MSARIMA and AutoMSARIMA

Multiple Seasonal ARIMA with fixed or automatically-selected orders.

MSARIMA

class smooth.MSARIMA(orders=None, lags=None, ar_order=0, i_order=1, ma_order=1, arima_select=False, constant=False, arma=None, initial='backcasting', initial_X=None, ic='AICc', loss='likelihood', h=None, holdout=False, bounds='usual', verbose=0, regressors='use', **kwargs)

Multiple Seasonal ARIMA in Single Source of Error state space form.

This class wraps ADAM with model="NNN" and distribution="dnorm" hardcoded, providing a clean interface for pure ARIMA (and SARIMA) models without ETS components. It mirrors R’s msarima() function.

The default specification is ARIMA(0,1,1), matching R’s default orders=list(ar=c(0), i=c(1), ma=c(1)).

Parameters:

  • orders (Optional[Dict[str, Any]], default=None) –

    R-style alternative to ar_order/i_order/ma_order. A dict with keys "ar", "i", "ma" (each an int or list of ints) and optionally "select" (bool). Example:

    orders={"ar": [1, 1], "i": [1, 1], "ma": [1, 1]}

    If ar_order, i_order, or ma_order are non-zero they take priority over orders.

  • lags (Optional[List[int]], default=None) – Seasonal period(s). E.g. lags=[1, 12] for monthly data. If None, defaults to [1] (non-seasonal).

  • ar_order (Union[int, List[int]], default=0) – Autoregressive order(s). Matches R default ar=c(0).

  • i_order (Union[int, List[int]], default=1) – Integration order(s). Matches R default i=c(1).

  • ma_order (Union[int, List[int]], default=1) – Moving average order(s). Matches R default ma=c(1).

  • arima_select (bool, default=False) – Whether to perform automatic ARIMA order selection. Equivalent to including "select": True in the orders dict.

  • constant (Union[bool, float], default=False) – Whether to include a constant (drift) term. True estimates it; a numeric value fixes it. The model name will show “with drift” when i_order > 0, or “with constant” otherwise. The fitted value is accessible via model.constant_value.

  • arma (Optional[Dict[str, Any]], default=None) – Fixed ARMA parameter values (not estimated). If None, all ARMA parameters are estimated.

  • initial (Union[str, Dict[str, Any]], default="backcasting") – Initialisation method or dict of fixed initial values. String options: "backcasting", "optimal", "complete", "two-stage".

  • initial_X (Optional[NDArray], default=None) – Initial values for regressor coefficients.

  • ic (Literal["AIC", "AICc", "BIC", "BICc"], default="AICc") – Information criterion for model selection.

  • loss (LOSS_OPTIONS, default="likelihood") – Loss function for parameter estimation.

  • h (Optional[int], default=None) – Forecast horizon. Can also be set in predict().

  • holdout (bool, default=False) – Whether to use a holdout sample for validation.

  • bounds (Literal["usual", "admissible", "none"], default="usual") – Parameter bounds type.

  • verbose (int, default=0) – Verbosity level. 0 = silent.

  • regressors (Literal["use", "select", "adapt"], default="use") – How to handle external regressors.

  • **kwargs – Additional arguments passed to ADAM.

See also

ADAM

Parent class with full attribute documentation.

ES

ETS-only wrapper (counterpart for pure ETS models).

Examples

Default ARIMA(0,1,1):

>>> from smooth import MSARIMA
>>> import numpy as np
>>> y = np.cumsum(np.random.randn(60)) + 100.0
>>> model = MSARIMA()
>>> model.fit(y)

ARIMA(1,1,1) with drift:

>>> model = MSARIMA(ar_order=1, i_order=1, ma_order=1, constant=True)
>>> model.fit(y)
>>> print(f"Drift: {model.constant_value:.4f}")

SARIMA(1,1,1)(1,1,1)[12] via R-style dict:

>>> model = MSARIMA(
...     orders={"ar": [1, 1], "i": [1, 1], "ma": [1, 1]},
...     lags=[1, 12],
... )
>>> model.fit(y)

References

MSARIMA fits a pure ARIMA model (no ETS components) with explicitly specified orders.

from smooth import MSARIMA

# ARIMA(0,1,1) — default
model = MSARIMA()
model.fit(y)
print(model)

# SARIMA(1,1,1)(1,1,1)[12]
model = MSARIMA(
    orders={"ar": [1, 1], "i": [1, 1], "ma": [1, 1]},
    lags=[1, 12],
)
model.fit(y)
fc = model.predict(h=12, interval="prediction", level=0.95)

# With drift term
model = MSARIMA(ar_order=1, i_order=1, ma_order=1, constant=True)
model.fit(y)

AutoMSARIMA

class smooth.AutoMSARIMA(lags=None, ar_order=[3, 3], i_order=[2, 1], ma_order=[3, 3], orders=None, constant=False, initial='backcasting', initial_X=None, ic='AICc', loss='likelihood', h=None, holdout=False, bounds='usual', regressors='use', outliers='ignore', level=0.99, verbose=0, **kwargs)

Automatic Multiple Seasonal ARIMA with order selection.

Wraps AutoADAM with model="NNN" and distribution="dnorm" fixed, providing automatic ARIMA order selection for pure ARIMA (and SARIMA) models without ETS components. Mirrors R’s auto.msarima().

Parameters:

  • lags (Optional[List[int]], default=None) – Seasonal period(s). E.g. lags=[1, 12] for monthly data. If None, defaults to [1] (non-seasonal).

  • ar_order (Union[int, List[int]], default=[3, 3]) – Maximum AR order(s) per lag level for selection. Matches R’s orders=list(ar=c(3,3)).

  • i_order (Union[int, List[int]], default=[2, 1]) – Maximum integration order(s) per lag level. Matches R’s orders=list(i=c(2,1)).

  • ma_order (Union[int, List[int]], default=[3, 3]) – Maximum MA order(s) per lag level for selection. Matches R’s orders=list(ma=c(3,3)).

  • orders (Optional[Dict[str, Any]], default=None) – R-style alternative to scalar max orders. A dict with keys "ar", "i", "ma" (each an int or list). When provided, ar_order/i_order/ma_order are ignored.

  • constant (Union[bool, float], default=False) – Whether to include a constant (drift) term.

  • initial (Union[str, Dict[str, Any]], default="backcasting") – Initialisation method or dict of fixed initial values. String options: "backcasting", "optimal", "complete", "two-stage".

  • initial_X (Optional[NDArray], default=None) – Initial values for regressor coefficients (equivalent to R’s initialX).

  • ic (Literal["AIC", "AICc", "BIC", "BICc"], default="AICc") – Information criterion for model comparison during selection.

  • loss (LOSS_OPTIONS, default="likelihood") – Loss function for parameter estimation.

  • h (Optional[int], default=None) – Forecast horizon. Can also be set in predict().

  • holdout (bool, default=False) – Whether to use a holdout sample.

  • bounds (Literal["usual", "admissible", "none"], default="usual") – Parameter bounds type.

  • regressors (Literal["use", "select", "adapt"], default="use") – How to handle external regressors.

  • outliers (Literal["ignore", "use", "select"], default="ignore") – Outlier handling mode (see AutoADAM).

  • level (float, default=0.99) – Confidence level for outlier detection.

  • verbose (int, default=0) – Verbosity level. 0 = silent.

  • **kwargs – Additional arguments forwarded to AutoADAM.

See also

AutoADAM

Full automatic model selection.

MSARIMA

Fixed-order MSARIMA wrapper.

Examples

Automatic non-seasonal ARIMA:

>>> from smooth import AutoMSARIMA
>>> import numpy as np
>>> y = np.cumsum(np.random.randn(100)) + 100.0
>>> model = AutoMSARIMA(lags=[1])
>>> model.fit(y)
>>> print(model)

Automatic seasonal ARIMA for monthly data:

>>> model = AutoMSARIMA(lags=[1, 12])
>>> model.fit(y)
>>> fc = model.predict(h=24)

References

AutoMSARIMA wraps AutoADAM with model="NNN" and distribution="dnorm" fixed, mirroring R’s auto.msarima(). The parameters model, distribution, and arima_select are fixed and cannot be overridden — passing them raises ValueError.

from smooth import AutoMSARIMA

# Automatic seasonal ARIMA
model = AutoMSARIMA(lags=[1, 12])
model.fit(y)
print(model)   # AutoMSARIMA: ARIMA([p,P],[d,D],[q,Q])

# Reduce search space for speed
model = AutoMSARIMA(
    lags=[1, 12],
    ar_order=[2, 1],
    i_order=[2, 1],
    ma_order=[2, 1],
)
model.fit(y)
fc = model.predict(h=24)

# With external regressors
model = AutoMSARIMA(lags=[1, 12], regressors="select")
model.fit(y, X=X)

See Also

  • AutoADAM — Full automatic ETS + ARIMA + distribution selection

  • ADAM — Base unified framework

  • msdecompose — Multiple seasonal decomposition