ES

ES - Exponential Smoothing

ES (Exponential Smoothing) is a wrapper around ADAM that provides a simplified interface for pure ETS (Error-Trend-Seasonal) models without ARIMA components.

Overview

The ES class implements the classical taxonomy of 30 ETS models in Single Source of Error (SSOE) state-space form. It uses normal distribution for errors by default and provides automatic model selection.

Mathematical Form

See ADAM for more details.

Model Taxonomy

All 30 ETS models are supported:

Trend \ Seasonal	N (None)	A (Additive)	M (Multiplicative)
N (None)	ANN, MNN	ANA, MNA	ANM, MNM
A (Additive)	AAN, MAN	AAA, MAA	AAM, MAM
Ad (Additive Damped)	AAdN, MAdN	AAdA, MAdA	AAdM, MAdM
M (Multiplicative)	AMN, MMN	AMA, MMA	AMM, MMM
Md (Multiplicative Damped)	AMdN, MMdN	AMdA, MMdA	AMdM, MMdM

Model Specification

See ADAM for more details.

Python Usage

from smooth import ES
import numpy as np

y = np.array([112, 118, 132, 129, 121, 135, 148, 148, 136, 119, 104, 118])

# Simple Exponential Smoothing
model = ES(model="ANN")
model.fit(y)
forecasts = model.predict(h=6)

# Holt-Winters with automatic selection
model = ES(model="ZXZ", lags=[12])
model.fit(y)
print(f"Selected model: {model.model_type_dict['model']}")

# Damped trend model
model = ES(model="AAdN")
model.fit(y)
print(f"Damping parameter: {model.phi_:.3f}")

R Usage

library(smooth)

# Simple Exponential Smoothing
model <- es(y, model="ANN")

# Automatic model selection
model <- es(y, model="ZZZ", lags=12)

# Holt-Winters additive
model <- es(AirPassengers, model="AAA", h=18, holdout=TRUE)
forecast(model, h=18)

# With holdout for validation
model <- es(y, model="ZXZ", h=12, holdout=TRUE)

With External Regressors

Note: Not yet supported in Python

# ETSX model
es(y, model="ZXZ", xreg=X)

# ETSX with regressors selection
es(y, model="ZXZ", xreg=X, regressors="select")

Parameters

Parameter	Type (R)	Type (Python)	Default	Description
model	character/vector	str/List[str]	"ZXZ"	ETS model specification
lags	numeric vector	List[int]/None	frequency(y)	Seasonal period(s)
persistence	list/vector	Dict[str, float]/None	NULL	Fixed smoothing parameters
phi	numeric	float/None	NULL	Damping parameter
initial	character/list	str/Dict/None	"backcasting"	Initialization method
ic	character	str	"AICc"	Information criterion
loss	character	str	"likelihood"	Loss function
bounds	character	str	"usual"	Parameter bounds
h	integer	int/None	0	Forecast horizon
holdout	logical	bool	FALSE	Use holdout validation

Fitted Attributes

Element (R)	Element (Python)	Type (R)	Type (Python)	Description
model	model_type_dict['model']	character	str	Name of the fitted model
persistence (alpha)	persistence_level_	numeric	float	Level smoothing parameter (α)
persistence (beta)	persistence_trend_	numeric	float	Trend smoothing parameter (β)
persistence (gamma)	persistence_seasonal_	numeric	List[float]	Seasonal smoothing parameter(s) (γ)
phi	phi_	numeric	float	Damping parameter (φ)
initial	initial_states_	list	NDArray	Initial state values
-	model_type_dict	-	Dict	Complete model specification
-	ic_selection	-	float	Information criterion value
states	adam_created['mat_vt']	matrix	NDArray	State matrix
fitted	prepared_model['y_fitted']	vector	NDArray	Fitted values
residuals	prepared_model['residuals']	vector	NDArray	Residuals
forecast	predict() result	vector	NDArray	Point forecasts

See ADAM for complete list of fitted attributes.

Relationship to ADAM

ES is a simplified interface to ADAM with:

• No ARIMA components (orders=NULL)
• Normal distribution for errors (distribution="dnorm")
• Focus on pure ETS models

For models combining ETS and ARIMA, use ADAM directly.

References

• Svetunkov, I. (2023). Forecasting and Analytics with the Augmented Dynamic Adaptive Model (ADAM). Online book: https://openforecast.org/adam/
• Svetunkov, I. (2023). Smooth forecasting with the smooth package in R. arXiv:2301.01790
• Hyndman, R.J., et al. (2008). Forecasting with Exponential Smoothing: The State Space Approach. Springer.

See Also

Related Functions

• ADAM - Full ADAM model with ARIMA support
• CES - Complex Exponential Smoothing
• SMA - Simple Moving Average

Parameter Documentation

• Model-Specification - Model string notation and selection
• Loss-Functions - Loss function options
• Initialisation - State initialization methods
• Persistence - Smoothing parameters
• Bounds - Parameter restrictions