smooth (version 1.4.4)

es: Exponential Smoothing in SSOE state-space model

Description

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

Usage

es(data, model="ZZZ", persistence=NULL, phi=NULL, initial=c("optimal","backcasting"), initialSeason=NULL, ic=c("AICc","AIC","BIC"), cfType=c("MSE","MAE","HAM","MLSTFE","MSTFE","MSEh"), h=10, holdout=FALSE, intervals=FALSE, level=0.95, intervalsType=c("parametric","semiparametric","nonparametric"), intermittent=c("none","auto","fixed","croston","tsb"), bounds=c("usual","admissible","none"), silent=c("none","all","graph","legend","output"), xreg=NULL, initialX=NULL, updateX=FALSE, persistenceX=NULL, transitionX=NULL, ...)

Arguments

data

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

model

The type of ETS model. Can consist of 3 or 4 chars: ANN, AAN, AAdN, AAA, AAdA, MAdM etc. ZZZ means that the model will be selected based on the chosen information criteria type. 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.

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 model and use all its parameters.

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

persistence

Vector of smoothing parameters. If NULL, the parameters are estimated.

phi

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

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

initialSeason

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

ic

Information criterion to use in model selection.

cfType

Type of Cost Function used in optimization. cfType can be: MSE (Mean Squared Error), MAE (Mean Absolute Error), HAM (Half Absolute Moment), MLSTFE - Mean Log Squared Trace Forecast Error, MSTFE - Mean Squared Trace Forecast Error and MSEh - optimisation using only h-steps ahead error. If cfType!="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, aMSTFE and aMLSTFE. These can be useful in cases of small samples.

h

Length of forecasting horizon.

holdout

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

intervals

If TRUE, prediction intervals are constructed.

level

Confidence level. Defines width of prediction interval.

intervalsType

Type of intervals to construct. First letter can be used instead of the whole word. This can be:

• parametric use state-space structure of ETS. For multiplicative models they are approximated using the same function as for additive. As a result they are a bit wider than should be but are still efficient. In case of mixed models this is done using simulations, which may take longer time than for the pure additive and pure multiplicative models.

semiparametric are based on covariance matrix of 1 to h steps ahead errors and assumption of normal distribution.

nonparametric intervals use 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.

intermittent

Defines type of intermittent model used. Can be: 1. none, meaning that the data should be considered as non-intermittent; 2. fixed, taking into account constant Bernoulli distribution of demand occurancies; 3. croston, based on Croston, 1972 method with SBA correction; 4. tsb, based on Teunter et al., 2011 method. 5. auto - automatic selection of intermittency type based on information criteria. The first letter can be used instead.

bounds

What type of bounds to use for smoothing parameters ("admissible" or "usual"). 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

Vector (either numeric or time series) or 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 Naive.

initialX

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

updateX

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

persistenceX

Persistence vector $g_X$, containing smoothing parameters for exogenous variables. If NULL, then estimated. Prerequisite - non-NULL xreg.

transitionX

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.

Value

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

• model - type of constructed model.
• 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.
• initialType - Typetof initial values used.
• initial - intial values of the state vector (non-seasonal).
• initialSeason - intial values of the seasonal part of state vector.
• nParam - number of estimated parameters.
• fitted - fitted values of ETS.
• forecast - point forecast of ETS.
• lower - lower bound of prediction interval. When intervals=FALSE then NA is returned.
• upper - higher bound of prediction interval. When intervals=FALSE 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 - standard deviation of the residuals (taking degrees of freedom into account).
• intervalsType - type of intervals asked by user.
• level - confidence level for intervals.
• actuals - original data.
• holdout - holdout part of the original data.
• iprob - fitted and forecasted values of the probability of demand occurrence.
• intermittent - type of intermittent model fitted to the data.
• xreg - provided vector or matrix of 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 and BIC.
• cf - Cost function value.
• cfType - Type of cost function used in the estimation.
• FI - Fisher Information. Equal to NULL if FI=FALSE or when FI is not provided at all.
• accuracy - vector or accuracy measures for the holdout sample. Includes MPE, MAPE, SMAPE, MASE, MAE/mean, RelMAE and Bias coefficient (based on complex numbers). Available only when holdout=TRUE.

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,
• intervalsType,
• level,
• actuals,
• holdout,
• iprob,
• intermittent,
• ICs - combined ic,
• ICw - ic weights used in the combination,
• cfType,
• xreg,
• accuracy.

Details

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

$y_[t] = o_[t] (w(v_[t-l]) + x_t a_[t-1] + r(v_[t-l]) \epsilon_[t])$

$v_[t] = f(v_[t-l]) + g(v_[t-l]) \epsilon_[t]$

$a_[t] = F_[X] a_[t-1] + g_[X] \epsilon_[t] / x_[t]$

Where $o_[t]$ is Bernoulli distributed random variable (in case of normal data it equals to 1 for all observations), $v_[t]$ is a state vector and $l$ is a vector of lags, $x_t$ vector of exogenous parameters. w(.) is measurement function, r(.) is an error function, f(.) is transition function and g(.) is persistence function.

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

References

1. Hyndman, R.J., Koehler, A.B., Ord, J.K., and Snyder, R.D. (2008) Forecasting with exponential smoothing: the state space approach, Springer-Verlag. http://www.exponentialsmoothing.net.
2. 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.
3. Kolassa, S. (2011) Combining exponential smoothing forecasts using Akaike weights. International Journal of Forecasting, 27, pp 238 - 251.
4. Teunter R., Syntetos A., Babai Z. (2011). Intermittent demand: Linking forecasting to inventory obsolescence. European Journal of Operational Research 214, 606-615.
5. Croston, J. (1972) Forecasting and stock control for intermittent demands. Operational Research Quarterly, 23(3), 289-303.

See Also

ets, forecast, ts, sim.es

Examples

library(Mcomp)

# See how holdout and trace parameters influence the forecast
es(M3$N1245$x,model="AAdN",h=8,holdout=FALSE,cfType="MSE")
## Not run: es(M3$N2568$x,model="MAM",h=18,holdout=TRUE,cfType="MSTFE")

# Model selection example
es(M3$N1245$x,model="ZZN",ic="AIC",h=8,holdout=FALSE,bounds="a")

# Model selection. Compare AICc of these two models:
## Not run: es(M3$N1683$x,"ZZZ",h=10,holdout=TRUE)
# es(M3$N1683$x,"MAdM",h=10,holdout=TRUE)## End(Not run)

# Combination example
## Not run: es(M3$N1245$x,model="CCC",h=8,holdout=TRUE)

# Model selection using a specified pool of models
test <- es(M3$N1587$x,model=c("ANN","AAM","AMdA"),h=18)

# Redo previous model and produce prediction intervals
es(M3$N1587$x,model=test,h=18,intervals=TRUE)

# Semiparametric intervals example
es(M3$N1587$x,h=18,holdout=TRUE,intervals=TRUE,intervalsType="s")

# Exogenous variables in ETS example
## Not run: x <- cbind(c(rep(0,25),1,rep(0,43)),c(rep(0,10),1,rep(0,58)))
# es(ts(c(M3$N1457$x,M3$N1457$xx),frequency=12),h=18,holdout=TRUE,xreg=x,cfType="aMSTFE")
# test <- es(ts(c(M3$N1457$x,M3$N1457$xx),frequency=12),h=18,holdout=TRUE,xreg=x,updateX=TRUE)## End(Not run)

# This will be the same model as in previous line but estimated on new portion of data
## Not run: es(ts(c(M3$N1457$x,M3$N1457$xx),frequency=12),model=test,h=18,holdout=FALSE)

# Intermittent data example
x <- rpois(100,0.2)
# Croston's method with the best ETS for demand sizes
es(x,"ZZN",intermittent="croston")
# TSB based on iETS(M,N,N)
es(x,"MNN",intermittent="tsb")
# Constant probability based on iETS(M,N,N)
es(x,"MNN",intermittent="fixed")
# Best type of intermittent model based on iETS(Z,Z,N)
test <- es(x,"ZZN",intermittent="auto")

summary(test)
forecast(test)
plot(forecast(test))