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Ivan Svetunkov edited this page Jun 16, 2026 · 10 revisions

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.

Function signatures

R

es(y, model = "ZXZ", lags = c(frequency(y)),
   persistence = NULL, phi = NULL,
   initial = c("backcasting", "optimal", "two-stage", "complete"),
   initialSeason = NULL,
   ic = c("AICc", "AIC", "BIC", "BICc"),
   loss = c("likelihood", "MSE", "MAE", "HAM",
            "MSEh", "TMSE", "GTMSE", "MSCE", "GPL"),
   h = 0, holdout = FALSE,
   bounds = c("usual", "admissible", "none"),
   silent = TRUE,
   xreg = NULL, regressors = c("use", "select"),
   initialX = NULL, ...)

Python

class ES(ADAM):
    def __init__(
        self,
        model: str | list[str] = "ZXZ",
        lags: list[int] | None = None,
        persistence: dict | None = None,
        phi: float | None = None,
        initial: str | dict | None = "backcasting",
        initial_season: NDArray | None = None,
        ic: Literal["AIC", "AICc", "BIC", "BICc"] = "AICc",
        loss: str = "likelihood",
        h: int | None = None,
        holdout: bool = False,
        bounds: Literal["usual", "admissible", "none"] = "usual",
        verbose: int = 0,
        regressors: Literal["use", "select"] = "use",
        initial_X: NDArray | None = None,
        **kwargs,
    ) -> None: ...

ES rejects ADAM-only kwargs (ar_order, i_order, ma_order, orders, arima_select, constant, ets) with a ValueError. Use ADAM for models with ARIMA components or a constant term.

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.

ES always uses the conventional ETS formulation (Hyndman et al. 2008) — the ets parameter is not exposed. If you need the "adam" ETS reformulation, use ADAM(ets="adam") directly.

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_name}")  # e.g., "ETS(AAA)"
print(f"ETS type: {model.model_type}")  # e.g., "AAA"

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

# Select best from a pool of models
model = ES(model=["ANN", "AAN", "AAA"], lags=[12])
model.fit(y)

# Model combination
model = ES(model="CCC", lags=[12])
model.fit(y)

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
modelName() model_name character str Full model name (e.g., "ETS(AAN)")
modelType() model_type character str ETS type code only (e.g., "AAN")
persistence persistence_vector vector Dict Smoothing parameters dict with keys persistence_level (α), persistence_trend (β), persistence_seasonal (γ)
phi phi_ numeric float/None Dampening parameter (φ), None if no dampening
initial initial_value list Dict Initial state values
states states matrix NDArray State matrix over time
transition transition matrix NDArray Transition matrix F
measurement measurement matrix NDArray Measurement matrix W
fitted fitted vector NDArray Fitted values
residuals residuals vector NDArray Residuals
forecast predict() result vector NDArray Point forecasts
logLik loglik numeric float Log-likelihood value
AICc aicc numeric float Corrected AIC
B coef vector NDArray All estimated parameters
distribution distribution_ character str Error distribution used
loss loss_ character str Loss function used
scale scale or sigma numeric float Scale parameter

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

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