## 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 from http://lib.stat.cmu.edu/jcgs/bell-p-w (29 January 2008). Own experiments with a Fortran 90 translation on 32-bit and 64-bit machines, where the numerical precision of real numbers can be adjusted conveniently, attest the robustness of the algorithm.

The Fast Fourier Transform or FFT is a numerical algorithm (Coo-ley and Tukey 1965) that reduces the number of operations from O (n2) to O (n log(n)). Data size n must be a power of two. The FFT was the technical basis of the scientific revolution that came with spectral analysis.

The F distribution with vy and vz degrees of freedom has following PDF:

f (x) = B (vy /2,VZ /2) ' (1+ x ■ vy /vZ )/2 , ^^

where x > 0 and B is the beta function (Section 2.7). It arises as the distribution of the ratio of two chi-squared variables (Section 3.9). Let Y and Z be independent and chi-squared distributed with vy and vZ degrees of freedom, respectively. Then X = [(Y/vy) ■ (Z/vZ)-1] is F-distributed (Eq. 5.39). See Johnson et al. (1995: Chapter 27 therein) for more details on the F distribution.

Lees and Park (1995) published a C subroutine for multitaper estimation. It can be obtained via http://www.iamg.org (29 January 2008). This software was used in an influential paper on signal detection against a red-noise background of climate spectra (Mann and Lees 1996).

Multitaper.zip is a Matlab implementation of multitaper estimation in the presence of missing data (Fodor and Stark 2000). It can be downloaded from http://www.stat.berkeley.edu/~stark/Code/ (29 January 2008).

mwlib is a Fortran 90 library of subroutines for multitaper estimation (Prieto et al. 2009). It is available at the internet address http://wwwprof.uniandes.edu.co/~gprieto/software/mwlib.html (11 December

SSA-MTM Toolkit is a compiled software that includes multitaper estimation in connection with SSA (http://www.atmos.ucla.edu/tcd/ssa/, 29 January 2008). Version exist for DEC, Linux, Macintosh, SGI and Sun systems.

CYSTRATI is a FORTRAN 77 package, developed and listed by Pardo-Iguzquiza et al. (1994), for cyclostratigraphic data analysis, including multitaper and maximum entropy spectrum estimation.

REDFIT is a Fortran 90 program (code, Windows binaries) for Lomb-Scargle spectrum estimation with bootstrap bias correction and test of the AR(1) red-noise alternative (Schulz and Mudelsee 2002). It is based on SPECTRUM (Schulz and Stattegger 1997), which has a graphical interface but no bias correction or red-noise test. An option is interactively working with SPECTRUM to find out suitable smoothing parameters and then performing with REDFIT the final calculations. RED2CON is a recent Matlab implementation of REDFIT with graphical interface. The core of the programs lies in the routines for lLs(fj) calculation (Scargle 1989). REDFIT, RED2CON and SPECTRUM are available at the site http://www.geo.uni-bremen.de/geomod/staff/mschulz/ (29 March 2010), REDFIT also at the web site for this book.

ENVELOPE is a DOS/Windows software implementing a windowed version of the harmonic filter (Section 5.2.4.3) for analysing slowly changing sinusoidal components (frequency f'). The time-dependent amplitude is (A2 + B2)1/2, see Eq. (5.33). It is estimated using a least-squares criterion (Ferraz-Mello 1981; Schulz 1996). The software can be ob tained from http://www.geo.uni-bremen.de/geomod/staff/mschulz/ (29 January 2008).

MATLAB Recipes for Earth Sciences is the title of a book (Trauth 2007) with software that includes Lomb-Scargle estimation.

AutoSignal is a commercial package containing spectral analysis tools, including multitaper and Lomb-Scargle estimation. It can be obtained from Systat (http://www.systat.com, 29 January 2008).

CLEAN is a deconvolution algorithm for switching between frequency and time domains while collecting iteratively the strongest spectral peaks and their time-domain representation, respectively (Roberts et al. 1987). It can be applied to unevenly spaced time series for spectrum estimation. A surrogate data resampling approach to derive significance levels (Heslop and Dekkers 2002) is available as Matlab package MC-CLEAN at http://www.geo.uu.nl/~forth/Software/mc_clean.zip (29 January 2008).

REDFITmc2 (Mudelsee et al. 2009) is an adaption of REDFIT, which implements Algorithms 5.5 and 5.6. See the web site for this book.

Chapter 6

### Extreme Value Time Series

Extreme value time series refer to the outlier component in the climate equation (Eq. 1.2). Quantifying the tail probability of the PDF of a climate variable—the risk of climate extremes—is of high socioeco-nomical relevance. In the context of climate change, it is important to move from stationary to nonstationary (time-dependent) models: with climate changes also risk changes may be associated.

Traditionally, extreme value data are evaluated in two forms: first, block extremes such as annual maxima, and second, exceedances of a high threshold. A stationary model of great flexibility for the first and the second form is the Generalized Extreme Value distribution and the generalized Pareto distribution, respectively. Classical estimation techniques based on maximum likelihood exist for both distributions.

Nonstationary models can be constructed parametrically, by writing the extreme value models with time-dependent parameters. Maximum likelihood estimation may impose numerical difficulties here. The in-homogeneous Poisson process constitutes an interesting nonparametric model of the time-dependence of the occurrence of an extreme. Here, bootstrap confidence bands can be constructed and hypothesis tests performed to assess the significance of trends in climate risk. A recent development is a hybrid, which estimates the time-dependence nonpara-metrically and, conditional on the occurrence of an extreme, models the extreme value parametrically.