smooth.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.

The default specification is ARIMA(0,1,1).

Parameters:

orders (Optional[Dict[str, Any]]) –
Dict-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]]) – Seasonal period(s). E.g. lags=[1, 12] for monthly data. If None, defaults to [1] (non-seasonal).
ar_order (Union[int, List[int]]) – Autoregressive order(s) per seasonal frequency in lags.
i_order (Union[int, List[int]]) – Integration order(s) per seasonal frequency in lags.
ma_order (Union[int, List[int]]) – Moving average order(s) per seasonal frequency in lags.
arima_select (bool) – Whether to perform automatic ARIMA order selection. Equivalent to including "select": True in the orders dict.
constant (Union[bool, float]) – 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]]) – Fixed ARMA parameter values (not estimated). If None, all ARMA parameters are estimated.
initial (Union[str, Dict[str, Any], None]) – Initialisation method or dict of fixed initial values. String options: "backcasting", "optimal", "complete", "two-stage".
initial_X (Optional[NDArray]) – Initial values for regressor coefficients.
ic (Literal['AIC', 'AICc', 'BIC', 'BICc']) – Information criterion for model selection.
loss (Literal['likelihood', 'GPL', 'MSE', 'MAE', 'HAM', 'MSEh', 'MAEh', 'HAMh', 'MSCE', 'MACE', 'CHAM', 'TMSE', 'TMAE', 'THAM', 'GTMSE', 'GTAME', 'GTHAM', 'LASSO', 'RIDGE']) – Loss function for parameter estimation.
h (Optional[int]) – Forecast horizon. Can also be set in predict().
holdout (bool) – Whether to use a holdout sample for validation.
bounds (Literal['usual', 'admissible', 'none']) – Parameter bounds type.
verbose (int) – Verbosity level. 0 = silent.
regressors (Literal['use', 'select', 'adapt']) – 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 the orders dict:

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

References

Svetunkov, I. (2023). Forecasting and Analytics with the Augmented Dynamic Adaptive Model. https://openforecast.org/adam/

Methods

`coefbootstrap`([nsim, size, replace, prob, ...])	Bootstrap the coefficient sampling distribution by refitting subsamples.
`confint`([parm, level, type, bootstrap, ...])	Confidence intervals for the estimated parameters.
`fit`(y[, X])	Fit the ADAM model to time series data.
`multicov`([type, h, nsim])	Covariance matrix of multi-step-ahead forecast errors.
`outlierdummy`([level, type])	Detect outliers and return a matrix of indicator dummy variables.
`plot`([which, level, legend, lowess])	Diagnostic plots for the fitted ADAM model (R: `plot.adam`).
`point_lik`([log])	Per-observation log-likelihood of the fitted model.
`predict`(h[, X, interval, level, side, ...])	Generate forecasts using the fitted ADAM model.
`predict_intervals`(h[, X, levels, side, nsim])	Generate prediction intervals using the fitted ADAM model.
`reapply`([nsim, type, bootstrap, heuristics, ...])	Re-run the model on the in-sample data for `nsim` parameter draws.
`reforecast`([h, X, occurrence, interval, ...])	Produce `h`-step-ahead forecasts via Monte-Carlo reforecasting.
`rmultistep`([h])	Return the (T-h) × h matrix of rolling in-sample multistep forecast errors.
`rstandard`()	Return standardised residuals.
`rstudent`()	Return studentised (leave-one-out) residuals.
`select_best_model`()	Select the best model based on information criteria and update model parameters.
`simulate`([nsim, seed, obs, randomizer])	Re-simulate `obs` observations from the fitted model.
`summary`([level, digits, type])	Generate a coefficient-table summary of the fitted model.
`vcov`([type, bootstrap, heuristics, step_size])	Variance-covariance matrix of the estimated parameters.

Attributes

`actuals`	Return original in-sample data.
`aic`	Return Akaike Information Criterion.
`aicc`	Return corrected Akaike Information Criterion.
`b_value`	$B).
`bic`	Return Bayesian Information Criterion.
`bicc`	Return corrected Bayesian Information Criterion.
`coef`	Return estimated coefficients (parameter vector B).
`coef_names`	Parameter names aligned with `coef` (the B vector).
`constant_value`	$constant).
`data`	$data).
`distribution_`	$distribution).
`error_type`	'A' (additive) or 'M' (multiplicative).
`fisher_information_`	Observed Fisher Information matrix at the estimated parameters.
`fitted`	Return in-sample fitted values.
`holdout_data`	$holdout).
`ic_weights`	$ICw).
`initial_type`	$initialType).
`initial_value`	$initial).
`is_combined`	Return True if model is a combination of multiple models.
`lags_used`	Return the vector of lags used in the model.
`loglik`	Return log-likelihood of the fitted model.
`loss_`	$loss).
`loss_value`	$lossValue).
`measurement`	$measurement).
`model_name`	modelName()).
`model_type`	Return ETS model type code (e.g., 'AAN', 'AAA', 'MAdM').
`models`	$models).
`n_param`	$nParam).
`nobs`	Return number of observations used for fitting.
`nparam`	Return number of estimated parameters.
`om_model`	Fitted occurrence model (OM / OMG / AutoOM), or None.
`orders`	Return ARIMA orders as dict with 'ar', 'i', 'ma' keys.
`persistence_vector`	$persistence).
`phi_`	$phi).
`profile`	$profile).
`residuals`	Return model residuals (errors from fitting).
`scale`	Internal optimisation scale of the error distribution (R: `adam_obj$scale`).
`sigma`	`sigma(adam_obj)`).
`states`	$states).
`time_elapsed`	Time taken to fit the model in seconds.
`transition`	$transition).