SMA

title: SMA — Simple Moving Average slug: SMA summary: Simple moving average in state-space form with automatic order selection. status: stable applies_to: [r, python] related: [ADAM]

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.

SMA is available in both R (sma()) and Python (SMA). Both share the same algorithm and options.

Function signatures

R

sma(y, order = NULL,
    ic = c("AICc", "AIC", "BIC", "BICc"),
    h = 10, holdout = FALSE,
    silent = TRUE, fast = TRUE, ...)

Python

class SMA(ADAM):
    def __init__(
        self,
        order: int | None = None,
        ic: Literal["AIC", "AICc", "BIC", "BICc"] = "AICc",
        h: int = 10,
        holdout: bool = False,
        fast: bool = True,
        verbose: int = 0,
        **kwargs,
    ) -> None: ...

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.

How it works (both R and Python). SMA(m) is implemented as a thin wrapper over ADAM with model="NNN", an AR(m) component, and every AR coefficient hard-fixed at 1/m. Because of this it inherits the full ADAM machinery — recursive multi-step forecasts, proper forecast variance, prediction intervals, and residual diagnostics. When the order is left unspecified it is selected automatically by an information criterion (a fast modified ternary search by default, or an exhaustive scan). In Python this is the SMA class (from smooth import SMA), which subclasses ADAM and therefore exposes the same fit() / predict() / diagnostics surface.

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"))

Python Usage

The Python examples below mirror the R ones one-to-one. The only structural difference is the scikit-learn-style split: options go to the SMA(...) constructor and the data is passed to .fit(y).

from smooth import SMA

# SMA with specific order
SMA(order=12, h=18, holdout=True).fit(y)

# Automatic order selection
SMA(h=18, holdout=True).fit(y)

# With specific information criterion
SMA(order=None, ic="BIC", h=18, holdout=True).fit(y)

# Fast search (modified ternary search)
SMA(h=18, holdout=True, fast=True).fit(y)

# Full search
SMA(h=18, holdout=True, fast=False).fit(y)

Forecasting

model = SMA(order=12, h=18, holdout=True)
model.fit(y)

# Point forecasts
fc = model.predict(h=18)
fc.mean

# With prediction intervals (R "parametric" -> Python "prediction")
model.predict(h=18, interval="prediction")
model.predict(h=18, interval="simulated")

Parameters

Parameter	Type (R)	Type (Python)	Default (R / Python)	Description
`y`	vector/ts	NDArray / pd.Series	-	Time series data. In Python it is passed to `.fit(y)`, not the constructor
`order`	integer/NULL	`int` / `None`	NULL / `None`	SMA order (NULL/None for auto-selection)
`ic`	character	`str`	"AICc"	Information criterion for order selection
`h`	integer	`int`	10	Forecast horizon
`holdout`	logical	`bool`	FALSE / False	Use holdout validation
`silent` (R) / `verbose` (Python)	logical	`int`	TRUE / 0	Suppress output. Python uses `verbose` (0 = silent)
`fast`	logical	`bool`	TRUE / 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". In Python, SMA subclasses ADAM, so the same elements are exposed as attributes/properties on the fitted object (Type (Python) columns below; — marks elements with no Python equivalent yet).

Model Information

Element	Type (R)	Type (Python)	Description
`model`	character	`str`	Model name, e.g. "SMA(12)" (`model.model`)
`timeElapsed`	difftime	`float`	Time elapsed for model construction (`model.time_elapsed_`)
`call`	call	—	The function call (R only)
`orders`	integer	`int`	The SMA order (R: `orders(model)`; Python: embedded in `model.model`)

State Space Components

Element	Type (R)	Type (Python)	Description
`states`	matrix	NDArray	State matrix, observations × states (`model.states`)
`transition`	matrix	NDArray	Transition matrix F, all elements = 1/n (`model.transition`)
`persistence`	numeric vector	NDArray	Persistence vector g, all elements = 1/n (`model.persistence`)
`measurement`	numeric vector	NDArray	Measurement vector w (`model.measurement`)
`initial`	numeric vector	dict	Initial state vector values (`model.initial_value`)
`initialType`	character	`str`	Type of initial values used (`model.initial_type`)

Fitted Values and Forecasts

Element	Type (R)	Type (Python)	Description
`fitted`	vector	pd.Series / NDArray	Fitted values (`model.fitted`)
`forecast`	vector	pd.Series	Point forecasts for h steps ahead (`model.predict(h).mean`)
`lower`	vector	pd.DataFrame	Lower bound of prediction interval (`model.predict(h, interval=...).lower`)
`upper`	vector	pd.DataFrame	Upper bound of prediction interval (`model.predict(h, interval=...).upper`)
`residuals`	vector	NDArray	Model residuals (`model.residuals`)
`errors`	matrix	—	Matrix of 1 to h steps ahead errors (for multistep losses)

Model Fit Statistics

Element	Type (R)	Type (Python)	Description
`s2`	numeric	`float`	Residual variance, adjusted for degrees of freedom (`model.sigma ** 2`)
`logLik`	numeric	`float`	Log-likelihood value (`model.loglik`)
`lossValue`	numeric	`float`	Cost function value (`model.loss_value`)
`loss`	character	`str`	Type of loss function used (`model.loss_`)
`nParam`	matrix	`int`	Number of estimated parameters (`model.nparam`)

Data

Element	Type (R)	Type (Python)	Description
`y`	vector/ts	NDArray	Original data (`model.actuals`)
`holdout`	vector/ts	NDArray	Holdout part of original data (`model.holdout_data`)
`interval`	character	`str`	Type of interval requested (`predict()` argument)
`level`	numeric	`float` / list	Confidence level for interval (`predict()` argument)
`cumulative`	logical	`bool`	Whether forecast was cumulative (`predict()` argument)

Order Selection (when `order=NULL` / `order=None`)

Element	Type (R)	Type (Python)	Description
`ICs`	named numeric vector	dict	IC values for each tested order (R: `model$ICs`; Python: `model.ICs_`)

Accuracy (when `holdout=TRUE`)

Element	Type (R)	Type (Python)	Description
`accuracy`	numeric vector	—	Accuracy measures (R only; not auto-computed in Python)

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

The same in Python (note the trailing-underscore fitted attributes):

model = SMA(h=18, holdout=True)
model.fit(y)

# Model name
model.model

# Fitted values and residuals
model.fitted
model.residuals

# Information criteria
model.aic
model.bic

# 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")

In Python (R "parametric" → Python "prediction"):

# Parametric intervals (normal assumption)
model.predict(h=12, interval="prediction")

# Simulated intervals (Monte Carlo)
model.predict(h=12, interval="simulated")

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

Note: for the Python SMA, interval="empirical" is not yet supported (use "prediction" or "simulated").

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

SMA is available in Python via the SMA class — see Python Usage above. It mirrors the R sma() function, including automatic order selection, multi-step forecasts, and prediction intervals. Unlike pandas' y.rolling(window=n).mean() (which only smooths in-sample), SMA produces proper recursive forecasts and uncertainty quantification.

from smooth import SMA

model = SMA(h=12)        # order auto-selected by AICc
model.fit(y)
print(model.model)       # e.g. "SMA(3)"
fc = model.predict(h=12)

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

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SMA

title: SMA — Simple Moving Average slug: SMA summary: Simple moving average in state-space form with automatic order selection. status: stable applies_to: [r, python] related: [ADAM]

SMA - Simple Moving Average

Function signatures

R

Python

Overview

R Usage

Forecasting

Python Usage

Forecasting

Parameters

Information Criteria

Order Selection

Output

Model Information

State Space Components

Fitted Values and Forecasts

Model Fit Statistics

Data

Order Selection (when order=NULL / order=None)

Accuracy (when holdout=TRUE)

Accessing Elements

Prediction Intervals

Comparison to Base R

Use Cases

Python

References

See Also

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Order Selection (when `order=NULL` / `order=None`)

Accuracy (when `holdout=TRUE`)