smooth (version 3.1.4)

# ssarima: State Space ARIMA

## Description

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

## Usage

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, ...)

## Arguments

y

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. 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.

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. The orders are set by a user. If you want the automatic order selection, then use auto.ssarima function instead.

constant

If TRUE, constant term is included in the model. Can also be a number (constant value).

AR

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.

MA

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.

ic

The information criterion used in the model selection procedure.

loss

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.

h

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 interval are produced instead of the normal ones. This is useful for inventory control systems.

interval

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).

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 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

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.

xregDo

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...

initialX

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.

## 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.

• 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 - 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.

## Details

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")

## 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.

• 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: 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. 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<U+2013>10. 10.1080/00207543.2019.1600764

auto.ssarima, orders, msarima, auto.msarima, sim.ssarima, adam

## 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,interval="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,interval="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 peculiar 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,loss="TMSE")
ssarima(rnorm(118,100,3),orders=list(ar=1,i=1,ma=1),lags=1,h=18,holdout=TRUE,loss="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))

summary(ourModel)
forecast(ourModel)
plot(forecast(ourModel))

# }


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