def test_innovations_algo_rtol():
ma = np.array([-0.9, 0.5])
acovf = np.array([1 + (ma ** 2).sum(), ma[0] + ma[1] * ma[0], ma[1]])
theta, sigma2 = innovations_algo(acovf, nobs=500)
theta_2, sigma2_2 = innovations_algo(acovf, nobs=500, rtol=1e-8)
assert_allclose(theta, theta_2)
assert_allclose(sigma2, sigma2_2)
def test_innovations_algo_rtol():
ma = np.array([-0.9, 0.5])
acovf = np.array([1 + (ma ** 2).sum(), ma[0] + ma[1] * ma[0], ma[1]])
theta, sigma2 = innovations_algo(acovf, nobs=500)
theta_2, sigma2_2 = innovations_algo(acovf, nobs=500, rtol=1e-8)
assert_allclose(theta, theta_2)
assert_allclose(sigma2, sigma2_2)
def test_innovations_algo_brockwell_davis():
ma = -0.9
acovf = np.array([1 + ma ** 2, ma])
theta, sigma2 = innovations_algo(acovf, nobs=4)
exp_theta = np.array([[0], [-.4972], [-.6606], [-.7404]])
assert_allclose(theta, exp_theta, rtol=1e-4)
assert_allclose(sigma2, [1.81, 1.3625, 1.2155, 1.1436], rtol=1e-4)
theta, sigma2 = innovations_algo(acovf, nobs=500)
assert_allclose(theta[-1, 0], ma)
assert_allclose(sigma2[-1], 1.0)
def test_innovations_algo_brockwell_davis():
ma = -0.9
acovf = np.array([1 + ma ** 2, ma])
theta, sigma2 = innovations_algo(acovf, nobs=4)
exp_theta = np.array([[0], [-.4972], [-.6606], [-.7404]])
assert_allclose(theta, exp_theta, rtol=1e-4)
assert_allclose(sigma2, [1.81, 1.3625, 1.2155, 1.1436], rtol=1e-4)
theta, sigma2 = innovations_algo(acovf, nobs=500)
assert_allclose(theta[-1, 0], ma)
assert_allclose(sigma2[-1], 1.0)
def test_innovations_algo_filter_kalman_filter(reset_randomstate):
# Test the innovations algorithm and filter against the Kalman filter
# for exact likelihood evaluation of an ARMA process
ar_params = np.array([0.5])
ma_params = np.array([0.2])
# TODO could generalize to sigma2 != 1, if desired, after #5324 is merged
# and there is a sigma2 argument to arma_acovf
# (but maybe this is not really necessary for the point of this test)
sigma2 = 1
endog = np.random.normal(size=10)
# Innovations algorithm approach
acovf = arma_acovf(np.r_[1, -ar_params], np.r_[1, ma_params],
nobs=len(endog))
theta, v = innovations_algo(acovf)
u = innovations_filter(endog, theta)
llf_obs = -0.5 * u ** 2 / (sigma2 * v) - 0.5 * np.log(2 * np.pi * v)
# Kalman filter apparoach
mod = SARIMAX(endog, order=(len(ar_params), 0, len(ma_params)))
res = mod.filter(np.r_[ar_params, ma_params, sigma2])
# Test that the two approaches are identical
atol = 1e-6 if PLATFORM_WIN else 0.0
assert_allclose(u, res.forecasts_error[0], rtol=1e-6, atol=atol)
assert_allclose(theta[1:, 0], res.filter_results.kalman_gain[0, 0, :-1],
atol=atol)
assert_allclose(llf_obs, res.llf_obs, atol=atol)
def test_innovations_algo_filter_kalman_filter(reset_randomstate):
# Test the innovations algorithm and filter against the Kalman filter
# for exact likelihood evaluation of an ARMA process
ar_params = np.array([0.5])
ma_params = np.array([0.2])
# TODO could generalize to sigma2 != 1, if desired, after #5324 is merged
# and there is a sigma2 argument to arma_acovf
# (but maybe this is not really necessary for the point of this test)
sigma2 = 1
endog = np.random.normal(size=10)
# Innovations algorithm approach
acovf = arma_acovf(np.r_[1, -ar_params], np.r_[1, ma_params],
nobs=len(endog))
theta, v = innovations_algo(acovf)
u = innovations_filter(endog, theta)
llf_obs = -0.5 * u**2 / (sigma2 * v) - 0.5 * np.log(2 * np.pi * v)
# Kalman filter apparoach
mod = SARIMAX(endog, order=(len(ar_params), 0, len(ma_params)))
res = mod.filter(np.r_[ar_params, ma_params, sigma2])
# Test that the two approaches are identical
atol = 1e-6 if PLATFORM_WIN else 0.0
assert_allclose(u, res.forecasts_error[0], atol=atol)
assert_allclose(theta[1:, 0], res.filter_results.kalman_gain[0, 0, :-1],
atol=atol)
assert_allclose(llf_obs, res.llf_obs, atol=atol)
def test_innovations_errors():
ma = -0.9
acovf = np.array([1 + ma ** 2, ma])
with pytest.raises(TypeError):
innovations_algo(acovf, nobs=2.2)
with pytest.raises(ValueError):
innovations_algo(acovf, nobs=-1)
with pytest.raises(ValueError):
innovations_algo(np.empty((2, 2)))
with pytest.raises(TypeError):
innovations_algo(acovf, rtol='none')
def test_innovations_errors():
ma = -0.9
acovf = np.array([1 + ma ** 2, ma])
with pytest.raises(ValueError):
innovations_algo(acovf, nobs=2.2)
with pytest.raises(ValueError):
innovations_algo(acovf, nobs=-1)
with pytest.raises(ValueError):
innovations_algo(np.empty((2, 2)))
with pytest.raises(ValueError):
innovations_algo(acovf, rtol='none')
def test_innovations_filter_errors():
ma = -0.9
acovf = np.array([1 + ma ** 2, ma])
theta, _ = innovations_algo(acovf, nobs=4)
with pytest.raises(ValueError):
innovations_filter(np.empty((2, 2)), theta)
with pytest.raises(ValueError):
innovations_filter(np.empty(4), theta[:-1])
with pytest.raises(ValueError):
innovations_filter(pd.DataFrame(np.empty((1, 4))), theta)
def test_innovations_filter_pandas(reset_randomstate):
ma = np.array([-0.9, 0.5])
acovf = np.array([1 + (ma**2).sum(), ma[0] + ma[1] * ma[0], ma[1]])
theta, _ = innovations_algo(acovf, nobs=10)
endog = np.random.randn(10)
endog_pd = pd.Series(endog, index=pd.date_range('2000-01-01', periods=10))
resid = innovations_filter(endog, theta)
resid_pd = innovations_filter(endog_pd, theta)
assert_allclose(resid, resid_pd.values)
assert_index_equal(endog_pd.index, resid_pd.index)
def test_innovations_filter_errors():
ma = -0.9
acovf = np.array([1 + ma ** 2, ma])
theta, _ = innovations_algo(acovf, nobs=4)
with pytest.raises(ValueError):
innovations_filter(np.empty((2, 2)), theta)
with pytest.raises(ValueError):
innovations_filter(np.empty(4), theta[:-1])
with pytest.raises(ValueError):
innovations_filter(pd.DataFrame(np.empty((1, 4))), theta)
def test_innovations_filter_pandas(reset_randomstate):
ma = np.array([-0.9, 0.5])
acovf = np.array([1 + (ma ** 2).sum(), ma[0] + ma[1] * ma[0], ma[1]])
theta, _ = innovations_algo(acovf, nobs=10)
endog = np.random.randn(10)
endog_pd = pd.Series(endog,
index=pd.date_range('2000-01-01', periods=10))
resid = innovations_filter(endog, theta)
resid_pd = innovations_filter(endog_pd, theta)
assert_allclose(resid, resid_pd.values)
assert_index_equal(endog_pd.index, resid_pd.index)
def test_innovations_filter_brockwell_davis(reset_randomstate):
ma = -0.9
acovf = np.array([1 + ma ** 2, ma])
theta, _ = innovations_algo(acovf, nobs=4)
e = np.random.randn(5)
endog = e[1:] + ma * e[:-1]
resid = innovations_filter(endog, theta)
expected = [endog[0]]
for i in range(1, 4):
expected.append(endog[i] - theta[i, 0] * expected[-1])
expected = np.array(expected)
assert_allclose(resid, expected)
def test_innovations_filter_brockwell_davis(reset_randomstate):
ma = -0.9
acovf = np.array([1 + ma ** 2, ma])
theta, _ = innovations_algo(acovf, nobs=4)
e = np.random.randn(5)
endog = e[1:] + ma * e[:-1]
resid = innovations_filter(endog, theta)
expected = [endog[0]]
for i in range(1, 4):
expected.append(endog[i] - theta[i, 0] * expected[-1])
expected = np.array(expected)
assert_allclose(resid, expected)
def innovations(endog, ma_order=0, demean=True):
"""
Estimate MA parameters using innovations algorithm.
Parameters
----------
endog : array_like or SARIMAXSpecification
Input time series array, assumed to be stationary.
ma_order : int, optional
Maximum moving average order. Default is 0.
demean : bool, optional
Whether to estimate and remove the mean from the process prior to
fitting the moving average coefficients. Default is True.
Returns
-------
parameters : list of SARIMAXParams objects
List elements correspond to estimates at different `ma_order`. For
example, parameters[0] is an `SARIMAXParams` instance corresponding to
`ma_order=0`.
other_results : Bunch
Includes one component, `spec`, containing the `SARIMAXSpecification`
instance corresponding to the input arguments.
Notes
-----
The primary reference is [1]_, section 5.1.3.
This procedure assumes that the series is stationary.
References
----------
.. [1] Brockwell, Peter J., and Richard A. Davis. 2016.
Introduction to Time Series and Forecasting. Springer.
"""
spec = max_spec = SARIMAXSpecification(endog, ma_order=ma_order)
endog = max_spec.endog
if demean:
endog = endog - endog.mean()
if not max_spec.is_ma_consecutive:
raise ValueError('Innovations estimation unavailable for models with'
' seasonal or otherwise non-consecutive MA orders.')
sample_acovf = acovf(endog, fft=True)
theta, v = innovations_algo(sample_acovf, nobs=max_spec.ma_order + 1)
ma_params = [theta[i, :i] for i in range(1, max_spec.ma_order + 1)]
sigma2 = v
out = []
for i in range(max_spec.ma_order + 1):
spec = SARIMAXSpecification(ma_order=i)
p = SARIMAXParams(spec=spec)
if i == 0:
p.params = sigma2[i]
else:
p.params = np.r_[ma_params[i - 1], sigma2[i]]
out.append(p)
# Construct other results
other_results = Bunch({
'spec': spec,
})
return out, other_results