forecTheta (version 2.0)

thetaModels: 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 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))
	
	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))
	
	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))
	
	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))
	
	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, quarterly and monthly time series are tested for statistically seasonal behaviour, with 95% of significance. Default is NULL.

par_ini

Vector of inicialization 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.

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
• $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

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.

References

Fioruci 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. Accepted Paper. https://www.researchgate.net/publication/294420765_Models_for_optimising_the_theta_method_and_their_relationship_to_state_space_models Fioruci J.A., Pellegrini T.R., Louzada F., Petropoulos F. (2015). The Optimised Theta Method. Free available at http://arxiv.org/abs/1503.03529. Assimakopoulos, V. and Nikolopoulos k. (2000). The theta model: a decomposition approach to forecasting. International Journal of Forecasting 16, 4, 521-530.

See Also

forecTheta-package, otm.arxiv

Examples

y1 = 2+ 0.15*(1:20) + rnorm(20,2)
y2 = y1[20]+ 0.3*(1:30) + rnorm(30,2)
y =  as.ts(c(y1,y2))

out <- dotm(y, h=10)
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){
#	 if(i \%\% 100 == 0){print(i);}
#	 x=M3[[i]]$x
#	 h=M3[[i]]$h
#	 out = dotm(x,h)
#	 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 )
########################################################