# auto.ces: Complex Exponential Smoothing Auto In config-i1/smooth: Forecasting Using State Space Models

## Description

Function estimates CES in state space form with information potential equal to errors with different seasonality types and chooses the one with the lowest IC value.

## Usage

 ```1 2 3 4 5 6 7 8``` ```auto.ces(y, models = c("none", "simple", "full"), 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. `models` The vector containing several types of seasonality that should be used in CES selection. See ces for more details about the possible types of seasonal models. `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 produce Fisher Information matrix, which then can be used to calculated variances of parameters of the model.

## Details

The function estimates several Complex Exponential Smoothing in the state space 2 described in Svetunkov, Kourentzes (2015) with the information potential equal to the approximation error using different types of seasonality and chooses the one with the lowest value of information criterion.

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

## Value

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

## Author(s)

Ivan Svetunkov, ivan@svetunkov.ru

## References

• Svetunkov, I., Kourentzes, N. (February 2015). Complex exponential smoothing. Working Paper of Department of Management Science, Lancaster University 2015:1, 1-31.

• Svetunkov I., Kourentzes N. (2017) Complex Exponential Smoothing for Time Series Forecasting. Not yet published.

```ces, es, forecast, ts```
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15``` ```y <- ts(rnorm(100,10,3),frequency=12) # CES with and without holdout auto.ces(y,h=20,holdout=TRUE) auto.ces(y,h=20,holdout=FALSE) # Selection between "none" and "full" seasonalities auto.ces(AirPassengers,h=8,holdout=TRUE, models=c("n","f"),interval="p",level=0.8,ic="AIC") ourModel <- auto.ces(AirPassengers,interval="sp") summary(ourModel) forecast(ourModel) plot(forecast(ourModel)) ```