In the previous chapter we examined long memory in financial time series for which there is a large and growing literature in econometrics. The origins of long memory series as a field of research lay elsewhere. Long memory time series were brought to prominence by Hurst (1951) through the study of geophysical time series, in particular through the study of hydrological series. Later, FGNs were introduced into applied statistics by Mandelbrot and van Ness (1968), in part, in an attempt to model the famous Nile River data. Mandelbrot and Wallis (1969) studied a wide range of geo- physical series and reported long memory to be pervasive in them. Thus geophysical series are a natural subject area to consider when trying to discriminate between true long memory and non-stationarity leading to spurious long memory. In this chapter we focus on geophysical time series and examine closely the long memory properties of these series.
Tree-ring sequences are often used in paleoclimatic studies both as proxies for past atmospheric conditions and as a basis for temperature reconstructions. These series are quite curious as there are a large number of known cyclic influences on climate, yet when spectral analysis is undertaken statistically significant periodicities are rarely found in them. For example, Thomson (1990) applied the newly developed multitaper
spectral analysis methods using discrete prolate spheroidal sequences to the Campito Mountain data (see Section 10.2 below for details). Thomson reported no statistically significant periodicities in the data longer than 5.5 years. Other authors have reported periodicities in tree ring series, for example Keqian and Butler (1998) and Raspopov et al. (2005), but Thomson pointed out that the widely used five percent significance levels were inappropriate for these data, a better level was 1−(1/N) where N is the number of data points. Often all these reported periodicities are, are a “bump” in the periodogram where one woulda priori expect to find a periodicity.
Spectral analysis of tree ring series is statistically difficult work as there are several solar cycles; the well-known Schwabe-Wolf cycle of approximately 11 years, the Hale cycle of approximately 22 years, the Gleissberg cycle of about 80-90 years, the Suess or de Vries cycle of around 200 years and the Halstatt cycle of about 2300 years. All of these influence the earth’s climate and hence growing conditions for trees. These cycles are not constant in period or amplitude. The “11-year” Schwabe-Wolf cycle ranges between a low of close to eight years to a high of around 14 years. This quasi-cyclic behaviour leads to high leakage even with sophisticated spectral analysis tools and so an indistinct peak appears in the estimated spectrum where genuine periodic phenomena exist.
Keqian and Butler (1998) studied solar cycles using tree ring data and reported the effects of changes in solar activity were smoothed out over time, probably because of the thermal inertia of the earth, and the amount of smoothing depended on altitude. In addition to these solar cycles, there are also quasi-regular changes in the circulation patterns in the earth’s atmosphere such as the Arctic and North Atlantic Oscillations, see Shindell et al. (2001). At very long time scales there are systematic changes in the earth’s orbital characteristics which are known as the Milankovitch cycles, see Imbrie and Imbrie (1979) for an easy introduction.
Some of the data sets we studied are of particular interest for specific aspects of paleoclimatology. For example, the Elk Lake varve sequence, see Section (10.2) for details. The region where Elk Lake is located is at the intersection of three airmasses;
the cold arctic airmass to the north, the dry Pacific airmass to the west, and the warmer and moister Gulf of Mexico-Atlantic airmass to the south and east. Variations occur on all time scales from seasonal to millennial. It is a climatically sensitive area making it ideal as a proxy for climate change. Also, as no streams flow into the lake the majority of the mineralized materials in the varves are interpreted as being deposited by wind. Thus the varves provide a record of wind conditions in the region.
The Elk Lake data is an example of a cyclostratigraphic time series. In the mono- graph of Weedon (2003) he pointed out (p84) that almost all cyclostratigraphic time series have a red noise background, which is characteristic of long memory time series, the origin of which is difficult to determine.
We examined a range of millennial scale temperature reconstructions. The partic- ular interest in these series arises because since the end of the last ice age the earth’s climate has enjoyed a period of relative stability. As the earth is now in a period of rising global temperatures a number of authors have considered the stochastic prop- erties of univariate time series of both atmospheric and oceanic temperatures from instrumental and proxy records on time scales of a few decades to several millenia in an effort to estimate the natural variability of the earth’s climate. This sets a baseline of variation which provides a context in which the observed temperature increases can be studied.
In the literature a number of authors have considered fractionally integrated se- ries as models for temperature and closely related time series. Bloomfield (1992) and Bloomfield and Nychka (1992) considered several time series models including FI(d) series to determine whether the observed global warming in instrumental records could be accounted for by natural fluctuations. Bloomfield and Nychka (1992), in partic- ular, concluded the observed rate of temperature rise could not be accounted for by a stationary FI(d) series. Beran (1994, pp29,30) summarized some studies of long memory in instrumental temperature records. Stephenson et al. (2000) considered FI(d) models among several others for the North Atlantic Oscillation and concluded an ARFIMA(1,0.15,0) model fitted the data best. Baillie and Chung (2002) considered
long memory in several tree ring series which are often used as temperature and precip- itation proxies and in climate reconstructions. Baillie and Chung found the series they examined to be very well described by FI(d) series with the exception of the period 1800AD to the present in two of their four data sets. Overland et al. (2006) considered three models of the North Pacific Ocean sea surface temperatures; AR(1), FI(d) and a square wave oscillator. Overland and his co-authors could not establish the statistical primacy of any of the three models. Mills (2007) considered in detail long memory in the Moberg et al. (2005) Northern Hemisphere temperature reconstruction. Mills tentatively suggested the evidence favoured a shifting trends in temperature model over true long memory.
Some of these authors were aware than statistical long memory could be caused by the series being non-stationary. However, as indicated above, distinguishing between a mean-reverting non-stationary series and true long memory was difficult with the statistical tools they had available.
In this chapter examine we examine a number of these series. For reasons of space we can only present one analysis in detail, some further results are presented in Appendix (B).