Theta Models: 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")
	
	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")
	
	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")
	
	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")
	
	stheta(y, h=5, s=NULL)

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.

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'.

Author

Jose Augusto Fiorucci, Francisco Louzada and Bao Yiqi

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

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. Accepted Paper. https://www.researchgate.net/publication/294420765_Models_for_optimising_the_theta_method_and_their_relationship_to_state_space_models

Fiorucci 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.

Examples

Run this code


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){
#	 if(i %% 100 == 0){print(i);}
#	 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 )
########################################################

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