ssarima: State-Space ARIMA

Description

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

Usage

ssarima(data, 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"), cfType = c("MSE", "MAE", "HAM",
  "MSEh", "TMSE", "GTMSE"), h = 10, holdout = FALSE, cumulative = FALSE,
  intervals = c("none", "parametric", "semiparametric", "nonparametric"),
  level = 0.95, intermittent = c("none", "auto", "fixed", "interval",
  "probability", "sba", "logistic"), imodel = "MNN",
  bounds = c("admissible", "none"), silent = c("all", "graph", "legend",
  "output", "none"), xreg = NULL, xregDo = c("use", "select"),
  initialX = NULL, updateX = FALSE, persistenceX = NULL,
  transitionX = NULL, ...)

Arguments

data

Vector or ts object, containing data needed to be forecasted.

orders

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.

lags

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.

constant

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.

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.

The information criterion used in the model selection procedure.

cfType

Type of Cost Function used in optimization. cfType can be: MSE (Mean Squared Error), MAE (Mean Absolute Error), HAM (Half Absolute Moment), GMSTFE - Mean Log Squared Trace Forecast Error, MSTFE - Mean Squared Trace Forecast Error and MSEh - optimisation using only h-steps ahead error, TFL - trace forecast likelihood. 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 aGMSTFE. These can be useful in cases of small samples.

Length of forecasting horizon.

holdout

If TRUE, holdout sample of size h is taken from the end of the data.

cumulative

If TRUE, then the cumulative forecast and prediction intervals are produced instead of the normal ones. This is useful for inventory control systems.

intervals

Type of intervals to construct. This can be:

none, aka n - do not produce prediction intervals.
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.
semiparametric, sp - intervals based on covariance matrix of 1 to h steps ahead errors and assumption of normal / log-normal distribution (depending on error type).
nonparametric, np - intervals 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 intervals are constructed, while the latter is equivalent to none.

level

Confidence level. Defines width of prediction interval.

intermittent

Defines type of intermittent model used. Can be: 1. none, meaning that the data should be considered as non-intermittent; 2. fixed, taking into account constant Bernoulli distribution of demand occurrences; 3. interval, Interval-based model, underlying Croston, 1972 method; 4. probability, Probability-based model, underlying Teunter et al., 2011 method. 5. auto - automatic selection of intermittency type based on information criteria. The first letter can be used instead. 6. "sba" - Syntetos-Boylan Approximation for Croston's method (bias correction) discussed in Syntetos and Boylan, 2005. 7. "logistic" - the probability is estimated based on logistic regression model principles.

imodel

Type of ETS model used for the modelling of the time varying probability. Object of the class "iss" can be provided here, and its parameters would be used in iETS model.

bounds

What type of bounds to use in the model estimation. The first letter can be used instead of the whole word.

silent

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").

xreg

Vector (either numeric or time series) or 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.

xregDo

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...

initialX

Vector of initial parameters for exogenous variables. Ignored if xreg is NULL.

updateX

If TRUE, transition matrix for exogenous variables is estimated, introducing non-linear interactions between parameters. Prerequisite - non-NULL xreg.

persistenceX

Persistence vector $g_{X}$ , containing smoothing parameters for exogenous variables. If NULL, then estimated. Prerequisite - non-NULL xreg.

transitionX

Transition matrix $F_{x}$ for exogenous variables. Can be provided as a vector. Matrix will be formed using the default matrix(transition,nc,nc), where nc is number of components in state vector. If NULL, then estimated. Prerequisite - non-NULL xreg.

...

Other non-documented parameters.

Vectors of orders can be passed here using ar.orders, i.orders and ma.orders. orders variable needs to be NULL in this case.

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.

Value

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.
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 of ETS.
forecast - the point forecast of ETS.
lower - the lower bound of prediction interval. When intervals="none" then NA is returned.
upper - the higher bound of prediction interval. When intervals="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).
intervals - type of intervals asked by user.
level - confidence level for intervals.
cumulative - whether the produced forecast was cumulative or not.
actuals - the original data.
holdout - the holdout part of the original data.
imodel - model of the class "iss" if intermittent model was estimated. If the model is non-intermittent, then imodel is NULL.
xreg - provided vector or matrix of exogenous variables. If xregDo="s", then this value will contain only selected 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.
logLik - log-likelihood of the function.
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 - 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.

Details

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 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):

$y_{t} = o_{t} (w^{'} v_{t - l} + x_{t} a_{t - 1} + ϵ_{t})$

$v_{t} = F v_{t - l} + g ϵ_{t}$

$a_{t} = F_{X} a_{t - 1} + g_{X} ϵ_{t} / x_{t}$

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...

References

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.

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. http://www.exponentialsmoothing.net.

Examples

Run this code

# NOT RUN {
# 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,intervals="p")

# The previous one is equivalent to:
# }
# NOT RUN {
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,intervals="p")
# }
# NOT RUN {
# 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]
# }
# NOT RUN {
ssarima(rnorm(118,100,3),orders=list(ar=c(2,1)),lags=c(1,4),h=18,holdout=TRUE)
# }
# NOT RUN {
# SARIMA(1,1,1)(0,0,1)[4] with different initialisations
# }
# NOT RUN {
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")
# }
# NOT RUN {
# SARIMA of a perculiar order on AirPassengers data
# }
# NOT RUN {
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)
# }
# NOT RUN {
# ARIMA(1,1,1) with Mean Squared Trace Forecast Error
# }
# NOT RUN {
ssarima(rnorm(118,100,3),orders=list(ar=1,i=1,ma=1),lags=1,h=18,holdout=TRUE,cfType="TMSE")
ssarima(rnorm(118,100,3),orders=list(ar=1,i=1,ma=1),lags=1,h=18,holdout=TRUE,cfType="aTMSE")
# }
# NOT RUN {
# 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))

# SARIMA(0,1,1) with exogenous variables with crazy estimation of xreg
# }
# NOT RUN {
ourModel <- ssarima(rnorm(118,100,3),orders=list(i=1,ma=1),h=18,holdout=TRUE,
                    xreg=c(1:118),updateX=TRUE)
# }
# NOT RUN {
summary(ourModel)
forecast(ourModel)
plot(forecast(ourModel))

# }

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