# auto.msarima: Automatic Multiple Seasonal ARIMA In config-i1/smooth: Forecasting Using State Space Models

## Description

Function selects the best State Space ARIMA based on information criteria, using fancy branch and bound mechanism. The resulting model can be not optimal in IC meaning, but it is usually reasonable. This mechanism is described in Svetunkov & Boylan (2019).

## Usage

 ```1 2 3 4 5 6 7 8 9``` ```auto.msarima(y, orders = list(ar = c(3, 3), i = c(2, 1), ma = c(3, 3)), lags = c(1, frequency(y)), combine = FALSE, fast = TRUE, constant = 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 maximum orders to check, containing vector variables `ar`, `i` and `ma`. 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). The length of `lags` must correspond to the length of `orders`. There is no restrictions on the length of `lags` vector. `combine` If `TRUE`, then resulting ARIMA is combined using AIC weights. `fast` If `TRUE`, then some of the orders of ARIMA are skipped. This is not advised for models with `lags` greater than 12. `constant` If `NULL`, then the function will check if constant is needed. if `TRUE`, then constant is forced in the model. Otherwise constant is not used. `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. For example `FI=TRUE` will make the function also produce Fisher Information matrix, which then can be used to calculated variances of parameters of the model. Maximum orders to check can also be specified separately, however `orders` variable must be set to `NULL`: `ar.orders` - Maximum order of AR term. Can be vector, defining max orders of AR, SAR etc. `i.orders` - Maximum order of I. Can be vector, defining max orders of I, SI etc. `ma.orders` - Maximum order of MA term. Can be vector, defining max orders of MA, SMA etc.

## Details

The function constructs bunch of ARIMAs in Single Source of Error state space form (see msarima documentation) and selects the best one based on information criterion. It works faster than auto.ssarima on large datasets and high frequency data.

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 time...

For some more information about the model and its implementation, see the vignette: `vignette("ssarima","smooth")`

## Value

Object of class "smooth" is returned. See msarima for details.

## Author(s)

Ivan Svetunkov, ivan@svetunkov.ru

## References

• 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. doi: 10.1007/978-3-540-71918-2.

• Svetunkov Ivan and Boylan John E. (2017). Multiplicative State-Space Models for Intermittent Time Series. Working Paper of Department of Management Science, Lancaster University, 2017:4 , 1-43.

• Teunter R., Syntetos A., Babai Z. (2011). Intermittent demand: Linking forecasting to inventory obsolescence. European Journal of Operational Research, 214, 606-615.

• Croston, J. (1972) Forecasting and stock control for intermittent demands. Operational Research Quarterly, 23(3), 289-303.

• Syntetos, A., Boylan J. (2005) The accuracy of intermittent demand estimates. International Journal of Forecasting, 21(2), 303-314.

• Svetunkov, I., & Boylan, J. E. (2019). State-space ARIMA for supply-chain forecasting. International Journal of Production Research, 0(0), 1–10. doi: 10.1080/00207543.2019.1600764

```es, ces, sim.es, gum, msarima```
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17``` ```x <- rnorm(118,100,3) # The best ARIMA for the data ourModel <- auto.msarima(x,orders=list(ar=c(2,1),i=c(1,1),ma=c(2,1)),lags=c(1,12), h=18,holdout=TRUE,interval="np") # The other one using optimised states auto.msarima(x,orders=list(ar=c(3,2),i=c(2,1),ma=c(3,2)),lags=c(1,12), initial="o",h=18,holdout=TRUE) # And now combined ARIMA auto.msarima(x,orders=list(ar=c(3,2),i=c(2,1),ma=c(3,2)),lags=c(1,12), combine=TRUE,h=18,holdout=TRUE) summary(ourModel) forecast(ourModel) plot(forecast(ourModel)) ```