# -*- coding: utf-8 -*-
#
# Author: Taylor Smith <taylor.smith@alkaline-ml.com>
#
# Tests for stationarity
from __future__ import absolute_import, division
from sklearn.base import BaseEstimator
from sklearn.utils.validation import column_or_1d, check_array
from sklearn.linear_model import LinearRegression
from sklearn.externals import six
from statsmodels import api as sm
from abc import ABCMeta, abstractmethod
import numpy as np
from ..compat.numpy import DTYPE
from ..utils.array import c, diff
from .approx import approx
# since the C import relies on the C code having been built with Cython,
# and since the platform might name the .so file something funky (like
# _arima.cpython-35m-darwin.so), import this absolutely and not relatively.
from ._arima import C_tseries_pp_sum
__all__ = [
'ADFTest',
'KPSSTest',
'PPTest'
]
class _BaseStationarityTest(six.with_metaclass(ABCMeta, BaseEstimator)):
@staticmethod
def _base_case(x):
# if x is empty, return false so the other methods return False
if (x is None) or (x.shape[0] == 0):
return False
return True
@staticmethod
def _embed(x, k):
# lag the vector and put the lags into columns
n = x.shape[0]
rows = [
# so, if k=2, it'll be (x[1:n], x[:n-1])
x[j:n - i] for i, j in enumerate(range(k - 1, -1, -1))
]
return np.asarray(rows)
# return np.array([x[1:], x[:m]])
class _DifferencingStationarityTest(six.with_metaclass(ABCMeta,
_BaseStationarityTest)):
"""Provides the base class for stationarity tests such as the
Kwiatkowski–Phillips–Schmidt–Shin, Augmented Dickey-Fuller and the
Phillips–Perron tests. These tests are used to determine whether a time
series is stationary.
"""
def __init__(self, alpha):
self.alpha = alpha
@abstractmethod
def is_stationary(self, x):
"""Test whether the time series is stationary.
Parameters
----------
x : array-like, shape=(n_samples,)
The time series vector.
Returns
-------
pval : float
The computed P-value of the test.
sig : bool
Whether the P-value is significant at the ``alpha`` level.
More directly, whether to difference the time series.
"""
[docs]class KPSSTest(_DifferencingStationarityTest):
"""Conduct a KPSS test for stationarity.
In econometrics, Kwiatkowski–Phillips–Schmidt–Shin (KPSS) tests are used
for testing a null hypothesis that an observable time series is stationary
around a deterministic trend (i.e. trend-stationary) against the
alternative of a unit root.
Parameters
----------
alpha : float, optional (default=0.05)
Level of the test
null : str, optional (default='level')
Whether to fit the linear model on the one vector, or an arange.
If ``null`` is 'trend', a linear model is fit on an arange, if
'level', it is fit on the one vector.
lshort : bool, optional (default=True)
Whether or not to truncate the ``l`` value in the C code.
Notes
-----
This test is generally used indirectly via the
:func:`pyramid.arima.ndiffs` function, which computes the
differencing term, ``d``.
References
----------
.. [1] R's tseries KPSS test source code: http://bit.ly/2eJP1IU
"""
_valid = {'trend', 'null'}
tablep = c(0.01, 0.025, 0.05, 0.10)
[docs] def __init__(self, alpha=0.05, null='level', lshort=True):
super(KPSSTest, self).__init__(alpha=alpha)
self.null = null
self.lshort = lshort
[docs] def is_stationary(self, x):
"""Test whether the time series is stationary.
Parameters
----------
x : array-like, shape=(n_samples,)
The time series vector.
Returns
-------
pval : float
The computed P-value of the test.
sig : bool
Whether the P-value is significant at the ``alpha`` level.
More directly, whether to difference the time series.
"""
if not self._base_case(x):
return np.nan, False
# ensure vector
x = column_or_1d(check_array(
x, ensure_2d=False, dtype=DTYPE,
force_all_finite=True)) # type: np.ndarray
n = x.shape[0]
# check on status of null
null = self.null
# fit a model on an arange to determine the residuals
if null == 'trend':
t = np.arange(n).reshape(n, 1)
# these numbers came out of the R code.. I've found 0 doc for these
table = c(0.216, 0.176, 0.146, 0.119)
elif null == 'level':
t = np.ones(n).reshape(n, 1)
# these numbers came out of the R code.. I've found 0 doc for these
table = c(0.739, 0.574, 0.463, 0.347)
else:
raise ValueError("null must be one of %r" % self._valid)
# fit the model
lm = LinearRegression().fit(t, x)
e = x - lm.predict(t) # residuals
s = np.cumsum(e)
eta = (s * s).sum() / (n**2)
s2 = (e * e).sum() / n
scalar, denom = 10, 14
if self.lshort:
scalar, denom = 3, 13
l_ = int(np.trunc(scalar * np.sqrt(n) / denom))
# compute the C subroutine
s2 = C_tseries_pp_sum(e, n, l_, s2)
stat = eta / s2
# do approximation
_, pval = approx(table, self.tablep, xout=stat, rule=2)
# R does a test for rule=1, but we don't want to do that, because they
# just do it to issue a warning in case the P-value is smaller/greater
# than the printed value is.
return pval[0], pval[0] < self.alpha
[docs]class ADFTest(_DifferencingStationarityTest):
"""Conduct an ADF test for stationarity.
In statistics and econometrics, an augmented Dickey–Fuller test (ADF)
tests the null hypothesis of a unit root is present in a time series
sample. The alternative hypothesis is different depending on which version
of the test is used, but is usually stationarity or trend-stationarity. It
is an augmented version of the Dickey–Fuller test for a larger and more
complicated set of time series models.
Parameters
----------
alpha : float, optional (default=0.05)
Level of the test
k : int, optional (default=None)
The drift parameter. If ``k`` is None, it will be set to:
``np.trunc(np.power(x.shape[0] - 1, 1 / 3.0))``
Notes
-----
This test is generally used indirectly via the
:func:`pyramid.arima.ndiffs` function, which computes the
differencing term, ``d``.
ADF test does not perform as close to the R code as do the KPSS and PP
tests. This is due to the fact that is has to use statsmodels OLS
regression for std err estimates rather than the more robust sklearn
LinearRegression.
References
----------
.. [1] https://wikipedia.org/wiki/Augmented_Dickey–Fuller_test
"""
table = -np.array([(4.38, 4.15, 4.04, 3.99, 3.98, 3.96),
(3.95, 3.80, 3.73, 3.69, 3.68, 3.66),
(3.60, 3.50, 3.45, 3.43, 3.42, 3.41),
(3.24, 3.18, 3.15, 3.13, 3.13, 3.12),
(1.14, 1.19, 1.22, 1.23, 1.24, 1.25),
(0.80, 0.87, 0.90, 0.92, 0.93, 0.94),
(0.50, 0.58, 0.62, 0.64, 0.65, 0.66),
(0.15, 0.24, 0.28, 0.31, 0.32, 0.33)]).T
tablen = table.shape[1]
tableT = c(25, 50, 100, 250, 500, 100000)
tablep = c(0.01, 0.025, 0.05, 0.10, 0.90, 0.95, 0.975, 0.99)
[docs] def __init__(self, alpha=0.05, k=None):
super(ADFTest, self).__init__(alpha=alpha)
self.k = k
if k is not None and k < 0:
raise ValueError('k must be a positive integer (>= 0)')
[docs] def is_stationary(self, x):
"""Test whether the time series is stationary.
Parameters
----------
x : array-like, shape=(n_samples,)
The time series vector.
Returns
-------
pval : float
The computed P-value of the test.
sig : bool
Whether the P-value is significant at the ``alpha`` level.
More directly, whether to difference the time series.
"""
if not self._base_case(x):
return np.nan, False
# ensure vector
x = column_or_1d(check_array(
x, ensure_2d=False, dtype=DTYPE,
force_all_finite=True)) # type: np.ndarray
# if k is none...
k = self.k
if k is None:
k = np.trunc(np.power(x.shape[0] - 1, 1 / 3.0))
k = int(k) + 1
y = diff(x)
n = y.shape[0]
z = self._embed(y, k)
yt = z[0, :]
tt = np.arange(k - 1, n)
# R does [k:n].. but that's 1-based indexing and inclusive on the tail
xt1 = x[tt]
# make tt inclusive again (it was used as a mask before)
tt += 1
# the array that will create the LM:
_n = xt1.shape[0]
X = np.hstack([xt1.reshape((_n, 1)),
np.ones(_n).reshape((_n, 1)),
tt.reshape((_n, 1))])
if k > 1:
yt1 = z[1:k, :] # R had 2:k
X = np.hstack([X, yt1.T])
# fit the linear regression - this one is a bit strange in that we
# are using OLS from statsmodels rather than LR from sklearn. This is
# because we need the std errors, and sklearn does not have a way to
# store them.
res = sm.OLS(yt, X).fit()
STAT = res.params[0] / res.HC0_se[0] # XXX: is the denom correct?...
tableipl = np.zeros(self.tablen)
for i in range(self.tablen):
_, pval = approx(self.tableT, self.table[:, i], xout=n, rule=2)
tableipl[i] = pval
# make sure to do 1 - x...
_, interpol = approx(tableipl, self.tablep, xout=STAT, rule=2)
pval = 1 - interpol[0]
# in the R code, here is where the P value warning is tested again...
return pval, pval < self.alpha
[docs]class PPTest(_DifferencingStationarityTest):
"""Conduct a PP test for stationarity.
In statistics, the Phillips–Perron test (named after Peter C. B.
Phillips and Pierre Perron) is a unit root test. It is used in time series
analysis to test the null hypothesis that a time series is integrated of
order 1. It builds on the Dickey–Fuller test of the null hypothesis
``p = 0``.
Parameters
----------
alpha : float, optional (default=0.05)
Level of the test
lshort : bool, optional (default=True)
Whether or not to truncate the ``l`` value in the C code.
Notes
-----
This test is generally used indirectly via the
:func:`pyramid.arima.ndiffs` function, which computes the
differencing term, ``d``.
The R code allows for two types of tests: 'Z(alpha)' and 'Z(t_alpha)'.
Since sklearn does not allow extraction of std errors from the linear
model fit, ``t_alpha`` is much more difficult to achieve, so we do not
allow that variant.
References
----------
.. [1] R's tseries PP test source code: http://bit.ly/2wbzx6V
"""
table = -np.array([
(22.5, 25.7, 27.4, 28.4, 28.9, 29.5),
(19.9, 22.4, 23.6, 24.4, 24.8, 25.1),
(17.9, 19.8, 20.7, 21.3, 21.5, 21.8),
(15.6, 16.8, 17.5, 18.0, 18.1, 18.3),
(3.66, 3.71, 3.74, 3.75, 3.76, 3.77),
(2.51, 2.60, 2.62, 2.64, 2.65, 2.66),
(1.53, 1.66, 1.73, 1.78, 1.78, 1.79),
(0.43, 0.65, 0.75, 0.82, 0.84, 0.87)
]).T
tablen = table.shape[1]
tableT = c(25, 50, 100, 250, 500, 100000).astype(DTYPE)
tablep = c(0.01, 0.025, 0.05, 0.10, 0.90, 0.95, 0.975, 0.99)
[docs] def __init__(self, alpha=0.05, lshort=True):
super(PPTest, self).__init__(alpha=alpha)
self.lshort = lshort
[docs] def is_stationary(self, x):
"""Test whether the time series is stationary.
Parameters
----------
x : array-like, shape=(n_samples,)
The time series vector.
Returns
-------
pval : float
The computed P-value of the test.
sig : bool
Whether the P-value is significant at the ``alpha`` level.
More directly, whether to difference the time series.
"""
if not self._base_case(x):
return np.nan, False
# ensure vector
x = column_or_1d(check_array(
x, ensure_2d=False, dtype=DTYPE,
force_all_finite=True)) # type: np.ndarray
# embed the vector. This is some funkiness that goes on in the R
# code... basically, make a matrix where the column (rows if not T)
# are lagged windows of x
z = self._embed(x, 2)
yt = z[0, :]
yt1 = z[1, :] # type: np.ndarray
# fit a linear model to a predictor matrix
n = yt.shape[0]
tt = (np.arange(n) + 1) - (n / 2.0)
X = np.array([np.ones(n), tt, yt1]).T
res = LinearRegression().fit(X, yt) # lm(yt ~ 1 + tt + yt1)
coef = res.coef_
# check for singularities - do we want to do this??? in the R code,
# it happens. but the very same lm in the R code is rank 3, and here
# it is rank 2. Should we just ignore?...
# if res.rank_ < 3:
# raise ValueError('singularities in regression')
u = yt - res.predict(X) # residuals
ssqru = (u * u).sum() / float(n)
scalar = 12 if not self.lshort else 4
l_ = int(np.trunc(scalar * np.power(n / 100.0, 0.25)))
ssqrtl = C_tseries_pp_sum(u, n, l_, ssqru)
# define trm vals
n2 = n * n
syt11n = (yt1 * (np.arange(n) + 1)).sum() # sum(yt1*(1:n))
trm1 = n2 * (n2 - 1) * (yt1 ** 2).sum() / 12.0
# R code: # n*sum(yt1*(1:n))^2
trm2 = n * (syt11n ** 2)
# R code: n*(n+1)*sum(yt1*(1:n))*sum(yt1)
trm3 = n * (n + 1) * syt11n * yt1.sum()
trm4 = (n * (n + 1) * (2 * n + 1) * (yt1.sum() ** 2)) / 6.0
dx = trm1 - trm2 + trm3 - trm4
# if self.typ == 'alpha':
alpha = coef[2] # it's the last col...
STAT = n * (alpha - 1) - (n ** 6) / (24.0 * dx) * (ssqrtl - ssqru)
tableipl = np.zeros(self.tablen)
for i in range(self.tablen):
_, pval = approx(self.tableT, self.table[:, i], xout=n, rule=2)
tableipl[i] = pval
# make sure to do 1 - x...
_, interpol = approx(tableipl, self.tablep, xout=STAT, rule=2)
pval = 1 - interpol[0]
# in the R code, here is where the P value warning is tested again...
return pval, pval < self.alpha