SMA

SMA - Simple Moving Average

SMA (Simple Moving Average) implements simple moving average in state-space form with automatic order selection. This provides a statistically rigorous foundation for the classic moving average method.

Note: SMA is currently available only in R.

Overview

The function constructs an AR model in state-space form based on the simple moving average concept:

yₜ = (1/n) Σⱼ₌₁ⁿ yₜ₋ⱼ

This is equivalent to an AR(n) process, allowing proper parameter estimation and prediction interval construction.

Note that the forecast from the SMA in the state space is not a straight line! It is a multistep conditional expectation from the respective AR(n) model with provided parameters. See Svetunkov & Petropoulos (2018) for more details.

R Usage

library(smooth)

# SMA with specific order
sma(y, order=12, h=18, holdout=TRUE)

# Automatic order selection
sma(y, h=18, holdout=TRUE)

# With specific information criterion
sma(y, order=NULL, ic="BIC", h=18, holdout=TRUE)

# Fast search (modified ternary search)
sma(y, h=18, holdout=TRUE, fast=TRUE)

# Full search
sma(y, h=18, holdout=TRUE, fast=FALSE)

Forecasting

model <- sma(y, order=12, h=18, holdout=TRUE)

# Point forecasts
forecast(model, h=18)

# With prediction intervals
plot(forecast(model, h=18, interval="empirical"))
plot(forecast(model, h=18, interval="parametric"))

Parameters

Parameter	Type (R)	Type (Python)	Default	Description
y	vector/ts	TBA	-	Time series data
order	integer/NULL	TBA	NULL	SMA order (NULL for auto-selection)
ic	character	TBA	"AICc"	Information criterion for order selection
h	integer	TBA	10	Forecast horizon
holdout	logical	TBA	FALSE	Use holdout validation
silent	logical	TBA	TRUE	Suppress output
fast	logical	TBA	TRUE	Use fast ternary search

Information Criteria

• "AICc": Corrected AIC (default)
• "AIC": Akaike Information Criterion
• "BIC": Bayesian Information Criterion
• "BICc": Corrected BIC

Order Selection

When order=NULL, the function automatically selects the optimal order:

• Fast mode (fast=TRUE): Modified ternary search - finds local minimum quickly
• Full mode (fast=FALSE): Exhaustive search - guarantees global minimum but slower

Output

The sma() function internally uses adam() with model="NNN" and AR coefficients all equal to 1/n. It returns an object of class "adam" containing:

Model Information

Element	Type (R)	Description
model	character	Model name (e.g., "SMA(12)")
timeElapsed	difftime	Time elapsed for model construction
call	call	The function call
orders	integer	The SMA order (accessible via orders(model))

State Space Components

Element	Type (R)	Description
states	matrix	State matrix (observations × states)
transition	matrix	Transition matrix F (all elements = 1/n)
persistence	numeric vector	Persistence vector g (all elements = 1/n)
measurement	numeric vector	Measurement vector w
initial	numeric vector	Initial state vector values
initialType	character	Type of initial values used

Fitted Values and Forecasts

Element	Type (R)	Description
fitted	vector	Fitted values
forecast	vector	Point forecasts for h steps ahead
lower	vector	Lower bound of prediction interval (NA if interval=FALSE)
upper	vector	Upper bound of prediction interval (NA if interval=FALSE)
residuals	vector	Model residuals
errors	matrix	Matrix of 1 to h steps ahead errors (for multistep losses)

Model Fit Statistics

Element	Type (R)	Description
s2	numeric	Residual variance (adjusted for degrees of freedom)
logLik	numeric	Log-likelihood value
lossValue	numeric	Cost function value
loss	character	Type of loss function used
nParam	matrix	Table of estimated/provided parameters

Data

Element	Type (R)	Description
y	vector/ts	Original data
holdout	vector/ts	Holdout part of original data
interval	character	Type of interval requested
level	numeric	Confidence level for interval
cumulative	logical	Whether forecast was cumulative

Order Selection (when order=NULL)

Element	Type (R)	Description
ICs	named numeric vector	IC values for each tested order (1 to maxOrder)

Accuracy (when holdout=TRUE)

Element	Type (R)	Description
accuracy	numeric vector	Accuracy measures if holdout is TRUE

Accessing Elements

model <- sma(y, h=18, holdout=TRUE)

# Model name
model$model

# Get the order
orders(model)

# Fitted values and residuals
fitted(model)
residuals(model)

# Information criteria
AIC(model)
BIC(model)

# If order was auto-selected, see all tested IC values
model$ICs

Prediction Intervals

SMA supports various interval types:

# Empirical intervals (bootstrap-based)
forecast(model, h=12, interval="empirical")

# Parametric intervals (normal assumption)
forecast(model, h=12, interval="parametric")

# No intervals
forecast(model, h=12, interval="none")

Comparison to Base R

Unlike stats::filter() which only provides smoothed values, smooth::sma():

• Provides proper prediction intervals
• Allows automatic order selection
• Returns state-space components
• Offers information criteria

Use Cases

SMA is appropriate when:

• Simple averaging is conceptually appropriate
• You want automatic order selection
• You need proper uncertainty quantification
• The series has no clear trend or seasonality

For trend or seasonal data, consider ES or ADAM.

Python Alternative

SMA isn't currently available in Python smooth.

For simple moving average smoothing without forecasting, use pandas:

y.rolling(window=n).mean()

References

• Svetunkov, I., & Petropoulos, F. (2018). Old dog, new tricks: a modelling view of simple moving averages. International Journal of Production Research, 56(18), 6034-6047. DOI: 10.1080/00207543.2017.1380326

See Also

• ES - Exponential Smoothing
• ADAM - Unified framework
• MSARIMA - ARIMA models