# Note: The information criteria add 1 to the number of parameters
# whenever the model has an AR or MA term since, in principle,
# the variance could be treated as a free parameter and restricted
# This code does not allow this, but it adds consistency with other
# packages such as gretl and X12-ARIMA
import copy
from datetime import datetime
import numpy as np
import pandas as pd
from numpy import dot, log, zeros, pi
from numpy.linalg import inv
from scipy import optimize
from scipy.signal import lfilter
from scipy.stats import norm
from statsmodels.compat.pandas import Appender
import statsmodels.base.wrapper as wrap
from statsmodels.regression.linear_model import yule_walker, OLS
from statsmodels.tools.decorators import cache_readonly
from statsmodels.tools.numdiff import approx_hess_cs, approx_fprime_cs
from statsmodels.tools.sm_exceptions import SpecificationWarning
from statsmodels.tools.validation import array_like, string_like
from statsmodels.tsa.ar_model import AutoReg, ar_select_order
from statsmodels.tsa.arima_process import arma2ma
from statsmodels.tsa.base import tsa_model
from statsmodels.tsa.kalmanf import KalmanFilter
from statsmodels.tsa.tsatools import (lagmat, add_trend,
_ar_transparams, _ar_invtransparams,
_ma_transparams, _ma_invtransparams,
unintegrate, unintegrate_levels)
from statsmodels.tsa.vector_ar import util
REPEATED_FIT_ERROR = """
Model has been fit using trend={0} and method={1}. These cannot be changed
in subsequent calls to `fit`. Instead, use a new instance of {mod}.
"""
_armax_notes = r"""
Notes
-----
If exogenous variables are given, then the model that is fit is
.. math::
\phi(L)(y_t - X_t\beta) = \theta(L)\epsilon_t
where :math:`\phi` and :math:`\theta` are polynomials in the lag
operator, :math:`L`. This is the regression model with ARMA errors,
or ARMAX model. This specification is used, whether or not the model
is fit using conditional sum of square or maximum-likelihood, using
the `method` argument in
:meth:`statsmodels.tsa.arima_model.%(Model)s.fit`. Therefore, for
now, `css` and `mle` refer to estimation methods only. This may
change for the case of the `css` model in future versions.
"""
_arma_params = """endog : array_like
The endogenous variable.
order : iterable
The (p,q) order of the model for the number of AR parameters,
and MA parameters to use.
exog : array_like, optional
An optional array of exogenous variables. This should *not* include a
constant or trend. You can specify this in the `fit` method."""
_arma_model = "Autoregressive Moving Average ARMA(p,q) Model"
_arima_model = "Autoregressive Integrated Moving Average ARIMA(p,d,q) Model"
_arima_params = """endog : array_like
The endogenous variable.
order : iterable
The (p,d,q) order of the model for the number of AR parameters,
differences, and MA parameters to use.
exog : array_like, optional
An optional array of exogenous variables. This should *not* include a
constant or trend. You can specify this in the `fit` method."""
_predict_notes = """
Notes
-----
Use the results predict method instead.
"""
_results_notes = """
Notes
-----
It is recommended to use dates with the time-series models, as the
below will probably make clear. However, if ARIMA is used without
dates and/or `start` and `end` are given as indices, then these
indices are in terms of the *original*, undifferenced series. Ie.,
given some undifferenced observations::
1970Q1, 1
1970Q2, 1.5
1970Q3, 1.25
1970Q4, 2.25
1971Q1, 1.2
1971Q2, 4.1
1970Q1 is observation 0 in the original series. However, if we fit an
ARIMA(p,1,q) model then we lose this first observation through
differencing. Therefore, the first observation we can forecast (if
using exact MLE) is index 1. In the differenced series this is index
0, but we refer to it as 1 from the original series.
"""
_predict = """
%(Model)s model in-sample and out-of-sample prediction
Parameters
----------
%(params)s
start : int, str, or datetime
Zero-indexed observation number at which to start forecasting, ie.,
the first forecast is start. Can also be a date string to
parse or a datetime type.
end : int, str, or datetime
Zero-indexed observation number at which to end forecasting, ie.,
the first forecast is start. Can also be a date string to
parse or a datetime type. However, if the dates index does not
have a fixed frequency, end must be an integer index if you
want out of sample prediction.
exog : array_like, optional
If the model is an ARMAX and out-of-sample forecasting is
requested, exog must be given. exog must be aligned so that exog[0]
is used to produce the first out-of-sample forecast. The number of
observation in exog should match the number of out-of-sample
forecasts produced. If the length of exog does not match the number
of forecasts, a SpecificationWarning is produced.
dynamic : bool, optional
The `dynamic` keyword affects in-sample prediction. If dynamic
is False, then the in-sample lagged values are used for
prediction. If `dynamic` is True, then in-sample forecasts are
used in place of lagged dependent variables. The first forecast
value is `start`.
%(extra_params)s
Returns
-------
%(returns)s
%(extra_section)s
"""
_predict_returns = """predict : array
The predicted values.
"""
_arma_predict = _predict % {"Model": "ARMA",
"params": """params : array_like
The fitted parameters of the model.""",
"extra_params": "",
"returns": _predict_returns,
"extra_section": _predict_notes}
_arma_results_predict = _predict % {"Model": "ARMA", "params": "",
"extra_params": "",
"returns": _predict_returns,
"extra_section": _results_notes}
_arima_extras = """typ : str {'linear', 'levels'}
- 'linear' : Linear prediction in terms of the differenced
endogenous variables.
- 'levels' : Predict the levels of the original endogenous
variables.\n"""
_arima_predict = _predict % {"Model": "ARIMA",
"params": """params : array_like
The fitted parameters of the model.""",
"extra_params": _arima_extras,
"returns": _predict_returns,
"extra_section": _predict_notes}
_arima_results_predict = _predict % {"Model": "ARIMA",
"params": "",
"extra_params": _arima_extras,
"returns": _predict_returns,
"extra_section": _results_notes}
_arima_plot_predict_example = """ Examples
--------
>>> import statsmodels.api as sm
>>> import matplotlib.pyplot as plt
>>> import pandas as pd
>>>
>>> dta = sm.datasets.sunspots.load_pandas().data[['SUNACTIVITY']]
>>> dta.index = pd.date_range(start='1700', end='2009', freq='A')
>>> res = sm.tsa.ARMA(dta, (3, 0)).fit()
>>> fig, ax = plt.subplots()
>>> ax = dta.loc['1950':].plot(ax=ax)
>>> fig = res.plot_predict('1990', '2012', dynamic=True, ax=ax,
... plot_insample=False)
>>> plt.show()
.. plot:: plots/arma_predict_plot.py
"""
_plot_extras = """alpha : float, optional
The confidence intervals for the forecasts are (1 - alpha)%
plot_insample : bool, optional
Whether to plot the in-sample series. Default is True.
ax : matplotlib.Axes, optional
Existing axes to plot with."""
_plot_predict = ("""
Plot forecasts
""" + '\n'.join(_predict.split('\n')[2:])) % {
"params": "",
"extra_params": _plot_extras,
"returns": """fig : matplotlib.Figure
The plotted Figure instance""",
"extra_section": ('\n' + _arima_plot_predict_example +
'\n' + _results_notes)
}
_arima_plot_predict = ("""
Plot forecasts
""" + '\n'.join(_predict.split('\n')[2:])) % {
"params": "",
"extra_params": _plot_extras,
"returns": """fig : matplotlib.Figure
The plotted Figure instance""",
"extra_section": ('\n' + _arima_plot_predict_example + '\n' +
'\n'.join(_results_notes.split('\n')[:3])
+ ("""
This is hard-coded to only allow plotting of the forecasts in levels.
""") + '\n'.join(_results_notes.split('\n')[3:]))
}
def cumsum_n(x, n):
for _ in range(n):
x = np.cumsum(x)
return x
def _prediction_adjust_exog(exog, row_labels, dynamic, end):
"""
Adjust exog if exog has dates that align with endog
Parameters
----------
exog : {array_like, None}
The exog values
row_labels : {pd.DatetimeIndex, None}
Row labels from endog
dynamic : bool
Flag indicating whether dynamic forecasts are expected
end : int
Index of final in-sample observation
"""
if exog is None:
return None
exog_start = 0
exog_index = getattr(exog, 'index', None)
exog_dates = isinstance(exog_index, pd.DatetimeIndex)
endog_dates = isinstance(row_labels, pd.DatetimeIndex)
date_adj = endog_dates and exog_dates and not dynamic
if date_adj and row_labels.isin(exog_index).all():
end_label = row_labels[end]
exog_start = exog.index.get_loc(end_label) + 1
exog = array_like(exog, 'exog', ndim=2)
return exog[exog_start:]
def _check_arima_start(start, k_ar, k_diff, method, dynamic):
if start < 0:
raise ValueError("The start index %d of the original series "
"has been differenced away" % start)
elif (dynamic or 'mle' not in method) and start < k_ar:
raise ValueError("Start must be >= k_ar for conditional MLE "
"or dynamic forecast. Got %d" % start)
def _get_predict_out_of_sample(endog, p, q, k_trend, k_exog, start, errors,
trendparam, exparams, arparams, maparams, steps,
method, exog=None):
"""
Returns endog, resid, mu of appropriate length for out of sample
prediction.
"""
if q:
resid = np.zeros(q)
if start and 'mle' in method or (start == p and not start == 0):
resid[:q] = errors[start - q:start]
elif start:
resid[:q] = errors[start - q - p:start - p]
else:
resid[:q] = errors[-q:]
else:
resid = None
y = endog
if k_trend == 1:
# use expectation not constant
if k_exog > 0:
# TODO: technically should only hold for MLE not
# conditional model. See #274.
# ensure 2-d for conformability
if np.ndim(exog) == 1 and k_exog == 1:
# have a 1d series of observations -> 2d
exog = exog[:, None]
elif np.ndim(exog) == 1:
# should have a 1d row of exog -> 2d
if len(exog) != k_exog:
raise ValueError("1d exog given and len(exog) != k_exog")
exog = exog[None, :]
X = lagmat(np.dot(exog, exparams), p, original='in', trim='both')
mu = trendparam * (1 - arparams.sum())
# arparams were reversed in unpack for ease later
mu = mu + (np.r_[1, -arparams[::-1]] * X).sum(1)[:, None]
else:
mu = trendparam * (1 - arparams.sum())
mu = np.array([mu] * steps)
elif k_exog > 0:
X = np.dot(exog, exparams)
X = lagmat(X, p, original='in', trim='both')
mu = (np.r_[1, -arparams[::-1]] * X).sum(1)[:, None]
else:
mu = np.zeros(steps)
endog = np.zeros(p + steps - 1)
if p and start:
endog[:p] = y[start - p:start]
elif p:
endog[:p] = y[-p:]
return endog, resid, mu
def _arma_predict_out_of_sample(params, steps, errors, p, q, k_trend, k_exog,
endog, exog=None, start=0, method='mle'):
(trendparam, exparams,
arparams, maparams) = _unpack_params(params, (p, q), k_trend,
k_exog, reverse=True)
endog, resid, mu = _get_predict_out_of_sample(endog, p, q, k_trend, k_exog,
start, errors, trendparam,
exparams, arparams,
maparams, steps, method,
exog)
forecast = np.zeros(steps)
if steps == 1:
if q:
return mu[0] + np.dot(arparams, endog[:p]) + np.dot(maparams,
resid[:q])
else:
return mu[0] + np.dot(arparams, endog[:p])
if q:
i = 0 # if q == 1
else:
i = -1
for i in range(min(q, steps - 1)):
fcast = (mu[i] + np.dot(arparams, endog[i:i + p]) +
np.dot(maparams[:q - i], resid[i:i + q]))
forecast[i] = fcast
endog[i + p] = fcast
for i in range(i + 1, steps - 1):
fcast = mu[i] + np.dot(arparams, endog[i:i + p])
forecast[i] = fcast
endog[i + p] = fcast
# need to do one more without updating endog
forecast[steps - 1] = mu[steps - 1] + np.dot(arparams, endog[steps - 1:])
return forecast
def _arma_predict_in_sample(start, end, endog, resid, k_ar, method):
"""
Pre- and in-sample fitting for ARMA.
"""
if 'mle' in method:
fittedvalues = endog - resid # get them all then trim
else:
fittedvalues = endog[k_ar:] - resid
fv_start = start
if 'mle' not in method:
fv_start -= k_ar # start is in terms of endog index
fv_end = min(len(fittedvalues), end + 1)
return fittedvalues[fv_start:fv_end]
def _unpack_params(params, order, k_trend, k_exog, reverse=False):
p, q = order
k = k_trend + k_exog
maparams = params[k + p:]
arparams = params[k:k + p]
trend = params[:k_trend]
exparams = params[k_trend:k]
if reverse:
return trend, exparams, arparams[::-1], maparams[::-1]
return trend, exparams, arparams, maparams
def _make_arma_names(data, k_trend, order, exog_names):
k_ar, k_ma = order
exog_names = exog_names or []
ar_lag_names = util.make_lag_names([data.ynames], k_ar, 0)
ar_lag_names = [''.join(('ar.', i)) for i in ar_lag_names]
ma_lag_names = util.make_lag_names([data.ynames], k_ma, 0)
ma_lag_names = [''.join(('ma.', i)) for i in ma_lag_names]
trend_name = util.make_lag_names('', 0, k_trend)
exog_names = trend_name + exog_names + ar_lag_names + ma_lag_names
return exog_names
def _make_arma_exog(endog, exog, trend):
k_trend = 1 # overwritten if no constant
if exog is None and trend == 'c': # constant only
exog = np.ones((len(endog), 1))
elif exog is not None and trend == 'c': # constant plus exogenous
exog = add_trend(exog, trend='c', prepend=True, has_constant='raise')
elif trend == 'nc':
k_trend = 0
return k_trend, exog
def _check_estimable(nobs, n_params):
if nobs <= n_params:
raise ValueError("Insufficient degrees of freedom to estimate")
[docs]class ARMA(tsa_model.TimeSeriesModel):
__doc__ = tsa_model._tsa_doc % {"model": _arma_model,
"params": _arma_params,
"extra_params": "",
"extra_sections":
_armax_notes % {"Model": "ARMA"}}
def __init__(self, endog, order, exog=None, dates=None, freq=None,
missing='none'):
super(ARMA, self).__init__(endog, exog, dates, freq, missing=missing)
# GH 2575
array_like(endog, 'endog')
exog = array_like(self.data.exog, 'exog', ndim=2, optional=True)
_check_estimable(len(self.endog), sum(order))
self.k_ar = k_ar = order[0]
self.k_ma = k_ma = order[1]
self.k_lags = max(k_ar, k_ma + 1)
if exog is not None:
k_exog = exog.shape[1] # number of exog. variables excl. const
else:
k_exog = 0
self.k_exog = k_exog
self._orig_exog_names = self.exog_names
self._fit_params = None
def _fit_start_params_hr(self, order, start_ar_lags=None):
"""
Get starting parameters for fit.
Parameters
----------
order : iterable
(p,q,k) - AR lags, MA lags, and number of exogenous variables
including the constant.
start_ar_lags : int, optional
If start_ar_lags is not None, rather than fitting an AR process
according to best BIC, fits an AR process with a lag length equal
to start_ar_lags.
Returns
-------
start_params : array
A first guess at the starting parameters.
Notes
-----
If necessary, fits an AR process with the laglength start_ar_lags, or
selected according to best BIC if start_ar_lags is None. Obtain the
residuals. Then fit an ARMA(p,q) model via OLS using these residuals
for a first approximation. Uses a separate OLS regression to find the
coefficients of exogenous variables.
References
----------
Hannan, E.J. and Rissanen, J. 1982. "Recursive estimation of mixed
autoregressive-moving average order." `Biometrika`. 69.1.
Durbin, J. 1960. "The Fitting of Time-Series Models."
`Review of the International Statistical Institute`. Vol. 28, No. 3
"""
p, q, k = order
start_params = zeros((p + q + k))
# make copy of endog because overwritten
endog = np.array(self.endog, np.float64)
exog = self.exog
if k != 0:
ols_params = OLS(endog, exog).fit().params
start_params[:k] = ols_params
endog -= np.dot(exog, ols_params).squeeze()
if q != 0:
if p != 0:
# make sure we do not run into small data problems in AR fit
nobs = len(endog)
if start_ar_lags is None:
maxlag = int(round(12 * (nobs / 100.) ** (1 / 4.)))
if maxlag >= nobs:
maxlag = nobs - 1
mod = ar_select_order(endog, maxlag, trend='n').model
armod = mod.fit()
else:
if start_ar_lags >= nobs:
start_ar_lags = nobs - 1
armod = AutoReg(endog, start_ar_lags, trend='n').fit()
arcoefs_tmp = armod.params
p_tmp = len(armod.ar_lags)
# it's possible in small samples that optimal lag-order
# does not leave enough obs. No consistent way to fix.
if p_tmp + q >= len(endog):
raise ValueError("Proper starting parameters cannot"
" be found for this order with this "
"number of observations. Use the "
"start_params argument, or set "
"start_ar_lags to an integer less than "
"len(endog) - q.")
resid = endog[p_tmp:] - np.dot(lagmat(endog, p_tmp,
trim='both'),
arcoefs_tmp)
if p < p_tmp + q:
endog_start = p_tmp + q - p
resid_start = 0
else:
endog_start = 0
resid_start = p - p_tmp - q
lag_endog = lagmat(endog, p, 'both')[endog_start:]
lag_resid = lagmat(resid, q, 'both')[resid_start:]
# stack ar lags and resids
X = np.column_stack((lag_endog, lag_resid))
coefs = OLS(endog[max(p_tmp + q, p):], X).fit().params
start_params[k:k + p + q] = coefs
else:
ar_coeffs = yule_walker(endog, order=q)[0]
ar = np.r_[[1], -ar_coeffs.squeeze()]
start_params[k + p:k + p + q] = arma2ma(ar, [1], lags=q+1)[1:]
if q == 0 and p != 0:
arcoefs = yule_walker(endog, order=p)[0]
start_params[k:k + p] = arcoefs
# check AR coefficients
if p and not np.all(np.abs(np.roots(np.r_[1, -start_params[k:k + p]]
)) < 1):
raise ValueError("The computed initial AR coefficients are not "
"stationary\nYou should induce stationarity, "
"choose a different model order, or you can\n"
"pass your own start_params.")
# check MA coefficients
elif q and not np.all(np.abs(np.roots(np.r_[1, start_params[k + p:]]
)) < 1):
raise ValueError("The computed initial MA coefficients are not "
"invertible\nYou should induce invertibility, "
"choose a different model order, or you can\n"
"pass your own start_params.")
# check MA coefficients
return start_params
def _fit_start_params(self, order, method, start_ar_lags=None):
if method != 'css-mle': # use Hannan-Rissanen to get start params
start_params = self._fit_start_params_hr(order, start_ar_lags)
else: # use CSS to get start params
def func(params):
return -self.loglike_css(params)
start_params = self._fit_start_params_hr(order, start_ar_lags)
if self.transparams:
start_params = self._invtransparams(start_params)
bounds = [(None,) * 2] * sum(order)
mlefit = optimize.fmin_l_bfgs_b(func, start_params,
approx_grad=True, m=12,
pgtol=1e-7, factr=1e3,
bounds=bounds, iprint=-1)
start_params = mlefit[0]
if self.transparams:
start_params = self._transparams(start_params)
return start_params
def score(self, params):
"""
Compute the score function at params.
Notes
-----
This is a numerical approximation.
"""
return approx_fprime_cs(params, self.loglike, args=(False,))
def hessian(self, params):
"""
Compute the Hessian at params,
Notes
-----
This is a numerical approximation.
"""
return approx_hess_cs(params, self.loglike, args=(False,))
def _transparams(self, params):
"""
Transforms params to induce stationarity/invertability.
Reference
---------
Jones(1980)
"""
k_ar, k_ma = self.k_ar, self.k_ma
k = self.k_exog + self.k_trend
newparams = np.zeros_like(params)
# just copy exogenous parameters
if k != 0:
newparams[:k] = params[:k]
# AR Coeffs
if k_ar != 0:
newparams[k:k + k_ar] = _ar_transparams(params[k:k + k_ar].copy())
# MA Coeffs
if k_ma != 0:
newparams[k + k_ar:] = _ma_transparams(params[k + k_ar:].copy())
return newparams
def _invtransparams(self, start_params):
"""
Inverse of the Jones reparameterization
"""
k_ar, k_ma = self.k_ar, self.k_ma
k = self.k_exog + self.k_trend
newparams = start_params.copy()
arcoefs = newparams[k:k + k_ar]
macoefs = newparams[k + k_ar:]
# AR coeffs
if k_ar != 0:
newparams[k:k + k_ar] = _ar_invtransparams(arcoefs)
# MA coeffs
if k_ma != 0:
newparams[k + k_ar:k + k_ar + k_ma] = _ma_invtransparams(macoefs)
return newparams
def _get_prediction_index(self, start, end, dynamic, index=None):
method = getattr(self, 'method', 'mle')
k_ar = getattr(self, 'k_ar', 0)
k_diff = getattr(self, 'k_diff', 0)
if start is None:
if 'mle' in method and not dynamic:
start = 0
else:
start = k_ar
start = self._index[start]
start, end, out_of_sample, prediction_index = (
super(ARMA, self)._get_prediction_index(start, end, index))
# This replaces the _validate() call
if 'mle' not in method and start < k_ar - k_diff:
raise ValueError("Start must be >= k_ar for conditional "
"MLE or dynamic forecast. Got %s" % start)
# Other validation
_check_arima_start(start, k_ar, k_diff, method, dynamic)
return start, end, out_of_sample, prediction_index
def geterrors(self, params):
"""
Get the errors of the ARMA process.
Parameters
----------
params : array_like
The fitted ARMA parameters
order : array_like
3 item iterable, with the number of AR, MA, and exogenous
parameters, including the trend
"""
# start, end, out_of_sample, prediction_index = (
# self._get_prediction_index(start, end, index))
params = np.asarray(params)
k_ar, k_ma = self.k_ar, self.k_ma
k = self.k_exog + self.k_trend
method = getattr(self, 'method', 'mle')
if 'mle' in method: # use KalmanFilter to get errors
(y, k, nobs, k_ar, k_ma, k_lags, newparams, Z_mat, m, R_mat,
T_mat, paramsdtype) = KalmanFilter._init_kalman_state(params,
self)
errors = KalmanFilter.geterrors(y, k, k_ar, k_ma, k_lags, nobs,
Z_mat, m, R_mat, T_mat,
paramsdtype)
else: # use scipy.signal.lfilter
y = self.endog.copy()
k = self.k_exog + self.k_trend
if k > 0:
y -= dot(self.exog, params[:k])
k_ar = self.k_ar
k_ma = self.k_ma
(trendparams, exparams,
arparams, maparams) = _unpack_params(params, (k_ar, k_ma),
self.k_trend, self.k_exog,
reverse=False)
b, a = np.r_[1, -arparams], np.r_[1, maparams]
zi = zeros((max(k_ar, k_ma)))
for i in range(k_ar):
zi[i] = sum(-b[:i + 1][::-1] * y[:i + 1])
e = lfilter(b, a, y, zi=zi)
errors = e[0][k_ar:]
return errors.squeeze()
@Appender(_arma_predict)
def predict(self, params, start=None, end=None, exog=None, dynamic=False,
**kwargs):
if kwargs and 'typ' not in kwargs:
raise TypeError('Unknown extra arguments')
if not (hasattr(self, 'k_ar') and hasattr(self, 'k_trend')):
raise RuntimeError('Model must be fit before calling predict')
params = array_like(params, 'params')
method = getattr(self, 'method', 'mle') # do not assume fit
# will return an index of a date
start, end, out_of_sample, _ = (
self._get_prediction_index(start, end, dynamic))
if out_of_sample and (exog is None and self.k_exog > 0):
raise ValueError("You must provide exog for ARMAX")
endog = self.endog
resid = self.geterrors(params)
k_ar = self.k_ar
# Adjust exog if exog has dates that align with endog
row_labels = self.data.row_labels
exog = _prediction_adjust_exog(exog, row_labels, dynamic, end)
if out_of_sample != 0 and self.k_exog > 0:
# we need the last k_ar exog for the lag-polynomial
if self.k_exog > 0 and k_ar > 0 and not dynamic:
# need the last k_ar exog for the lag-polynomial
exog = np.vstack((self.exog[-k_ar:, self.k_trend:], exog))
if dynamic:
if self.k_exog > 0:
# need the last k_ar exog for the lag-polynomial
exog_insample = self.exog[start - k_ar:, self.k_trend:]
if exog is not None:
exog = np.vstack((exog_insample, exog))
else:
exog = exog_insample
# TODO: now that predict does dynamic in-sample it should
# also return error estimates and confidence intervals
# but how? len(endog) is not tot_obs
out_of_sample += end - start + 1
return _arma_predict_out_of_sample(params, out_of_sample, resid,
k_ar, self.k_ma, self.k_trend,
self.k_exog, endog, exog,
start, method)
predictedvalues = _arma_predict_in_sample(start, end, endog, resid,
k_ar, method)
if out_of_sample:
forecastvalues = _arma_predict_out_of_sample(params, out_of_sample,
resid, k_ar,
self.k_ma,
self.k_trend,
self.k_exog, endog,
exog,
method=method)
if (exog is not None and
(exog.shape[0] - k_ar) != forecastvalues.shape[0]):
import warnings
msg = """
The number of observations in exog does not match the number of out-of-sample
observations. This might indicate that exog is not correctly aligned. exog
should be aligned so that the exog[0] is used for the first out-of-sample
forecast, and exog[-1] is used for the last out-of-sample forecast.
exog is not used for in-sample observations which are the fitted values.
To silence this warning, ensure the number of observation in exog ({0})
matches the number of out-of-sample forecasts ({1})'
"""
msg = msg.format(exog.shape[0], forecastvalues.shape[0])
warnings.warn(msg, SpecificationWarning)
predictedvalues = np.r_[predictedvalues, forecastvalues]
return predictedvalues
def loglike(self, params, set_sigma2=True):
"""
Compute the log-likelihood for ARMA(p,q) model
Notes
-----
Likelihood used depends on the method set in fit
"""
method = self.method
if method in ['mle', 'css-mle']:
return self.loglike_kalman(params, set_sigma2)
elif method == 'css':
return self.loglike_css(params, set_sigma2)
else:
raise ValueError("Method %s not understood" % method)
def loglike_kalman(self, params, set_sigma2=True):
"""
Compute exact loglikelihood for ARMA(p,q) model by the Kalman Filter.
"""
return KalmanFilter.loglike(params, self, set_sigma2)
def loglike_css(self, params, set_sigma2=True):
"""
Conditional Sum of Squares likelihood function.
"""
k_ar = self.k_ar
k_ma = self.k_ma
k = self.k_exog + self.k_trend
y = self.endog.copy().astype(params.dtype)
nobs = self.nobs
# how to handle if empty?
if self.transparams:
newparams = self._transparams(params)
else:
newparams = params
if k > 0:
y -= dot(self.exog, newparams[:k])
# the order of p determines how many zeros errors to set for lfilter
b, a = np.r_[1, -newparams[k:k + k_ar]], np.r_[1, newparams[k + k_ar:]]
zi = np.zeros((max(k_ar, k_ma)), dtype=params.dtype)
for i in range(k_ar):
zi[i] = sum(-b[:i + 1][::-1] * y[:i + 1])
errors = lfilter(b, a, y, zi=zi)[0][k_ar:]
ssr = np.dot(errors, errors)
sigma2 = ssr / nobs
if set_sigma2:
self.sigma2 = sigma2
llf = -nobs / 2. * (log(2 * pi) + log(sigma2)) - ssr / (2 * sigma2)
return llf
def fit(self, start_params=None, trend='c', method="css-mle",
transparams=True, solver='lbfgs', maxiter=500, full_output=1,
disp=5, callback=None, start_ar_lags=None, **kwargs):
"""
Fits ARMA(p,q) model using exact maximum likelihood via Kalman filter.
Parameters
----------
start_params : array_like, optional
Starting parameters for ARMA(p,q). If None, the default is given
by ARMA._fit_start_params. See there for more information.
transparams : bool, optional
Whether or not to transform the parameters to ensure stationarity.
Uses the transformation suggested in Jones (1980). If False,
no checking for stationarity or invertibility is done.
method : str {'css-mle','mle','css'}
This is the loglikelihood to maximize. If "css-mle", the
conditional sum of squares likelihood is maximized and its values
are used as starting values for the computation of the exact
likelihood via the Kalman filter. If "mle", the exact likelihood
is maximized via the Kalman Filter. If "css" the conditional sum
of squares likelihood is maximized. All three methods use
`start_params` as starting parameters. See above for more
information.
trend : str {'c','nc'}
Whether to include a constant or not. 'c' includes constant,
'nc' no constant.
solver : str or None, optional
Solver to be used. The default is 'lbfgs' (limited memory
Broyden-Fletcher-Goldfarb-Shanno). Other choices are 'bfgs',
'newton' (Newton-Raphson), 'nm' (Nelder-Mead), 'cg' -
(conjugate gradient), 'ncg' (non-conjugate gradient), and
'powell'. By default, the limited memory BFGS uses m=12 to
approximate the Hessian, projected gradient tolerance of 1e-8 and
factr = 1e2. You can change these by using kwargs.
maxiter : int, optional
The maximum number of function evaluations. Default is 500.
tol : float
The convergence tolerance. Default is 1e-08.
full_output : bool, optional
If True, all output from solver will be available in
the Results object's mle_retvals attribute. Output is dependent
on the solver. See Notes for more information.
disp : int, optional
If True, convergence information is printed. For the default
l_bfgs_b solver, disp controls the frequency of the output during
the iterations. disp < 0 means no output in this case.
callback : function, optional
Called after each iteration as callback(xk) where xk is the current
parameter vector.
start_ar_lags : int, optional
Parameter for fitting start_params. When fitting start_params,
residuals are obtained from an AR fit, then an ARMA(p,q) model is
fit via OLS using these residuals. If start_ar_lags is None, fit
an AR process according to best BIC. If start_ar_lags is not None,
fits an AR process with a lag length equal to start_ar_lags.
See ARMA._fit_start_params_hr for more information.
**kwargs
See Notes for keyword arguments that can be passed to fit.
Returns
-------
statsmodels.tsa.arima_model.ARMAResults class
See Also
--------
statsmodels.base.model.LikelihoodModel.fit : for more information
on using the solvers.
ARMAResults : results class returned by fit
Notes
-----
If fit by 'mle', it is assumed for the Kalman Filter that the initial
unknown state is zero, and that the initial variance is
P = dot(inv(identity(m**2)-kron(T,T)),dot(R,R.T).ravel('F')).reshape(r,
r, order = 'F')
"""
trend = string_like(trend, 'trend', options=('nc', 'c'))
if self._fit_params is not None:
fp = (trend, method)
if self._fit_params != fp:
raise RuntimeError(REPEATED_FIT_ERROR.format(*fp, mod='ARMA'))
k_ar = self.k_ar
k_ma = self.k_ma
# enforce invertibility
self.transparams = transparams
endog, exog = self.endog, self.exog
k_exog = self.k_exog
self.nobs = len(endog) # this is overwritten if method is 'css'
# (re)set trend and handle exogenous variables
# always pass original exog
if hasattr(self, 'k_trend'):
k_trend = self.k_trend
exog = self.exog
else:
# Ensures only call once per ARMA instance
k_trend, exog = _make_arma_exog(endog, self.exog, trend)
# Check has something to estimate
if k_ar == 0 and k_ma == 0 and k_trend == 0 and k_exog == 0:
raise ValueError("Estimation requires the inclusion of least one "
"AR term, MA term, a constant or an exogenous "
"variable.")
# check again now that we know the trend
_check_estimable(len(endog), k_ar + k_ma + k_exog + k_trend)
self.k_trend = k_trend
self.exog = exog # overwrites original exog from __init__
# (re)set names for this model
self.exog_names = _make_arma_names(self.data, k_trend,
(k_ar, k_ma), self._orig_exog_names)
k = k_trend + k_exog
# choose objective function
if k_ma == 0 and k_ar == 0:
method = "css" # Always CSS when no AR or MA terms
self.method = method = method.lower()
# adjust nobs for css
if method == 'css':
self.nobs = len(self.endog) - k_ar
if start_params is not None:
start_params = array_like(start_params, 'start_params')
else: # estimate starting parameters
start_params = self._fit_start_params((k_ar, k_ma, k), method,
start_ar_lags)
if transparams: # transform initial parameters to ensure invertibility
start_params = self._invtransparams(start_params)
if solver == 'lbfgs':
kwargs.setdefault('pgtol', 1e-8)
kwargs.setdefault('factr', 1e2)
kwargs.setdefault('m', 12)
kwargs.setdefault('approx_grad', True)
mlefit = super(ARMA, self).fit(start_params, method=solver,
maxiter=maxiter,
full_output=full_output, disp=disp,
callback=callback, **kwargs)
params = mlefit.params
if transparams: # transform parameters back
params = self._transparams(params)
self.transparams = False # so methods do not expect transf.
normalized_cov_params = None # TODO: fix this
armafit = ARMAResults(copy.copy(self), params, normalized_cov_params)
armafit.mle_retvals = mlefit.mle_retvals
armafit.mle_settings = mlefit.mle_settings
# Save core fit parameters for future checks
self._fit_params = (trend, method)
return ARMAResultsWrapper(armafit)
@classmethod
def from_formula(cls, formula, data, subset=None, drop_cols=None, *args,
**kwargs):
raise NotImplementedError("from_formula is not supported"
" for ARMA models.")
# TODO: the length of endog changes when we give a difference to fit
# so model methods are not the same on unfit models as fit ones
# starting to think that order of model should be put in instantiation...
[docs]class ARIMA(ARMA):
__doc__ = tsa_model._tsa_doc % {"model": _arima_model,
"params": _arima_params,
"extra_params": "",
"extra_sections":
_armax_notes % {"Model": "ARIMA"}}
def __new__(cls, endog, order, exog=None, dates=None, freq=None,
missing='none'):
p, d, q = order
if d == 0: # then we just use an ARMA model
return ARMA(endog, (p, q), exog, dates, freq, missing)
else:
mod = super(ARIMA, cls).__new__(cls)
mod.__init__(endog, order, exog, dates, freq, missing)
return mod
def __getnewargs__(self):
# using same defaults as in __init__
dates = getattr(self, 'dates', None)
freq = getattr(self, 'freq', None)
missing = getattr(self, 'missing', 'none')
return ((self.endog),
(self.k_lags, self.k_diff, self.k_ma),
self.exog, dates, freq, missing)
def __init__(self, endog, order, exog=None, dates=None, freq=None,
missing='none'):
p, d, q = order
if d > 2:
# TODO: to make more general, need to address the d == 2 stuff
# in the predict method
raise ValueError("d > 2 is not supported")
super(ARIMA, self).__init__(endog, (p, q), exog, dates, freq, missing)
self.k_diff = d
self._first_unintegrate = unintegrate_levels(self.endog[:d], d)
self.endog = np.diff(self.endog, n=d)
# NOTE: will check in ARMA but check again since differenced now
_check_estimable(len(self.endog), p + q)
if exog is not None:
self.exog = self.exog[d:]
if d == 1:
self.data.ynames = 'D.' + self.endog_names
else:
self.data.ynames = 'D{0:d}.'.format(d) + self.endog_names
# what about exog, should we difference it automatically before
# super call?
# Reset index
orig_length = len(self._index)
new_length = self.endog.shape[0]
if self.data.row_labels is not None:
self.data._cache['row_labels'] = (
self.data.row_labels[orig_length - new_length:])
if self._index is not None:
if self._index_generated:
self._index = self._index[:-(orig_length - new_length)]
else:
self._index = self._index[orig_length - new_length:]
def _get_prediction_index(self, start, end, dynamic, index=None):
method = getattr(self, 'method', 'mle')
k_ar = getattr(self, 'k_ar', 0)
k_diff = getattr(self, 'k_diff', 0)
if start is None:
if 'mle' in method and not dynamic:
start = 0
else:
start = k_ar
start = self._index[start]
elif isinstance(start, (int, np.integer)):
start -= k_diff
if start < 0:
raise ValueError('The start index %d of the original series '
' has been differenced away' % start)
if isinstance(end, (int, np.integer)):
end -= k_diff
start, end, out_of_sample, prediction_index = (
super(ARIMA, self)._get_prediction_index(start, end, index))
# From _get_predict_end
if 'mle' not in self.method and not dynamic:
end -= k_ar
# This replaces the _validate() call
if 'mle' not in method and start < k_ar - k_diff:
raise ValueError("Start must be >= k_ar for conditional "
"MLE or dynamic forecast. Got %s" % start)
# Other validation
_check_arima_start(start, k_ar, k_diff, method, dynamic)
return start, end, out_of_sample, prediction_index
def fit(self, start_params=None, trend='c', method="css-mle",
transparams=True, solver='lbfgs', maxiter=500, full_output=1,
disp=5, callback=None, start_ar_lags=None, **kwargs):
"""
Fits ARIMA(p,d,q) model by exact maximum likelihood via Kalman filter.
Parameters
----------
start_params : array_like, optional
Starting parameters for ARMA(p,q). If None, the default is given
by ARMA._fit_start_params. See there for more information.
transparams : bool, optional
Whether or not to transform the parameters to ensure stationarity.
Uses the transformation suggested in Jones (1980). If False,
no checking for stationarity or invertibility is done.
method : str {'css-mle','mle','css'}
This is the loglikelihood to maximize. If "css-mle", the
conditional sum of squares likelihood is maximized and its values
are used as starting values for the computation of the exact
likelihood via the Kalman filter. If "mle", the exact likelihood
is maximized via the Kalman Filter. If "css" the conditional sum
of squares likelihood is maximized. All three methods use
`start_params` as starting parameters. See above for more
information.
trend : str {'c','nc'}
Whether to include a constant or not. 'c' includes constant,
'nc' no constant.
solver : str or None, optional
Solver to be used. The default is 'lbfgs' (limited memory
Broyden-Fletcher-Goldfarb-Shanno). Other choices are 'bfgs',
'newton' (Newton-Raphson), 'nm' (Nelder-Mead), 'cg' -
(conjugate gradient), 'ncg' (non-conjugate gradient), and
'powell'. By default, the limited memory BFGS uses m=12 to
approximate the Hessian, projected gradient tolerance of 1e-8 and
factr = 1e2. You can change these by using kwargs.
maxiter : int, optional
The maximum number of function evaluations. Default is 500.
tol : float
The convergence tolerance. Default is 1e-08.
full_output : bool, optional
If True, all output from solver will be available in
the Results object's mle_retvals attribute. Output is dependent
on the solver. See Notes for more information.
disp : int, optional
If True, convergence information is printed. For the default
l_bfgs_b solver, disp controls the frequency of the output during
the iterations. disp < 0 means no output in this case.
callback : function, optional
Called after each iteration as callback(xk) where xk is the current
parameter vector.
start_ar_lags : int, optional
Parameter for fitting start_params. When fitting start_params,
residuals are obtained from an AR fit, then an ARMA(p,q) model is
fit via OLS using these residuals. If start_ar_lags is None, fit
an AR process according to best BIC. If start_ar_lags is not None,
fits an AR process with a lag length equal to start_ar_lags.
See ARMA._fit_start_params_hr for more information.
**kwargs
See Notes for keyword arguments that can be passed to fit.
Returns
-------
`statsmodels.tsa.arima.ARIMAResults` class
See Also
--------
statsmodels.base.model.LikelihoodModel.fit : for more information
on using the solvers.
ARIMAResults : results class returned by fit
Notes
-----
If fit by 'mle', it is assumed for the Kalman Filter that the initial
unknown state is zero, and that the initial variance is
P = dot(inv(identity(m**2)-kron(T,T)),dot(R,R.T).ravel('F')).reshape(r,
r, order = 'F')
"""
mlefit = super(ARIMA, self).fit(start_params, trend,
method, transparams, solver,
maxiter, full_output, disp,
callback, start_ar_lags, **kwargs)
normalized_cov_params = None # TODO: fix this?
arima_fit = ARIMAResults(self, mlefit._results.params,
normalized_cov_params)
arima_fit.k_diff = self.k_diff
arima_fit.mle_retvals = mlefit.mle_retvals
arima_fit.mle_settings = mlefit.mle_settings
return ARIMAResultsWrapper(arima_fit)
@Appender(_arima_predict)
def predict(self, params, start=None, end=None, exog=None, typ='linear',
dynamic=False):
if not (hasattr(self, 'k_ar') and hasattr(self, 'k_trend')):
raise RuntimeError('Model must be fit before calling predict')
# go ahead and convert to an index for easier checking
if isinstance(start, (str, datetime)):
start, _, _ = self._get_index_label_loc(start)
if isinstance(start, slice):
start = start.start
# Adjustment since _index was already changed to fit the
# differenced endog.
start += self.k_diff
if typ == 'linear':
if not dynamic or (start != self.k_ar + self.k_diff and
start is not None):
return super(ARIMA, self).predict(params, start, end, exog,
dynamic)
else:
# need to assume pre-sample residuals are zero
# do this by a hack
q = self.k_ma
self.k_ma = 0
predictedvalues = super(ARIMA, self).predict(params, start,
end, exog,
dynamic)
self.k_ma = q
return predictedvalues
elif typ == 'levels':
endog = self.data.endog
if not dynamic:
predict = super(ARIMA, self).predict(params, start, end, exog,
dynamic)
start, end, out_of_sample, _ = (
self._get_prediction_index(start, end, dynamic))
d = self.k_diff
if 'mle' in self.method:
start += d - 1 # for case where d == 2
end += d - 1
# add each predicted diff to lagged endog
if out_of_sample:
fv = predict[:-out_of_sample] + endog[start:end + 1]
if d == 2: # TODO: make a general solution to this
fv += np.diff(endog[start - 1:end + 1])
levels = unintegrate_levels(endog[-d:], d)
fv = np.r_[fv,
unintegrate(predict[-out_of_sample:],
levels)[d:]]
else:
fv = predict + endog[start:end + 1]
if d == 2:
fv += np.diff(endog[start - 1:end + 1])
else:
k_ar = self.k_ar
if out_of_sample:
fv = (predict[:-out_of_sample] +
endog[max(start, self.k_ar - 1):end + k_ar + 1])
if d == 2:
fv += np.diff(endog[start - 1:end + 1])
levels = unintegrate_levels(endog[-d:], d)
fv = np.r_[fv,
unintegrate(predict[-out_of_sample:],
levels)[d:]]
else:
fv = predict + endog[max(start, k_ar):end + k_ar + 1]
if d == 2:
fv += np.diff(endog[start - 1:end + 1])
else:
# IFF we need to use pre-sample values assume pre-sample
# residuals are zero, do this by a hack
if start == self.k_ar + self.k_diff or start is None:
# do the first k_diff+1 separately
p = self.k_ar
q = self.k_ma
k_exog = self.k_exog
k_trend = self.k_trend
k_diff = self.k_diff
(trendparam, exparams,
arparams, maparams) = _unpack_params(params, (p, q),
k_trend,
k_exog,
reverse=True)
# this is the hack
self.k_ma = 0
predict = super(ARIMA, self).predict(params, start, end,
exog, dynamic)
if not start:
start, _, _, _ = self._get_prediction_index(
start, end, dynamic)
start += k_diff
self.k_ma = q
return endog[start - 1] + np.cumsum(predict)
else:
predict = super(ARIMA, self).predict(params, start, end,
exog, dynamic)
return endog[start - 1] + np.cumsum(predict)
return fv
else: # pragma : no cover
raise ValueError("typ %s not understood" % typ)
[docs]class ARMAResults(tsa_model.TimeSeriesModelResults):
"""
Class to hold results from fitting an ARMA model.
Parameters
----------
model : ARMA instance
The fitted model instance
params : array
Fitted parameters
normalized_cov_params : array, optional
The normalized variance covariance matrix
scale : float, optional
Optional argument to scale the variance covariance matrix.
Attributes
----------
aic : float
Akaike Information Criterion
:math:`-2*llf+2* df_model`
where `df_model` includes all AR parameters, MA parameters, constant
terms parameters on constant terms and the variance.
arparams : array
The parameters associated with the AR coefficients in the model.
arroots : array
The roots of the AR coefficients are the solution to
(1 - arparams[0]*z - arparams[1]*z**2 -...- arparams[p-1]*z**k_ar) = 0
Stability requires that the roots in modulus lie outside the unit
circle.
bic : float
Bayes Information Criterion
-2*llf + log(nobs)*df_model
Where if the model is fit using conditional sum of squares, the
number of observations `nobs` does not include the `p` pre-sample
observations.
bse : array
The standard errors of the parameters. These are computed using the
numerical Hessian.
df_model : array
The model degrees of freedom = `k_exog` + `k_trend` + `k_ar` + `k_ma`
df_resid : array
The residual degrees of freedom = `nobs` - `df_model`
fittedvalues : array
The predicted values of the model.
hqic : float
Hannan-Quinn Information Criterion
-2*llf + 2*(`df_model`)*log(log(nobs))
Like `bic` if the model is fit using conditional sum of squares then
the `k_ar` pre-sample observations are not counted in `nobs`.
k_ar : int
The number of AR coefficients in the model.
k_exog : int
The number of exogenous variables included in the model. Does not
include the constant.
k_ma : int
The number of MA coefficients.
k_trend : int
This is 0 for no constant or 1 if a constant is included.
llf : float
The value of the log-likelihood function evaluated at `params`.
maparams : array
The value of the moving average coefficients.
maroots : array
The roots of the MA coefficients are the solution to
(1 + maparams[0]*z + maparams[1]*z**2 + ... + maparams[q-1]*z**q) = 0
Stability requires that the roots in modules lie outside the unit
circle.
model : ARMA instance
A reference to the model that was fit.
nobs : float
The number of observations used to fit the model. If the model is fit
using exact maximum likelihood this is equal to the total number of
observations, `n_totobs`. If the model is fit using conditional
maximum likelihood this is equal to `n_totobs` - `k_ar`.
n_totobs : float
The total number of observations for `endog`. This includes all
observations, even pre-sample values if the model is fit using `css`.
params : array
The parameters of the model. The order of variables is the trend
coefficients and the `k_exog` exogenous coefficients, then the
`k_ar` AR coefficients, and finally the `k_ma` MA coefficients.
pvalues : array
The p-values associated with the t-values of the coefficients. Note
that the coefficients are assumed to have a Student's T distribution.
resid : array
The model residuals. If the model is fit using 'mle' then the
residuals are created via the Kalman Filter. If the model is fit
using 'css' then the residuals are obtained via `scipy.signal.lfilter`
adjusted such that the first `k_ma` residuals are zero. These zero
residuals are not returned.
scale : float
This is currently set to 1.0 and not used by the model or its results.
sigma2 : float
The variance of the residuals. If the model is fit by 'css',
sigma2 = ssr/nobs, where ssr is the sum of squared residuals. If
the model is fit by 'mle', then sigma2 = 1/nobs * sum(v**2 / F)
where v is the one-step forecast error and F is the forecast error
variance. See `nobs` for the difference in definitions depending on the
fit.
"""
_cache = {}
def __init__(self, model, params, normalized_cov_params=None, scale=1.):
super(ARMAResults, self).__init__(model, params, normalized_cov_params,
scale)
self.sigma2 = model.sigma2
nobs = model.nobs
self.nobs = nobs
k_exog = model.k_exog
self.k_exog = k_exog
k_trend = model.k_trend
self.k_trend = k_trend
k_ar = model.k_ar
self.k_ar = k_ar
self.n_totobs = len(model.endog)
k_ma = model.k_ma
self.k_ma = k_ma
df_model = k_exog + k_trend + k_ar + k_ma
self._ic_df_model = df_model + 1
self.df_model = df_model
self.df_resid = self.nobs - df_model
self._cache = {}
@cache_readonly
def arroots(self):
return np.roots(np.r_[1, -self.arparams]) ** -1
@cache_readonly
def maroots(self):
return np.roots(np.r_[1, self.maparams]) ** -1
@cache_readonly
def arfreq(self):
r"""
Returns the frequency of the AR roots.
This is the solution, x, to z = abs(z)*exp(2j*np.pi*x) where z are the
roots.
"""
z = self.arroots
return np.arctan2(z.imag, z.real) / (2 * pi)
@cache_readonly
def mafreq(self):
r"""
Returns the frequency of the MA roots.
This is the solution, x, to z = abs(z)*exp(2j*np.pi*x) where z are the
roots.
"""
z = self.maroots
return np.arctan2(z.imag, z.real) / (2 * pi)
@cache_readonly
def arparams(self):
k = self.k_exog + self.k_trend
return self.params[k:k + self.k_ar]
@cache_readonly
def maparams(self):
k = self.k_exog + self.k_trend
k_ar = self.k_ar
return self.params[k + k_ar:]
@cache_readonly
def llf(self):
return self.model.loglike(self.params)
@cache_readonly
def bse(self):
params = self.params
hess = self.model.hessian(params)
if len(params) == 1: # cannot take an inverse, ensure 1d
return np.sqrt(-1. / hess[0])
return np.sqrt(np.diag(-inv(hess)))
@cache_readonly
def cov_params_default(self):
hess = self.model.hessian(self.params)
return -inv(hess)
@cache_readonly
def aic(self):
return -2 * self.llf + 2 * self._ic_df_model
@cache_readonly
def bic(self):
nobs = self.nobs
return -2 * self.llf + np.log(nobs) * self._ic_df_model
@cache_readonly
def hqic(self):
nobs = self.nobs
return -2 * self.llf + 2 * np.log(np.log(nobs)) * self._ic_df_model
@cache_readonly
def fittedvalues(self):
model = self.model
endog = model.endog.copy()
k_ar = self.k_ar
exog = model.exog # this is a copy
if exog is not None:
if model.method == "css" and k_ar > 0:
exog = exog[k_ar:]
if model.method == "css" and k_ar > 0:
endog = endog[k_ar:]
fv = endog - self.resid
return fv
@cache_readonly
def resid(self):
return self.model.geterrors(self.params)
[docs] @Appender(_arma_results_predict)
def predict(self, start=None, end=None, exog=None, dynamic=False,
**kwargs):
return self.model.predict(self.params, start, end, exog, dynamic,
**kwargs)
def _forecast_error(self, steps):
sigma2 = self.sigma2
ma_rep = arma2ma(np.r_[1, -self.arparams],
np.r_[1, self.maparams], lags=steps)
fcasterr = np.sqrt(sigma2 * np.cumsum(ma_rep ** 2))
return fcasterr
def _forecast_conf_int(self, forecast, fcasterr, alpha):
const = norm.ppf(1 - alpha / 2.)
conf_int = np.c_[forecast - const * fcasterr,
forecast + const * fcasterr]
return conf_int
def forecast(self, steps=1, exog=None, alpha=.05):
"""
Out-of-sample forecasts
Parameters
----------
steps : int
The number of out of sample forecasts from the end of the
sample.
exog : array
If the model is an ARMAX, you must provide out of sample
values for the exogenous variables. This should not include
the constant. The number of observation in exog must match the
value of steps.
alpha : float
The confidence intervals for the forecasts are (1 - alpha) %
Returns
-------
forecast : array
Array of out of sample forecasts
stderr : array
Array of the standard error of the forecasts.
conf_int : array
2d array of the confidence interval for the forecast
"""
if exog is not None:
exog = array_like(exog, 'exog', maxdim=2)
if self.k_exog == 1 and exog.ndim == 1:
exog = exog[:, None]
elif exog.ndim == 1:
if len(exog) != self.k_exog:
raise ValueError("1d exog given and len(exog) != k_exog")
exog = exog[None, :]
if exog.shape[0] != steps:
raise ValueError("new exog needed for each step")
if self.k_exog != exog.shape[1]:
raise ValueError('exog must contain the same number of '
'variables as in the estimated model.')
# prepend in-sample exog observations
if self.k_ar > 0:
exog = np.vstack((self.model.exog[-self.k_ar:, self.k_trend:],
exog))
else:
if self.k_exog:
raise ValueError('Forecast values for exog are required when '
'the model contains exogenous regressors.')
forecast = _arma_predict_out_of_sample(self.params,
steps, self.resid, self.k_ar,
self.k_ma, self.k_trend,
self.k_exog, self.model.endog,
exog, method=self.model.method)
# compute the standard errors
fcasterr = self._forecast_error(steps)
conf_int = self._forecast_conf_int(forecast, fcasterr, alpha)
return forecast, fcasterr, conf_int
[docs] def summary(self, alpha=.05):
"""Summarize the Model
Parameters
----------
alpha : float, optional
Significance level for the confidence intervals.
Returns
-------
smry : Summary instance
This holds the summary table and text, which can be printed or
converted to various output formats.
See Also
--------
statsmodels.iolib.summary.Summary
"""
from statsmodels.iolib.summary import Summary
model = self.model
title = model.__class__.__name__ + ' Model Results'
method = model.method
# get sample TODO: make better sample machinery for estimation
k_diff = getattr(self, 'k_diff', 0)
if 'mle' in method:
start = k_diff
else:
start = k_diff + self.k_ar
if self.data.dates is not None:
dates = self.data.dates
sample = [dates[start].strftime('%m-%d-%Y')]
sample += ['- ' + dates[-1].strftime('%m-%d-%Y')]
else:
sample = str(start) + ' - ' + str(len(self.data.orig_endog))
k_ar, k_ma = self.k_ar, self.k_ma
if not k_diff:
order = str((k_ar, k_ma))
else:
order = str((k_ar, k_diff, k_ma))
top_left = [('Dep. Variable:', None),
('Model:', [model.__class__.__name__ + order]),
('Method:', [method]),
('Date:', None),
('Time:', None),
('Sample:', [sample[0]]),
('', [sample[1]])
]
top_right = [
('No. Observations:', [str(len(self.model.endog))]),
('Log Likelihood', ["%#5.3f" % self.llf]),
('S.D. of innovations', ["%#5.3f" % self.sigma2 ** .5]),
('AIC', ["%#5.3f" % self.aic]),
('BIC', ["%#5.3f" % self.bic]),
('HQIC', ["%#5.3f" % self.hqic])]
smry = Summary()
smry.add_table_2cols(self, gleft=top_left, gright=top_right,
title=title)
smry.add_table_params(self, alpha=alpha, use_t=False)
# Make the roots table
from statsmodels.iolib.table import SimpleTable
if k_ma and k_ar:
arstubs = ["AR.%d" % i for i in range(1, k_ar + 1)]
mastubs = ["MA.%d" % i for i in range(1, k_ma + 1)]
stubs = arstubs + mastubs
roots = np.r_[self.arroots, self.maroots]
freq = np.r_[self.arfreq, self.mafreq]
elif k_ma:
mastubs = ["MA.%d" % i for i in range(1, k_ma + 1)]
stubs = mastubs
roots = self.maroots
freq = self.mafreq
elif k_ar:
arstubs = ["AR.%d" % i for i in range(1, k_ar + 1)]
stubs = arstubs
roots = self.arroots
freq = self.arfreq
else: # 0,0 model
stubs = []
if len(stubs): # not 0, 0
modulus = np.abs(roots)
data = np.column_stack((roots.real, roots.imag, modulus, freq))
roots_table = SimpleTable([('%17.4f' % row[0],
'%+17.4fj' % row[1],
'%17.4f' % row[2],
'%17.4f' % row[3]) for row in data],
headers=[' Real',
' Imaginary',
' Modulus',
' Frequency'],
title="Roots",
stubs=stubs)
smry.tables.append(roots_table)
return smry
[docs] def summary2(self, title=None, alpha=.05, float_format="%.4f"):
"""
Experimental summary function for ARIMA Results
Parameters
----------
title : str, optional
Title for the top table. If not None, then this replaces the
default title
alpha : float, optional
significance level for the confidence intervals
float_format : str, optional
print format for floats in parameters summary
Returns
-------
smry : Summary instance
This holds the summary table and text, which can be printed or
converted to various output formats.
See Also
--------
statsmodels.iolib.summary2.Summary : class to hold summary
results
"""
from pandas import DataFrame
# get sample TODO: make better sample machinery for estimation
k_diff = getattr(self, 'k_diff', 0)
if 'mle' in self.model.method:
start = k_diff
else:
start = k_diff + self.k_ar
if self.data.dates is not None:
dates = self.data.dates
sample = [dates[start].strftime('%m-%d-%Y')]
sample += [dates[-1].strftime('%m-%d-%Y')]
else:
sample = str(start) + ' - ' + str(len(self.data.orig_endog))
k_ar, k_ma = self.k_ar, self.k_ma
# Roots table
if k_ma and k_ar:
arstubs = ["AR.%d" % i for i in range(1, k_ar + 1)]
mastubs = ["MA.%d" % i for i in range(1, k_ma + 1)]
stubs = arstubs + mastubs
roots = np.r_[self.arroots, self.maroots]
freq = np.r_[self.arfreq, self.mafreq]
elif k_ma:
mastubs = ["MA.%d" % i for i in range(1, k_ma + 1)]
stubs = mastubs
roots = self.maroots
freq = self.mafreq
elif k_ar:
arstubs = ["AR.%d" % i for i in range(1, k_ar + 1)]
stubs = arstubs
roots = self.arroots
freq = self.arfreq
else: # 0, 0 order
stubs = []
if len(stubs):
modulus = np.abs(roots)
data = np.column_stack((roots.real, roots.imag, modulus, freq))
data = DataFrame(data)
data.columns = ['Real', 'Imaginary', 'Modulus', 'Frequency']
data.index = stubs
# Summary
from statsmodels.iolib import summary2
smry = summary2.Summary()
# Model info
model_info = summary2.summary_model(self)
model_info['Method:'] = self.model.method
model_info['Sample:'] = sample[0]
model_info[' '] = sample[-1]
model_info['S.D. of innovations:'] = "%#5.3f" % self.sigma2 ** .5
model_info['HQIC:'] = "%#5.3f" % self.hqic
model_info['No. Observations:'] = str(len(self.model.endog))
# Parameters
params = summary2.summary_params(self)
smry.add_dict(model_info)
smry.add_df(params, float_format=float_format)
if len(stubs):
smry.add_df(data, float_format="%17.4f")
smry.add_title(results=self, title=title)
return smry
@Appender(_plot_predict)
def plot_predict(self, start=None, end=None, exog=None, dynamic=False,
alpha=.05, plot_insample=True, ax=None):
from statsmodels.graphics.utils import _import_mpl, create_mpl_ax
_ = _import_mpl()
fig, ax = create_mpl_ax(ax)
# use predict so you set dates
forecast = self.predict(start, end, exog, dynamic)
# doing this twice. just add a plot keyword to predict?
start, end, out_of_sample, _ = (
self.model._get_prediction_index(start, end, dynamic=False))
if out_of_sample:
steps = out_of_sample
fc_error = self._forecast_error(steps)
conf_int = self._forecast_conf_int(forecast[-steps:], fc_error,
alpha)
if hasattr(self.data, "predict_dates"):
from pandas import Series
forecast = Series(forecast, index=self.data.predict_dates)
ax = forecast.plot(ax=ax, label='forecast')
else:
ax.plot(forecast)
x = ax.get_lines()[-1].get_xdata()
if out_of_sample:
label = "{0:.0%} confidence interval".format(1 - alpha)
ax.fill_between(x[-out_of_sample:], conf_int[:, 0], conf_int[:, 1],
color='gray', alpha=.5, label=label)
if plot_insample:
ax.plot(x[:end + 1 - start], self.model.endog[start:end + 1],
label=self.model.endog_names)
ax.legend(loc='best')
return fig
class ARMAResultsWrapper(wrap.ResultsWrapper):
_attrs = {}
_wrap_attrs = wrap.union_dicts(
tsa_model.TimeSeriesResultsWrapper._wrap_attrs, _attrs)
_methods = {}
_wrap_methods = wrap.union_dicts(
tsa_model.TimeSeriesResultsWrapper._wrap_methods, _methods)
wrap.populate_wrapper(ARMAResultsWrapper, ARMAResults) # noqa:E305
[docs]class ARIMAResults(ARMAResults):
@Appender(_arima_results_predict)
def predict(self, start=None, end=None, exog=None, typ='linear',
dynamic=False):
return self.model.predict(self.params, start, end, exog, typ, dynamic)
def _forecast_error(self, steps):
sigma2 = self.sigma2
ma_rep = arma2ma(np.r_[1, -self.arparams],
np.r_[1, self.maparams], lags=steps)
fcerr = np.sqrt(np.cumsum(cumsum_n(ma_rep, self.k_diff) ** 2) * sigma2)
return fcerr
def _forecast_conf_int(self, forecast, fcerr, alpha):
const = norm.ppf(1 - alpha / 2.)
conf_int = np.c_[forecast - const * fcerr, forecast + const * fcerr]
return conf_int
def forecast(self, steps=1, exog=None, alpha=.05):
"""
Out-of-sample forecasts
Parameters
----------
steps : int
The number of out of sample forecasts from the end of the
sample.
exog : ndarray
If the model is an ARIMAX, you must provide out of sample
values for the exogenous variables. This should not include
the constant. The number of observation in exog must match the
value of steps.
alpha : float
The confidence intervals for the forecasts are (1 - alpha) %
Returns
-------
forecast : array
Array of out of sample forecasts
stderr : array
Array of the standard error of the forecasts.
conf_int : array
2d array of the confidence interval for the forecast
Notes
-----
Prediction is done in the levels of the original endogenous variable.
If you would like prediction of differences in levels use `predict`.
"""
if exog is not None:
exog = array_like(exog, 'exog', ndim=2)
if exog.shape[0] != steps:
raise ValueError("new exog needed for each step")
if self.k_exog != exog.shape[1]:
raise ValueError('exog must contain the same number of '
'variables as in the estimated model.')
# prepend in-sample exog observations
if self.k_ar > 0:
exog = np.vstack((self.model.exog[-self.k_ar:, self.k_trend:],
exog))
else:
if self.k_exog:
raise ValueError('Forecast values for exog are required when '
'the model contains exogenous regressors.')
forecast = _arma_predict_out_of_sample(self.params, steps, self.resid,
self.k_ar, self.k_ma,
self.k_trend, self.k_exog,
self.model.endog,
exog, method=self.model.method)
d = self.model.k_diff
endog = self.model.data.endog[-d:]
forecast = unintegrate(forecast, unintegrate_levels(endog, d))[d:]
# get forecast errors
fcerr = self._forecast_error(steps)
conf_int = self._forecast_conf_int(forecast, fcerr, alpha)
return forecast, fcerr, conf_int
@Appender(_arima_plot_predict)
def plot_predict(self, start=None, end=None, exog=None, dynamic=False,
alpha=.05, plot_insample=True, ax=None):
from statsmodels.graphics.utils import _import_mpl, create_mpl_ax
_ = _import_mpl()
fig, ax = create_mpl_ax(ax)
# use predict so you set dates
forecast = self.predict(start, end, exog, 'levels', dynamic)
# doing this twice. just add a plot keyword to predict?
start, end, out_of_sample, _ = (
self.model._get_prediction_index(start, end, dynamic))
if out_of_sample:
steps = out_of_sample
fc_error = self._forecast_error(steps)
conf_int = self._forecast_conf_int(forecast[-steps:], fc_error,
alpha)
if hasattr(self.data, "predict_dates"):
from pandas import Series
forecast = Series(forecast, index=self.data.predict_dates)
ax = forecast.plot(ax=ax, label='forecast')
else:
ax.plot(forecast)
x = ax.get_lines()[-1].get_xdata()
if out_of_sample:
label = "{0:.0%} confidence interval".format(1 - alpha)
ax.fill_between(x[-out_of_sample:], conf_int[:, 0], conf_int[:, 1],
color='gray', alpha=.5, label=label)
if plot_insample:
import re
k_diff = self.k_diff
label = re.sub(r"D\d*\.", "", self.model.endog_names)
levels = unintegrate(self.model.endog,
self.model._first_unintegrate)
ax.plot(x[:end + 1 - start],
levels[start + k_diff:end + k_diff + 1], label=label)
ax.legend(loc='best')
return fig
class ARIMAResultsWrapper(ARMAResultsWrapper):
pass
wrap.populate_wrapper(ARIMAResultsWrapper, ARIMAResults) # noqa:E305