ges: General Exponential Smoothing

Description

Function constructs General Exponential Smoothing, estimating matrices F, w, vector g and initial parameters.

Usage

ges(data, orders=c(1,1), lags=c(1,frequency(data)), persistence=NULL, transition=NULL, measurement=NULL, initial=c("optimal","backcasting"), 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("admissible","none"), silent=c("none","all","graph","legend","output"), xreg=NULL, initialX=NULL, updateX=FALSE, persistenceX=NULL, transitionX=NULL, ...)

Arguments

data

The data that needs to be forecasted.

orders

Order of the model. Specified as vector of number of states with different lags. For example, orders=c(1,1) means that there are two states: one of the first lag type, the second of the second type.

lags

Defines lags for the corresponding orders. If, for example, orders=c(1,1) and lags are defined as lags=c(1,12), then the model will have two states: the first will have lag 1 and the second will have lag 12. The length of lags must correspond to the length of orders.

persistence

Persistence vector $g$, containing smoothing parameters. If NULL, then estimated.

transition

Transition matrix $F$. Can be provided as a vector. Matrix will be formed using the default matrix(transition,nc,nc), where nc is the number of components in state vector. If NULL, then estimated.

measurement

Measurement vector $w$. If NULL, then 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.

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.

The forecasting horizon.

holdout

If TRUE, holdout 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 parameter model can accept a previously estimated GES 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:

model - name of the estimated model.
timeElapsed - time elapsed for the construction of the model.
states - matrix of fuzzy components of GES, where rows correspond to time and cols to states.
initialType - Typetof initial values used.
initial - initial values of state vector (extracted from states).
nParam - number of estimated parameters.
measurement - matrix w.
transition - matrix F.
persistence - persistence vector. This is the place, where smoothing parameters live.
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 - the residuals of the estimated model.
errors - matrix of 1 to h steps ahead errors.
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.

Details

The function estimates the Single Source of Error State-space model of the following type:

$y_[t] = o_[t] (w' v_[t-l] + x_t a_[t-1] + \epsilon_[t])$

$v_[t] = F v_[t-l] + g \epsilon_[t]$

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

Where $o_[t]$ is Bernoulli distributed random variable (in case of normal data equal to 1), $v_[t]$ is a state vector (defined using orders) and $l$ is a vector of lags, $x_t$ vector of exogenous parameters.

References

No references yet.

Examples

Run this code

# Something simple:
ges(rnorm(118,100,3),orders=c(1),lags=c(1),h=18,holdout=TRUE,bounds="a",intervals=TRUE)

# A more complicated model with seasonality
## Not run: test <- ges(rnorm(118,100,3),orders=c(2,1),lags=c(1,4),h=18,holdout=TRUE)

# Redo previous model on a new data and produce prediction intervals
## Not run: ges(rnorm(118,100,3),model=test,h=18,intervals=TRUE)

# Produce something crazy with optimal initials (not recommended)
## Not run: ges(rnorm(118,100,3),orders=c(1,1,1),lags=c(1,3,5),h=18,holdout=TRUE,initial="o")

# Simpler model estiamted using trace forecast error cost function and its analytical analogue
## Not run: ges(rnorm(118,100,3),orders=c(1),lags=c(1),h=18,holdout=TRUE,bounds="n",cfType="MSTFE")
# ges(rnorm(118,100,3),orders=c(1),lags=c(1),h=18,holdout=TRUE,bounds="n",cfType="aMSTFE")## End(Not run)

# Introduce exogenous variables
## Not run: ges(rnorm(118,100,3),orders=c(1),lags=c(1),h=18,holdout=TRUE,xreg=c(1:118))

# Ask for their update
## Not run: ges(rnorm(118,100,3),orders=c(1),lags=c(1),h=18,holdout=TRUE,xreg=c(1:118),updateX=TRUE)

# Do the same but now let's shrink parameters...
## Not run: ges(rnorm(118,100,3),orders=c(1),lags=c(1),h=18,xreg=c(1:118),updateX=TRUE,cfType="MSTFE")
# 
# test <- ges(rnorm(118,100,3),orders=c(1),lags=c(1),h=18,holdout=TRUE,cfType="aMSTFE")
# 
# summary(test)
# forecast(test)
# plot(forecast(test))## End(Not run)

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