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.
auto.ssarima(y, orders = list(ar = c(3, 3), i = c(2, 1), ma = c(3, 3)),
lags = c(1, frequency(y)), combine = FALSE, fast = TRUE,
constant = NULL, initial = c("backcasting", "optimal"), ic = c("AICc",
"AIC", "BIC", "BICc"), loss = c("likelihood", "MSE", "MAE", "HAM", "MSEh",
"TMSE", "GTMSE", "MSCE"), h = 10, holdout = FALSE, cumulative = FALSE,
interval = c("none", "parametric", "likelihood", "semiparametric",
"nonparametric"), level = 0.95, bounds = c("admissible", "none"),
silent = c("all", "graph", "legend", "output", "none"), xreg = NULL,
regressors = c("use", "select"), initialX = NULL, ...)Object of class "smooth" is returned. See ssarima for details.
Vector or ts object, containing data needed to be forecasted.
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.
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.
If TRUE, then resulting ARIMA is combined using AIC
weights.
If TRUE, then some of the orders of ARIMA are
skipped. This is not advised for models with lags greater than 12.
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.
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.
The information criterion used in the model selection procedure.
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.
Length of forecasting horizon.
If TRUE, holdout sample of size h is taken from
the end of the data.
If TRUE, then the cumulative forecast and prediction
interval are produced instead of the normal ones. This is useful for
inventory control systems.
Type of interval to construct. This can be:
"none", aka "n" - do not produce prediction
interval.
"parametric", "p" - use state-space structure of ETS. In
case of mixed models this is done using simulations, which may take longer
time than for the pure additive and pure multiplicative models. This type
of interval relies on unbiased estimate of in-sample error variance, which
divides the sume of squared errors by T-k rather than just T.
"likelihood", "l" - these are the same as "p", but
relies on the biased estimate of variance from the likelihood (division by
T, not by T-k).
"semiparametric", "sp" - interval based on covariance
matrix of 1 to h steps ahead errors and assumption of normal / log-normal
distribution (depending on error type).
"nonparametric", "np" - interval based on values from a
quantile regression on error matrix (see Taylor and Bunn, 1999). The model
used in this process is e[j] = a j^b, where j=1,..,h.
The parameter also accepts TRUE and FALSE. The former means that
parametric interval are constructed, while the latter is equivalent to
none.
If the forecasts of the models were combined, then the interval are combined
quantile-wise (Lichtendahl et al., 2013).
Confidence level. Defines width of prediction interval.
What type of bounds to use in the model estimation. The first letter can be used instead of the whole word.
If silent="none", then nothing is silent, everything is
printed out and drawn. silent="all" means that nothing is produced or
drawn (except for warnings). In case of silent="graph", no graph is
produced. If silent="legend", then legend of the graph is skipped.
And finally silent="output" means that nothing is printed out in the
console, but the graph is produced. silent also accepts TRUE
and FALSE. In this case silent=TRUE is equivalent to
silent="all", while silent=FALSE is equivalent to
silent="none". The parameter also accepts first letter of words ("n",
"a", "g", "l", "o").
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.
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.
"combine" will be available at some point in future...
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.
Ivan Svetunkov, ivan@svetunkov.com
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")
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. tools:::Rd_expr_doi("10.1007/978-3-540-71918-2").
Svetunkov Ivan and Boylan John E. (2017). Multiplicative State-Space Models for Intermittent Time Series. Working Paper of Department of Management Science, Lancaster University, 2017:4 , 1-43.
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.
Syntetos, A., Boylan J. (2005) The accuracy of intermittent demand estimates. International Journal of Forecasting, 21(2), 303-314.
Svetunkov, I., & Boylan, J. E. (2019). State-space ARIMA for supply-chain forecasting. International Journal of Production Research, 0(0), 1–10. tools:::Rd_expr_doi("10.1080/00207543.2019.1600764")
es, ces,
sim.es, gum, ssarima
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,interval="np")
# 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="o",h=18,holdout=TRUE)
# And now combined ARIMA
auto.ssarima(x,orders=list(ar=c(3,2),i=c(2,1),ma=c(3,2)),lags=c(1,12),
combine=TRUE,h=18,holdout=TRUE)
summary(ourModel)
forecast(ourModel)
plot(forecast(ourModel))
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