ssarima: State Space ARIMA
In smooth: Forecasting Using State Space Models

ssarima

R Documentation

State Space ARIMA

Description

Function constructs State Space ARIMA, estimating AR, MA terms and initial states.

Function selects the best State Space ARIMA based on information criteria, using fancy branch and bound mechanism. The resulting model can be not optimal in IC meaning, but it is usually reasonable.

Usage

ssarima(y, orders = list(ar = c(0), i = c(1), ma = c(1)), lags = c(1,
  frequency(y)), constant = FALSE, arma = NULL, model = NULL,
  initial = c("backcasting", "optimal", "two-stage", "complete"),
  loss = c("likelihood", "MSE", "MAE", "HAM", "MSEh", "TMSE", "GTMSE",
  "MSCE", "GPL"), h = 0, holdout = FALSE, bounds = c("admissible",
  "usual", "none"), silent = TRUE, xreg = NULL, regressors = c("use",
  "select", "adapt"), initialX = NULL, ...)

auto.ssarima(y, orders = list(ar = c(3, 3), i = c(2, 1), ma = c(3, 3)),
  lags = c(1, frequency(y)), fast = TRUE, constant = NULL,
  initial = c("backcasting", "optimal", "two-stage", "complete"),
  loss = c("likelihood", "MSE", "MAE", "HAM", "MSEh", "TMSE", "GTMSE",
  "MSCE", "GPL"), ic = c("AICc", "AIC", "BIC", "BICc"), h = 0,
  holdout = FALSE, bounds = c("admissible", "usual", "none"),
  silent = TRUE, xreg = NULL, regressors = c("use", "select", "adapt"),
  ...)

Arguments

`y`	Vector or ts object, containing data needed to be forecasted.
`orders`	List of maximum orders to check, containing vector variables `ar`, `i` and `ma`. If a variable is not provided in the list, then it is assumed to be equal to zero. At least one variable should have the same length as `lags`.
`lags`	Defines lags for the corresponding orders (see examples). The length of `lags` must correspond to the length of `orders`. There is no restrictions on the length of `lags` vector.
`constant`	If `NULL`, then the function will check if constant is needed. if `TRUE`, then constant is forced in the model. Otherwise constant is not used.
`arma`	Either the named list or a vector with AR / MA parameters ordered lag-wise. The number of elements should correspond to the specified orders e.g. `orders=list(ar=c(1,1),ma=c(1,1)), lags=c(1,4), arma=list(ar=c(0.9,0.8),ma=c(-0.3,0.3))`
`model`	A previously estimated ssarima model, if provided, the function will not estimate anything and will use all its parameters.
`initial`	Can be either character or a vector of initial states. If it is character, then it can be `"optimal"`, meaning that the initial states are optimised, or `"backcasting"`, meaning that the initials are produced using backcasting procedure.
`loss`	The type of Loss Function used in optimization. `loss` can be: `likelihood` (assuming Normal distribution of error term), `MSE` (Mean Squared Error), `MAE` (Mean Absolute Error), `HAM` (Half Absolute Moment), `TMSE` - Trace Mean Squared Error, `GTMSE` - Geometric Trace Mean Squared Error, `MSEh` - optimisation using only h-steps ahead error, `MSCE` - Mean Squared Cumulative Error. If `loss!="MSE"`, then likelihood and model selection is done based on equivalent `MSE`. Model selection in this cases becomes not optimal. There are also available analytical approximations for multistep functions: `aMSEh`, `aTMSE` and `aGTMSE`. These can be useful in cases of small samples. Finally, just for fun the absolute and half analogues of multistep estimators are available: `MAEh`, `TMAE`, `GTMAE`, `MACE`, `TMAE`, `HAMh`, `THAM`, `GTHAM`, `CHAM`.
`h`	Length of forecasting horizon.
`holdout`	If `TRUE`, holdout sample of size `h` is taken from the end of the data.
`bounds`	What type of bounds to use in the model estimation. The first letter can be used instead of the whole word. In case of `ssarima()`, the "usual" means restricting AR and MA parameters to lie between -1 and 1.
`silent`	accepts `TRUE` and `FALSE`. If FALSE, the function will print its progress and produce a plot at the end.
`xreg`	The vector (either numeric or time series) or the matrix (or data.frame) of exogenous variables that should be included in the model. If matrix included than columns should contain variables and rows - observations. Note that `xreg` should have number of observations equal either to in-sample or to the whole series. If the number of observations in `xreg` is equal to in-sample, then values for the holdout sample are produced using es function.
`regressors`	The variable defines what to do with the provided xreg: `"use"` means that all of the data should be used, while `"select"` means that a selection using `ic` should be done.
`initialX`	The vector of initial parameters for exogenous variables. Ignored if `xreg` is NULL.
`...`	Other non-documented parameters. For example `FI=TRUE` will make the function also produce Fisher Information matrix, which then can be used to calculated variances of parameters of the model. Maximum orders to check can also be specified separately, however `orders` variable must be set to `NULL`: `ar.orders` - Maximum order of AR term. Can be vector, defining max orders of AR, SAR etc. `i.orders` - Maximum order of I. Can be vector, defining max orders of I, SI etc. `ma.orders` - Maximum order of MA term. Can be vector, defining max orders of MA, SMA etc.
`fast`	If `TRUE`, then some of the orders of ARIMA are skipped. This is not advised for models with `lags` greater than 12.
`ic`	The information criterion to use in the model selection.

Details

The model, implemented in this function, is discussed in Svetunkov & Boylan (2019).

The basic ARIMA(p,d,q) used in the function has the following form:

(1 - B)^d (1 - a_1 B - a_2 B^2 - ... - a_p B^p) y_[t] = (1 + b_1 B + b_2 B^2 + ... + b_q B^q) \epsilon_[t] + c

where y_[t] is the actual values, \epsilon_[t] is the error term, a_i, b_j are the parameters for AR and MA respectively and c is the constant. In case of non-zero differences c acts as drift.

This model is then transformed into ARIMA in the Single Source of Error State space form (proposed in Snyder, 1985):

y_{t} = w' v_{t-l} + \epsilon_{t}

v_{t} = F v_{t-l} + g_t \epsilon_{t}

where v_{t} is the state vector (defined based on orders) and l is the vector of lags, w_t is the measurement vector (with explanatory variables if provided), F is the transition matrix, g_t is the persistence vector (which includes explanatory variables if they were used).

Due to the flexibility of the model, multiple seasonalities can be used. For example, something crazy like this can be constructed: SARIMA(1,1,1)(0,1,1)[24](2,0,1)[24*7](0,0,1)[24*30], but the estimation may take a lot of time... If you plan estimating a model with more than one seasonality, it is recommended to use msarima instead.

The model selection for SSARIMA is done by the auto.ssarima function.

For some more information about the model and its implementation, see the vignette: vignette("ssarima","smooth")

The function constructs bunch of ARIMAs in Single Source of Error state space form (see ssarima documentation) and selects the best one based on information criterion. The mechanism is described in Svetunkov & Boylan (2019).

Due to the flexibility of the model, multiple seasonalities can be used. For example, something crazy like this can be constructed: SARIMA(1,1,1)(0,1,1)[24](2,0,1)[24*7](0,0,1)[24*30], but the estimation may take a lot of time... It is recommended to use auto.msarima in cases with more than one seasonality and high frequencies.

For some more information about the model and its implementation, see the vignette: vignette("ssarima","smooth")

Value

Object of class "adam" is returned with similar elements to the adam function.

Object of class "smooth" is returned. See ssarima for details.

Author(s)

Ivan Svetunkov, ivan@svetunkov.com

References

Svetunkov I. (2023) Smooth forecasting with the smooth package in R. arXiv:2301.01790. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.48550/arXiv.2301.01790")}.
Svetunkov I. (2015 - Inf) "smooth" package for R - series of posts about the underlying models and how to use them: https://openforecast.org/category/r-en/smooth/.

Svetunkov, I., 2023. Smooth Forecasting with the Smooth Package in R. arXiv. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.48550/arXiv.2301.01790")}
Snyder, R. D., 1985. Recursive Estimation of Dynamic Linear Models. Journal of the Royal Statistical Society, Series B (Methodological) 47 (2), 272-276.
Hyndman, R.J., Koehler, A.B., Ord, J.K., and Snyder, R.D. (2008) Forecasting with exponential smoothing: the state space approach, Springer-Verlag. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1007/978-3-540-71918-2")}.

Svetunkov, I., 2023. Smooth Forecasting with the Smooth Package in R. arXiv. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.48550/arXiv.2301.01790")}
Snyder, R. D., 1985. Recursive Estimation of Dynamic Linear Models. Journal of the Royal Statistical Society, Series B (Methodological) 47 (2), 272-276.
Hyndman, R.J., Koehler, A.B., Ord, J.K., and Snyder, R.D. (2008) Forecasting with exponential smoothing: the state space approach, Springer-Verlag. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1007/978-3-540-71918-2")}.

Svetunkov, I., Boylan, J.E., 2023a. iETS: State Space Model for Intermittent Demand Forecastings. International Journal of Production Economics. 109013. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1016/j.ijpe.2023.109013")}
Teunter R., Syntetos A., Babai Z. (2011). Intermittent demand: Linking forecasting to inventory obsolescence. European Journal of Operational Research, 214, 606-615.
Croston, J. (1972) Forecasting and stock control for intermittent demands. Operational Research Quarterly, 23(3), 289-303.

Svetunkov, I., & Boylan, J. E. (2019). State-space ARIMA for supply-chain forecasting. International Journal of Production Research, 0(0), 1–10. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1080/00207543.2019.1600764")}

Examples

# ARIMA(1,1,1) fitted to some data
ourModel <- ssarima(rnorm(118,100,3),orders=list(ar=c(1),i=c(1),ma=c(1)),lags=c(1))

# Model with the same lags and orders, applied to a different data
ssarima(rnorm(118,100,3),orders=orders(ourModel),lags=lags(ourModel))

# The same model applied to a different data
ssarima(rnorm(118,100,3),model=ourModel)

# Example of SARIMA(2,0,0)(1,0,0)[4]
ssarima(rnorm(118,100,3),orders=list(ar=c(2,1)),lags=c(1,4))

# SARIMA(1,1,1)(0,0,1)[4] with different initialisations
ssarima(rnorm(118,100,3),orders=list(ar=c(1),i=c(1),ma=c(1,1)),
        lags=c(1,4),h=18,holdout=TRUE,initial="backcasting")


set.seed(41)
x <- rnorm(118,100,3)

# The best ARIMA for the data
ourModel <- auto.ssarima(x,orders=list(ar=c(2,1),i=c(1,1),ma=c(2,1)),lags=c(1,12),
                                   h=18,holdout=TRUE)

# The other one using optimised states
auto.ssarima(x,orders=list(ar=c(3,2),i=c(2,1),ma=c(3,2)),lags=c(1,12),
                       initial="two",h=18,holdout=TRUE)

summary(ourModel)
forecast(ourModel)
plot(forecast(ourModel))

smooth documentation built on Feb. 5, 2026, 5:08 p.m.