# smoothCombine: Combination of forecasts of state space models In config-i1/smooth: Forecasting Using State Space Models

## Description

Function constructs ETS, SSARIMA, CES, GUM and SMA and combines their forecasts using IC weights.

## Usage

 ```1 2 3 4 5 6 7 8``` ```smoothCombine(y, models = NULL, initial = c("optimal", "backcasting"), ic = c("AICc", "AIC", "BIC", "BICc"), loss = c("MSE", "MAE", "HAM", "MSEh", "TMSE", "GTMSE", "MSCE"), h = 10, holdout = FALSE, cumulative = FALSE, interval = c("none", "parametric", "likelihood", "semiparametric", "nonparametric"), level = 0.95, bins = 200, intervalCombine = c("quantile", "probability"), 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` List of the estimated smooth models to use in the combination. If `NULL`, then all the models are estimated in the function. `initial` 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. `bins` The number of bins for the prediction interval. The lower value means faster work of the function, but less precise estimates of the quantiles. This needs to be an even number. `intervalCombine` How to average the prediction interval: quantile-wise (`"quantile"`) or probability-wise (`"probability"`). `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. `...` This currently determines nothing. `timeElapsed` - time elapsed for the construction of the model. `initialType` - type of the initial values used. `fitted` - fitted values of ETS. `quantiles` - the 3D array of produced quantiles if `interval!="none"` with the dimensions: (number of models) x (bins) x (h). `forecast` - point forecast of ETS. `lower` - lower bound of prediction interval. When `interval="none"` then NA is returned. `upper` - higher bound of prediction interval. When `interval="none"` then NA is returned. `residuals` - residuals of the estimated model. `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` - original data. `holdout` - 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. `ICs` - values of information criteria of the model. Includes AIC, AICc, BIC and BICc. `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.

## Details

The combination of these models using information criteria weights is possible because they are all formulated in Single Source of Error framework. Due to the the complexity of some of the models, the estimation process may take some time. So be patient.

The prediction interval are combined either probability-wise or quantile-wise (Lichtendahl et al., 2013), which may take extra time, because we need to produce all the distributions for all the models. This can be sped up with the smaller value for bins parameter, but the resulting interval may be imprecise.

## 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.

• Kolassa, S. (2011) Combining exponential smoothing forecasts using Akaike weights. International Journal of Forecasting, 27, pp 238 - 251.

• 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: doi: 10.1287/mnsc.1120.1667

## See Also

```es, auto.ssarima, auto.ces, auto.gum, sma```

## Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13``` ```library(Mcomp) ourModel <- smoothCombine(M3[],interval="p") plot(ourModel) # models parameter accepts either previously estimated smoothCombine # or a manually formed list of smooth models estimated in sample: smoothCombine(M3[],models=ourModel) ## Not run: models <- list(es(M3[]), sma(M3[])) smoothCombine(M3[],models=models) ## End(Not run) ```

config-i1/smooth documentation built on June 16, 2021, 2:13 p.m.