# -*- coding: utf-8 -*-
from __future__ import division
from statsmodels.compat.python import iteritems, range, string_types, lmap, long
import numpy as np
from numpy import dot, identity
from numpy.linalg import inv, slogdet
from scipy.stats import norm
from statsmodels.regression.linear_model import OLS
from statsmodels.tsa.tsatools import (lagmat, add_trend,
_ar_transparams, _ar_invtransparams)
import statsmodels.tsa.base.tsa_model as tsbase
import statsmodels.base.model as base
from statsmodels.tools.decorators import (resettable_cache,
cache_readonly, cache_writable)
from statsmodels.tools.numdiff import approx_fprime, approx_hess
from statsmodels.tsa.kalmanf.kalmanfilter import KalmanFilter
import statsmodels.base.wrapper as wrap
from statsmodels.tsa.vector_ar import util
__all__ = ['AR']
def sumofsq(x, axis=0):
"""Helper function to calculate sum of squares along first axis"""
return np.sum(x**2, axis=axis)
def _check_ar_start(start, k_ar, method, dynamic):
if (method == 'cmle' or dynamic) and start < k_ar:
raise ValueError("Start must be >= k_ar for conditional MLE "
"or dynamic forecast. Got %d" % start)
def _ar_predict_out_of_sample(y, params, p, k_trend, steps, start=0):
mu = params[:k_trend] or 0 # only have to worry about constant
arparams = params[k_trend:][::-1] # reverse for dot
# dynamic endogenous variable
endog = np.zeros(p + steps) # this is one too big but doesn't matter
if start:
endog[:p] = y[start-p:start]
else:
endog[:p] = y[-p:]
forecast = np.zeros(steps)
for i in range(steps):
fcast = mu + np.dot(arparams, endog[i:i+p])
forecast[i] = fcast
endog[i + p] = fcast
return forecast
[docs]class AR(tsbase.TimeSeriesModel):
__doc__ = tsbase._tsa_doc % {"model" : "Autoregressive AR(p) model",
"params" : """endog : array-like
1-d endogenous response variable. The independent variable.""",
"extra_params" : base._missing_param_doc,
"extra_sections" : ""}
def __init__(self, endog, dates=None, freq=None, missing='none'):
super(AR, self).__init__(endog, None, dates, freq, missing=missing)
endog = self.endog # original might not have been an ndarray
if endog.ndim == 1:
endog = endog[:, None]
self.endog = endog # to get shapes right
elif endog.ndim > 1 and endog.shape[1] != 1:
raise ValueError("Only the univariate case is implemented")
[docs] def initialize(self):
pass
def _transparams(self, params):
"""
Transforms params to induce stationarity/invertability.
Reference
---------
Jones(1980)
"""
p = self.k_ar
k = self.k_trend
newparams = params.copy()
newparams[k:k+p] = _ar_transparams(params[k:k+p].copy())
return newparams
def _invtransparams(self, start_params):
"""
Inverse of the Jones reparameterization
"""
p = self.k_ar
k = self.k_trend
newparams = start_params.copy()
newparams[k:k+p] = _ar_invtransparams(start_params[k:k+p].copy())
return newparams
def _presample_fit(self, params, start, p, end, y, predictedvalues):
"""
Return the pre-sample predicted values using the Kalman Filter
Notes
-----
See predict method for how to use start and p.
"""
k = self.k_trend
# build system matrices
T_mat = KalmanFilter.T(params, p, k, p)
R_mat = KalmanFilter.R(params, p, k, 0, p)
# Initial State mean and variance
alpha = np.zeros((p, 1))
Q_0 = dot(inv(identity(p**2)-np.kron(T_mat, T_mat)),
dot(R_mat, R_mat.T).ravel('F'))
Q_0 = Q_0.reshape(p, p, order='F') # TODO: order might need to be p+k
P = Q_0
Z_mat = KalmanFilter.Z(p)
for i in range(end): # iterate p-1 times to fit presample
v_mat = y[i] - dot(Z_mat, alpha)
F_mat = dot(dot(Z_mat, P), Z_mat.T)
Finv = 1./F_mat # inv. always scalar
K = dot(dot(dot(T_mat, P), Z_mat.T), Finv)
# update state
alpha = dot(T_mat, alpha) + dot(K, v_mat)
L = T_mat - dot(K, Z_mat)
P = dot(dot(T_mat, P), L.T) + dot(R_mat, R_mat.T)
#P[0,0] += 1 # for MA part, R_mat.R_mat.T above
if i >= start - 1: # only record if we ask for it
predictedvalues[i + 1 - start] = dot(Z_mat, alpha)
def _get_prediction_index(self, start, end, dynamic, index=None):
method = getattr(self, 'method', 'mle')
k_ar = getattr(self, 'k_ar', 0)
if start is None:
if method == 'mle' and not dynamic:
start = 0
else: # can't do presample fit for cmle or dynamic
start = k_ar
start = self._index[start]
if end is None:
end = self._index[-1]
start, end, out_of_sample, prediction_index = (
super(AR, self)._get_prediction_index(start, end, index))
# This replaces the _validate() call
if 'mle' not in method and start < k_ar:
raise ValueError("Start must be >= k_ar for conditional MLE or "
"dynamic forecast. Got %s" % start)
# Other validation
_check_ar_start(start, k_ar, method, dynamic)
return start, end, out_of_sample, prediction_index
[docs] def predict(self, params, start=None, end=None, dynamic=False):
"""
Returns in-sample and out-of-sample prediction.
Parameters
----------
params : array
The fitted model parameters.
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.
dynamic : bool
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 forecasted
value is `start`.
Returns
-------
predicted values : array
Notes
-----
The linear Gaussian Kalman filter is used to return pre-sample fitted
values. The exact initial Kalman Filter is used. See Durbin and Koopman
in the references for more information.
"""
# will return an index of a date
start, end, out_of_sample, _ = (
self._get_prediction_index(start, end, dynamic))
if start - end > 1:
raise ValueError("end is before start")
k_ar = self.k_ar
k_trend = self.k_trend
method = self.method
endog = self.endog.squeeze()
if dynamic:
out_of_sample += end - start + 1
return _ar_predict_out_of_sample(endog, params, k_ar,
k_trend, out_of_sample, start)
predictedvalues = np.zeros(end + 1 - start)
# fit pre-sample
if method == 'mle': # use Kalman Filter to get initial values
if k_trend:
mu = params[0]/(1-np.sum(params[k_trend:]))
else:
mu = 0
# modifies predictedvalues in place
if start < k_ar:
self._presample_fit(params, start, k_ar, min(k_ar-1, end),
endog[:k_ar] - mu, predictedvalues)
predictedvalues[:k_ar-start] += mu
if end < k_ar:
return predictedvalues
# just do the whole thing and truncate
fittedvalues = dot(self.X, params)
pv_start = max(k_ar - start, 0)
fv_start = max(start - k_ar, 0)
fv_end = min(len(fittedvalues), end-k_ar+1)
predictedvalues[pv_start:] = fittedvalues[fv_start:fv_end]
if out_of_sample:
forecastvalues = _ar_predict_out_of_sample(endog, params,
k_ar, k_trend,
out_of_sample)
predictedvalues = np.r_[predictedvalues, forecastvalues]
return predictedvalues
def _presample_varcov(self, params):
"""
Returns the inverse of the presample variance-covariance.
Notes
-----
See Hamilton p. 125
"""
k = self.k_trend
p = self.k_ar
p1 = p+1
# get inv(Vp) Hamilton 5.3.7
params0 = np.r_[-1, params[k:]]
Vpinv = np.zeros((p, p), dtype=params.dtype)
for i in range(1, p1):
Vpinv[i-1, i-1:] = np.correlate(params0, params0[:i],)[:-1]
Vpinv[i-1, i-1:] -= np.correlate(params0[-i:], params0,)[:-1]
Vpinv = Vpinv + Vpinv.T - np.diag(Vpinv.diagonal())
return Vpinv
def _loglike_css(self, params):
"""
Loglikelihood of AR(p) process using conditional sum of squares
"""
nobs = self.nobs
Y = self.Y
X = self.X
ssr = sumofsq(Y.squeeze() - np.dot(X, params))
sigma2 = ssr/nobs
return (-nobs/2 * (np.log(2 * np.pi) + np.log(sigma2)) -
ssr/(2 * sigma2))
def _loglike_mle(self, params):
"""
Loglikelihood of AR(p) process using exact maximum likelihood
"""
nobs = self.nobs
X = self.X
endog = self.endog
k_ar = self.k_ar
k_trend = self.k_trend
# reparameterize according to Jones (1980) like in ARMA/Kalman Filter
if self.transparams:
params = self._transparams(params)
# get mean and variance for pre-sample lags
yp = endog[:k_ar].copy()
if k_trend:
c = [params[0]] * k_ar
else:
c = [0]
mup = np.asarray(c / (1 - np.sum(params[k_trend:])))
diffp = yp - mup[:, None]
# get inv(Vp) Hamilton 5.3.7
Vpinv = self._presample_varcov(params)
diffpVpinv = np.dot(np.dot(diffp.T, Vpinv), diffp).item()
ssr = sumofsq(endog[k_ar:].squeeze() - np.dot(X, params))
# concentrating the likelihood means that sigma2 is given by
sigma2 = 1./nobs * (diffpVpinv + ssr)
self.sigma2 = sigma2
logdet = slogdet(Vpinv)[1] # TODO: add check for singularity
loglike = -1/2. * (nobs * (np.log(2 * np.pi) + np.log(sigma2)) -
logdet + diffpVpinv / sigma2 + ssr / sigma2)
return loglike
[docs] def loglike(self, params):
"""
The loglikelihood of an AR(p) process
Parameters
----------
params : array
The fitted parameters of the AR model
Returns
-------
llf : float
The loglikelihood evaluated at `params`
Notes
-----
Contains constant term. If the model is fit by OLS then this returns
the conditonal maximum likelihood.
.. math:: \\frac{\\left(n-p\\right)}{2}\\left(\\log\\left(2\\pi\\right)+\\log\\left(\\sigma^{2}\\right)\\right)-\\frac{1}{\\sigma^{2}}\\sum_{i}\\epsilon_{i}^{2}
If it is fit by MLE then the (exact) unconditional maximum likelihood
is returned.
.. math:: -\\frac{n}{2}log\\left(2\\pi\\right)-\\frac{n}{2}\\log\\left(\\sigma^{2}\\right)+\\frac{1}{2}\\left|V_{p}^{-1}\\right|-\\frac{1}{2\\sigma^{2}}\\left(y_{p}-\\mu_{p}\\right)^{\\prime}V_{p}^{-1}\\left(y_{p}-\\mu_{p}\\right)-\\frac{1}{2\\sigma^{2}}\\sum_{t=p+1}^{n}\\epsilon_{i}^{2}
where
:math:`\\mu_{p}` is a (`p` x 1) vector with each element equal to the
mean of the AR process and :math:`\\sigma^{2}V_{p}` is the (`p` x `p`)
variance-covariance matrix of the first `p` observations.
"""
#TODO: Math is on Hamilton ~pp 124-5
if self.method == "cmle":
return self._loglike_css(params)
else:
return self._loglike_mle(params)
[docs] def score(self, params):
"""
Return the gradient of the loglikelihood at params.
Parameters
----------
params : array-like
The parameter values at which to evaluate the score function.
Notes
-----
Returns numerical gradient.
"""
loglike = self.loglike
return approx_fprime(params, loglike, epsilon=1e-8)
[docs] def hessian(self, params):
"""
Returns numerical hessian for now.
"""
loglike = self.loglike
return approx_hess(params, loglike)
def _stackX(self, k_ar, trend):
"""
Private method to build the RHS matrix for estimation.
Columns are trend terms then lags.
"""
endog = self.endog
X = lagmat(endog, maxlag=k_ar, trim='both')
k_trend = util.get_trendorder(trend)
if k_trend:
X = add_trend(X, prepend=True, trend=trend)
self.k_trend = k_trend
return X
[docs] def select_order(self, maxlag, ic, trend='c', method='mle'):
"""
Select the lag order according to the information criterion.
Parameters
----------
maxlag : int
The highest lag length tried. See `AR.fit`.
ic : str {'aic','bic','hqic','t-stat'}
Criterion used for selecting the optimal lag length.
See `AR.fit`.
trend : str {'c','nc'}
Whether to include a constant or not. 'c' - include constant.
'nc' - no constant.
Returns
-------
bestlag : int
Best lag according to IC.
"""
endog = self.endog
# make Y and X with same nobs to compare ICs
Y = endog[maxlag:]
self.Y = Y # attach to get correct fit stats
X = self._stackX(maxlag, trend) # sets k_trend
self.X = X
k = self.k_trend # k_trend set in _stackX
k = max(1, k) # handle if startlag is 0
results = {}
if ic != 't-stat':
for lag in range(k, maxlag+1):
# have to reinstantiate the model to keep comparable models
endog_tmp = endog[maxlag-lag:]
fit = AR(endog_tmp).fit(maxlag=lag, method=method,
full_output=0, trend=trend,
maxiter=100, disp=0)
results[lag] = eval('fit.'+ic)
bestic, bestlag = min((res, k) for k, res in iteritems(results))
else: # choose by last t-stat.
stop = 1.6448536269514722 # for t-stat, norm.ppf(.95)
for lag in range(maxlag, k - 1, -1):
# have to reinstantiate the model to keep comparable models
endog_tmp = endog[maxlag - lag:]
fit = AR(endog_tmp).fit(maxlag=lag, method=method,
full_output=0, trend=trend,
maxiter=35, disp=-1)
bestlag = 0
if np.abs(fit.tvalues[-1]) >= stop:
bestlag = lag
break
return bestlag
[docs] def fit(self, maxlag=None, method='cmle', ic=None, trend='c',
transparams=True, start_params=None, solver='lbfgs', maxiter=35,
full_output=1, disp=1, callback=None, **kwargs):
"""
Fit the unconditional maximum likelihood of an AR(p) process.
Parameters
----------
maxlag : int
If `ic` is None, then maxlag is the lag length used in fit. If
`ic` is specified then maxlag is the highest lag order used to
select the correct lag order. If maxlag is None, the default is
round(12*(nobs/100.)**(1/4.))
method : str {'cmle', 'mle'}, optional
cmle - Conditional maximum likelihood using OLS
mle - Unconditional (exact) maximum likelihood. See `solver`
and the Notes.
ic : str {'aic','bic','hic','t-stat'}
Criterion used for selecting the optimal lag length.
aic - Akaike Information Criterion
bic - Bayes Information Criterion
t-stat - Based on last lag
hqic - Hannan-Quinn Information Criterion
If any of the information criteria are selected, the lag length
which results in the lowest value is selected. If t-stat, the
model starts with maxlag and drops a lag until the highest lag
has a t-stat that is significant at the 95 % level.
trend : str {'c','nc'}
Whether to include a constant or not. 'c' - include constant.
'nc' - no constant.
The below can be specified if method is 'mle'
transparams : bool, optional
Whether or not to transform the parameters to ensure stationarity.
Uses the transformation suggested in Jones (1980).
start_params : array-like, optional
A first guess on the parameters. Default is cmle estimates.
solver : str or None, optional
Solver to be used if method is 'mle'. 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'.
maxiter : int, optional
The maximum number of function evaluations. Default is 35.
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 : bool, optional
If True, convergence information is output.
callback : function, optional
Called after each iteration as callback(xk) where xk is the current
parameter vector.
kwargs
See Notes for keyword arguments that can be passed to fit.
References
----------
Jones, R.H. 1980 "Maximum likelihood fitting of ARMA models to time
series with missing observations." `Technometrics`. 22.3.
389-95.
See also
--------
statsmodels.base.model.LikelihoodModel.fit
"""
method = method.lower()
if method not in ['cmle', 'yw', 'mle']:
raise ValueError("Method %s not recognized" % method)
self.method = method
self.trend = trend
self.transparams = transparams
nobs = len(self.endog) # overwritten if method is 'cmle'
endog = self.endog
if maxlag is None:
maxlag = int(round(12*(nobs/100.)**(1/4.)))
k_ar = maxlag # stays this if ic is None
# select lag length
if ic is not None:
ic = ic.lower()
if ic not in ['aic', 'bic', 'hqic', 't-stat']:
raise ValueError("ic option %s not understood" % ic)
k_ar = self.select_order(k_ar, ic, trend, method)
self.k_ar = k_ar # change to what was chosen by ic
# redo estimation for best lag
# make LHS
Y = endog[k_ar:, :]
# make lagged RHS
X = self._stackX(k_ar, trend) # sets self.k_trend
k_trend = self.k_trend
self.exog_names = util.make_lag_names(self.endog_names, k_ar, k_trend)
self.Y = Y
self.X = X
if method == "cmle": # do OLS
arfit = OLS(Y, X).fit()
params = arfit.params
self.nobs = nobs - k_ar
self.sigma2 = arfit.ssr/arfit.nobs # needed for predict fcasterr
elif method == "mle":
solver = solver.lower()
self.nobs = nobs
if start_params is None:
start_params = OLS(Y, X).fit().params
else:
if len(start_params) != k_trend + k_ar:
raise ValueError("Length of start params is %d. There"
" are %d parameters." %
(len(start_params), k_trend + k_ar))
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(AR, self).fit(start_params=start_params,
method=solver, maxiter=maxiter,
full_output=full_output, disp=disp,
callback=callback, **kwargs)
params = mlefit.params
if self.transparams:
params = self._transparams(params)
self.transparams = False # turn off now for other results
# don't use yw, because we can't estimate the constant
#elif method == "yw":
# params, omega = yule_walker(endog, order=maxlag,
# method="mle", demean=False)
# how to handle inference after Yule-Walker?
# self.params = params #TODO: don't attach here
# self.omega = omega
pinv_exog = np.linalg.pinv(X)
normalized_cov_params = np.dot(pinv_exog, pinv_exog.T)
arfit = ARResults(self, params, normalized_cov_params)
if method == 'mle' and full_output:
arfit.mle_retvals = mlefit.mle_retvals
arfit.mle_settings = mlefit.mle_settings
return ARResultsWrapper(arfit)
[docs]class ARResults(tsbase.TimeSeriesModelResults):
"""
Class to hold results from fitting an AR model.
Parameters
----------
model : AR Model instance
Reference to the model that is fit.
params : array
The fitted parameters from the AR Model.
normalized_cov_params : array
inv(dot(X.T,X)) where X is the lagged values.
scale : float, optional
An estimate of the scale of the model.
Returns
-------
**Attributes**
aic : float
Akaike Information Criterion using Lutkephol's definition.
:math:`log(sigma) + 2*(1 + k_ar + k_trend)/nobs`
bic : float
Bayes Information Criterion
:math:`\\log(\\sigma) + (1 + k_ar + k_trend)*\\log(nobs)/nobs`
bse : array
The standard errors of the estimated parameters. If `method` is 'cmle',
then the standard errors that are returned are the OLS standard errors
of the coefficients. If the `method` is 'mle' then they are computed
using the numerical Hessian.
fittedvalues : array
The in-sample predicted values of the fitted AR model. The `k_ar`
initial values are computed via the Kalman Filter if the model is
fit by `mle`.
fpe : float
Final prediction error using Lütkepohl's definition
((n_totobs+k_trend)/(n_totobs-k_ar-k_trend))*sigma
hqic : float
Hannan-Quinn Information Criterion.
k_ar : float
Lag length. Sometimes used as `p` in the docs.
k_trend : float
The number of trend terms included. 'nc'=0, 'c'=1.
llf : float
The loglikelihood of the model evaluated at `params`. See `AR.loglike`
model : AR model instance
A reference to the fitted AR model.
nobs : float
The number of available observations `nobs` - `k_ar`
n_totobs : float
The number of total observations in `endog`. Sometimes `n` in the docs.
params : array
The fitted parameters of the model.
pvalues : array
The p values associated with the standard errors.
resid : array
The residuals of the model. If the model is fit by 'mle' then the
pre-sample residuals are calculated using fittedvalues from the Kalman
Filter.
roots : array
The roots of the AR process 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.
scale : float
Same as sigma2
sigma2 : float
The variance of the innovations (residuals).
trendorder : int
The polynomial order of the trend. 'nc' = None, 'c' or 't' = 0,
'ct' = 1, etc.
tvalues : array
The t-values associated with `params`.
"""
_cache = {} # for scale setter
def __init__(self, model, params, normalized_cov_params=None, scale=1.):
super(ARResults, self).__init__(model, params, normalized_cov_params,
scale)
self._cache = resettable_cache()
self.nobs = model.nobs
n_totobs = len(model.endog)
self.n_totobs = n_totobs
self.X = model.X # copy?
self.Y = model.Y
k_ar = model.k_ar
self.k_ar = k_ar
k_trend = model.k_trend
self.k_trend = k_trend
trendorder = None
if k_trend > 0:
trendorder = k_trend - 1
self.trendorder = trendorder
#TODO: cmle vs mle?
self.df_model = k_ar + k_trend
self.df_resid = self.model.df_resid = n_totobs - self.df_model
[docs] @cache_writable()
def sigma2(self):
model = self.model
if model.method == "cmle": # do DOF correction
return 1. / self.nobs * sumofsq(self.resid)
else:
return self.model.sigma2
[docs] @cache_writable() # for compatability with RegressionResults
def scale(self):
return self.sigma2
[docs] @cache_readonly
def bse(self): # allow user to specify?
if self.model.method == "cmle": # uses different scale/sigma def.
resid = self.resid
ssr = np.dot(resid, resid)
ols_scale = ssr / (self.nobs - self.k_ar - self.k_trend)
return np.sqrt(np.diag(self.cov_params(scale=ols_scale)))
else:
hess = approx_hess(self.params, self.model.loglike)
return np.sqrt(np.diag(-np.linalg.inv(hess)))
[docs] @cache_readonly
def pvalues(self):
return norm.sf(np.abs(self.tvalues))*2
[docs] @cache_readonly
def aic(self):
#JP: this is based on loglike with dropped constant terms ?
# Lutkepohl
#return np.log(self.sigma2) + 1./self.model.nobs * self.k_ar
# Include constant as estimated free parameter and double the loss
return np.log(self.sigma2) + 2 * (1 + self.df_model)/self.nobs
# Stata defintion
#nobs = self.nobs
#return -2 * self.llf/nobs + 2 * (self.k_ar+self.k_trend)/nobs
[docs] @cache_readonly
def hqic(self):
nobs = self.nobs
# Lutkepohl
# return np.log(self.sigma2)+ 2 * np.log(np.log(nobs))/nobs * self.k_ar
# R uses all estimated parameters rather than just lags
return (np.log(self.sigma2) + 2 * np.log(np.log(nobs))/nobs *
(1 + self.df_model))
# Stata
#nobs = self.nobs
#return -2 * self.llf/nobs + 2 * np.log(np.log(nobs))/nobs * \
# (self.k_ar + self.k_trend)
[docs] @cache_readonly
def fpe(self):
nobs = self.nobs
df_model = self.df_model
#Lutkepohl
return ((nobs+df_model)/(nobs-df_model))*self.sigma2
[docs] @cache_readonly
def bic(self):
nobs = self.nobs
# Lutkepohl
#return np.log(self.sigma2) + np.log(nobs)/nobs * self.k_ar
# Include constant as est. free parameter
return np.log(self.sigma2) + (1 + self.df_model) * np.log(nobs)/nobs
# Stata
# return -2 * self.llf/nobs + np.log(nobs)/nobs * (self.k_ar + \
# self.k_trend)
[docs] @cache_readonly
def resid(self):
#NOTE: uses fittedvalues because it calculate presample values for mle
model = self.model
endog = model.endog.squeeze()
if model.method == "cmle": # elimate pre-sample
return endog[self.k_ar:] - self.fittedvalues
else:
return model.endog.squeeze() - self.fittedvalues
#def ssr(self):
# resid = self.resid
# return np.dot(resid, resid)
[docs] @cache_readonly
def roots(self):
k = self.k_trend
return np.roots(np.r_[1, -self.params[k:]]) ** -1
[docs] @cache_readonly
def fittedvalues(self):
return self.model.predict(self.params)
[docs] def predict(self, start=None, end=None, dynamic=False):
params = self.params
predictedvalues = self.model.predict(params, start, end, dynamic)
return predictedvalues
# start, end, out_of_sample, prediction_index = (
# self.model._get_prediction_index(start, end, index))
##TODO: return forecast errors and confidence intervals
#from statsmodels.tsa.arima_process import arma2ma
#ma_rep = arma2ma(np.r_[1,-params[::-1]], [1], out_of_sample)
#fcasterr = np.sqrt(self.sigma2 * np.cumsum(ma_rep**2))
preddoc = AR.predict.__doc__.split('\n')
extra_doc = (""" confint : bool, float
Whether to return confidence intervals. If `confint` == True,
95 % confidence intervals are returned. Else if `confint` is a
float, then it is assumed to be the alpha value of the confidence
interval. That is confint == .05 returns a 95% confidence
interval, and .10 would return a 90% confidence interval."""
).split('\n')
#ret_doc = """
# fcasterr : array-like
# confint : array-like
#"""
predict.__doc__ = '\n'.join(preddoc[:5] + preddoc[7:20] + extra_doc +
preddoc[20:])
class ARResultsWrapper(wrap.ResultsWrapper):
_attrs = {}
_wrap_attrs = wrap.union_dicts(tsbase.TimeSeriesResultsWrapper._wrap_attrs,
_attrs)
_methods = {}
_wrap_methods = wrap.union_dicts(tsbase.TimeSeriesResultsWrapper._wrap_methods,
_methods)
wrap.populate_wrapper(ARResultsWrapper, ARResults)
if __name__ == "__main__":
import statsmodels.api as sm
sunspots = sm.datasets.sunspots.load()
# Why does R demean the data by defaut?
ar_ols = AR(sunspots.endog)
res_ols = ar_ols.fit(maxlag=9)
ar_mle = AR(sunspots.endog)
res_mle_bfgs = ar_mle.fit(maxlag=9, method="mle", solver="bfgs",
maxiter=500, gtol=1e-10)
# res_mle2 = ar_mle.fit(maxlag=1, method="mle", maxiter=500, penalty=True,
# tol=1e-13)
# ar_yw = AR(sunspots.endog)
# res_yw = ar_yw.fit(maxlag=4, method="yw")
# # Timings versus talkbox
# from timeit import default_timer as timer
# print "Time AR fit vs. talkbox"
# # generate a long series of AR(2) data
#
# nobs = 1000000
# y = np.empty(nobs)
# y[0:2] = 0
# for i in range(2,nobs):
# y[i] = .25 * y[i-1] - .75 * y[i-2] + np.random.rand()
#
# mod_sm = AR(y)
# t = timer()
# res_sm = mod_sm.fit(method="yw", trend="nc", demean=False, maxlag=2)
# t_end = timer()
# print str(t_end - t) + " seconds for sm.AR with yule-walker, 2 lags"
# try:
# import scikits.talkbox as tb
# except:
# raise ImportError("You need scikits.talkbox installed for timings")
# t = timer()
# mod_tb = tb.lpc(y, 2)
# t_end = timer()
# print str(t_end - t) + " seconds for talkbox.lpc"
# print """For higher lag lengths ours quickly fills up memory and starts
#thrashing the swap. Should we include talkbox C code or Cythonize the
#Levinson recursion algorithm?"""
## Try with a pandas series
import pandas
import scikits.timeseries as ts
d1 = ts.Date(year=1700, freq='A')
#NOTE: have to have yearBegin offset for annual data until parser rewrite
#should this be up to the user, or should it be done in TSM init?
#NOTE: not anymore, it's end of year now
ts_dr = ts.date_array(start_date=d1, length=len(sunspots.endog))
pandas_dr = pandas.DatetimeIndex(start=d1.datetime,
periods=len(sunspots.endog),
freq='A-DEC')
#pandas_dr = pandas_dr.shift(-1, pandas.datetools.yearBegin)
dates = np.arange(1700, 1700 + len(sunspots.endog))
dates = ts.date_array(dates, freq='A')
#sunspots = pandas.Series(sunspots.endog, index=dates)
#NOTE: pandas only does business days for dates it looks like
import datetime
dt_dates = np.asarray(lmap(datetime.datetime.fromordinal,
ts_dr.toordinal().astype(int)))
sunspots = pandas.Series(sunspots.endog, index=dt_dates)
#NOTE: pandas can't handle pre-1900 dates
mod = AR(sunspots, freq='A')
res = mod.fit(method='mle', maxlag=9)
# some data for an example in Box Jenkins
IBM = np.asarray([460, 457, 452, 459, 462, 459, 463, 479, 493, 490.])
w = np.diff(IBM)
theta = .5
