Description Usage Arguments Details Value Author(s) References See Also Examples
Function constructs State Space ARIMA, estimating AR, MA terms and initial states.
1 2 3 4 5 6 7 8 9  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, xregDo = c("use", "select"),
initialX = NULL, ...)

y 
Vector or ts object, containing data needed to be forecasted. 
orders 
List of orders, containing vector variables 
lags 
Defines lags for the corresponding orders (see examples above).
The length of 
constant 
If 
AR 
Vector or matrix of AR parameters. The order of parameters should be lagwise. 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. 
MA 
Vector or matrix of MA parameters. The order of parameters should be lagwise. 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. 
initial 
Can be either character or a vector of initial states. If it
is character, then it can be 
ic 
The information criterion used in the model selection procedure. 
loss 
The type of Loss Function used in optimization. There are also available analytical approximations for multistep functions:
Finally, just for fun the absolute and half analogues of multistep estimators
are available: 
h 
Length of forecasting horizon. 
holdout 
If 
cumulative 
If 
interval 
Type of interval to construct. This can be:
The parameter also accepts 
level 
Confidence level. Defines width of prediction interval. 
bounds 
What type of bounds to use in the model estimation. The first letter can be used instead of the whole word. 
silent 
If 
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 
xregDo 
The variable defines what to do with the provided xreg:

initialX 
The vector of initial parameters for exogenous variables.
Ignored if 
... 
Other nondocumented parameters. Parameter

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) ε_[t] + c
where y_[t] is the actual values, ε_[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 nonzero 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_{tl} + x_t a_{t1} + ε_{t})
v_{t} = F v_{tl} + g ε_{t}
a_{t} = F_{X} a_{t1} + g_{X} ε_{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")
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.
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
xregDo="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
 loglikelihood 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 nonintermittent 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.
Ivan Svetunkov, ivan@svetunkov.ru
Taylor, J.W. and Bunn, D.W. (1999) A Quantile Regression Approach to Generating Prediction Intervals. Management Science, Vol 45, No 2, pp 225237.
Lichtendahl Kenneth C., Jr., GrushkaCockayne Yael, Winkler Robert L., (2013) Is It Better to Average Probabilities or Quantiles? Management Science 59(7):15941611. DOI: 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), 272276.
Hyndman, R.J., Koehler, A.B., Ord, J.K., and Snyder, R.D. (2008) Forecasting with exponential smoothing: the state space approach, SpringerVerlag. doi: 10.1007/9783540719182.
Svetunkov, I., & Boylan, J. E. (2019). Statespace ARIMA for supplychain forecasting. International Journal of Production Research, 0(0), 1–10. doi: 10.1080/00207543.2019.1600764
auto.ssarima, orders,
msarima, auto.msarima,
sim.ssarima, adam
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37  # 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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