smooth

hex-sticker of the smooth package for Python

Python implementation of the smooth package for forecasting and time series analysis using Single Source of Error (SSOE) state-space models.

Every wheel published to PyPI is signed via Sigstore on the exact GitHub Actions runner that built it and ships with PEP 740 attestations (SLSA Build Level 3 provenance). Verify a downloaded wheel client-side with pypi-attestations:

pip install pypi-attestations
pypi-attestations verify pypi --repository https://github.com/openforecast-org/smooth smooth-*.whl

The package includes the following models:

ADAM - Augmented Dynamic Adaptive Model, uniting exponential smoothing, ARIMA and regression, implemented in the ADAM class.
ETS - Exponential Smoothing in the SSOE state space form, implemented in the ES class.
CES - Complex Exponential Smoothing with complex-valued smoothing parameters, implemented in the CES class (fixed seasonality type) and AutoCES class (automatic seasonality selection).
MSARIMA - Multiple seasonal ARIMA in state space form, implemented in the MSARIMA class (fixed orders) and AutoMSARIMA class (automatic order selection).
OM - Occurrence Model for intermittent demand, implemented in the OM class (plus OMG for the general two-component model and AutoOM for automatic type selection).
SMA - Simple Moving Average in state-space form (an AR(m) model with fixed coefficients), implemented in the SMA class with automatic order selection.

The package also provides standalone data generators that mirror R's sim.* family — sim_es, sim_ssarima, sim_ces, sim_gum, sim_sma, and sim_oes — plus a .simulate() method on fitted ADAM, OM, and OMG objects. See Simulation Functions.

All of these are implemented with the support of the following features:

Automatic components selection in ETS and forecasts combination
Explanatory variables
Multiple seasonal models (e.g. for high frequency data)
Advanced loss functions
Fine tuning of any elements of ADAM/ETS/ARIMA/Regression
A variety of prediction interval construction methods

Like the R version, the Python smooth depends on the greybox package for distributions, information criteria, regressor selection, and the LOWESS smoother. It is installed automatically as a dependency.

Installation

From PyPI (recommended):

pip install smooth

From source (development):

pip install "git+https://github.com/openforecast-org/smooth.git@master#subdirectory=python"

See the Installation Guide for platform-specific instructions.

System Requirements

If installing from source, this package requires compilation of C++ extensions. Before installing, ensure you have:

C++ compiler (g++, clang++, or MSVC)
CMake >= 3.25
Armadillo linear algebra library

Quick Example

import numpy as np
from smooth import ADAM

# Sample data
y = np.array([10, 12, 15, 13, 16, 18, 20, 19, 22, 25, 28, 30,
              11, 13, 16, 14, 17, 19, 21, 20, 23, 26, 29, 31])

# Fit ADAM model with additive error, no trend, no seasonality
model = ADAM(model="ANN")
model.fit(y)

# Generate forecasts
forecasts = model.predict(h=12)

# With seasonal component (monthly data, annual seasonality)
model = ADAM(model="ANA", lags=[1, 12])
model.fit(y)
forecasts = model.predict(h=12)

ADAMX — ADAM with Explanatory Variables

This also works with the exponential smoothing (ETSX) via the ES() class.

import numpy as np
from smooth import ADAM

# Simulate data where y depends on two external regressors
rng = np.random.default_rng(42)
n = 120
X = rng.standard_normal((n, 2))
y = 10 + 2 * X[:, 0] - 1.5 * X[:, 1] + rng.standard_normal(n)

# Fit ETSX(AAN) — use all regressors with fixed coefficients
model = ADAM(model="AAN", regressors="use")
model.fit(y, X)
print(model)           # shows fitted coefficients including xreg

# Forecast 12 steps ahead with future regressor values
X_future = rng.standard_normal((12, 2))
fc = model.predict(h=12, X=X_future)
print(fc.mean)

# Automatic variable selection (drops insignificant regressors)
model_sel = ADAM(model="AAN", regressors="select")
model_sel.fit(y, X)

# Adaptive (time-varying) regressor coefficients
model_adp = ADAM(model="AAN", regressors="adapt")
model_adp.fit(y, X)

X accepts a NumPy array or a pandas DataFrame (column names are preserved as regressor names). regressors controls treatment: "use" (fixed coefficients), "select" (stepwise selection via greybox), or "adapt" (ETS-style time-varying coefficients).

AutoMSARIMA — Automatic ARIMA Order Selection

AutoMSARIMA selects the best ARIMA orders automatically using information criteria, mirroring R's auto.msarima(). It fixes distribution="dnorm" and uses pure ARIMA (no ETS components).

import numpy as np
from smooth import AutoMSARIMA

# Monthly time series (e.g. AirPassengers, 144 observations)
y = np.array([
    112, 118, 132, 129, 121, 135, 148, 148, 136, 119, 104, 118,
    # ... remaining observations
], dtype=float)

# Automatic seasonal ARIMA — searches up to ARIMA(3,2,3)(3,1,3)[12]
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],   # max AR: p≤2 at lag 1, P≤1 at lag 12
    i_order=[2, 1],    # max I:  d≤2 at lag 1, D≤1 at lag 12
    ma_order=[2, 1],   # max MA: q≤2 at lag 1, Q≤1 at lag 12
)
model.fit(y)
fc = model.predict(h=24)

CES — Complex Exponential Smoothing

CES and AutoCES mirror R's ces() / auto.ces(). CES uses complex-valued smoothing parameters to capture both the level and the "potential" of a series, covering four seasonality modes: "none", "simple", "partial", "full".

import numpy as np
from smooth import CES, AutoCES

y = np.array([
    112, 118, 132, 129, 121, 135, 148, 148, 136, 119, 104, 118,
    115, 126, 141, 135, 125, 149, 170, 170, 158, 133, 114, 140,
], dtype=float)

# CES with a fixed seasonality type
model = CES(seasonality="partial", lags=[1, 12], h=6, holdout=True)
model.fit(y)
print(model.model_name)         # e.g. "CES(partial)"
print(model.a_, model.b_)       # complex smoothing parameters
fc = model.predict(h=6)
print(fc.mean)

# AutoCES — select the best seasonality type by information criterion
auto = AutoCES(lags=[1, 12], h=6, holdout=True, ic="AICc")
auto.fit(y)
print(auto.best_model_.model_name)

Note: strict R-parity in the two-stage NLopt path requires nlopt>=2.10.0; older versions still fit, but the BOBYQA stage-1 trajectory may differ slightly from R.

Documentation

GitHub Wiki - Full documentation
ADAM - Main unified ETS/ARIMA framework
Installation Guide - Dependencies and troubleshooting

The pages below document the models and their Python classes:

ADAM — Augmented Dynamic Adaptive Model — unified ETS/ARIMA/regression framework
AutoADAM — Automatic ADAM with distribution and ARIMA order selection
ES — Exponential Smoothing (ETS) wrapper for ADAM
CES — Complex Exponential Smoothing (CES, AutoCES)
MSARIMA — Multiple Seasonal ARIMA (fixed orders) and automatic selection (AutoMSARIMA)
OM — Occurrence Model for intermittent demand (OM, OMG, AutoOM)
SMA — Simple Moving Average in state-space form with automatic order selection
Simulation Functions — sim_es, sim_ssarima, sim_ces, sim_gum, sim_sma, sim_oes, and the .simulate() method on fitted models

Book: Svetunkov, I. (2023). Forecasting and Analytics with the Augmented Dynamic Adaptive Model (ADAM). Chapman and Hall/CRC. Online: https://openforecast.org/adam/

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1.0.8

Aug 27, 2026 14 files

This release

1.0.7 This release

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1.0.3

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1.0.1

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1.0.0

Apr 9, 2026 17 files

smooth 1.0.7

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