Theta Models {forecTheta}

Theta Models

Description

Functions for forecast univariate time series using the Dynamic Optimised Theta Model, Dynamic Standard Theta Model, Optimised Theta Model and Standard Theta Model (Fiorucci et al, 2016). We also provide an implementation for the Standard Theta Method (STheta) of Assimakopoulos and Nikolopoulos (2000).

Usage

	dotm(y, h=5, level=c(80,90,95), s=NULL, par_ini=c(y[1]/2, 0.5, 2),
	estimation=TRUE, lower=c(-1e+10, 0.1, 1.0), upper=c(1e+10, 0.99, 1e+10),
	opt.method="Nelder-Mead", xreg=NULL)

	dstm(y, h=5, level=c(80,90,95), s=NULL, par_ini=c(y[1]/2, 0.5), estimation=TRUE,
	lower=c(-1e+10, 0.1), upper=c(1e+10, 0.99), opt.method="Nelder-Mead", xreg=NULL)

	otm(y, h=5, level=c(80,90,95), s=NULL, par_ini=c(y[1]/2, 0.5, 2),
	estimation=TRUE, lower=c(-1e+10, 0.1, 1.0), upper=c(1e+10, 0.99, 1e+10),
	opt.method="Nelder-Mead", xreg=NULL)

	stm(y, h=5, level=c(80,90,95), s=NULL, par_ini=c(y[1]/2, 0.5), estimation=TRUE,
	lower=c(-1e+10, 0.1), upper=c(1e+10, 0.99), opt.method="Nelder-Mead", xreg=NULL)

	stheta(y, h=5, s=NULL)

Arguments

y

Object of time series class.

h

Number of required forecasting periods.

level

Levels for prediction intervals.

s

If TRUE, the multiplicative seasonal decomposition is used. If NULL and frequency(y)>=4 the time series is tested for statistically seasonal behaviour, with 90% of significance. If s='additive' or close zero values been find in the multiplicative decomposition, the additive decomposition is performed hatter than multiplicative. Default is NULL.

par_ini

Vector of initialization for (ell, alpha, theta) parameters.

estimation

If TRUE, the optim() function is consider for compute the minimum square estimator of parameters. If FALSE, the models/methods are computed for par_ini values.

lower

The lower limit of parametric space.

upper

The upper limit of parametric space.

opt.method

The numeric optimisation method for optim() function. Choose one among 'Nelder-Mead', 'L-BFGS-B', 'SANN'.

xreg

A matrix with the regressor variables including the out-of-sample data.

Details

By default (s=NULL), the 90% significance seasonal Z-test, used by Assimakopoulos and Nikolopoulos (2000), is applied for quarterly and monthly time series.

For details of each model see Fiorucci et al, 2016. If you are looking for the methods presented in the arXiv paper (Fiorucci et al, 2015), see otm.arxiv() function.

Value

An object of thetaModel class with one list containing the elements:

$method

The name of the model/method

$y

The original time series.

$s

A binary indication for seasonal decomposition.

type

Classical seasonal decomposition type.

opt.method

The optimisation method used in the optim() function.

$par

The estimated values for (ell, alpha, theta) parameters

$weights

The estimated weights values.

$fitted

A time series element with the fitted points.

$residuals

A time series element with the residual points.

$mean

The forecasting values.

$level

The levels for prediction intervals.

$lower

Lower limits for prediction intervals.

$upper

Upper limits for prediction intervals.

$tests

The p.value of Teraesvirta Neural Network test applied on unseasoned time series and the p.value of Shapiro-Wilk test applied on unseasoned residuals.

Author(s)

Jose Augusto Fiorucci, Francisco Louzada

References

Fiorucci J.A., Pellegrini T.R., Louzada F., Petropoulos F., Koehler, A. (2016). Models for optimising the theta method and their relationship to state space models, International Journal of Forecasting, 32 (4), 1151–1161, <doi:10.1016/j.ijforecast.2016.02.005>.

Assimakopoulos, V. and Nikolopoulos k. (2000). The theta model: a decomposition approach to forecasting. International Journal of Forecasting 16, 4, 521–530, <doi:10.1016/S0169-2070(00)00066-2>.

See Also

forecTheta-package, otm.arxiv

Examples

y1 = 2+ 0.15*(1:20) + rnorm(20)
y2 = y1[20]+ 0.3*(1:30) + rnorm(30)
y =  as.ts(c(y1,y2))
out <- dotm(y, h=10)
summary(out)
plot(out)

#### additive seasonal decomposition ###
x = sin(2*pi*seq(0,9,len=300)) + exp((1:300)/150) + rnorm(mean=0,sd=0.5,n=300)
y = ts(x, frequency=33)
out <- dotm(y, h=50, s='additive')
summary(out)
plot(out)

# #########################################################
# ######### Reproducing the M3 results by DOTM ############
# #########################################################
#
# library(Mcomp)
# data(M3)
#
# forec = matrix(NA, nrow=3003, ncol=18)
# obs = matrix(NA, nrow=3003, ncol=18) #matrix of the out-sample values
# meanDiff <- rep(1, 3003)
#
# for(i in 1:3003){
#   x=M3[[i]]$x
#   h=M3[[i]]$h
#   out = dotm(x,h,level=NULL)
#   forec[i,1:h] = out$mean
#   obs[i,1:h] = M3[[i]]$xx
#   meanDiff[i] = mean(abs(diff(x, lag = frequency(x))))
# }
#
# ############## sMAPE ###################
# sAPE_matrix = errorMetric(obs=obs, forec=forec, type="sAPE", statistic="N")
# #### Yearly ###
# mean( sAPE_matrix[1:645, 1:6] )
# #### QUARTERLY ###
# mean( sAPE_matrix[646:1401, 1:8] )
# #### MONTHLY ###
# mean( sAPE_matrix[1402:2829, 1:18] )
# #### Other ###
# mean( sAPE_matrix[2830:3003, 1:8] )
# #### ALL ###
# mean( sAPE_matrix, na.rm=TRUE )
# #
# ############# MASE ######################
# AE_matrix = errorMetric(obs=obs, forec=forec, type="AE", statistic="N")
# ASE_matrix=AE_matrix/meanDiff
# #### Yearly ###
# mean( ASE_matrix[1:645, 1:6] )
# #### QUARTERLY ###
# mean( ASE_matrix[646:1401, 1:8] )
# #### MONTHLY ###
# mean( ASE_matrix[1402:2829, 1:18] )
# #### Other ###
# mean( ASE_matrix[2830:3003, 1:8] )
# #### ALL ###
# mean( ASE_matrix, na.rm=TRUE )
# ########################################################