Automatically discover the optimal order for an ARIMA model.
The auto-ARIMA process seeks to identify the most optimal
parameters for an
ARIMA
model, settling on a single fitted ARIMA model.
This process is based on the commonly-used R function,
forecast::auto.arima
[3].
Auto-ARIMA works by conducting differencing tests (i.e.,
Kwiatkowski–Phillips–Schmidt–Shin, Augmented Dickey-Fuller or
Phillips–Perron) to determine the order of differencing,
d
, and then
fitting models within ranges of defined
start_p
,
max_p
,
start_q
,
max_q
ranges. If the
seasonal
optional is enabled,
auto-ARIMA also seeks to identify the optimal
P
and
Q
hyper-
parameters after conducting the Canova-Hansen to determine the optimal
order of seasonal differencing,
D
.
In order to find the best model, auto-ARIMA optimizes for a given
information_criterion
, one of (‘aic’, ‘aicc’, ‘bic’, ‘hqic’, ‘oob’)
(Akaike Information Criterion, Corrected Akaike Information Criterion,
Bayesian Information Criterion, Hannan-Quinn Information Criterion, or
“out of bag”–for validation scoring–respectively) and returns the ARIMA
which minimizes the value.
Note that due to stationarity issues, auto-ARIMA might not find a
suitable model that will converge. If this is the case, a
ValueError
will be thrown suggesting stationarity-inducing measures be taken prior
to re-fitting or that a new range of
order
values be selected. Non-
stepwise (i.e., essentially a grid search) selection can be slow,
especially for seasonal data. Stepwise algorithm is outlined in Hyndman and
Khandakar (2008).
Parameters:
y
: array-like or iterable, shape=(n_samples,)
The time-series to which to fit the
ARIMA
estimator. This may
either be a Pandas
Series
object (statsmodels can internally
use the dates in the index), or a numpy array. This should be a
one-dimensional array of floats, and should not contain any
np.nan
or
np.inf
values.
X
: array-like, shape=[n_obs, n_vars], optional (default=None)
An optional 2-d array of exogenous variables. If provided, these
variables are used as additional features in the regression
operation. This should not include a constant or trend. Note that
if an
ARIMA
is fit on exogenous features, it must be provided
exogenous features for making predictions.
start_p
: int, optional (default=2)
The starting value of
p
, the order (or number of time lags)
of the auto-regressive (“AR”) model. Must be a positive integer.
d
: int, optional (default=None)
The order of first-differencing. If None (by default), the value
will automatically be selected based on the results of the
test
(i.e., either the Kwiatkowski–Phillips–Schmidt–Shin, Augmented
Dickey-Fuller or the Phillips–Perron test will be conducted to find
the most probable value). Must be a positive integer or None. Note
that if
d
is None, the runtime could be significantly longer.
start_q
: int, optional (default=2)
The starting value of
q
, the order of the moving-average
(“MA”) model. Must be a positive integer.
max_p
: int, optional (default=5)
The maximum value of
p
, inclusive. Must be a positive integer
greater than or equal to
start_p
.
max_d
: int, optional (default=2)
The maximum value of
d
, or the maximum number of non-seasonal
differences. Must be a positive integer greater than or equal to
d
.
max_q
: int, optional (default=5)
The maximum value of
q
, inclusive. Must be a positive integer
greater than
start_q
.
start_P
: int, optional (default=1)
The starting value of
P
, the order of the auto-regressive portion
of the seasonal model.
D
: int, optional (default=None)
The order of the seasonal differencing. If None (by default, the value
will automatically be selected based on the results of the
seasonal_test
. Must be a positive integer or None.
start_Q
: int, optional (default=1)
The starting value of
Q
, the order of the moving-average portion
of the seasonal model.
max_P
: int, optional (default=2)
The maximum value of
P
, inclusive. Must be a positive integer
greater than
start_P
.
max_D
: int, optional (default=1)
The maximum value of
D
. Must be a positive integer greater
than
D
.
max_Q
: int, optional (default=2)
The maximum value of
Q
, inclusive. Must be a positive integer
greater than
start_Q
.
max_order
: int, optional (default=5)
Maximum value of p+q+P+Q if model selection is not stepwise.
If the sum of
p
and
q
is >=
max_order
, a model will
not
be fit with those parameters, but will progress to the next
combination. Default is 5. If
max_order
is None, it means there
are no constraints on maximum order.
m
: int, optional (default=1)
The period for seasonal differencing,
m
refers to the number of
periods in each season. For example,
m
is 4 for quarterly data, 12
for monthly data, or 1 for annual (non-seasonal) data. Default is 1.
Note that if
m
== 1 (i.e., is non-seasonal),
seasonal
will be
set to False. For more information on setting this parameter, see
Setting m
.
seasonal
: bool, optional (default=True)
Whether to fit a seasonal ARIMA. Default is True. Note that if
seasonal
is True and
m
== 1,
seasonal
will be set to False.
stationary
: bool, optional (default=False)
Whether the time-series is stationary and
d
should be set to zero.
information_criterion
: str, optional (default=’aic’)
The information criterion used to select the best ARIMA model. One of
pmdarima.arima.auto_arima.VALID_CRITERIA
, (‘aic’, ‘bic’, ‘hqic’,
‘oob’).
alpha
: float, optional (default=0.05)
Level of the test for testing significance.
test
: str, optional (default=’kpss’)
Type of unit root test to use in order to detect stationarity if
stationary
is False and
d
is None. Default is ‘kpss’
(Kwiatkowski–Phillips–Schmidt–Shin).
seasonal_test
: str, optional (default=’ocsb’)
This determines which seasonal unit root test is used if
seasonal
is True and
D
is None. Default is ‘OCSB’.
stepwise
: bool, optional (default=True)
Whether to use the stepwise algorithm outlined in Hyndman and Khandakar
(2008) to identify the optimal model parameters. The stepwise algorithm
can be significantly faster than fitting all (or a
random
subset
of) hyper-parameter combinations and is less likely to over-fit
the model.
n_jobs
: int, optional (default=1)
The number of models to fit in parallel in the case of a grid search
(
stepwise=False
). Default is 1, but -1 can be used to designate
“as many as possible”.
start_params
: array-like, optional (default=None)
Starting parameters for
ARMA(p,q)
. If None, the default is given
by
ARMA._fit_start_params
.
method
: str, optional (default=’lbfgs’)
The
method
determines which solver from
scipy.optimize
is used, and it can be chosen from among the following strings:
‘newton’ for Newton-Raphson
‘nm’ for Nelder-Mead
‘bfgs’ for Broyden-Fletcher-Goldfarb-Shanno (BFGS)
‘lbfgs’ for limited-memory BFGS with optional box constraints
‘powell’ for modified Powell’s method
‘cg’ for conjugate gradient
‘ncg’ for Newton-conjugate gradient
‘basinhopping’ for global basin-hopping solver
The explicit arguments in
fit
are passed to the solver,
with the exception of the basin-hopping solver. Each
solver has several optional arguments that are not the same across
solvers. These can be passed as
**
fit_kwargs
trend
: str or None, optional (default=None)
The trend parameter. If
with_intercept
is True,
trend
will be
used. If
with_intercept
is False, the trend will be set to a no-
intercept value.
maxiter
: int, optional (default=50)
The maximum number of function evaluations. Default is 50.
offset_test_args
: dict, optional (default=None)
The args to pass to the constructor of the offset (
d
) test. See
pmdarima.arima.stationarity
for more details.
seasonal_test_args
: dict, optional (default=None)
The args to pass to the constructor of the seasonal offset (
D
)
test. See
pmdarima.arima.seasonality
for more details. Examples of
valid kwargs will vary based on the test. For the
OCSBTest
(default) they include:
‘lag_method’
‘max_lag’
suppress_warnings
: bool, optional (default=True)
Many warnings might be thrown inside of statsmodels. If
suppress_warnings
is True, all of the warnings coming from
ARIMA
will be squelched. Note that this will not suppress
UserWarnings created by bad argument combinations.
error_action
: str, optional (default=’warn’)
If unable to fit an
ARIMA
for whatever reason, this controls the
error-handling behavior. Model fits can fail for linear algebra errors,
convergence errors, or any number of problems related to stationarity
or input data.
‘warn’: Warns when an error is encountered (default)
‘raise’: Raises when an error is encountered
‘ignore’: Ignores errors (not recommended)
Whether to print status on the fits. A value of False will print no
debugging information. A value of True will print some. Integer values
exceeding 1 will print increasing amounts of debug information at each
random
: bool, optional (default=False)
Similar to grid searches,
auto_arima
provides the capability to
perform a “random search” over a hyper-parameter space. If
random
is True, rather than perform an exhaustive search or
stepwise
search, only
n_fits
ARIMA models will be fit (
stepwise
must be
False for this option to do anything).
random_state
: int, long or numpy
RandomState
, optional (default=None)
The PRNG for when
random=True
. Ensures replicable testing and
results.
n_fits
: int, optional (default=10)
If
random
is True and a “random search” is going to be performed,
n_fits
is the number of ARIMA models to be fit.
return_valid_fits
: bool, optional (default=False)
If True, will return all valid ARIMA fits in a list. If False (by
default), will only return the best fit.
out_of_sample_size
: int, optional (default=0)
The
ARIMA
class can fit only a portion of the data if specified,
in order to retain an “out of bag” sample score. This is the
number of examples from the tail of the time series to hold out
and use as validation examples. The model will not be fit on these
samples, but the observations will be added into the model’s
endog
and
exog
arrays so that future forecast values originate from the
end of the endogenous vector.
For instance:
y = [0, 1, 2, 3, 4, 5, 6]
out_of_sample_size = 2
> Fit on: [0, 1, 2, 3, 4]
> Score on: [5, 6]
> Append [5, 6] to end of self.arima_res_.data.endog values
scoring : str, optional (default=’mse’)
If performing validation (i.e., if out_of_sample_size > 0), the
metric to use for scoring the out-of-sample data. One of (‘mse’, ‘mae’)
scoring_args : dict, optional (default=None)
A dictionary of key-word arguments to be passed to the scoring
metric.
with_intercept : bool or str, optional (default=”auto”)
Whether to include an intercept term. Default is “auto” which behaves
like True until a point in the search where the sum of differencing
terms will explicitly set it to True or False.
sarimax_kwargs : dict or None, optional (default=None)
Keyword arguments to pass to the ARIMA constructor.
**fit_args : dict, optional (default=None)
A dictionary of keyword arguments to pass to the ARIMA.fit()
method.
