Identification of dynamical transitions in marine palaeoclimate records by recurrence network analysis

The analysis of palaeoclimate time series is usually affected by severe methodological problems, resulting primarily from non-equidistant sampling and uncertain age models. As an alternative to existing methods of time series analysis, in this paper we argue that the statistical properties of recurrence networks - a recently developed approach - are promising candidates for characterising the system's nonlinear dynamics and quantifying structural changes in its reconstructed phase space as time evolves. In a first order approximation, the results of recurrence network analysis are invariant to changes in the age model and are not directly affected by non-equidistant sampling of the data. Specifically, we investigate the behaviour of recurrence network measures for both paradigmatic model systems with non-stationary parameters and four marine records of long-term palaeoclimate variations. We show that the obtained results are qualitatively robust under changes of the relevant parameters of our method, including detrending, size of the running window used for analysis, and embedding delay. We demonstrate that recurrence network analysis is able to detect relevant regime shifts in synthetic data as well as in problematic geoscientific time series. This suggests its application as a general exploratory tool of time series analysis complementing existing methods

Is there long-range memory in solar activity on time scales shorter than the sunspot period?

The sunspot number (SSN), the total solar irradiance (TSI), a TSI reconstruction, and the solar flare index (SFI), are analyzed for long-range persistence (LRP). Standard Hurst analysis yields $H \approx 0.9$, which suggests strong LRP. However, solar activity time series are non-stationary due to the almost periodic 11 year smooth component, and the analysis does not give the correct $H$ for the stochastic component. Better estimates are obtained by detrended fluctuations analysis (DFA), but estimates are biased and errors are large due to the short time records. These time series can be modeled as a stochastic process of the form $x(t)=y(t)+\sigma \sqrt{y(t)}\, w_H(t)$, where $y(t)$ is the smooth component, and $w_H(t)$ is a stationary fractional noise with Hurst exponent $H$. From ensembles of numerical solutions to the stochastic model, and application of Bayes' theorem, we can obtain bias and error bars on $H$ and also a test of the hypothesis that a process is uncorrelated ($H=1/2$). The conclusions from the present data sets are that SSN, TSI and TSI reconstruction almost certainly are long-range persistent, but with most probable value $H\approx 0.7$. The SFI process, however, is either very weakly persistent ($H<0.6$) or completely uncorrelated. Some differences between stochastic properties of the TSI and its reconstruction indicate some error in the reconstruction scheme.Comment: 23 pages, 12 figure

On the record properties of integrated time series

This paper compares the statistical properties of the records from i.i.d. time series with those of time series containing a single unit root. It is shown that there are important differences in both the limiting distributions and the convergence rates of the associated record counting processes. Since the record properties of i.i.d. time series are shared by a large class of stationary time series, the reported differences underline the possibility of using record-based statistics for robust resting procedures of the unit root hypothesis

Statistics of Extreme Values in Time Series with Intermediate-Term Correlations

It will be discussed the statistics of the extreme values in time series characterized by finite-term correlations with non-exponential decay. Precisely, it will be considered the results of numerical analyses concerning the return intervals of extreme values of the fluctuations of resistance and defect-fraction displayed by a resistor with granular structure in a nonequilibrium stationary state. The resistance and defect-fraction are calculated as a function of time by Monte Carlo simulations using a resistor network approach. It will be shown that when the auto-correlation function of the fluctuations displays a non-exponential and non-power-law decay, the distribution of the return intervals of extreme values is a stretched exponential, with exponent largely independent of the threshold. Recently, a stretched exponential distribution of the return intervals of extreme values has been identified in long-term correlated time series by Bunde et al. (2003) and Altmann and Kantz (2005). Thus, the present results show that the stretched exponential distribution of the return intervals is not an exclusive feature of long-term correlated time series.Comment: 6 pages, 7 figures, conference paper, in Noise and Stochastics in Complex Systems and Finance, ed. by J. Kertez, S. Bornhold, R. N. Mantegna, Procs. of SPIE, vol. 6601, 19, 200

Untenable nonstationarity: An assessment of the fitness for purpose of trend tests in hydrology

The detection and attribution of long-term patterns in hydrological time series have been important research topics for decades. A significant portion of the literature regards such patterns as ‘deterministic components’ or ‘trends’ even though the complexity of hydrological systems does not allow easy deterministic explanations and attributions. Consequently, trend estimation techniques have been developed to make and justify statements about tendencies in the historical data, which are often used to predict future events. Testing trend hypothesis on observed time series is widespread in the hydro-meteorological literature mainly due to the interest in detecting consequences of human activities on the hydrological cycle. This analysis usually relies on the application of some null hypothesis significance tests (NHSTs) for slowly-varying and/or abrupt changes, such as Mann-Kendall, Pettitt, or similar, to summary statistics of hydrological time series (e.g., annual averages, maxima, minima, etc.). However, the reliability of this application has seldom been explored in detail. This paper discusses misuse, misinterpretation, and logical flaws of NHST for trends in the analysis of hydrological data from three different points of view: historic-logical, semantic-epistemological, and practical. Based on a review of NHST rationale, and basic statistical definitions of stationarity, nonstationarity, and ergodicity, we show that even if the empirical estimation of trends in hydrological time series is always feasible from a numerical point of view, it is uninformative and does not allow the inference of nonstationarity without assuming a priori additional information on the underlying stochastic process, according to deductive reasoning. This prevents the use of trend NHST outcomes to support nonstationary frequency analysis and modeling. We also show that the correlation structures characterizing hydrological time series might easily be underestimated, further compromising the attempt to draw conclusions about trends spanning the period of records. Moreover, even though adjusting procedures accounting for correlation have been developed, some of them are insufficient or are applied only to some tests, while some others are theoretically flawed but still widely applied. In particular, using 250 unimpacted stream flow time series across the conterminous United States (CONUS), we show that the test results can dramatically change if the sequences of annual values are reproduced starting from daily stream flow records, whose larger sizes enable a more reliable assessment of the correlation structures

Distribution of Return Intervals of Extreme Events

The distribution of return intervals of extreme events is studied in time series characterized by finite-term correlations with non-exponential decay. Precisely, it has been analyzed the statistics of the return intervals of extreme values of the resistance fluctuations displayed by resistors with granular structure in nonequilibrium stationary states. The resistance fluctuations are calculated by Monte Carlo simulations using a resistor network approach. It has been found that for highly disordered networks, when the auto-correlation function displays a non-exponential and non-power-law decay, the distribution of return intervals of the extreme values is a stretched exponential, with exponent independent of the threshold.Comment: 10 pages, 6 figures, Next-SigmaPhi Int. Conference, News Expectations and Trends in Statistical Physics, 13-18 August 2005, Kolymbari - Crete (Greece

Cointegration tests based on record counting statistics

This paper presents of number of cointegration tests that exploit the statistical properties of the records from the original time series variables. We prove their consistency and obtain their asymptotic null distributions. Among the advantages of this novel methodology, the new tests are invariant with respect to the individual series' variances and also with respect to monotonic transformations applied to these series. In addition, these tests are robust against the presence of level breaks as long as the number of these breaks increases slowly enough with the sample size. Finally, an alternative scheme is proposed to deal with additive outliers, which prevent them from causing size distortions