tmpyr {arfima}

Temperature Data

Description

Central England mean yearly temperatures from 1659 to 1976

Format

A ts tmpyr

Details

Hosking notes that while the ARFIMA(1, d, 1) has a lower AIC, it is not much lower than the AIC of the ARFIMA(1, d, 0).

Bhansali and Kobozka find: muHat = 9.14, d = 0.28, phi = -0.77, and theta = -0.66 for the ARFIMA(1, d, 1), which is close to our result, although our result reveals trimodality if numeach is large enough. The third mode is close to Hosking's fit of an ARMA(1, 1) to these data, while the second is very antipersistent.

Our package gives a very close result to Hosking for the ARFIMA(1, d, 0) case, although there is also a second mode. Given how close it is to the boundary, it may or may not be spurious. A check with dmean = FALSE shows that it is not the optimized mean giving a spurious mode.

If, however, we use whichopt = 1, we only have one mode. Note that Nelder-Mead sometimes does take out non-spurious modes, or add spurious modes to the surface.

Source

https://hadleyserver.metoffice.gov.uk/hadobs/hadcet/

References

Parker, D.E., Legg, T.P., and Folland, C.K. (1992). A new daily Central England Temperature Series, 1772-1991. Int. J. Clim., Vol 12, pp 317-342

Manley,G. (1974). Central England Temperatures: monthly means 1659 to 1973. Q.J.R. Meteorol. Soc., Vol 100, pp 389-405.

Hosking, J. R. M. (1984). Modeling persistence in hydrological time series using fractional differencing, Water Resour. Res., 20(12)

Bhansali, R. J. and Koboszka, P. S. (2003) Prediction of Long-Memory Time Series In Doukhan, P., Oppenheim, G. and Taqqu, M. S. (Eds) Theory and Applications of Long-Range Dependence (pp355-368) Birkhauser Boston Inc.

Veenstra, J.Q. Persistence and Antipersistence: Theory and Software (PhD Thesis)

Examples

data(tmpyr)

fit <- arfima(tmpyr, order = c(1, 0, 1), numeach = c(3, 3), dmean = TRUE, back=TRUE)
fit
##suspect that fourth mode may be spurious, even though not close to a boundary
##may be an induced mode from the optimization of the mean

fit <- arfima(tmpyr, order = c(1, 0, 1), numeach = c(3, 3), dmean = FALSE, back=TRUE)
fit

##perhaps so


plot(tacvf(fit), maxlag = 30, tacf = TRUE)

fit1 <- arfima(tmpyr, order = c(1, 0, 0), dmean = TRUE, back=TRUE)
fit1

fit2 <- arfima(tmpyr, order = c(1, 0, 0), dmean = FALSE, back=TRUE)
fit2  ##still bimodal.  Second mode may or may not be spurious.

fit3 <- arfima(tmpyr, order = c(1, 0, 0), dmean = FALSE, whichopt = 1, numeach = c(3, 3))
fit3  ##Unimodal.  So the second mode was likely spurious.

plot(tacvf(fit2), maxlag = 30, tacf = TRUE)
##maybe not spurious.  Hard to tell without visualizing the surface.

##compare to plotted tacf of fit1:  looks alike
plot(tacvf(fit1), maxlag = 30, tacf = TRUE)

tacfplot(list(fit1, fit2))