The historical records of the low- and high-water levels of the Nile River are among the longest climatic records that have near-annual resolution. There are few gaps in the first part of the records (A.D. 622-1470) and larger gaps later (A.D. 1471-1922). We apply advanced spectral methods, Singular-Spectrum Analysis (SSA) and the Multi-Taper Method (MTM), to fill the gaps and to locate interannual and interdecadal periodicities. The gap filling uses a novel, iterative version of SSA. Our analysis reveals several statistically significant features of the records: a nonlinear, data-adaptive trend that includes a 256-year cycle, a quasi-quadriennial (4.2-year) and a quasi-biennial (2.2-year) mode, as well as additional periodicities of 64, 19, 12, and, most strikingly, 7 years. The quasi-quadriennial and quasi-biennial modes support the long-established connection between the Nile River discharge and the El-Niño/Southern Oscillation (ENSO) phenomenon in the Indo-Pacific Ocean. The longest periods might be of astronomical origin. The 7-year periodicity, possibly related to the biblical cycle of lean and fat years, seems to be due to North Atlantic influences.

We distinguish between two dimensions of a dynamical system given by experimental time series. Statistical dimension gives a theoretical upper bound for the minimal number of degrees of freedom required to describe tje attractor up to the accuracy of the data, taking into account sampling and noise problems. The dynamical dimension is the intrinsic dimension of the attractor and does not depend on the quality of the data. Singular Spectrum Analysis (SSA) provides estimates of the statistical dimension. SSA also describes the main physical phenomena reflected by the data. It gives adaptive spectral filters associated with the dominant oscillations of the system and clarifies the noise characteristics of the data. We apply SSA to four paleoclimatic records. The principal climatic oscillations, and the regime changes in their amplitude are detected. About 10 degrees of freedom are statistically significant in the data. Large noise and insufficient sample length do not allow reliable estimates of the dynamical dimension.

This chapter presents a modeling framework for macroeconomic growth dynamics; it is motivated by recent attempts to formulate and study “integrated models” of the coupling between natural and socioeconomic phe­ nomena. The challenge is to describe the interfaces between human activities and the functioning of the earth system. We examine the way in which this interface works in the presence of endogenous business cycle dynam­ ics, based on a nonequilibrium dynamic model. Recent findings about the macroeconomic response to natural disasters in such a nonequilibrium setting have shown a more severe response to natural disasters during expan­ sions than during recessions. These findings raise questions about the assessment of climate change damages or natural disaster losses that are based purely on long-term growth models. In order to compare the theoretical findings with observational data, we analyze cyclic behavior in the U.S. economy, based on multivariate singular spectrum analysis. We analyze a total of nine aggregate indicators in a 52 year interval (1954–2005) and demon­ strate that the behavior of the U.S. economy changes significantly between intervals of growth and recession, with higher volatility during expansions.

This study aims at a global description of climatic phenomena that exhibit some regularity during the twentieth century. Multi-channel singular spectrum analysis is used to extract long-term trends and quasi-regular oscillations of global sea-surface temperature (SST) fields since 1901. Regional analyses are also performed on the Pacific, (Northern and Southern) Atlantic, and Indian Ocean basins. The strongest climatic signal is the irregular long-term trend, characterized by overall warming during 1910–1940 and since 1975, with cooling (especially of the Northern Hemisphere) between these two warming intervals. Substantial cooling prevailed in the North Pacific between 1950 and 1980, and continues in the North Atlantic today. Both cooling and warming are preceded by SST anomalies of the same sign in the subpolar North Atlantic. Near-decadal oscillations are present primarily over the North Atlantic, but also over the South Atlantic and the Indian Ocean. A 13–15-y oscillation exhibits a seesaw pattern between the Gulf-Stream region and the North-Atlantic Drift and affects also the tropical Atlantic. Another 7–8-y oscillation involves the entire double-gyre circulation of the North Atlantic, being mostly of one sign across the basin, with a minor maximum of opposite sign in the subpolar gyre and the major maximum in the northwestern part of the subtropical gyre. Three distinct interannual signals are found, with periods of about 60–65, 45 and 24–30 months. All three are strongest in the tropical Eastern Pacific. The first two extend throughout the whole Pacific and still exhibit some consistent, albeit weak, patterns in other ocean basins. The latter is weaker overall and has no consistent signature outside the Pacific. The 60-month oscillation obtains primarily before the 1960s and the 45-month oscillation afterwards.

Using multi-scale ideas from wavelet analysis, we extend singular-spectrum analysis (SSA) to the study of nonstationary time series, including the case where intermittency gives rise to the divergence of their variance. The wavelet transform resembles a local Fourier transform within a finite moving window whose width W, proportional to the major period of interest, is varied to explore a broad range of such periods. SSA, on the other hand, relies on the construction of the lag-correlation matrix C on M lagged copies of the time series over a fixed window width W to detect the regular part of the variability in that window in terms of the minimal number of oscillatory components; here W=M[Delta]t with [Delta]t as the time step. The proposed multi-scale SSA is a local SSA analysis within a moving window of width M<=W<=N, where N is the length of the time series. Multi-scale SSA varies W, while keeping a fixed W/M ratio, and uses the eigenvectors of the corresponding lag-correlation matrix C(M) as data-adaptive wavelets; successive eigenvectors of C(M) correspond approximately to successive derivatives of the first mother wavelet in standard wavelet analysis. Multi-scale SSA thus solves objectively the delicate problem of optimizing the analyzing wavelet in the time-frequency domain by a suitable localization of the signal's correlation matrix. We present several examples of application to synthetic signals with fractal or power-law behavior which mimic selected features of certain climatic or geophysical time series. The method is applied next to the monthly values of the Southern Oscillation Index (SOI) for 1933-1996; the SOI time series is widely believed to capture major features of the El Niño/Southern Oscillation (ENSO) in the Tropical Pacific. Our methodology highlights an abrupt periodicity shift in the SOI near 1960. This abrupt shift between 5 and 3 years supports the Devil's staircase scenario for the ENSO phenomenon (preliminary results of this study were presented at the XXII General Assembly of the European Geophysical Society, Vienna, May 1997, and at the Fall Meeting of the American Geophysical Union, San Francisco, December 1997).

Singular spectrum analysis (SSA) along with its multivariate extension (M-SSA) provides an efficient way to identify weak oscillatory behavior in high-dimensional data. To prevent the misinterpretation of stochastic fluctuations in short time series as oscillations, Monte Carlo (MC)–type hypothesis tests provide objective criteria for the statistical significance of the oscillatory behavior. Procrustes target rotation is introduced here as a key method for refining previously available MC tests. The proposed modification helps reduce the risk of type-I errors, and it is shown to improve the test’s discriminating power. The reliability of the proposed methodology is examined in an idealized setting for a cluster of harmonic oscillators immersed in red noise. Furthermore, the common method of data compression into a few leading principal components, prior to M-SSA, is reexamined, and its possibly negative effects are discussed. Finally, the generalized Procrustes test is applied to the analysis of interannual variability in the North Atlantic’s sea surface temperature and sea level pressure fields. The results of this analysis provide further evidence for shared mechanisms of variability between the Gulf Stream and the North Atlantic Oscillation in the interannual frequency band.

Understanding the natural variability of climate is important for predicting its near-term evolution. Models of the oceans' thermohaline and wind-driven circulation show low-frequency oscillations. Long instrumental records can help validate the oscillatory behavior of these models. Singular spectrum analysis applied to the 335-year-long central England temperature (CET) record has identified climate oscillations with interannual (7- to 8-year) and interdecadal (15- and 25-year) periods, probably related to the North Atlantic's wind-driven and thermohaline circulation, respectively. Statistical prediction of oscillatory variability shows CETs decreasing toward the end of this decade and rising again into the middle of the next.

Evaluating the effects of climate variation on ecosystems is of paramount importance for our ability to forecast and mitigate the consequences of global change. However, the ways in which complex food webs respond to climate variations remain poorly understood. Here, we use long-term time series to investigate the effects of temperature variation on the intraguild-predation (IGP) system of Windermere (UK), a lake where pike (Esox lucius, top predator) feed on small-sized perch (Perca fluviatilis) but compete with large-sized perch for the same food sources. Spectral analyses of time series reveal that pike recruitment dynamics are temperature controlled. In 1976, expansion of a size-truncating perch pathogen into the lake severely impacted large perch and favoured pike as the IGP-dominant species. This pathogen-induced regime shift to a pike-dominated IGP apparently triggered a temperature-controlled trophic cascade passing through pike down to dissolved nutrients. In simple food chains, warming is predicted to strengthen top–down control by accelerating metabolic rates in ectothermic consumers, while pathogens of top consumers are predicted to dampen this top–down control. In contrast, the local IGP structure in Windermere made warming and pathogens synergistic in their top–down effects on ecosystem functioning. More generally, our results point to top predators as major mediators of community response to global change, and show that size-selective agents (e.g. pathogens, fishers or hunters) may change the topological architecture of food webs and alter whole ecosystem sensitivity to climate variation.

Global sea surface temperature (SST) evolution is analyzed by constructing predictive models that best describe the dataset’s statistics. These inverse models assume that the system’s variability is driven by spatially coherent, additive noise that is white in time and are constructed in the phase space of the dataset’s leading empirical orthogonal functions. Multiple linear regression has been widely used to obtain inverse stochastic models; it is generalized here in two ways. First, the dynamics is allowed to be nonlinear by using polynomial regression. Second, a multilevel extension of classic regression allows the additive noise to be correlated in time; to do so, the residual stochastic forcing at a given level is modeled as a function of variables at this level and the preceding ones. The number of variables, as well as the order of nonlinearity, is determined by optimizing model performance. The two-level linear and quadratic models have a better El Niño–Southern Oscillation (ENSO) hindcast skill than their one-level counterparts. Estimates of skewness and kurtosis of the models’ simulated Niño-3 index reveal that the quadratic model reproduces better the observed asymmetry between the positive El Niño and negative La Niña events. The benefits of the quadratic model are less clear in terms of its overall, cross-validated hindcast skill; this model outperforms, however, the linear one in predicting the magnitude of extreme SST anomalies. Seasonal ENSO dependence is captured by incorporating additive, as well as multiplicative forcing with a 12-month period into the first level of each model. The quasi-quadrennial ENSO oscillatory mode is robustly simulated by all models. The “spring barrier” of ENSO forecast skill is explained by Floquet and singular vector analysis, which show that the leading ENSO mode becomes strongly damped in summer, while nonnormal optimum growth has a strong peak in December.

We apply multivariate singular spectrum analysis to the study of U.S. business cycle dynamics. This method provides a robust way to identify and reconstruct oscillations, whether intermittent or modulated. We show such oscillations to be associated with comovements across the entire economy. The problem of spurious cycles generated by the use of detrending filters is addressed and we present a Monte Carlo test to extract significant oscillations. The behavior of the U.S. economy is shown to change significantly from one phase of the business cycle to another: the recession phase is dominated by a five-year mode, while the expansion phase exhibits more complex dynamics, with higher-frequency modes coming into play. We show that the variations so identified cannot be generated by random shocks alone, as assumed in ‘real’ business-cycle models, and that endogenous, deterministically generated variability has to be involved.