smooth (version 2.6.0)

es: Exponential Smoothing in SSOE state space model


Function constructs ETS model and returns forecast, fitted values, errors and matrix of states.


es(y, model = "ZZZ", persistence = NULL, phi = NULL,
  initial = c("optimal", "backcasting"), initialSeason = NULL,
  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, occurrence = c("none",
  "auto", "fixed", "general", "odds-ratio", "inverse-odds-ratio", "direct"),
  oesmodel = "MNN", bounds = c("usual", "admissible", "none"),
  silent = c("all", "graph", "legend", "output", "none"), xreg = NULL,
  xregDo = c("use", "select"), initialX = NULL, updateX = FALSE,
  persistenceX = NULL, transitionX = NULL, ...)



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


The type of ETS model. The first letter stands for the type of the error term ("A" or "M"), the second (and sometimes the third as well) is for the trend ("N", "A", "Ad", "M" or "Md"), and the last one is for the type of seasonality ("N", "A" or "M"). So, the function accepts words with 3 or 4 characters: ANN, AAN, AAdN, AAA, AAdA, MAdM etc. ZZZ means that the model will be selected based on the chosen information criteria type. Models pool can be restricted with additive only components. This is done via model="XXX". For example, making selection between models with none / additive / damped additive trend component only (i.e. excluding multiplicative trend) can be done with model="ZXZ". Furthermore, selection between multiplicative models (excluding additive components) is regulated using model="YYY". This can be useful for positive data with low values (for example, slow moving products). Finally, if model="CCC", then all the models are estimated and combination of their forecasts using AIC weights is produced (Kolassa, 2011). This can also be regulated. For example, model="CCN" will combine forecasts of all non-seasonal models and model="CXY" will combine forecasts of all the models with non-multiplicative trend and non-additive seasonality with either additive or multiplicative error. Not sure why anyone would need this thing, but it is available.

The parameter model can also be a vector of names of models for a finer tuning (pool of models). For example, model=c("ANN","AAA") will estimate only two models and select the best of them.

Also model can accept a previously estimated ES or ETS (from forecast package) model and use all its parameters.

Keep in mind that model selection with "Z" components uses Branch and Bound algorithm and may skip some models that could have slightly smaller information criteria.


Persistence vector \(g\), containing smoothing parameters. If NULL, then estimated.


Value of damping parameter. If NULL then it is estimated.


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 (advised for data with high frequency). If character, then initialSeason will be estimated in the way defined by initial.


Vector of initial values for seasonal components. If NULL, they are estimated during optimisation.


The information criterion used in the model selection procedure.


The type of Loss Function used in optimization. loss can be: 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.


Length of forecasting horizon.


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


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


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).


Confidence level. Defines width of prediction interval.


The type of model used in probability estimation. Can be "none" - none, "fixed" - constant probability, "general" - the general Beta model with two parameters, "odds-ratio" - the Odds-ratio model with b=1 in Beta distribution, "inverse-odds-ratio" - the model with a=1 in Beta distribution, "direct" - the TSB-like (Teunter et al., 2011) probability update mechanism a+b=1, "auto" - the automatically selected type of occurrence model.


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


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


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").


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.


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...


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


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


The persistence vector \(g_X\), containing smoothing parameters for exogenous variables. If NULL, then estimated. Prerequisite - non-NULL xreg.


The 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. For example FI=TRUE will make the function also produce Fisher Information matrix, which then can be used to calculated variances of smoothing parameters and initial states of the model. Parameters B, lb and ub can be passed via ellipsis as well. In this case they will be used for optimisation. B sets the initial values before the optimisation, lb and ub define lower and upper bounds for the search inside of the specified bounds. These values should have length equal to the number of parameters to estimate. You can also pass two parameters to the optimiser: 1. maxeval - maximum number of evaluations to carry on; 2. xtol_rel - the precision of the optimiser. The default values used in es() are maxeval=500 and xtol_rel=1e-8. You can read more about these parameters in the documentation of nloptr function.


Object of class "smooth" is returned. It contains the list of the following values for classical ETS models:

  • model - type of constructed model.

  • formula - mathematical formula, describing interactions between components of es() and exogenous variables.

  • timeElapsed - time elapsed for the construction of the model.

  • states - matrix of the components of ETS.

  • persistence - persistence vector. This is the place, where smoothing parameters live.

  • phi - value of damping parameter.

  • transition - transition matrix of the model.

  • measurement - measurement vector of the model.

  • initialType - type of the initial values used.

  • initial - initial values of the state vector (non-seasonal).

  • initialSeason - initial values of the seasonal part of state vector.

  • 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 - fitted values of ETS. In case of the intermittent model, the fitted are multiplied by the probability of occurrence.

  • 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.

  • errors - trace forecast in-sample errors, returned as a matrix. In the case of trace forecasts this is the matrix used in optimisation. In non-trace estimations it is returned just for the information.

  • s2 - variance of the residuals (taking degrees of freedom into account). This is an unbiased estimate of variance.

  • 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.

  • occurrence - model of the class "oes" if the occurrence model was estimated. If the model is non-intermittent, then occurrence 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, BIC and BICc.

  • logLik - concentrated log-likelihood of the function.

  • lossValue - loss 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.

If combination of forecasts is produced (using model="CCC"), then a shorter list of values is returned:

  • model,

  • timeElapsed,

  • initialType,

  • fitted,

  • forecast,

  • lower,

  • upper,

  • residuals,

  • s2 - variance of additive error of combined one-step-ahead forecasts,

  • interval,

  • level,

  • cumulative,

  • y,

  • holdout,

  • occurrence,

  • ICs - combined ic,

  • ICw - ic weights used in the combination,

  • loss,

  • xreg,

  • accuracy.


Function estimates ETS in a form of the Single Source of Error state space model of the following type:

$$y_{t} = o_t (w(v_{t-l}) + h(x_t, a_{t-1}) + r(v_{t-l}) \epsilon_{t})$$

$$v_{t} = f(v_{t-l}) + g(v_{t-l}) \epsilon_{t}$$

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

Where \(o_{t}\) is the Bernoulli distributed random variable (in case of normal data it equals to 1 for all observations), \(v_{t}\) is the state vector and \(l\) is the vector of lags, \(x_t\) is the vector of exogenous variables. w(.) is the measurement function, r(.) is the error function, f(.) is the transition function, g(.) is the persistence function and h(.) is the explanatory variables function. \(a_t\) is the vector of parameters for exogenous variables, \(F_{X}\) is the transitionX matrix and \(g_{X}\) is the persistenceX matrix. Finally, \(\epsilon_{t}\) is the error term.

For the details see Hyndman et al.(2008).

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

Also, there are posts about the functions of the package smooth on the website of Ivan Svetunkov: - they explain the underlying models and how to use the functions.


  • 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.

  • 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.

  • 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: [10.1287/mnsc.1120.1667](

See Also

ets, forecast, ts,



# See how holdout and trace parameters influence the forecast
# }
# }
# Model selection example

# Model selection. Compare AICc of these two models:
# }
# }
# Model selection, excluding multiplicative trend
# }
# }
# Combination example
# }
# }
# Model selection using a specified pool of models
ourModel <- es(M3$N1587$x,model=c("ANN","AAM","AMdA"),h=18)

# Redo previous model and produce prediction interval

# Semiparametric interval example
# }
# }
# Exogenous variables in ETS example
# }
x <- cbind(c(rep(0,25),1,rep(0,43)),c(rep(0,10),1,rep(0,58)))
y <- ts(c(M3$N1457$x,M3$N1457$xx),frequency=12)
ourModel <- es(ts(c(M3$N1457$x,M3$N1457$xx),frequency=12),h=18,holdout=TRUE,xreg=x,updateX=TRUE)
# }
# This will be the same model as in previous line but estimated on new portion of data
# }
# }
# Intermittent data example
x <- rpois(100,0.2)
# Odds ratio model with the best ETS for demand sizes
# Inverse odds ratio model (underlies Croston) on iETS(M,N,N)
# Constant probability based on iETS(M,N,N)
# Best type of occurrence model based on iETS(Z,Z,N)
ourModel <- es(x,"ZZN",occurrence="auto")


# }