ssarima(data, ar.orders=c(0), i.orders=c(1), ma.orders=c(1), lags=c(1), constant=FALSE, AR=NULL, MA=NULL, initial=c("backcasting","optimal"), cfType=c("MSE","MAE","HAM","MLSTFE","MSTFE","MSEh"), h=10, holdout=FALSE, intervals=FALSE, level=0.95, intervalsType=c("parametric","semiparametric","nonparametric"), intermittent=c("none","auto","fixed","croston","tsb"), bounds=c("admissible","none"), silent=c("none","all","graph","legend","output"), xreg=NULL, initialX=NULL, updateX=FALSE, persistenceX=NULL, transitionX=NULL, ...)ar.orders=c(1,1) with lags=c(1,12) will lead to model with AR(1), SAR(1).
i.orders=c(0,1) with lags=c(1,12) will lead to model with I(0), SI(1).
i.orders=c(1,2) with lags=c(1,12) will lead to model with MA(1), SMA(2).
lags must correspond to the length of either ar.orders or i.orders or ma.orders. There is no restrictions on the length of lags vector. It is recommended to order lags ascending.
TRUE, constant term is included in the model.
"optimal", meaning that the initial states are optimised, or "backcasting", meaning that the initials are produced using backcasting procedure (advised for data with high frequency). If character, then initial.season will be estimated in the way defined by initial.
cfType can be: MSE (Mean Squared Error), MAE (Mean Absolute Error), HAM (Half Absolute Moment), MLSTFE - Mean Log Squared Trace Forecast Error, MSTFE - Mean Squared Trace Forecast Error and MSEh - optimisation using only h-steps ahead error. If cfType!="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, aMSTFE and aMLSTFE. These can be useful in cases of small samples.
TRUE, the holdout of the size h is taken from the end of the data.
TRUE, the prediction interval is constructed.
parametric use state-space structure of ETS. For multiplicative models they are approximated using the same function as for additive. As a result they are a bit wider than should be but are still efficient. In case of mixed models this is done using simulations, which may take longer time than for the pure additive and pure multiplicative models.
semiparametric are based on covariance matrix of 1 to h steps ahead errors and assumption of normal distribution.
nonparametric intervals use 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.
none, meaning that the data should be considered as non-intermittent; 2. fixed, taking into account constant Bernoulli distribution of demand occurancies; 3. croston, based on Croston, 1972 method with SBA correction; 4. tsb, based on Teunter et al., 2011 method. 5. auto - automatic selection of intermittency type based on information criteria. The first letter can be used instead.
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").
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 Naive.
xreg is NULL.
TRUE, transition matrix for exogenous variables is estimated, introducing non-linear interractions between parameters. Prerequisite - non-NULL xreg.
NULL, then estimated. Prerequisite - non-NULL xreg.
matrix(transition,nc,nc), where nc is number of components in state vector. If NULL, then estimated. Prerequisite - non-NULL xreg.
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.
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.
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 - Typetof initial values used.
initial - the initial values of the state vector (extracted from states).
nParam - number of estimated parameters.
fitted - the fitted values of ETS.
forecast - the point forecast of ETS.
lower - the lower bound of prediction interval. When intervals=FALSE then NA is returned.
upper - the higher bound of prediction interval. When intervals=FALSE then NA is returned.
residuals - the residuals of the estimated model.
errors - The matrix of 1 to h steps ahead errors.
s2 - standard deviation of the residuals (taking degrees of freedom into account).
intervalsType - type of intervals asked by user.
level - confidence level for intervals.
actuals - the original data.
holdout - the holdout part of the original data.
iprob - the fitted and forecasted values of the probability of demand occurrence.
intermittent - type of intermittent model fitted to the data.
xreg - the provided vector or matrix of exogenous variables.
updateX - boolean, defining, if the states of exogenous variables were estimated as well.
initialX - initial values for parameters of exogenous variables.
persistenceX - persistence vector g for exogenous variables.
transitionX - transition matrix F for exogenous variables.
ICs - values of information criteria of the model. Includes AIC, AICc and BIC.
cf - Cost function value.
cfType - Type of cost function used in the estimation.
FI - Fisher Information. Equal to NULL if FI=FALSE or when FI is not provided at all.
accuracy - the vector or accuracy measures for the holdout sample. Includes MPE, MAPE, SMAPE, MASE, MAE/mean, RelMAE and Bias coefficient (based on complex numbers). Available only when holdout=TRUE.
$y_[t] = o_[t] (w' v_[t-l] + x_t a_[t-1] + \epsilon_[t])$
$v_[t] = F v_[t-1] + g \epsilon_[t]$
$a_[t] = F_[X] a_[t-1] + g_[X] \epsilon_[t] / x_[t]$
Where $o_[t]$ is Bernoulli distributed random variable (in case of normal data equals to 1 for all observations), $v_[t]$ is a state vector (defined using ar.orders and i.orders), $x_t$ vector of exogenous parameters.
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...
ets, es, ces, sim.es, ges
# ARIMA(1,1,1)
test <- ssarima(rnorm(118,100,3),ar.orders=c(1),lags=c(1),h=18,holdout=TRUE,intervals=TRUE)
# The same model applied to a different data
ssarima(rnorm(118,100,3),model=test,h=18,holdout=TRUE)
# SARIMA(1,0,0)(1,0,0)[4]
## Not run: ssarima(rnorm(118,100,3),ar.orders=c(2,1),lags=c(1,4),h=18,holdout=TRUE)
# SARIMA(1,1,1)(0,0,1)[4]
## Not run: ssarima(rnorm(118,100,3),ar.orders=c(1),i.orders=c(1),ma.orders=c(1,1),
# lags=c(1,4),h=18,holdout=TRUE)
# ssarima(rnorm(118,100,3),ar.orders=c(1),i.orders=c(1),ma.orders=c(1,1),
# lags=c(1,4),h=18,holdout=TRUE,initial="o")## End(Not run)
# SARIMA of a perculiar order on AirPassengers data
## Not run: ssarima(AirPassengers,ar.orders=c(1,0,3),i.orders=c(1,0,1),ma.orders=c(0,1,2),lags=c(1,6,12),
# h=10,holdout=TRUE)## End(Not run)
# ARIMA(1,1,1) with Mean Squared Trace Forecast Error
## Not run: ssarima(rnorm(118,100,3),ar.orders=1,lags=1,h=18,holdout=TRUE,cfType="MSTFE")
# ssarima(rnorm(118,100,3),ar.orders=1,lags=1,h=18,holdout=TRUE,cfType="aMSTFE")## End(Not run)
# SARIMA(0,1,1) with exogenous variables
ssarima(rnorm(118,100,3),ar.orders=c(1),h=18,holdout=TRUE,xreg=c(1:118))
# SARIMA(0,1,1) with exogenous variables with crazy estimation of xreg
test <- ssarima(rnorm(118,100,3),ar.orders=c(1),h=18,holdout=TRUE,xreg=c(1:118),updateX=TRUE)
summary(test)
forecast(test)
plot(forecast(test))
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