Function estimates CES in state space form with information potential equal to errors and returns several variables.

```
ces(y, seasonality = c("none", "simple", "partial", "full"),
initial = c("backcasting", "optimal"), a = NULL, b = NULL,
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, ...)
```

y

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

seasonality

The type of seasonality used in CES. Can be: `none`

- No seasonality; `simple`

- Simple seasonality, using lagged CES
(based on `t-m`

observation, where `m`

is the seasonality lag);
`partial`

- Partial seasonality with real seasonal components
(equivalent to additive seasonality); `full`

- Full seasonality with
complex seasonal components (can do both multiplicative and additive
seasonality, depending on the data). First letter can be used instead of
full words. Any seasonal CES can only be constructed for time series
vectors.

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.

a

First complex smoothing parameter. Should be a complex number.

NOTE! CES is very sensitive to a and b values so it is advised either to leave them alone, or to use values from previously estimated model.

b

Second complex smoothing parameter. Can be real if
`seasonality="partial"`

. In case of `seasonality="full"`

must be
complex number.

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 parameter
`model`

can accept a previously estimated CES model and use all its
parameters. `FI=TRUE`

will make the function produce Fisher
Information matrix, which then can be used to calculated variances of
parameters of the model.

Object of class "smooth" is returned. It contains the list of the following values:

`model`

- type of constructed model.`timeElapsed`

- time elapsed for the construction of the model.`states`

- the matrix of the components of CES. The included minimum is "level" and "potential". In the case of seasonal model the seasonal component is also included. In the case of exogenous variables the estimated coefficients for the exogenous variables are also included.`a`

- complex smoothing parameter in the form a0 + ia1`b`

- smoothing parameter for the seasonal component. Can either be real (if`seasonality="P"`

) or complex (if`seasonality="F"`

) in a form b0 + ib1.`persistence`

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

- transition matrix of the model.`measurement`

- measurement vector of the model.`initialType`

- Type of the initial values used.`initial`

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

- the fitted values of CES.`forecast`

- the point forecast of CES.`lower`

- the lower bound of prediction interval. When`interval="none"`

then NA is returned.`upper`

- the upper bound of prediction interval. When`interval="none"`

then NA is returned.`residuals`

- the residuals of the estimated model.`errors`

- The matrix of 1 to h steps ahead errors.`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`

- The data provided in the call of the function.`holdout`

- the 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. exogenous variables were estimated as well.`initialX`

- initial values for parameters of exogenous variables.`ICs`

- values of information criteria of the model. Includes AIC, AICc, BIC and BICc.`logLik`

- log-likelihood of the function.`lossValue`

- Cost 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.

The function estimates Complex Exponential Smoothing in the state space 2 described in Svetunkov, Kourentzes (2017) with the information potential equal to the approximation error. The estimation of initial states of xt is done using backcast.

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

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.

```
# NOT RUN {
y <- rnorm(100,10,3)
ces(y,h=20,holdout=TRUE)
ces(y,h=20,holdout=FALSE)
y <- 500 - c(1:100)*0.5 + rnorm(100,10,3)
ces(y,h=20,holdout=TRUE,interval="p",bounds="a")
ces(Mcomp::M3[[740]],h=8,holdout=TRUE,seasonality="s",interval="sp",level=0.8)
# }
# NOT RUN {
ces(Mcomp::M3[[1683]],h=18,holdout=TRUE,seasonality="s",interval="sp")
ces(Mcomp::M3[[1683]],h=18,holdout=TRUE,seasonality="p",interval="np")
ces(Mcomp::M3[[1683]],h=18,holdout=TRUE,seasonality="f",interval="p")
# }
# NOT RUN {
# }
# NOT RUN {
x <- cbind(c(rep(0,25),1,rep(0,43)),c(rep(0,10),1,rep(0,58)))
ces(Mcomp::M3[[1457]],holdout=TRUE,interval="np",xreg=x,loss="TMSE")
# }
# NOT RUN {
# }
```

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