Function constructs State Space ARIMA, estimating AR, MA terms and initial states.
ssarima(y, orders = list(ar = c(0), i = c(1), ma = c(1)), lags = c(1),
constant = FALSE, AR = NULL, MA = 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. It contains the list of the following values:
model - the name of the estimated model.
timeElapsed - time elapsed for the construction of the model.
states - the matrix of the fuzzy components of ssarima, where
rows correspond to time and cols to states.
transition - matrix F.
persistence - the persistence vector. This is the place, where
smoothing parameters live.
measurement - measurement vector of the model.
AR - the matrix of coefficients of AR terms.
I - the matrix of coefficients of I terms.
MA - the matrix of coefficients of MA terms.
constant - the value of the constant term.
initialType - Type of the initial values used.
initial - the initial values of the state vector (extracted
from states).
nParam - table with the number of estimated / provided parameters.
If a previous model was reused, then its initials are reused and the number of
provided parameters will take this into account.
fitted - the fitted values.
forecast - the point forecast.
lower - the lower bound of prediction interval. When
interval="none" then NA is returned.
upper - the higher bound of prediction interval. When
interval="none" then NA is returned.
residuals - the residuals of the estimated model.
errors - The matrix of 1 to h steps ahead errors. Only returned when the
multistep losses are used and semiparametric interval is needed.
s2 - variance of the residuals (taking degrees of freedom into
account).
interval - type of interval asked by user.
level - confidence level for interval.
cumulative - whether the produced forecast was cumulative or not.
y - the original data.
holdout - the holdout part of the original data.
xreg - provided vector or matrix of exogenous variables. If
regressors="s", then this value will contain only selected exogenous
variables.
initialX - initial values for parameters of exogenous
variables.
ICs - values of information criteria of the model. Includes
AIC, AICc, BIC and BICc.
logLik - log-likelihood of the function.
lossValue - Cost function value.
loss - Type of loss function used in the estimation.
FI - Fisher Information. Equal to NULL if FI=FALSE
or when FI is not provided at all.
accuracy - vector of accuracy measures for the holdout sample.
In case of non-intermittent data includes: MPE, MAPE, SMAPE, MASE, sMAE,
RelMAE, sMSE and Bias coefficient (based on complex numbers). In case of
intermittent data the set of errors will be: sMSE, sPIS, sCE (scaled
cumulative error) and Bias coefficient. This is available only when
holdout=TRUE.
B - the vector of all the estimated parameters.
Vector or ts object, containing data needed to be forecasted.
List of orders, containing vector variables ar,
i and ma. Example:
orders=list(ar=c(1,2),i=c(1),ma=c(1,1,1)). 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. Another option is to
specify orders as a vector of a form orders=c(p,d,q). The non-seasonal
ARIMA(p,d,q) is constructed in this case.
Defines lags for the corresponding orders (see examples above).
The length of lags must correspond to the length of either ar,
i or ma in orders variable. There is no restrictions on
the length of lags vector. It is recommended to order lags
ascending.
The orders are set by a user. If you want the automatic order selection,
then use auto.ssarima function instead.
If TRUE, constant term is included in the model. Can
also be a number (constant value).
Vector or matrix of AR parameters. The order of parameters should be lag-wise. This means that first all the AR parameters of the firs lag should be passed, then for the second etc. AR of another ssarima can be passed here.
Vector or matrix of MA parameters. The order of parameters should be lag-wise. This means that first all the MA parameters of the firs lag should be passed, then for the second etc. MA of another ssarima can be passed here.
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.
Parameter model can accept a previously estimated SSARIMA model and
use all its parameters.
FI=TRUE will make the function produce Fisher Information matrix,
which then can be used to calculated variances of parameters of the model.
Ivan Svetunkov, ivan@svetunkov.ru
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):
\(y_{t} = o_{t} (w' v_{t-l} + x_t a_{t-1} + \epsilon_{t})\)
\(v_{t} = F v_{t-l} + g \epsilon_{t}\)
\(a_{t} = F_{X} a_{t-1} + g_{X} \epsilon_{t} / x_{t}\)
Where \(o_{t}\) is the Bernoulli distributed random variable (in case of
normal data equal to 1), \(v_{t}\) is the state vector (defined based on
orders) and \(l\) is the vector of lags, \(x_t\) is the
vector of exogenous parameters. \(w\) is the measurement vector,
\(F\) is the transition matrix, \(g\) is the persistence
vector, \(a_t\) is the vector of parameters for exogenous variables,
\(F_{X}\) is the transitionX matrix and \(g_{X}\) is the
persistenceX matrix.
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 some finite time... If you plan estimating a model with more than one seasonality, it is recommended to consider doing it using msarima.
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")
Taylor, J.W. and Bunn, D.W. (1999) A Quantile Regression Approach to Generating Prediction Intervals. Management Science, Vol 45, No 2, pp 225-237.
Lichtendahl Kenneth C., Jr., Grushka-Cockayne Yael, Winkler Robert L., (2013) Is It Better to Average Probabilities or Quantiles? Management Science 59(7):1594-1611. DOI: tools:::Rd_expr_doi("10.1287/mnsc.1120.1667")
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, 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")
auto.ssarima, orders,
msarima, auto.msarima,
sim.ssarima, adam
# 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),h=18,
holdout=TRUE,interval="p")
# The previous one is equivalent to:
ourModel <- ssarima(rnorm(118,100,3),ar.orders=c(1),i.orders=c(1),ma.orders=c(1),
lags=c(1),h=18,holdout=TRUE,interval="p")
# Model with the same lags and orders, applied to a different data
ssarima(rnorm(118,100,3),orders=orders(ourModel),lags=lags(ourModel),h=18,holdout=TRUE)
# The same model applied to a different data
ssarima(rnorm(118,100,3),model=ourModel,h=18,holdout=TRUE)
# 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),h=18,holdout=TRUE)
# 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)
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="o")
# SARIMA of a peculiar order on AirPassengers data
ssarima(AirPassengers,orders=list(ar=c(1,0,3),i=c(1,0,1),ma=c(0,1,2)),
lags=c(1,6,12),h=10,holdout=TRUE)
# ARIMA(1,1,1) with Mean Squared Trace Forecast Error
ssarima(rnorm(118,100,3),orders=list(ar=1,i=1,ma=1),lags=1,h=18,holdout=TRUE,loss="TMSE")
ssarima(rnorm(118,100,3),orders=list(ar=1,i=1,ma=1),lags=1,h=18,holdout=TRUE,loss="aTMSE")
# SARIMA(0,1,1) with exogenous variables
ssarima(rnorm(118,100,3),orders=list(i=1,ma=1),h=18,holdout=TRUE,xreg=c(1:118))
summary(ourModel)
forecast(ourModel)
plot(forecast(ourModel))
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