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, ...)
```

y

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

model

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

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

, then 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

The information criterion used in the model selection procedure.

loss

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`

.

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.

occurrence

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.

oesmodel

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.

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.

updateX

If `TRUE`

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

.

persistenceX

The persistence vector \(g_X\), containing smoothing
parameters for exogenous variables. If `NULL`

, then estimated.
Prerequisite - non-NULL `xreg`

.

transitionX

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: https://forecasting.svetunkov.ru/en/tag/smooth/ - 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. http://dx.doi.org/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.

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](https://doi.org/10.1287/mnsc.1120.1667)

# NOT RUN { library(Mcomp) # See how holdout and trace parameters influence the forecast es(M3$N1245$x,model="AAdN",h=8,holdout=FALSE,loss="MSE") # } # NOT RUN { es(M3$N2568$x,model="MAM",h=18,holdout=TRUE,loss="TMSE") # } # NOT RUN { # 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) # } # NOT RUN { # Model selection, excluding multiplicative trend # } # NOT RUN { es(M3$N1245$x,model="ZXZ",h=8,holdout=TRUE) # } # NOT RUN { # Combination example # } # NOT RUN { es(M3$N1245$x,model="CCN",h=8,holdout=TRUE) # } # NOT RUN { # 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 es(M3$N1587$x,model=ourModel,h=18,interval="p") # Semiparametric interval example # } # NOT RUN { es(M3$N1587$x,h=18,holdout=TRUE,interval="sp") # } # NOT RUN { # 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))) y <- ts(c(M3$N1457$x,M3$N1457$xx),frequency=12) es(y,h=18,holdout=TRUE,xreg=x,loss="aTMSE",interval="np") ourModel <- es(ts(c(M3$N1457$x,M3$N1457$xx),frequency=12),h=18,holdout=TRUE,xreg=x,updateX=TRUE) # } # 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=ourModel,h=18,holdout=FALSE) # } # NOT RUN { # Intermittent data example x <- rpois(100,0.2) # Odds ratio model with the best ETS for demand sizes es(x,"ZZN",occurrence="o") # Inverse odds ratio model (underlies Croston) on iETS(M,N,N) es(x,"MNN",occurrence="i") # Constant probability based on iETS(M,N,N) es(x,"MNN",occurrence="fixed") # Best type of occurrence model based on iETS(Z,Z,N) ourModel <- es(x,"ZZN",occurrence="auto") par(mfcol=c(2,2)) plot(ourModel) summary(ourModel) forecast(ourModel) plot(forecast(ourModel)) # }