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smooth Wiki

Welcome to the smooth package wiki! The smooth package implements Single Source of Error (SSOE) state-space models for forecasting and time series analysis, available for both R and Python (under development).

Quick Links

• CRAN Package
• PyPI Package
• GitHub Repository
• ADAM Online Book

Main Forecasting Functions

Function	Description	Python	R
ADAM	Augmented Dynamic Adaptive Model - unified ETS/ARIMA/regression framework	Yes	Yes
AutoADAM	Automatic ADAM with distribution and ARIMA order selection	Yes	Yes
ES	Exponential Smoothing (ETS) wrapper for ADAM	Yes	Yes
CES	Complex Exponential Smoothing	TBA	Yes
SSARIMA	State Space ARIMA	TBA	Yes
MSARIMA	Multiple Seasonal ARIMA (fixed orders) and automatic selection (AutoMSARIMA)	Yes	Yes
GUM	Generalised Univariate Model	TBA	Yes
SMA	Simple Moving Average	TBA	Yes
OES	Occurrence ETS for intermittent demand	TBA	Yes

Utility Functions

Function	Description	Python	R
msdecompose	Multiple seasonal decomposition (used for ADAM/ES initialization)	Yes	Yes
lowess	Scatter plot smoothing from Cleveland, W. S. (1979)	Yes	in stats package

lowess was implemented in Python to exactly reproduce R's stats::lowess behaviour. The implementation is written in C++ based on R's original C code, producing identical results while running faster than other Python implementations.

Getting Started

Installation

See Installation for details.

Python

from smooth import ADAM, AutoADAM, ES, msdecompose

# Automatic ETS model selection
model = ADAM(model="ZXZ", lags=12)
model.fit(y)
forecasts = model.predict(h=12)

# Automatic distribution and ARIMA selection
model = AutoADAM(model="ZZZ", lags=[1, 12],
                 orders={"ar": 2, "i": 2, "ma": 2})
model.fit(y)

# Simple Exponential Smoothing
model = ES(model="ZXZ")
model.fit(y)

# Time series decomposition
result = msdecompose(y, lags=[12], type='additive')

R

library(smooth)

# Automatic model selection
model <- adam(y, model="ZXZ", lags=12)
forecast(model, h=12)

# Automatic distribution and ARIMA selection
model <- auto.adam(y, model="ZZZ",
                   orders=list(ar=2, i=2, ma=2, select=TRUE),
                   distribution=c("dnorm","dlaplace","ds"))

# Exponential Smoothing
model <- es(y, model="ZXZ", h=12)

Recommended Function

ADAM is the recommended function for most forecasting tasks. It provides:

• Unified ETS and ARIMA framework
• Multiple seasonality support
• Various error distributions
• Intermittent demand handling
• External regressors
• Automatic model selection
• ... and more

Common Parameters

These pages document parameters shared across multiple functions:

• Model-Specification - The model parameter, ETS taxonomy, and model type methods
• Orders-and-Lags - ARIMA orders and seasonal lags
• Loss-Functions - Loss functions for estimation (MSE, MAE, likelihood, etc.)
• Explanatory-Variables - External regressors via formula/xreg (R) and X argument (Python)
• Initialisation - The initial parameter and state initialization
• Persistence - Smoothing parameters (alpha, beta, gamma) and phi
• Bounds - Parameter bounds and stability restrictions

Methods and Tools

Output and Visualisation

• Visualisation-and-Output - plot(), print(), summary(), xtable()

Parameters and Forecasts

• Coefficients-and-Parameters - coef(), confint(), vcov(), coefbootstrap()
• Fitted-Values-and-Forecasts - fitted(), actuals(), predict(), forecast()

Residuals and Diagnostics

• Residuals-and-Errors - residuals(), rstandard(), rmultistep(), outlierdummy()

Model Comparison

• Likelihood-and-Information-Criteria - logLik(), AIC(), BIC(), accuracy(), pls()

Model Information

• Model-Information - nobs(), nparam(), sigma(), extractScale()

Advanced Methods

• Simulation-Functions - simulate(), sim.es(), sim.ssarima(), sim.ces(), sim.gum(), sim.sma()
• Refitting-and-Reforecasting - reapply(), reforecast()
• Scale-Model - sm() for modelling heteroscedasticity

R vs Python Comparison

Main Functions

Model	R	Python
ADAM	adam()	ADAM
AutoADAM	auto.adam()	AutoADAM
ES	es()	ES
CES	ces()	TBA
GUM	gum()	TBA
SSARIMA	ssarima()	TBA
MSARIMA	msarima()	MSARIMA
MSARIMA	auto.msarima()	AutoMSARIMA
SMA	sma()	TBA
OES	oes()	TBA
msdecompose	msdecompose()	msdecompose()
lowess	stats::lowess()	lowess()

Available functionality

Functionality	Description	Wiki
Estimate ETS	Estimate a specific type of ETS model	ADAM, ES
Default selection	Select the best model from an automatically created pool	ADAM, ES
Selection from a pool	Select the best model from the defined pool	ADAM, ES
Combination	AIC-weighted combination of forecasts	ADAM, ES
Combination from a pool	AIC-weighted combination of forecasts from the pool of models	ADAM, ES
ADAMX (fixed regressors)	ADAM/ETS/ARIMA with external regressors (fixed coefficients)	Explanatory-Variables
ADAMX (selection)	ADAM/ETS/ARIMA with automatic stepwise regressor selection	Explanatory-Variables
ADAMX (adaptive)	ADAM/ETS/ARIMA with time-varying regressor coefficients	Explanatory-Variables
Multiple seasonal ETS	ETS with multiple seasonal patterns (e.g. daily + weekly)	ADAM, ES
Point forecasts	Generate mean or median point forecasts	Fitted-Values-and-Forecasts
Prediction intervals	Construct intervals via analytical, simulated, or semiparametric methods	Fitted-Values-and-Forecasts
Cumulative forecasts	Aggregate forecasts over the horizon (e.g. total demand over lead time)	Fitted-Values-and-Forecasts
ARIMA (fixed orders)	Estimate a pure ARIMA model with specified orders	MSARIMA, ADAM
ARIMA (automatic selection)	Select optimal AR/I/MA orders using information criteria	AutoADAM, MSARIMA
Multiple seasonal ARIMA	ARIMA with multiple seasonal patterns	MSARIMA, ADAM
ETS + ARIMA	Combine ETS and ARIMA components in a single model	ADAM, AutoADAM
Advanced loss functions	Estimate using MSE, MAE, likelihood, HAM, and other loss functions	Loss-Functions
Distribution (fixed)	Estimate model with a specified error distribution	ADAM, ES
Distribution (automatic)	Select the best error distribution using information criteria	AutoADAM
Outlier detection (use)	Detect outliers and include all as dummy regressors	AutoADAM, ADAM
Outlier detection (select)	Detect outliers and select significant ones via stepwise	AutoADAM, ADAM
Occurrence model	Model the probability of non-zero demand (intermittent series); R only, Python TBA	OES
Scale model	Model heteroscedasticity via a separate scale equation; R only, Python TBA	Scale-Model
Model diagnostics via plots	Diagnostic plots (residuals, ACF, states, etc.); R only, Python TBA	Visualisation-and-Output
Multi-step forecast errors	Extract and analyse multi-step forecast errors via rmultistep(); R only, Python TBA	Residuals-and-Errors

Methods

Fitting and Forecasting:

Method	R	Python	Description
fit	adam()	.fit()	Fit the model
predict	forecast() / predict()	.predict()	Generate forecasts
fitted	fitted()	.fitted	Fitted values
actuals	actuals()	.actuals	Original data
simulate	simulate()	TBA	Simulate from model

Output and Visualisation:

Method	R	Python	Description
summary	summary()	TBA	Model summary with confidence intervals etc
print	print()	print()	Basic output
plot	plot()	TBA	Diagnostic plots
xtable	xtable()	TBA	LaTeX tables

Coefficients and Parameters:

Method	R	Python	Description
coef	coef()	.coef	Extract coefficients
confint	confint()	TBA	Confidence intervals
vcov	vcov()	TBA	Variance-covariance matrix
coefbootstrap	coefbootstrap()	TBA	Bootstrap coefficients

Residuals and Errors:

Method	R	Python	Description
residuals	residuals()	.residuals	Model residuals
rstandard	rstandard()	.rstandard()	Standardised residuals
rstudent	rstudent()	.rstudent()	Studentised residuals
rmultistep	rmultistep()	TBA	Multi-step forecast errors
multicov	multicov()	TBA	Multi-step error covariance
outlierdummy	outlierdummy()	.outlierdummy()	Outlier dummy variables

Likelihood and Information Criteria:

Method	R	Python	Description
logLik	logLik()	.loglik	Log-likelihood
pointLik	pointLik()	TBA	Point log-likelihoods per observation
AIC / BIC	AIC() / BIC()	.aic / .bic	Information criteria
AICc / BICc	AICc() / BICc()	.aicc / .bicc	Corrected IC
pAIC / pBIC	pAIC() / pBIC()	TBA	Point information criteria
accuracy	accuracy()	TBA	Forecast accuracy measures

Model Information:

Method	R	Python	Description
nobs	nobs()	.nobs	Number of observations
nparam	nparam()	.nparam	Number of parameters
sigma	sigma()	.sigma	Residual standard deviation
extractScale	extractScale()	TBA	Scale parameter (time-varying if sm used)
errorType	errorType()	.error_type	Error type (A/M)
modelType	modelType()	.model_type	Model specification
modelName	modelName()	.model_name	Full model name
orders	orders()	.orders	ARIMA orders
lags	lags()	.lags	Model lags

Advanced Methods:

Method	R	Python	Description
reapply	reapply()	TBA	Refit with perturbed parameters
reforecast	reforecast()	TBA	Forecast with parameter uncertainty
sm	sm()	TBA	Scale model (heteroscedasticity)
implant	implant()	TBA	Merge location and scale models

Simulation Functions:

Method	R	Python	Description
sim.es	sim.es()	TBA	Simulate from ETS models
sim.ssarima	sim.ssarima()	TBA	Simulate from ARIMA models
sim.ces	sim.ces()	TBA	Simulate from CES models
sim.gum	sim.gum()	TBA	Simulate from GUM models
sim.sma	sim.sma()	TBA	Simulate from SMA models
sim.oes	sim.oes()	TBA	Simulate occurrence probabilities

Additional Resources

• Installation - Installation instructions and troubleshooting
• Model-Estimation - Optimisation and estimation parameters
• Resources - Publications and DOIs for each function