smooth (version 1.9.0)

ces: Complex Exponential Smoothing

Description

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

Usage

ces(data, seasonality = c("none", "simple", "partial", "full"),
  initial = c("optimal", "backcasting"), A = NULL, B = NULL,
  ic = c("AICc", "AIC", "BIC"), cfType = c("MSE", "MAE", "HAM", "MLSTFE",
  "MSTFE", "MSEh"), h = 10, holdout = FALSE, intervals = c("none",
  "parametric", "semiparametric", "nonparametric"), level = 0.95,
  intermittent = c("none", "auto", "fixed", "croston", "tsb", "sba"),
  bounds = c("admissible", "none"), silent = c("none", "all", "graph",
  "legend", "output"), xreg = NULL, xregDo = c("use", "select"),
  initialX = NULL, updateX = FALSE, persistenceX = NULL,
  transitionX = NULL, ...)

Arguments

data

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.

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

Type of intervals to construct. This can be:

• none, aka n - do not produce prediction intervals.
• 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.
• semiparametric, sp - intervals based on covariance matrix of 1 to h steps ahead errors and assumption of normal / log-normal distribution (depending on error type).
• nonparametric, np - intervals 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. Former means that parametric intervals are constructed, while latter is equivalent to none.

level

Confidence level. Defines width of prediction interval.

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. 6. "sba" - Syntetos-Boylan Approximation for Croston's method (bias correction) discussed in Syntetos and Boylan, 2005.

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 es function.

xregDo

Variable defines what to do with the provided xreg: "use" means that all of the data should be used, whilie "select" means that a selection using ic should be done. "combine" will be available at some point in future...

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 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.

Value

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.
• initialType - Typetof initial values used.
• initial - the intial values of the state vector (non-seasonal).
• nParam - number of estimated parameters.
• fitted - the fitted values of CES.
• forecast - the point forecast of CES.
• lower - the lower bound of prediction interval. When intervals="none" then NA is returned.
• upper - the upper bound of prediction interval. When intervals="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).
• intervals - type of intervals asked by user.
• level - confidence level for intervals.
• actuals - The data provided in the call of the function.
• holdout - the holdout part of the original data.
• iprob - the 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. 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 CIC (Complex IC).
• logLik - log-likelihood of the function.
• 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 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.

Details

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.

References

• 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.

See Also

ets, forecast, ts, auto.ces

Examples

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,intervals="p",bounds="a")

library("Mcomp")
y <- ts(c(M3$N0740$x,M3$N0740$xx),start=start(M3$N0740$x),frequency=frequency(M3$N0740$x))
ces(y,h=8,holdout=TRUE,seasonality="s",intervals="sp",level=0.8)

## Not run: ------------------------------------
# y <- ts(c(M3$N1683$x,M3$N1683$xx),start=start(M3$N1683$x),frequency=frequency(M3$N1683$x))
# ces(y,h=18,holdout=TRUE,seasonality="s",intervals="sp")
# ces(y,h=18,holdout=TRUE,seasonality="p",intervals="np")
# ces(y,h=18,holdout=TRUE,seasonality="f",intervals="p")
## ---------------------------------------------

## Not run: ------------------------------------
# x <- cbind(c(rep(0,25),1,rep(0,43)),c(rep(0,10),1,rep(0,58)))
# ces(ts(c(M3$N1457$x,M3$N1457$xx),frequency=12),h=18,holdout=TRUE,
#     intervals="np",xreg=x,cfType="MSTFE")
## ---------------------------------------------

# Exogenous variables in CES
## Not run: ------------------------------------
# x <- cbind(c(rep(0,25),1,rep(0,43)),c(rep(0,10),1,rep(0,58)))
# ces(ts(c(M3$N1457$x,M3$N1457$xx),frequency=12),h=18,holdout=TRUE,xreg=x)
# ourModel <- ces(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
# ces(ts(c(M3$N1457$x,M3$N1457$xx),frequency=12),model=ourModel,h=18,holdout=FALSE)
## ---------------------------------------------

# Intermittent data example
x <- rpois(100,0.2)
# Best type of intermittent model based on iETS(Z,Z,N)
ourModel <- ces(x,intermittent="auto")

summary(ourModel)
forecast(ourModel)
plot(forecast(ourModel))