def test_innovations_algo_direct_filter_kalman_filter(ar_params, ma_params,
sigma2):
# Test the innovations algorithm and filter against the Kalman filter
# for exact likelihood evaluation of an ARMA process, using the direct
# function.
endog = np.random.normal(size=10)
# Innovations algorithm approach
u, r = arma_innovations.arma_innovations(endog, ar_params, ma_params,
sigma2)
v = np.array(r) * sigma2
u = np.array(u)
llf_obs = -0.5 * u**2 / 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
assert_allclose(u, res.forecasts_error[0])
# assert_allclose(theta[1:, 0], res.filter_results.kalman_gain[0, 0, :-1])
assert_allclose(llf_obs, res.llf_obs)
# Get llf_obs directly
llf_obs2 = _arma_innovations.darma_loglikeobs_fast(
endog, ar_params, ma_params, sigma2)
assert_allclose(llf_obs2, res.llf_obs)
def test_innovations_algo_direct_filter_kalman_filter(ar_params, ma_params,
sigma2):
# Test the innovations algorithm and filter against the Kalman filter
# for exact likelihood evaluation of an ARMA process, using the direct
# function.
endog = np.random.normal(size=10)
# Innovations algorithm approach
u, r = arma_innovations.arma_innovations(endog, ar_params, ma_params,
sigma2)
v = np.array(r) * sigma2
u = np.array(u)
llf_obs = -0.5 * u**2 / 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
assert_allclose(u, res.forecasts_error[0])
# assert_allclose(theta[1:, 0], res.filter_results.kalman_gain[0, 0, :-1])
assert_allclose(llf_obs, res.llf_obs)
# Get llf_obs directly
llf_obs2 = _arma_innovations.darma_loglikeobs_fast(
endog, ar_params, ma_params, sigma2)
assert_allclose(llf_obs2, res.llf_obs)
def test_itsmr():
# This is essentially a high precision version of
# test_brockwell_davis_example_517, where the desired values were computed
# from R itsmr::hannan; see results/results_hr.R
endog = lake.copy()
hr, _ = hannan_rissanen(endog,
ar_order=1,
ma_order=1,
demean=True,
initial_ar_order=22,
unbiased=False)
assert_allclose(hr.ar_params, [0.69607715], atol=1e-4)
assert_allclose(hr.ma_params, [0.3787969217], atol=1e-4)
# Because our fast implementation of the innovations algorithm does not
# allow for non-stationary processes, the estimate of the variance returned
# by `hannan_rissanen` is based on the residuals from the least-squares
# regression, rather than (as reported by BD) based on the innovations
# algorithm output. Since the estimates here do correspond to a stationary
# series, we can compute the innovations variance manually to check
# against BD.
u, v = arma_innovations(endog - endog.mean(),
hr.ar_params,
hr.ma_params,
sigma2=1)
tmp = u / v**0.5
assert_allclose(np.inner(tmp, tmp) / len(u), 0.4773580109, atol=1e-4)
def test_itsmr_with_fixed_params(fixed_params):
# This test is a variation of test_itsmr where we fix 1 or more parameters
# for Example 5.1.7 in Brockwell and Davis (2016) and check that free
# parameters are still correct'.
endog = lake.copy()
hr, _ = hannan_rissanen(
endog, ar_order=1, ma_order=1, demean=True,
initial_ar_order=22, unbiased=False,
fixed_params=fixed_params
)
assert_allclose(hr.ar_params, [0.69607715], atol=1e-4)
assert_allclose(hr.ma_params, [0.3787969217], atol=1e-4)
# Because our fast implementation of the innovations algorithm does not
# allow for non-stationary processes, the estimate of the variance returned
# by `hannan_rissanen` is based on the residuals from the least-squares
# regression, rather than (as reported by BD) based on the innovations
# algorithm output. Since the estimates here do correspond to a stationary
# series, we can compute the innovations variance manually to check
# against BD.
u, v = arma_innovations(endog - endog.mean(), hr.ar_params, hr.ma_params,
sigma2=1)
tmp = u / v**0.5
assert_allclose(np.inner(tmp, tmp) / len(u), 0.4773580109, atol=1e-4)
def test_brockwell_davis_example_517():
# Get the lake data
endog = lake.copy()
# BD do not implement the "bias correction" third step that they describe,
# so we can't use their results to test that. Thus here `unbiased=False`.
# Note: it's not clear why BD use initial_order=22 (and they don't mention
# that they do this), but it is the value that allows the test to pass.
hr, _ = hannan_rissanen(endog,
ar_order=1,
ma_order=1,
demean=True,
initial_ar_order=22,
unbiased=False)
assert_allclose(hr.ar_params, [0.6961], atol=1e-4)
assert_allclose(hr.ma_params, [0.3788], atol=1e-4)
# Because our fast implementation of the innovations algorithm does not
# allow for non-stationary processes, the estimate of the variance returned
# by `hannan_rissanen` is based on the residuals from the least-squares
# regression, rather than (as reported by BD) based on the innovations
# algorithm output. Since the estimates here do correspond to a stationary
# series, we can compute the innovations variance manually to check
# against BD.
u, v = arma_innovations(endog - endog.mean(),
hr.ar_params,
hr.ma_params,
sigma2=1)
tmp = u / v**0.5
assert_allclose(np.inner(tmp, tmp) / len(u), 0.4774, atol=1e-4)
def check_innovations_ma_itsmr(lake):
# Test against R itsmr::ia; see results/results_innovations.R
ia, _ = innovations(lake, 10, demean=True)
desired = [
1.0816255264, 0.7781248438, 0.5367164430, 0.3291559246, 0.3160039850,
0.2513754550, 0.2051536531, 0.1441070313, 0.3431868340, 0.1827400798]
assert_allclose(ia[10].ma_params, desired)
# itsmr::ia returns the innovations algorithm estimate of the variance
u, v = arma_innovations(np.array(lake) - np.mean(lake),
ma_params=ia[10].ma_params, sigma2=1)
desired_sigma2 = 0.4523684344
assert_allclose(np.sum(u**2 / v) / len(u), desired_sigma2)
def test_nonstationary_series():
# Test against R stats::ar.burg; see results/results_burg.R
endog = np.arange(1, 12) * 1.0
res, _ = burg(endog, 2, demean=False)
desired_ar_params = [1.9669331547, -0.9892846679]
assert_allclose(res.ar_params, desired_ar_params)
desired_sigma2 = 0.02143066427
assert_allclose(res.sigma2, desired_sigma2)
# With var.method = 1, stats::ar.burg also returns something equivalent to
# the innovations algorithm estimate of sigma2
u, v = arma_innovations(endog, ar_params=res.ar_params, sigma2=1)
desired_sigma2 = 0.02191056906
assert_allclose(np.sum(u**2 / v) / len(u), desired_sigma2)
def check_itsmr(lake):
# Test against R itsmr::yw; see results/results_yw_dl.R
yw, _ = yule_walker(lake, 5)
desired = [
1.08213598501, -0.39658257147, 0.11793957728, -0.03326633983,
0.06209208707
]
assert_allclose(yw.ar_params, desired)
# stats::ar.yw return the innovations algorithm estimate of the variance
u, v = arma_innovations(np.array(lake) - np.mean(lake),
ar_params=yw.ar_params,
sigma2=1)
desired_sigma2 = 0.4716322564
assert_allclose(np.sum(u**2 / v) / len(u), desired_sigma2)
def check_itsmr(lake):
# Test against R itsmr::burg; see results/results_burg.R
res, _ = burg(lake, 10, demean=True)
desired_ar_params = [
1.05853631096, -0.32639150878, 0.04784765122, 0.02620476111,
0.04444511374, -0.04134010262, 0.02251178970, -0.01427524694,
0.22223486915, -0.20935524387
]
assert_allclose(res.ar_params, desired_ar_params)
# itsmr always returns the innovations algorithm estimate of sigma2,
# whereas we return Burg's estimate
u, v = arma_innovations(np.array(lake) - np.mean(lake),
ar_params=res.ar_params,
sigma2=1)
desired_sigma2 = 0.4458956354
assert_allclose(np.sum(u**2 / v) / len(u), desired_sigma2)
def check_itsmr(lake):
# Test against R itsmr::yw; see results/results_yw_dl.R
dl, _ = durbin_levinson(lake, 5)
assert_allclose(dl[0].params, np.var(lake))
assert_allclose(dl[1].ar_params, [0.8319112104])
assert_allclose(dl[2].ar_params, [1.0538248798, -0.2667516276])
desired = [1.0887037577, -0.4045435867, 0.1307541335]
assert_allclose(dl[3].ar_params, desired)
desired = [1.08425065810, -0.39076602696, 0.09367609911, 0.03405704644]
assert_allclose(dl[4].ar_params, desired)
desired = [
1.08213598501, -0.39658257147, 0.11793957728, -0.03326633983,
0.06209208707
]
assert_allclose(dl[5].ar_params, desired)
# itsmr::yw returns the innovations algorithm estimate of the variance
# we'll just check for p=5
u, v = arma_innovations(np.array(lake) - np.mean(lake),
ar_params=dl[5].ar_params,
sigma2=1)
desired_sigma2 = 0.4716322564
assert_allclose(np.sum(u**2 / v) / len(u), desired_sigma2)
def run(csv_list):
time, data = get_data(csv_list)
data_plot(time, data, 'Time', 'Discharge', 'Plot of raw data', n=40)
ar = init_ar(time, data)
raw = ar.auto_correlation(ar.data, 40)
data_plot(range(len(raw)),
raw,
'Lag',
'Correlation',
'Plot of ACF of raw data',
n=40,
plt_type='bar',
significance=(1.96 / (len(raw))**0.5))
period = check_seasonality()
if period:
ar, non_seasonal = do_diff(ar, period)
else:
non_seasonal = ar.data
d = 0
diff = non_seasonal
while True:
is_needed = check_diff()
if is_needed:
d += 1
ar, diff = do_diff(ar, 1)
else:
break
ar.data = diff
pacf = ar.yule_walker_pacf(100)
data_plot(range(len(pacf)),
pacf,
'Lags',
'Correlation',
'PACF',
plt_type='bar',
n=40,
significance=(1.96 / (len(pacf))**0.5))
p = int(input('Enter value of p: '))
phi = pacf[:p]
print(f'The AR({p}) equation is')
print(
' + '.join([
f'({round(phi_i, 4)}) * y(t-{i + 1})'
for i, phi_i in enumerate(phi)
]), '+ residues')
ar.data = ar.predict(phi)
out = ar.difference(period, rev=True)
plt.plot(time[:len(out)], data[:len(out)])
plt.plot(time[:len(out)], out)
plt.show()
# o = AR(data, time)
# ou = o.fit()
# print(ou.params)
# plt.plot(time, data)
# plt.plot(time, list(o.predict(ou.params)) + [0] * 15)
# plt.show()
# ma params
q = int(input('Enter value of q: '))
if q:
ma_params, mse = arma_innovations(out)
theta = ma_params[:q]
print('MA Equation is:')
print(' + '.join([
f'({round(theta_i, 4)}) * e(t-{i + 1})'
for i, theta_i in enumerate(theta)
]))
# Calculate Error
residues = []
for y, y_cap in zip(data, out):
residues.append(y - y_cap)
data_plot(time[:len(residues)],
residues,
'Time',
'Residues',
'Plot of residues',
n=40)
# Final Predicted Output
# final_output = predict_using_arima(data, phi, residues, theta)
pass
def gls(endog,
exog=None,
order=(0, 0, 0),
seasonal_order=(0, 0, 0, 0),
include_constant=None,
n_iter=None,
max_iter=50,
tolerance=1e-8,
arma_estimator='innovations_mle',
arma_estimator_kwargs=None):
"""
Estimate ARMAX parameters by GLS.
Parameters
----------
endog : array_like
Input time series array.
exog : array_like, optional
Array of exogenous regressors. If not included, then `include_constant`
must be True, and then `exog` will only include the constant column.
order : tuple, optional
The (p,d,q) order of the ARIMA model. Default is (0, 0, 0).
seasonal_order : tuple, optional
The (P,D,Q,s) order of the seasonal ARIMA model.
Default is (0, 0, 0, 0).
include_constant : bool, optional
Whether to add a constant term in `exog` if it's not already there.
The estimate of the constant will then appear as one of the `exog`
parameters. If `exog` is None, then the constant will represent the
mean of the process. Default is True if the specified model does not
include integration and False otherwise.
n_iter : int, optional
Optionally iterate feasible GSL a specific number of times. Default is
to iterate to convergence. If set, this argument overrides the
`max_iter` and `tolerance` arguments.
max_iter : int, optional
Maximum number of feasible GLS iterations. Default is 50. If `n_iter`
is set, it overrides this argument.
tolerance : float, optional
Tolerance for determining convergence of feasible GSL iterations. If
`iter` is set, this argument has no effect.
Default is 1e-8.
arma_estimator : str, optional
The estimator used for estimating the ARMA model. This option should
not generally be used, unless the default method is failing or is
otherwise unsuitable. Not all values will be valid, depending on the
specified model orders (`order` and `seasonal_order`). Possible values
are:
* 'innovations_mle' - can be used with any specification
* 'statespace' - can be used with any specification
* 'hannan_rissanen' - can be used with any ARMA non-seasonal model
* 'yule_walker' - only non-seasonal consecutive
autoregressive (AR) models
* 'burg' - only non-seasonal, consecutive autoregressive (AR) models
* 'innovations' - only non-seasonal, consecutive moving
average (MA) models.
The default is 'innovations_mle'.
arma_estimator_kwargs : dict, optional
Arguments to pass to the ARMA estimator.
Returns
-------
parameters : SARIMAXParams object
Contains the parameter estimates from the final iteration.
other_results : Bunch
Includes eight components: `spec`, `params`, `converged`,
`differences`, `iterations`, `arma_estimator`, 'arma_estimator_kwargs',
and `arma_results`.
Notes
-----
The primary reference is [1]_, section 6.6. In particular, the
implementation follows the iterative procedure described in section 6.6.2.
Construction of the transformed variables used to compute the GLS estimator
described in section 6.6.1 is done via an application of the innovations
algorithm (rather than explicit construction of the transformation matrix).
Note that if the specified model includes integration, both the `endog` and
`exog` series will be differenced prior to estimation and a warning will
be issued to alert the user.
References
----------
.. [1] Brockwell, Peter J., and Richard A. Davis. 2016.
Introduction to Time Series and Forecasting. Springer.
"""
# Handle n_iter
if n_iter is not None:
max_iter = n_iter
tolerance = np.inf
# Default for include_constant is True if there is no integration and
# False otherwise
integrated = order[1] > 0 or seasonal_order[1] > 0
if include_constant is None:
include_constant = not integrated
elif include_constant and integrated:
raise ValueError('Cannot include a constant in an integrated model.')
# Handle including the constant (need to do it now so that the constant
# parameter can be included in the specification as part of `exog`.)
if include_constant:
exog = np.ones_like(endog) if exog is None else add_constant(exog)
# Create the SARIMAX specification
spec = SARIMAXSpecification(endog,
exog=exog,
order=order,
seasonal_order=seasonal_order)
endog = spec.endog
exog = spec.exog
# Handle integration
if spec.is_integrated:
# TODO: this is the approach suggested by BD (see Remark 1 in
# section 6.6.2 and Example 6.6.3), but maybe there are some cases
# where we don't want to force this behavior on the user?
warnings.warn('Provided `endog` and `exog` series have been'
' differenced to eliminate integration prior to GLS'
' parameter estimation.')
endog = diff(endog,
k_diff=spec.diff,
k_seasonal_diff=spec.seasonal_diff,
seasonal_periods=spec.seasonal_periods)
exog = diff(exog,
k_diff=spec.diff,
k_seasonal_diff=spec.seasonal_diff,
seasonal_periods=spec.seasonal_periods)
augmented = np.c_[endog, exog]
# Validate arma_estimator
spec.validate_estimator(arma_estimator)
if arma_estimator_kwargs is None:
arma_estimator_kwargs = {}
# Step 1: OLS
mod_ols = OLS(endog, exog)
res_ols = mod_ols.fit()
exog_params = res_ols.params
resid = res_ols.resid
# 0th iteration parameters
p = SARIMAXParams(spec=spec)
p.exog_params = exog_params
if spec.max_ar_order > 0:
p.ar_params = np.zeros(spec.k_ar_params)
if spec.max_seasonal_ar_order > 0:
p.seasonal_ar_params = np.zeros(spec.k_seasonal_ar_params)
if spec.max_ma_order > 0:
p.ma_params = np.zeros(spec.k_ma_params)
if spec.max_seasonal_ma_order > 0:
p.seasonal_ma_params = np.zeros(spec.k_seasonal_ma_params)
p.sigma2 = res_ols.scale
ar_params = p.ar_params
seasonal_ar_params = p.seasonal_ar_params
ma_params = p.ma_params
seasonal_ma_params = p.seasonal_ma_params
sigma2 = p.sigma2
# Step 2 - 4: iterate feasible GLS to convergence
arma_results = [None]
differences = [None]
parameters = [p]
converged = False if n_iter is None else None
i = 0
for i in range(1, max_iter + 1):
prev = exog_params
# Step 2: ARMA
# TODO: allow estimator-specific kwargs?
if arma_estimator == 'yule_walker':
p_arma, res_arma = yule_walker(resid,
ar_order=spec.ar_order,
demean=False,
**arma_estimator_kwargs)
elif arma_estimator == 'burg':
p_arma, res_arma = burg(resid,
ar_order=spec.ar_order,
demean=False,
**arma_estimator_kwargs)
elif arma_estimator == 'innovations':
out, res_arma = innovations(resid,
ma_order=spec.ma_order,
demean=False,
**arma_estimator_kwargs)
p_arma = out[-1]
elif arma_estimator == 'hannan_rissanen':
p_arma, res_arma = hannan_rissanen(resid,
ar_order=spec.ar_order,
ma_order=spec.ma_order,
demean=False,
**arma_estimator_kwargs)
else:
# For later iterations, use a "warm start" for parameter estimates
# (speeds up estimation and convergence)
start_params = (None if i == 1 else np.r_[ar_params, ma_params,
seasonal_ar_params,
seasonal_ma_params,
sigma2])
# Note: in each case, we do not pass in the order of integration
# since we have already differenced the series
tmp_order = (spec.order[0], 0, spec.order[2])
tmp_seasonal_order = (spec.seasonal_order[0], 0,
spec.seasonal_order[2],
spec.seasonal_order[3])
if arma_estimator == 'innovations_mle':
p_arma, res_arma = innovations_mle(
resid,
order=tmp_order,
seasonal_order=tmp_seasonal_order,
demean=False,
start_params=start_params,
**arma_estimator_kwargs)
else:
p_arma, res_arma = statespace(
resid,
order=tmp_order,
seasonal_order=tmp_seasonal_order,
include_constant=False,
start_params=start_params,
**arma_estimator_kwargs)
ar_params = p_arma.ar_params
seasonal_ar_params = p_arma.seasonal_ar_params
ma_params = p_arma.ma_params
seasonal_ma_params = p_arma.seasonal_ma_params
sigma2 = p_arma.sigma2
arma_results.append(res_arma)
# Step 3: GLS
# Compute transformed variables that satisfy OLS assumptions
# Note: In section 6.1.1 of Brockwell and Davis (2016), these
# transformations are developed as computed by left multiplcation
# by a matrix T. However, explicitly constructing T and then
# performing the left-multiplications does not scale well when nobs is
# large. Instead, we can retrieve the transformed variables as the
# residuals of the innovations algorithm (the `normalize=True`
# argument applies a Prais-Winsten-type normalization to the first few
# observations to ensure homoskedasticity). Brockwell and Davis
# mention that they also take this approach in practice.
tmp, _ = arma_innovations.arma_innovations(augmented,
ar_params=ar_params,
ma_params=ma_params,
normalize=True)
u = tmp[:, 0]
x = tmp[:, 1:]
# OLS on transformed variables
mod_gls = OLS(u, x)
res_gls = mod_gls.fit()
exog_params = res_gls.params
resid = endog - np.dot(exog, exog_params)
# Construct the parameter vector for the iteration
p = SARIMAXParams(spec=spec)
p.exog_params = exog_params
if spec.max_ar_order > 0:
p.ar_params = ar_params
if spec.max_seasonal_ar_order > 0:
p.seasonal_ar_params = seasonal_ar_params
if spec.max_ma_order > 0:
p.ma_params = ma_params
if spec.max_seasonal_ma_order > 0:
p.seasonal_ma_params = seasonal_ma_params
p.sigma2 = sigma2
parameters.append(p)
# Check for convergence
difference = np.abs(exog_params - prev)
differences.append(difference)
if n_iter is None and np.all(difference < tolerance):
converged = True
break
else:
if n_iter is None:
warnings.warn('Feasible GLS failed to converge in %d iterations.'
' Consider increasing the maximum number of'
' iterations using the `max_iter` argument or'
' reducing the required tolerance using the'
' `tolerance` argument.' % max_iter)
# Construct final results
p = parameters[-1]
other_results = Bunch({
'spec': spec,
'params': parameters,
'converged': converged,
'differences': differences,
'iterations': i,
'arma_estimator': arma_estimator,
'arma_estimator_kwargs': arma_estimator_kwargs,
'arma_results': arma_results,
})
return p, other_results
def test_innovations_nonstationary(ar_params):
np.random.seed(42)
endog = np.random.normal(size=100)
with pytest.raises(ValueError, match="The model's autoregressive"):
arma_innovations.arma_innovations(endog, ar_params=ar_params)
def innovations(data):
"""
Fallback for MA innovation algorithm
"""
return arma_innovations(data)
