CES

class CES(model='N', season_length=1, alpha_real=None, alpha_imag=None, beta_real=None, beta_imag=None, initial_level=None, initial_level_imag=None, initial_seasonal_real=None, initial_seasonal_imag=None, alpha_real_bounds=(0.01, 1.8), alpha_imag_bounds=(0.01, 1.9), beta_real_bounds=(0.01, 1.5), beta_imag_bounds=(0.01, 1.5), initial_level_bounds=(-10000000000.0, 10000000000.0), max_iter=1000, tol=0.0001)

Bases: BaseForecaster, IterativeForecastingMixin

Complex Exponential Smoothing (CES) forecaster.

Implements the CES family proposed by Svetunkov and Kourentzes [1]. CES extends exponential smoothing by using complex-valued smoothing parameters and two-component (real + imaginary “correction”) state.

Four model variants are implemented:

• model="N" (non-seasonal): a single two-component state \((\ell_{1,t}, \ell_{2,t})\) with complex smoothing parameter \(\tilde{\alpha} = \alpha_0 + i\,\alpha_1\). The recurrence is \(\hat{y}_t = \ell_{1,t-1}\), \(\ell_{1,t} = \ell_{1,t-1} + (\alpha_1-1)\,\ell_{2,t-1} + (\alpha_0-\alpha_1)\,\varepsilon_t\), \(\ell_{2,t} = \ell_{1,t-1} + (1-\alpha_0)\,\ell_{2,t-1} + (\alpha_0+\alpha_1)\,\varepsilon_t\). The persistence vector (\alpha_0 - \alpha_1, \alpha_0 + \alpha_1) matches the smooth/ADAM and StatsForecast AutoCES parameterisation so fitted alphas transfer directly between libraries.

• model="S" (simple seasonal): a two-component state is updated at seasonal lag season_length. The one-step forecast uses the state from the same position in the previous seasonal cycle.

• model="P" (partial seasonal): combines the non-seasonal two-component state with a one-component additive seasonal state updated at seasonal lag season_length.

• model="F" (full seasonal): adds a seasonal complex-state pair \((l_{1,t}, c_{1,t})\) lagged by season_length and a second complex smoothing parameter \(\tilde{\beta} = \beta_0 + i\,\beta_1\). The observation is \(\hat{y}_t = l_{0,t-1} + l_{1,t-m}\), and the seasonal state has its own CES-style update driven by the same error series.

The estimator uses deterministic initial states and a backfitting pass following StatsForecast’s AutoCES implementation, so aeon and StatsForecast forecasts can be compared directly for N/S/P/F and AutoCES model="Z" style selection.

Parameters:

model{“N”, “S”, “P”, “F”, “none”, “simple”, “partial”, “full”}

Model code. "N" is the default. Seasonal models require season_length >= 2.

season_lengthint, default=1

Seasonal period. 1 means no seasonality. Required > 1 for any seasonal model.

alpha_real, alpha_imagfloat or None, default=None

Components of the non-seasonal complex smoothing parameter. If None they are estimated.

beta_real, beta_imagfloat or None, default=None

Components of the seasonal smoothing parameter. beta_real is used by "P" and "F"; beta_imag is used only by "F". If None the relevant components are estimated.

initial_levelfloat or None, default=None

Initial value for the non-seasonal real-state component ell_{1,0} / l_{0,0}. If None it is seeded deterministically from the data.

initial_level_imagfloat or None, default=None

Initial value for the non-seasonal imaginary-state component ell_{2,0} / c_{0,0}. If None it is seeded deterministically from the data.

initial_seasonal_real, initial_seasonal_imagarray-like or None

Length-season_length initial seasonal state. If None the seasonal seed is computed deterministically from the data (per- position mean minus overall mean; correction set to zero). The seasonal seed is deterministic and is not optimised.

alpha_real_boundstuple of float, default=(0.01, 1.8)

Box bounds for the optimisation of alpha_real. Match StatsForecast’s AutoCES non-seasonal optimiser bounds. These are practical optimisation bounds, not a complete CES admissibility region.

alpha_imag_boundstuple of float, default=(0.01, 1.9)

Box bounds for the optimisation of alpha_imag.

beta_real_bounds, beta_imag_boundstuple of float

Box bounds for the seasonal smoothing parameters; default to StatsForecast’s seasonal CES bounds.

initial_level_boundstuple of float, default=(-1e10, 1e10)

Box bounds for both real and imaginary initial-state components.

max_iterint, default=1000

Maximum number of optimiser iterations.

tolfloat, default=1e-4

Optimiser convergence tolerance.

Attributes:

model_str

Canonical model code actually used ("N", "S", "P", or "F").

season_length_int

Seasonal period used (1 for non-seasonal).

alpha_real_, alpha_imag_float

Fitted non-seasonal smoothing parameters.

complex_alpha_complex

Convenience alpha_real_ + 1j * alpha_imag_.

beta_real_, beta_imag_float

Fitted seasonal smoothing parameters (NaN for non-seasonal).

complex_beta_complex

Convenience beta_real_ + 1j * beta_imag_ (NaN for non-seasonal).

initial_level_, initial_level_imag_float

Fitted initial non-seasonal state components.

initial_seasonal_real_, initial_seasonal_imag_np.ndarray or None

Seasonal state seed (length season_length_). None for non-seasonal.

level_real_, level_imag_float

Final non-seasonal state components after fitting.

seasonal_real_, seasonal_imag_np.ndarray or None

Final seasonal-state buffers after fitting (length season_length_). None for non-seasonal.

fitted_values_np.ndarray

In-sample one-step-ahead fitted values.

residuals_np.ndarray

y - fitted_values_.

forecast_float

Stored one-step-ahead forecast.

sse_float

Sum of squared in-sample residuals.

n_params_int

Number of parameters used for IC calculations.

n_free_params_int

Number of smoothing parameters optimised by the fit.

optimization_success_bool

Optimiser success flag.

optimization_message_str

Optimiser termination message.

n_iter_int

Optimiser iteration count.

Notes

Capabilities

Missing Values	No
Multithreading	No
Univariate	Yes
Multivariate	No
Horizon	No
Exogenous	No

References

[1]

Svetunkov, I. and Kourentzes, N. (2018). Complex exponential smoothing for time series forecasting. Naval Research Logistics, 65(8), 685-704.

Examples

>>> import numpy as np
>>> from aeon.forecasting.stats import CES
>>> y = np.array([2.1, 2.4, 2.8, 3.0, 3.6, 4.1, 4.4, 4.9, 5.3, 5.9])
>>> f = CES()
>>> f.iterative_forecast(y, prediction_horizon=2).shape
(2,)

Methods

clone([random_state])	Obtain a clone of the object with the same hyperparameters.
fit(y[, exog, axis])	Fit forecaster to series y.
forecast(y[, exog, axis])	Forecast the next horizon steps ahead of y.
get_class_tag(tag_name[, raise_error, ...])	Get tag value from estimator class (only class tags).
get_class_tags()	Get class tags from estimator class and all its parent classes.
get_fitted_params([deep])	Get fitted parameters.
get_params([deep])	Get parameters for this estimator.
get_tag(tag_name[, raise_error, ...])	Get tag value from estimator class.
get_tags()	Get tags from estimator.
iterative_forecast(y, prediction_horizon[, ...])	Fit CES on y and produce prediction_horizon step forecasts.
predict(y[, exog, axis])	Predict the next horizon steps ahead.
reset([keep])	Reset the object to a clean post-init state.
set_fit_request(*[, axis, exog])	Configure whether metadata should be requested to be passed to the fit method.
set_params(**params)	Set the parameters of this estimator.
set_predict_request(*[, axis, exog])	Configure whether metadata should be requested to be passed to the predict method.
set_tags(**tag_dict)	Set dynamic tags to given values.

clone(random_state=None)

Obtain a clone of the object with the same hyperparameters.

A clone is a different object without shared references, in post-init state. This function is equivalent to returning sklearn.clone of self. Equal in value to type(self)(**self.get_params(deep=False)).

Parameters:

random_stateint, RandomState instance, or None, default=None

Sets the random state of the clone. If None, the random state is not set. If int, random_state is the seed used by the random number generator. If RandomState instance, random_state is the random number generator.

Returns:

estimatorobject

Instance of type(self), clone of self (see above)

fit(y, exog=None, axis=1)

Fit forecaster to series y.

Fit a forecaster to predict self.horizon steps ahead using y.

Parameters:

ynp.ndarray

A time series on which to learn a forecaster to predict horizon ahead.

exognp.ndarray, default =None

Optional target-time exogenous values aligned with y.

Returns:

self

Fitted BaseForecaster.

forecast(y, exog=None, axis=1) → float

Forecast the next horizon steps ahead of y.

By default this is simply fit followed by predict.

Parameters:

ynp.ndarray

A time series to predict the next horizon value for. Must be of shape (n_channels, n_timepoints) if a multivariate time series.

exognp.ndarray, default =None

Optional target-time exogenous values aligned with y.

Returns:

float

single prediction self.horizon steps ahead of y.

classmethod get_class_tag(tag_name, raise_error=True, tag_value_default=None)

Get tag value from estimator class (only class tags).

Parameters:

tag_namestr

Name of tag value.

raise_errorbool, default=True

Whether a ValueError is raised when the tag is not found.

tag_value_defaultany type, default=None

Default/fallback value if tag is not found and error is not raised.

Returns:

tag_value

Value of the tag_name tag in cls. If not found, returns an error if raise_error is True, otherwise it returns tag_value_default.

Raises:

ValueError

if raise_error is True and tag_name is not in self.get_tags().keys()

Examples

>>> from aeon.classification import DummyClassifier
>>> DummyClassifier.get_class_tag("capability:multivariate")
True

classmethod get_class_tags()

Get class tags from estimator class and all its parent classes.

Returns:

collected_tagsdict

Dictionary of tag name and tag value pairs. Collected from _tags class attribute via nested inheritance. These are not overridden by dynamic tags set by set_tags or class __init__ calls.

get_fitted_params(deep=True)

Get fitted parameters.

State required:

Requires state to be “fitted”.

Parameters:

deepbool, default=True

If True, will return the fitted parameters for this estimator and contained subobjects that are estimators.

Returns:

fitted_paramsdict

Fitted parameter names mapped to their values.

get_params(deep=True)

Get parameters for this estimator.

Parameters:

deepbool, default=True

If True, will return the parameters for this estimator and contained subobjects that are estimators.

Returns:

paramsdict

Parameter names mapped to their values.

get_tag(tag_name, raise_error=True, tag_value_default=None)

Get tag value from estimator class.

Includes dynamic and overridden tags.

Parameters:

tag_namestr

Name of tag to be retrieved.

raise_errorbool, default=True

Whether a ValueError is raised when the tag is not found.

tag_value_defaultany type, default=None

Default/fallback value if tag is not found and error is not raised.

Returns:

tag_value

Value of the tag_name tag in self. If not found, returns an error if raise_error is True, otherwise it returns tag_value_default.

Raises:

ValueError

if raise_error is True and tag_name is not in self.get_tags().keys()

Examples

>>> from aeon.classification import DummyClassifier
>>> d = DummyClassifier()
>>> d.get_tag("capability:multivariate")
True

get_tags()

Get tags from estimator.

Includes dynamic and overridden tags.

Returns:

collected_tagsdict

Dictionary of tag name and tag value pairs. Collected from _tags class attribute via nested inheritance and then any overridden and new tags from __init__ or set_tags.

iterative_forecast(y, prediction_horizon, exog=None, *, future_exog=None)

Fit CES on y and produce prediction_horizon step forecasts.

exog and future_exog are accepted for signature compatibility with IterativeForecastingMixin only; passing either raises NotImplementedError.

predict(y, exog=None, axis=1) → float

Predict the next horizon steps ahead.

Parameters:

ynp.ndarray

A time series to predict the next horizon value for.

exognp.ndarray, default =None

Optional exogenous values for the prediction target. For models fitted with exogenous variables, this should contain the exogenous values needed to make the next prediction.

Returns:

float

single prediction self.horizon steps ahead of y.

reset(keep=None)

Reset the object to a clean post-init state.

After a self.reset() call, self is equal or similar in value to type(self)(**self.get_params(deep=False)), assuming no other attributes were kept using keep.

Detailed behaviour:

removes any object attributes, except:

hyper-parameters (arguments of __init__) object attributes containing double-underscores, i.e., the string “__”

runs __init__ with current values of hyperparameters (result of get_params)

Not affected by the reset are:

object attributes containing double-underscores class and object methods, class attributes any attributes specified in the keep argument

Parameters:

keepNone, str, or list of str, default=None

If None, all attributes are removed except hyperparameters. If str, only the attribute with this name is kept. If list of str, only the attributes with these names are kept.

Returns:

selfobject

Reference to self.

Raises:

TypeError

If ‘keep’ is not a string or a list of strings.

set_fit_request(*, axis: bool | None | str = '$UNCHANGED$', exog: bool | None | str = '$UNCHANGED$') → CES

Configure whether metadata should be requested to be passed to the fit method.

Note that this method is only relevant when this estimator is used as a sub-estimator within a meta-estimator and metadata routing is enabled with enable_metadata_routing=True (see sklearn.set_config). Please check the User Guide on how the routing mechanism works.

The options for each parameter are:

• True: metadata is requested, and passed to fit if provided. The request is ignored if metadata is not provided.

• False: metadata is not requested and the meta-estimator will not pass it to fit.

• None: metadata is not requested, and the meta-estimator will raise an error if the user provides it.

• str: metadata should be passed to the meta-estimator with this given alias instead of the original name.

The default (sklearn.utils.metadata_routing.UNCHANGED) retains the existing request. This allows you to change the request for some parameters and not others.

Added in version 1.3.

Parameters:

axisstr, True, False, or None, default=sklearn.utils.metadata_routing.UNCHANGED

Metadata routing for axis parameter in fit.

exogstr, True, False, or None, default=sklearn.utils.metadata_routing.UNCHANGED

Metadata routing for exog parameter in fit.

Returns:

selfobject

The updated object.

set_params(**params)

Set the parameters of this estimator.

The method works on simple estimators as well as on nested objects (such as Pipeline). The latter have parameters of the form <component>__<parameter> so that it’s possible to update each component of a nested object.

Parameters:

**paramsdict

Estimator parameters.

Returns:

selfestimator instance

Estimator instance.

set_predict_request(*, axis: bool | None | str = '$UNCHANGED$', exog: bool | None | str = '$UNCHANGED$') → CES

Configure whether metadata should be requested to be passed to the predict method.

Note that this method is only relevant when this estimator is used as a sub-estimator within a meta-estimator and metadata routing is enabled with enable_metadata_routing=True (see sklearn.set_config). Please check the User Guide on how the routing mechanism works.

The options for each parameter are:

• True: metadata is requested, and passed to predict if provided. The request is ignored if metadata is not provided.

• False: metadata is not requested and the meta-estimator will not pass it to predict.

• None: metadata is not requested, and the meta-estimator will raise an error if the user provides it.

• str: metadata should be passed to the meta-estimator with this given alias instead of the original name.

The default (sklearn.utils.metadata_routing.UNCHANGED) retains the existing request. This allows you to change the request for some parameters and not others.

Added in version 1.3.

Parameters:

axisstr, True, False, or None, default=sklearn.utils.metadata_routing.UNCHANGED

Metadata routing for axis parameter in predict.

exogstr, True, False, or None, default=sklearn.utils.metadata_routing.UNCHANGED

Metadata routing for exog parameter in predict.

Returns:

selfobject

The updated object.

set_tags(**tag_dict)

Set dynamic tags to given values.

Parameters:

**tag_dictdict

Dictionary of tag name and tag value pairs.

Returns:

selfobject

Reference to self.