## H

Is the autocorrelation estimator and R(h) Xnoise(i) Xnoise(i - h), h 0, 1, 2, (2.18) Figure 2.4. Regions of asymptotic stationarity for the AR(2) process (Eq. 2.14) (shaded). The region for complex roots (dark shaded), which allows quasi-periodic behaviour, lies below the parabolic, a2 < a 4. Figure 2.4. Regions of asymptotic stationarity for the AR(2) process (Eq. 2.14) (shaded). The region for complex roots (dark shaded), which allows quasi-periodic behaviour, lies below the parabolic, a2...

## Age ka

Ramp regression of the marine S18O record ODP 846 (Fig. ). The long-term trend (thick line) documents the Northern Hemisphere Glaciation. The estimated change-points ( standard errors) of this climate transition are t1 2462 129 ka, x1 3.63 0.04 , t2 3700 119 ka and x2 2.99 0.04 . The estimates were obtained by WLS using iteratively updated variability. Also S(i) was given a ramp form (S(i) 0.27 o for t(i) < 2600 ka, 0.18 o for t(i) > 3550 ka and linearly connected between these...

## Info

Where h is the time lag, E is the expectation operator and VAR is the variance operator, is given by (Priestley 1981 Section 3.5 therein) p(h) a h , h 0, 1, 2, (2.3) For a > 0, this behaviour may be referred to as exponentially decreasing memory (Fig. 2.2). Figure 2.2. Autocorrelation function of the AR(1) process, a > 0. In the case of even spacing (Section 2.1.1) p(h) is given by a h exp h d r , in the case of uneven spacing (Section 2.1.2) by exp - T(i + h) T(i) r . In both cases, the...

## Sejj [Cjj12 421

J 0,1, where the matrix C is given by The GLS estimators are under the above assumptions also unbiased (Sen and Srivastava 1990) j 0,1. The properties of the GLS regression parameter estimators hold of course also for WLS (a special case of GLS where V is diagonal with unequal diagonal elements) and OLS (a special case of GLS where V is diagonal with equal diagonal elements). In the case of OLS, the standard errors can be written in short explicit form (Montgomery and Peck 1992) f 2 r n n 2 1 2...

## Ri

It is, for Gaussian X(i), F-distributed with v and 2K v degrees of freedom (Section 5.4). For testing at a pre-defined frequency, v 2, but if frequency is estimated as well, v 3. An alternative denominator for Eq. (5.27) can be used by integrating (and perhaps weighting) that expression over a frequency range of width 2W (Thomson 1990a Section 5.2 therein). Obtaining the avantage of a possibly higher accuracy of the background power estimate may come at the cost of missing two line components...

## [1 aX 1 aY12

This model requires the autocorrelation parameters ax and aY to have the same sign. The bivariate AR(1) process for even spacing (Eq. 7.25) is strictly stationary. Its properties are VAR X (i) VAR Y (i) 1 (7.28) CORR X (i), Y (i) pxy PE. (7.29) The bivariate AR(1) process for uneven time spacing is obtained in the usual manner replace aX by exp T(i) T(i 1) tx and aY by exp T(i) T(i 1) ty . This leads to heteroscedastic innovation terms, as already noticed in the univariate case. The model is...

## List of Algorithms

3.1 Moving block bootstrap algorithm (MBB) 79 3.2 Block length selector after Biihlmann and Kiinsch 3.3 MBB for realistic climate processes 82 3.4 Autoregressive bootstrap algorithm (ARB), even 3.5 Autoregressive bootstrap algorithm (ARB), uneven spacing 85 3.6 Surrogate data approach 87 4.1 Linear weighted least-squares regression, unknown variability 115 4.2 Construction of classical confidence intervals, Prais-Winsten procedure 122 4.3 Construction of bootstrap confidence intervals,...

## N K

SSQ ( A,, Bj IK l) X(i) - A, cos (2n T(i)) (5.16) This is in fact a regression and does not require even time spacing. However, the solution is simple if the spacing (d) is constant, n is even and fj 1 (nd), 2 (nd), , 1 (2d). Then A (2 n) Y X(i) cos (2nfjT(i)) (5.17) Bj (2 n) Y X(i) sin(2nfjT(i)) . (5.18) For other frequencies, these expressions are approximate to O (1 n). If the frequencies and other parameters of the harmonic process are unknown, which is more realistic, then we may try to...

## Spectral Analysis

Spectral analysis investigates the noise component in the climate equation (Eq. 1.2). A Fourier transformation into the frequency domain makes it possible to separate short-term from long-term variations and to distinguish between cyclical forcing mechanisms of the climate system and broad-band resonances. Spectral analysis allows to learn about the climate physics. The task is to estimate the spectral density function, and to test for harmonic (cyclical) signals. This poses more difficulties...

## List of Tables

1.1 Main types of climate archives, covered time ranges and absolute dating methods 1.2 Climate archives and variables studied in this book (selection) 1.3 Measurement and proxy errors in selected climate time series 2.1 Result of DFA study, estimated power-law exponents a 3.1 Monte Carlo experiment, mean estimation of a Gaussian 3.2 Monte Carlo experiment, standard deviation estimation of a Gaussian purely random process 71 3.3 Monte Carlo experiment, mean and median estimation of a lognormal...

## Future Directions

What changes may bring the future to climate time series analysis First we outline (Sections 9.1, 9.2 and 9.3) more short-term objectives of normal science (Kuhn 1970), extensions of previous material (Chapters 1, 2, 3, 4, 5, 6, 7 and 8). Then we take a chance (Sections 9.4 and 9.5) and look on paradigm changes in climate data analysis that may be effected by virtue of strongly increased computing power (and storage capacity). Whether this technological achievement comes in the form of grid...

## Pxy x PX y Py y Py2

PX and pY is the mean, SX and SY the variance of the univariate processes X(i) and Y(i), respectively pXY pE is the correlation coefficient. The PDF is slanted for pxy 0 (Fig. 7.9). See Priestley (1981 Section 2.12.9 therein), Patel and Read (1996 Chapter 9 therein) and Kotz et al. (2000 Chapter 46 therein) for more details on the binormal distribution. The bivariate lognormal distribution is in the more general case, with shape parameters oX and oy and scale parameters bX and bY, given by X(i)...

## Frequency a1Frequency a1

Monsoon spectrum, influence of timescale errors. Focus is on two portions (a, b) of the original spectrum of the proxy record Q5 (Fig. 5.8). Errors from the age determination (Fleitmann et al. 2003 Table S1 therein) were used for timescale resampling (Section 4.1.7.2) of a piecewise linear age-depth model. The 90 bootstrap bound for the red noise (the lower of the grey lines in a and b), obtained from B 10,000 simulations, is higher than the 90 bound (the lowest of the four black,...

## Bootstrap Confidence Intervals

In statistical analysis of climate time series, our aim (Chapter 1) is to estimate parameters of Xtrend(T), Xout(T), S(T) and Xnoise(T). Denote in general such a parameter as 9. An estimator, 0, is a recipe how to calculate 9 from a set of data. The data, discretely sampled time series t(i), x(i) 1, are influenced by measurement and proxy errors of x(i), outliers, dating errors of t(i) and climatic noise. Therefore, 0 cannot be expected to equal 9. The accuracy of 0, how close it comes to 9, is...

## A dfCOV MA dTA 1 COV M

On the sample level, the parameter estimates and the elements of the estimated covariance matrix are plugged in. Declustering records prior to fitting a GP distribution discards excess data and loses information, as noted by Coles (2001b). A more efficient GP estimation may come from retaining all excess data (also those within a cluster) and modelling the serial dependence. Fawcett and Walshaw (2006) present Monte Carlo evidence supporting this approach and...

## Sampling and time spacing

The sampling of a climate archive (Fig. 1.13) can influence the de-tectability of extreme events. Table 6.5 lists the notation for this section. Consider the case that the spacing, d(i), is large compared with the sample duration, D(i), or its diffusion-extended form, D'(i), and also large compared with the persistence time, t. It may then be that the time series fails to record information about an extreme event, Xout(i). This would render the series useless for the purpose of risk analysis....

## Technical issues

The calculation of the dpss multitapers (Section 5.2.3) can be done in various ways. Percival and Walden (1993 Chapter 8 therein) note numerical integration and bypassing the problem by using substitutes in form of trigonometric polynomials, but they favour two other calculation types, namely via a tridiagonal formulation or directly from the defining eigenvalue problem (Eq. 5.24). For solving the latter, Bell et al. (1993) developed an iterative algorithm, written in FORTRAN 77 and available...

## References

Abarbanel HDI, Brown R, Sidorowich JJ, Tsimring LS 1993 The analysis of observed chaotic data in physical systems. Reviews of Modern Physics 65 4 1331-1392. Abraham B, Wei WWS 1984 Inferences about the parameters of a time series model with changing variance. Metrika 31 3-4 183-194. Abram NJ, Gagan MK, Cole JE, Hantoro WS, Mudelsee M 2008 Recent intensification of tropical climate variability in the Indian Ocean. Nature Geoscience 1 12 849-853. Abram NJ, Mulvaney R, Wolff EW, Mudelsee M 2007...