yval[0], expected_output)
def test_replicate(self):
xval, yval = self.ss1.replicate(T=100, num_reps=5000)
assert_allclose(xval, yval)
assert_(xval.size == 5000)
assert_(abs(np.mean(xval)) <= .05)
def test_replicate_with_seed(self):
xval, yval = self.ss1.replicate(T=100, num_reps=5, random_state=5)
expected_output = np.array([0.10498898, 0.02892168, 0.04915998,
0.18568489, 0.04541764])
assert_allclose(xval[0], expected_output)
assert_allclose(yval[0], expected_output)
def whitener_lss(self):
r"""
This function takes the linear state space system
that is an input to the Kalman class and it converts
that system to the time-invariant whitener represenation
given by
\tilde{x}_{t+1}^* = \tilde{A} \tilde{x} + \tilde{C} v
a = \tilde{G} \tilde{x}
where
\tilde{x}_t = [x+{t}, \hat{x}_{t}, v_{t}]
and
\tilde{A} = [A 0 0
KG A-KG KH
0 0 0]
\tilde{C} = [C 0
0 0
0 I]
\tilde{G} = [G -G H]
with A, C, G, H coming from the linear state space system
that defines the Kalman instance
Returns
-------
whitened_lss : LinearStateSpace
This is the linear state space system that represents
the whitened system
"""
# Check for steady state Sigma and K
if self.K_infinity is None:
Sig, K = self.stationary_values()
self.Sigma_infinity = Sig
self.K_infinity = K
else:
K = self.K_infinity
# Get the matrix sizes
n, k, m, l = self.ss.n, self.ss.k, self.ss.m, self.ss.l
A, C, G, H = self.ss.A, self.ss.C, self.ss.G, self.ss.H
Atil = np.vstack([
np.hstack([A, np.zeros((n, n)),
np.zeros((n, l))]),
np.hstack([dot(K, G), A - dot(K, G),
dot(K, H)]),
np.zeros((l, 2 * n + l))
])
Ctil = np.vstack([
np.hstack([C, np.zeros((n, l))]),
np.zeros((n, m + l)),
np.hstack([np.zeros((l, m)), np.eye(l)])
])
Gtil = np.hstack([G, -G, H])
whitened_lss = LinearStateSpace(Atil, Ctil, Gtil)
self.whitened_lss = whitened_lss
return whitened_lss