<figure><img src="image_17.png" /><figcaption>**Fig. 16.** The graph of the activity-level function $A(m; s^*, u^*, \mu^*, v^*)$ for the prediction of US economic recessions. Parameter values are $(s^*, u^*, \mu^*, v^*) = (6, 48, 0.4, 12)$ and they were calculated from the Federal Reserve Bank of St. Louis (FRBSL) monthly series of industrial production total (IPT; black curve). Official economic recessions in the US are shown as grey strips, and alarms declared by the algorithm as magenta bars of duration $\tau = 12$ months; see Sect. 7.2.3 and Appendix D for details.</figcaption></figure>

with $s^* = 6$ months and $u^* = 48$ months, to detect the second of the two recessions. Note that declaring an alarm 2 months after the start of a recession may, nevertheless, be of interest for potential end users because the NBER announcements of a recession are usually delayed by 5 months or more.

The algorithm declared, in real time, a current alarm for a recession to start in June 2008 (Fig. 16). According to the NBER, the recession started in January 2008, but the announcement was only made in December 2008.

*Fast acceleration of unemployment (FAU).* In this application, we consider the data on monthly unemployment rates for the US civilian labor force in the time interval from January 1959 to July 2010 as given by the US Department of Labor (USDL, http://stats.bls.gov/). The index $f(m)$ is now LHELL, as defined and explained at the end of the previous subsection, and we use the values from January 1959 to May 2008 that are available from the Economic Research Section of the FRBSL, http://research.stlouisfed.org/fred2/series/INDPRO/. The values of this index are not available after May 2008.

In the present case, the parameter values ($s^*, u^*, \mu^*, v^*, L_1^*, L_2^*, \tau^*$) are the same as above, except that now $\mu^* = 1$. The resulting graph of $A(m; s^*, u^*, \mu^*, v^*)$ is given in Fig. 17 (solid black line), along with the time intervals of FAU (grey vertical strips) and the alarms declared by the algorithm (magenta boxes). Each FAU interval here is determined, following Keilis-Borok et al. (2005), as an interval between the local minimum and the next local maximum of the smoothed monthly unemployment rate.

The FAU intervals that start in 1967, 1979 and 2006 are well predicted by the algorithm. The alarm in 1989 is identified in the same month as the start of the FAU. The other two

<figure><img src="image_18.png" /><figcaption>**Fig. 17.** Same as Fig. 16, but for the prediction of periods of fast acceleration of unemployment (FAU). Parameter values were calculated from the FRBSL monthly series of the index of help wanted advertising (LHELL); they are the same as for the economic recession prediction, except for $\mu^* = 1.0$. Same conventions as in the previous figure.</figcaption></figure>

alarms appear after the beginning of the corresponding FAU: specifically, the alarms are delayed by 2 months in 1969 and by 7 months in 1973. The 1981 alarm is an immediate continuation of the longest FAU (1979–1983) that occurred during the time of data vailability, while those in 1986 and 1995 are evidently false alarms. The overall statistics of this application suggests that the algorithm does have some predictive capacity, although the two false alarms raise legitimate doubts. At the same time, the last FAU interval, which started in December 2006, has been predicted in advance by means of this algorithm (Fig. 17), as well as by the algorithm already suggested by Keilis-Borok et al. (2005).

*Homicides surges in megacities.* We describe here the application for the City of Los Angeles, with its 3.8 million inhabitants (in 2003), and for New York City, with its 8.1 million inhabitants (likewise in 2003). The index $f$ in both cases is GTAAO, as defined and explained at the end of Sect. 7.2.2. For the City of Los Angeles, the data used are for January 1974 to December 2003; they are available from the website of the National Archive of Criminal Justice Data (NACJD), http://www.icpsr.umich.edu/NACJD/. Based on this dataset, the following parameter values were chosen:

$$ (s^*, u^*, \mu^*, v^*, L_1^*, L_2^*, \tau^*) = (6, 48, 184, 12, 1.0, 2.0, 12). $$

The resulting graph of $A(m; s^*, u^*, \mu^*, v^*)$ is given in Fig. 18a (solid black line) along with the periods of the homicide surge (grey vertical strips) and the alarms declared by the algorithm (magenta boxes). The periods of homicide surge are determined, following Keilis-Borok et al. (2003), as periods of a lasting homicide raise after smoothing seasonal variations.

The homicide surge in 1977 falls outside the range of the $A(m; s^*, u^*, \mu^*, v^*)$ definition; as explained for the economic recessions, the range in which $A(m)$ is defined starts only in
<figure><img src="image_19.png" /><figcaption>**Fig. 18.** Same as Fig. 16, but for the prediction of homicide surges in (a) the City of Los Angeles, and (b) New York City. In both cases, data on the grand total of all actual offences (GTAAO) from the National Archive of Criminal Justice Data (NACJD) were used. The NACJD data on crime statistics in (a) the City of Los Angeles were available for January 1974 to December 2003, and in (b) New York City for January 1967 to May 2002. Based on these data, the parameter values chosen for both megacities were $(s^*, u^*, v^*) = (6, 48, 12)$, while $\mu = 184$ in panel (a) and $\mu = 1000$ in panel (b). Same conventions as in Figs. 16 and 17.</figcaption></figure>

July 1979. Each of the other four periods of homicide surge is predicted. The alarm in 1991 appears inside the longest homicide surge period for the city, which extended from mid-1988 till early 1993.

The NACJD data set on crime statistics in New York City contains the GTAAO index from January 1967 to May 2002. It clearly makes sense to exclude the victims of September 11, 2001. The resulting parameter values for New York City are

$$ (s^*, u^*, \mu^*, v^*, L_1^*, L_2^*, \tau^*) = (6, 48, 1000, 12, 0.0, 3.0, 12). $$

The results for New York City are summarized in Fig. 18b. The two homicide surges in 1974 and 1985 are predicted by the algorithm, while the alarm associated with the remaining one in 1978 is delayed by 4 months.

<figure><img src="image_20.png" /><figcaption>**Fig. 19.** Schematic diagram of the prediction algorithm. The four panels illustrate the steps in the transition from the original time series $f(m)$ to the sequence of background-activity events of size $\mu^f(m_j; s, u)$ to the event-counting, activity-level function $A(m; s, u, \mu, v)$ and on to the alarms of length $\tau$ at times of significant change in the activity-level function $A(m)$. The parameters $s, u, \mu$ are introduced in panel (b), $L_1, L_2$ in panel (c) and $\tau$ in panel (d). See Appendix D for details.</figcaption></figure>

## 8 Concluding remarks

The results of the investigations reviewed herein have clearly made significant contributions to the various aspects of the overall E2-C2 project and to the study of extreme events in general. Given the length of this review, though, it seems most appropriate to conclude with a number of questions, rather than with a summary and with definitive conclusions, if any.

The key question for the description, understanding and prediction of extreme events can be formulated as a paraphrase to F. Scott Fitzgerald's assertion in "The Great Gatsby": "The rich are like you and me, only richer". Is this true of extreme events, i.e., are they just other events in the life of a given system, only bigger? If so, one can – as one often does – extrapolate from the numerous small ones to the few large ones. This approach allows one to jump from
the climate sciences because of its conceptual simplicity, its easy implementation and its good performance for most distributions encountered in the geosciences. Smith (1985) studied and implemented the maximum-likelihood estimation (MLE) method for the GEV density. According to Hosking et al. (1985), the PWM approach is superior to the MLE for small GEV-distributed samples.

Coles and Dixon (1999) argued that the PWM method makes a priori assumptions about the shape parameter that are equivalent to assuming a finite mean for the distribution under investigation. To integrate a similar condition in the MLE approach, these authors proposed a penalized MLE scheme with the constraint $\xi < 1$. If this condition is satisfied, then the MLE approach is as competitive as the PWM one, even for small samples. Still, the debate over the advantages and drawbacks of these two estimation methods is not closed and new estimators are still being proposed.

Diebolt et al. (2008) introduced the concept of generalized probability-weighted moments (GPWM) for the GEV. It broadens the domain of validity of the PWM approach, as it allows heavier tails to be fitted. Despite this advantage, the MLE method has kept one strong advantage over the PWM approach, namely its inherent flexibility in a non-stationary context. To complete this appendix, we would like to emphasize that, besides the three aforementioned estimation methods – MLE, PWM and GPWM – there exist other variants and extensions. For example, Hosking (1990) proposed and studied the L-moments, Zhang (2007) derived a new and interesting method-of-moments, and Bayesian estimation has also generated a lot of interest in extreme value analysis (e.g., Lye et al., 1993; Coles, 2001; Cooley et al., 2007; Coles and Powell, 1996).

# Appendix B

## Bivariate point processes

Equation (19) yields

$$P(Z(x) \le u \text{ and } Z(x+h) \le u) = \exp[-\theta(h)/u],$$

where

$$\theta(h) = \int \max\{g(s, x), g(s, x+h)\} \delta(ds)$$

is called the extremal coefficient and has been used in many dependence studies for bivariate vectors (Fougères, 2004; Ledford and Tawn, 1997). The coefficient $\theta(h)$ lies in the interval $1 \le \theta \le 2$ and it gives partial information about the dependence structure of the bivariate pair $(Z(x), Z(x+h))$. If the two variables of this pair are independent, then $\theta(h) = 2$; at the other end of the spectrum of possibilities, the variables are equal in probability, and $\theta(h) = 1$. Hence, the extremal coefficient value $\theta(h)$ provides some dependence information as a function of the distance between two locations. For

example, the bivariate distribution for the Schlather model, $P(Z(x) \le s, Z(x+h) \le t)$, corresponds to

$$\exp\left\{-\frac{1}{2}\left(\frac{1}{t} + \frac{1}{s}\right)\left(1 + \left[1 - 2(\rho(h) + 1)\frac{st}{(s+t)^2}\right]^{1/2}\right)\right\}$$

where $\rho(h)$ is the covariance function of the underlying Gaussian process. This yields an extremal coefficient of $\theta(h) = 1 + \{1 - \frac{1}{2}(\rho(h) + 1)\}^{1/2}$. Naveau et al. (2009) proposed nonparametric techniques to estimate such bivariate structures. As an application, Vannitsem and Naveau (2007) analyzed Belgium precipitation maxima. Again, to work with exceedances – rather than modeling maxima, as done here – requires one to develop another strategy.

# Appendix C

## The M8 earthquake prediction algorithm

This intermediate-term earthquake prediction method was originally designed by the retrospective analysis of the dynamics associated with seismic activity that preceded the great earthquakes – i.e., with magnitude $M \simeq 8.0$ – worldwide, hence its name. Since the announcement of its prototype in 1984 and the original version in 1986, the M8 algorithm has been tested in retrospective prediction, as well as in ongoing real-time applications; see, for instance, Kossobokov and Shebalin (2003), Kossobokov (2006), and references therein.

Algorithm M8 is based on a simple pattern-recognition scheme of prediction aimed at earthquakes in a magnitude range designated by $M_0+$, where $M_0+ = [M_0, M_0 + \Delta M)$ and $\Delta M$ is a prescribed constant. Overlapping circles of investigation with diameters $D(M_0)$, starting with $D$ from 5 to 10 times larger than those of the target earthquakes, are used to scan the territory under study. Within each circle, one considers the sequence of earthquakes that occur, with aftershocks removed.

Denote this sequence by $\{(t_i, m_i, h_i, b_i(e)) : i = 1, 2, ...\}$, where $t_i$ are the times of occurrence, with $t_i \le t_{i+1}$; $m_i$ is the magnitude; $h_i$ is focal depth; and $b_i(e)$ is the number of aftershocks of magnitude $M_{aft}$ or greater during the first $e$ days after $t_i$. Several running counts are computed for this sequence in the trailing time window $(t - s, t)$ and magnitude range $M \le m_i < M_0$. These counts correspond to different measures of seismic flux, its deviations from the long-term trend, and clustering of earthquakes. The averages include:

- $N(t) = N(t|M, s)$, the number of main shocks of magnitude $M$ or larger in $(t - s, t)$;

- $L(t) = L(t|M, s, s^*)$, the deviation of $N(t)$ from the longer term trend, i.e.,

$$L(t) = N(t) - N(t - s|M, s^*)(t - s^*)/(t - s - s^*);$$
- $Z(t) = Z(t|M, M_0 - g, s, \alpha, \beta)$, the linear concentration of main shocks $\{i\}$ that lie in the magnitude range $[M, M_0 - g)$ and the time interval $(t - s, t)$, while the parameters $(\alpha, \beta)$ are used in determining the average source diameter for earthquakes in this magnitude range; and

- $B(t) = B(t|M_0 - p, M_0 - q, s, M_{aft}, e) = \max_i\{b_i\}$, the maximum number of aftershocks - i.e., a measure of earthquake clustering - with the sequence $\{i\}$ considered in the trailing time window $(t - s, t)$ and in the magnitude range $[M_0 - p, M_0 - q)$.

Each of the functions $N, L$, and $Z$ is calculated twice, with $M$ providing the long-term average $\nu$ of the annual number of earthquakes in the sequence, with $\nu = 20$ and with $\nu = 10$, respectively. As a result, the earthquake sequence is given a robust description by seven functions: $N, L, Z$ (twice each), and $B$ (once only). Threshold values are identified for each function from the condition that they exceed the $Q$th percentile, i.e., that they exceed $Q\%$ of the encountered values.

An alarm or time of increased probability (TIP) is declared for $\Delta = 5$ yr when at least six out of the seven functions, including $B$, are larger than the selected threshold within a narrow time window $(t - u, t)$. To stabilize prediction, this criterion is checked at two consecutive moments, namely at $t$ and at $t + 0.5$ yr, and the TIP is declared only after the second test agrees with the first. In the course of a real-time application, the alarm can extend beyond or terminate in less than 5 yr when updating of the catalogue causes changes in the magnitude cutoffs or the percentiles.

The following standard values of the parameters listed above are used in the M8 algorithm: $D(M_0) = \exp(M_0 - 5.6) + 1$ in degrees of meridian, $s = 6$ yr, $s^* = 30$ yr, $s = 1$ yr, $g = 0.5$, $p = 2$, $q = 0.2$, and $u = 3$ yr, while $Q = 75\%$ for $B$ and $90\%$ for the other six functions. The linear concentration is estimated as the ratio of the average source diameter $l$ to the average distance $r$ between sources. Usually, $l = n^{-1} \Sigma_i 10^{\beta(M_i - \alpha)}$, where $n$ is the number of main shocks in the catalogue $\{i\}$, and one takes $r \sim n^{-1/3}$, while $\beta = 0.46$ is chosen to represent the linear dimension of the source, and $\alpha = 0$ without loss of generality,

Running averages are defined in a robust way, so that a reasonable variation of parameters does not affect predictions. At the same time, the point-process character of seismic data and the strict usage of the preset thresholds result in a certain discreteness of the alarms.

The M8 algorithm uses a fairly traditional description of a rather complex dynamical system, by merely adding dimensionless concentration $Z$ and a characteristic measure of clustering $B$ to the more commonly used phase space coordinates of rate $N$ and rate differential $L$. The algorithm defines certain thresholds in these phase space coordinates to capture the proximity of a singularity in the system's dynamics. When a trajectory enters the region defined by these threshold values, the probability of an extreme event increases to

a level sufficient to predict its actual occurrence. The choice of the M8 thresholds focuses on a specific intermediate-term rise of seismic activity.

## Appendix D

### The pattern-based prediction algorithm

We consider monthly time series of indices that describe the aggregated activity of a given complex system. Let $f(m)$ be a sample series of the index $f$ and $W^f(l; q, p)$ be the local linear least-squares regression of $f(m)$ estimated on an interval $[q, p]$:

$$W^f(l; q, p) = K^f(q, p)l + B^f(q, p), \quad (D1)$$

where $q \le l \le p$, $K^f(q, p)$ and $B^f(q, p)$ are the regression coefficients.

We introduce $S^f(m; s, u) = K^f(m - s, m) - K^f(m - s - u, m - s)$, where $s$ and $u$ are positive integers. This function reflects the difference between the two trends of $f(m)$ in the $s$ months after $m - s$ and in the $u$ months before $m - s$. The value of $S^f(m; s, u)$ is referred to month $m$ so that to avoid using information from the future.

Let us assume that extreme events in the system considered are associated with a decline in the trend of the $f$-index. Mark an event that reflects background activity ("broken windows") of magnitude $\mu^f(m; s, u) = |S^f(m; s, u)|$ as happening at month $m$ if $S^f(m; s, u) < 0$, as shown in Fig. 20a. When, to the contrary, the extreme events are associated with a rise of the trend in $f$, mark an event at $m$ if $S^f(m; s, u) > 0$, as shown in Fig. 20b.

Given the values of the parameters $s$ and $u$, the procedure described above transforms a monthly series of $f(m)$ into a unique sequence of events at $m_1, m_2, \dots$ with the corresponding magnitudes $\mu^f(m_1; s, u), \mu^f(m_2; s, u), \dots$. Note that, if a series of $f(m)$ is given on the interval $[m_0, m_T]$, then the occurrences of events fall on the interval $[m_0 + s + u, m_T]$.

Consider the events of background activity determined as above, on the basis of the index $f(m)$ and parameters $s$ and $u$. Let $A(m; s, u, \mu, v)$ be the number of events with magnitudes $\mu^f(m_i; s, u) \ge \mu$ that occurred within the preceding $v$ months, i.e., $m - v + 1 \le m_i \le m$. Evidently, the increase of $A(m; s, u, \mu, v)$ characterizes a rise of activity. To give a formal definition, we introduce the two thresholds $L_1 < L_2$ and identify a significant rise in background activity at $m$, provided $A(m - 1; s, u, \mu, v) \le L_1$ and $L_2 \le A(m; s, u, \mu, v)$.

The following general scheme for developing a prediction algorithm adapted to a given data set $\{f(m): m_0 \le m \le m_T\}$ appears to be rather natural. The scheme below and Fig. 19 that illustrates it are given on the assumption that the rise of background activity is a precursory pattern for the occurrence of extreme events. The changes to the scheme and figure in the case it is a decrease that is premonitory are obvious.
1. Given $s$ and $u$, transform a monthly series $f(m)$, as shown in Fig. 20b, into a sequence of events $\{\mu^f(m_j; s, u) : 0 \le j \le J\}$, with $J \le m_T - m_0$.

2. Given $\mu$ and $v$, evaluate $A(m; s, u, \mu, v)$.

3. Given $L_1$ and $L_2$, identify from $A(m; s, u, \mu, v)$ the months $\{m_k : 0 \le k \le K\}$ of background activity rise, with $K \le J$.

4. Given $\tau$, declare $\tau$ months of alarm for an extreme event each time the rise of background activity is identified.

5. Iterate the selection of the parameters $s, u, \mu, v, L_1, L_2, \tau$ to optimize the algorithm's performance and robustness.

The phrase “With three parameters I can fit an elephant, and with four I can have it wag its tail” has been attributed to a number of famous mathematicians and physicists: it is supposed to highlight the problems generated by using too large a number of parameters to fit a given data set. The pattern recognition approach allows considerable freedom in the retrospective development of a prediction algorithm, like the one outlined above. To avoid the overfitting problems highlighted by the “elephant-fitting” quote, such an algorithm has to be insensitive to the variations in its adjustable parameters, choice of premonitory phenomena, definition of alarms, etc. The sensitivity tests have to comprise an exhaustive set of numerical experiments, which represent a major portion of the effort in the algorithmus development.

Molchan (1997) introduced a particular type of error diagrams to evaluate earthquake prediction algorithms. The definition of such an error diagram is the following: Consider the outcomes of prediction during a time interval of length $T$. During that time interval, $N$ strong events occurred and $N_F$ of them were not predicted. The number of declared alarms was $A$, with $A_F$ of them being false alarms, while the total duration of alarms was $D$.

Obviously if $D = T$ there will be no strong events missed, while if $D = 0$ there will be no false alarms. The error diagram of Molchan (1997) shows the trade-off between the relative duration of alarms $\tau = D/T$, the rate of failures to predict $n = N_F/N$, and the rate of false alarms $f = A_F/A$. In the $(n, \tau)$-plane, the straight line $n + \tau = 1$ corresponds to a random binomial prediction – at each epoch an alarm is declared with probability $\tau$ and is not declared with probability $1 - \tau$. Outcomes have to improve upon this straight line in order for the algorithm to be competitive; see also Zaliapin et al. (2003b) for the use of this diagram in the idealized world of a BDE model that can generate an arbitrarily long catalog of earthquakes. Similar considerations apply when adapting the algorithm above to other complex systems.

## Appendix E

### Table of acronyms.

<table><tr><td>ACF</td><td>Auto-correlation function</td></tr><tr><td>AIC</td><td>Akaike information criterion</td></tr><tr><td>AR</td><td>Auto regressive</td></tr><tr><td>ARIMA</td><td>Autoregressive integrated moving average</td></tr><tr><td>ARMA</td><td>Autoregressive moving average</td></tr><tr><td>BDE</td><td>Boolean delay equation</td></tr><tr><td>CA</td><td>Cellular automaton</td></tr><tr><td>DDE</td><td>Delay differential equation</td></tr><tr><td>dDS</td><td>Discrete dynamical system</td></tr><tr><td>DDS</td><td>Differentiable dynamical system</td></tr><tr><td>DFA</td><td>Detrended fluctuation analysis</td></tr><tr><td>DOF</td><td>Degrees of freedom</td></tr><tr><td>E2-C2</td><td>Extreme Events: Causes and Consequences</td></tr><tr><td>EnBC</td><td>Endogenous business cycle</td></tr><tr><td>ENSO</td><td>El Niño–Southern Oscillation</td></tr><tr><td>EOF</td><td>Empirical orthogonal function</td></tr><tr><td>ETCCDI</td><td>Expert Team on Climate Change Detection and Indices</td></tr><tr><td>EVT</td><td>Extreme value theory</td></tr><tr><td>FARIMA</td><td>Fractional Autoregressive integrated moving average</td></tr><tr><td>FAU</td><td>Fast acceleration of unemployment</td></tr><tr><td>FDE</td><td>Functional differential equation</td></tr><tr><td>fGn</td><td>Fractional Gaussian noise</td></tr><tr><td>FRBSL</td><td>Federal Reserve Bank of St. Louis</td></tr><tr><td>GAM</td><td>Generalized additive model</td></tr><tr><td>GARCH</td><td>Generalized autoregressive conditional heteroskedasticity</td></tr><tr><td>GCM</td><td>General circulation model</td></tr><tr><td>GDP</td><td>Gross domestic product</td></tr><tr><td>GEV</td><td>Generalized Extreme Value (distribution)</td></tr><tr><td>GLM</td><td>Generalized Linear Model</td></tr><tr><td>GPD</td><td>Generalized Pareto distribution</td></tr><tr><td>GPH</td><td>Geweke and Porter-Hudak (estimator)</td></tr><tr><td>GPWM</td><td>Generalized probability-weighted moments</td></tr><tr><td>GTAAO</td><td>Grand total of all actual offences</td></tr><tr><td>IPT</td><td>Industrial production total</td></tr><tr><td>i.i.d.</td><td>independent and identically distributed (random variables)</td></tr><tr><td>LAM</td><td>Limited-area model</td></tr><tr><td>LHELL</td><td>Index of help wanted advertising</td></tr><tr><td>LRD</td><td>Long-range dependence</td></tr><tr><td>MLE</td><td>Maximum likelihood estimation</td></tr><tr><td>MTM</td><td>Multi-taper method</td></tr><tr><td>NACJD</td><td>National Archive of Criminal Justice Data</td></tr><tr><td>NBER</td><td>National Bureau of Economic Research</td></tr><tr><td>NEIC</td><td>National Earthquake Information Center</td></tr><tr><td>NEdYM</td><td>Non-Equilibrium Dynamical model</td></tr><tr><td>ODE</td><td>Ordinary differential equation</td></tr><tr><td>OΔE</td><td>Ordinary difference equation</td></tr><tr><td>PC</td><td>Principal component</td></tr><tr><td>PDE</td><td>Partial differential equation</td></tr><tr><td>PΔE</td><td>Partial difference equation</td></tr><tr><td>PSP</td><td>Premonitory seismic pattern</td></tr><tr><td>POT</td><td>Peak over threshold</td></tr><tr><td>PWM</td><td>Probability-weighted moments</td></tr><tr><td>QBO</td><td>Quasi-biennial oscillation</td></tr><tr><td>QQO</td><td>Quasi-quadrrennial oscillation</td></tr><tr><td>RBC</td><td>Real business cycle</td></tr><tr><td>RC</td><td>Reconstructed component</td></tr><tr><td>R/S</td><td>Rescaled-range statistic</td></tr><tr><td>SDM</td><td>Statistical dynamical model</td></tr><tr><td>SOI</td><td>Southern Oscillation Index</td></tr><tr><td>SRD</td><td>Short-range dependence</td></tr><tr><td>SSA</td><td>Singular spectrum analysis</td></tr><tr><td>SST</td><td>Sea surface temperature</td></tr><tr><td>SWG</td><td>Stochastic weather generator</td></tr><tr><td>TIP</td><td>Time of increased probability</td></tr><tr><td>VGAM</td><td>Vector Generalized Additive Model</td></tr><tr><td>VGLM</td><td>Vector Generalized Linear Model</td></tr></table>
**Acknowledgements.** It is a pleasure to thank all our collaborators in the E2-C2 project and all the participants in the project's several open conferences, special sessions of professional society meetings (American Geophysical Union and European Geosciences Union), and smaller workshops. R. Blender and two anonymous reviewers read carefully the paper's original version and provided useful suggestions, while V. Lucarini, S. Vaienti and R. Vitolo suggested improvements to Sect. 4.2. M.-C. Lanceau provided precious help in editing the merged list of references. This review paper reflects work carried out under the European Commission's NEST project "Extreme Events: Causes and Consequences (E2-C2)."

Edited by: J. Kurths

Reviewed by: R. Blender and two anonymous referees

<figure><img src="image_21.png" /></figure>

INSU
Institut national des sciences de l'Univers

The publication of this article is financed by CNRS-INSU.

## References

Abaimov, S. G., Turcotte, D. L., Shcherbakov, R., and Rundle, J. B.: Recurrence and interoccurrence behavior of self-organized complex phenomena, Nonlin. Processes Geophys., 14, 455–464, doi:10.5194/npg-14-455-2007, 2007.

Abarbanel, H. D. I.: Analysis of Observed Chaotic Data, Springer-Verlag, New York, 272 pp., 1996.

Albeverio, S., Jentsch, V., and Kantz, H.: Extreme Events in Nature and Society, Birkhäuser, Basel/Boston, 352 pp., 2005.

Allègre, C. J., Lemouel, J. L., and Provost, A.: Scaling rules in rock fracture and possible implications for earthquake prediction, Nature, 297, 47–49, 1982.

Allen, M.: Interactions Between the Atmosphere and Oceans on Time Scales of Weeks to Years, Ph.D. thesis, St. John's College, Oxford, 352 pp., 1992.

Allen, M. and Smith, L.: Monte Carlo SSA: Detecting irregular oscillations in the presence of colored noise, J. Climate, 9, 3373–3404, 1996.

Altmann, E. G. and Kantz, H.: Recurrence time analysis, long-term correlations, and extreme events, Phys. Rev. E, 71, 2005.

Anh, V., Yu, Z.-G., and Wanliss, J. A.: Analysis of global geo-magnetic variability, Nonlin. Processes Geophys., 14, 701–708, doi:10.5194/npg-14-701-2007, 2007.

Arnol'd, V. I.: Geometrical Methods in the Theory of Ordinary Differential Equations, Springer-Verlag, New York, 386 pp., 1983.

Ausloos, M. and Ivanova, K.: Power-law correlations in the Southern-Oscillation-Index fluctuations characterizing El Niño, Phys. Rev. E, 63, doi:10.1103/PhysRevE.63.047201, 2001.

Avnir, D., Biham, O., Lidar, D., and Malcai, O.: Is the geometry of nature fractal?, Science, 279, 39–40, 1998.

Bak, P., Tang, C., and Wiesenfeld, K.: Self-organized criticality, Phys. Rev. A, 38, 364–374, 1988.

Balakrishnan, V., Nicolis, C., and Nicolis, G.: Extreme-value distributions in chaotic dynamics, J. Stat. Phys., 80, 307–336, 1995.

Bardet, J.-M., Lang, G., Oppenheim, G., Philippe, A., and Taququ, M. S.: Generators of Long-Range Dependent Processes: A Survey, in: Theory and Applications of Long-Range Dependence, edited by: Doukhan, P., Oppenheim, G., and Taququ, M. S., 579–623, Birkhäuser, Boston, 750 pp., 2003.

Barnett, T. and Preisendorfer, R.: Multifield analog prediction of short-term climate fluctuations using a climate state vector, J. Atmos. Sci., 35, 1771–1787, 1978.

Beirlant, J., Goegebeur, Y., Segers, J., and Teugels, J.: Statistics of Extremes: Theory and Applications, Probability and Statistics, Wiley, 522 pp., 2004.

Benhabib, J. and Nishimura, K.: The Hopf-bifurcation and the existence of closed orbits in multi-sectoral models of optimal economic growth, J. Econ. Theory, 21, 421–444, 1979.

Benson, C. and Clay, E.: Understanding the Economic and Financial Impact of Natural Disasters, The International Bank for Reconstruction and Development, The World Bank, Washington, D.C., 2004.

Beran, J.: Statistics for Long-Memory Processes, Monographs on Statistics and Applied Probability, Chapman & Hall, 315 pp., 1994.

Beran, J., Bhansali, R. J., and Ocker, D.: On unified model selection for stationary and nonstationary short- and long-memory autoregressive processes, Biometrika, 85, 921–934, 1998.

Bernacchia, A. and Naveau, P.: Detecting spatial patterns with the cumulant function – Part 1: The theory, Nonlin. Processes Geophys., 15, 159–167, doi:10.5194/npg-15-159-2008, 2008.

Bernacchia, A., Naveau, P., Vrac, M., and Yiou, P.: Detecting spatial patterns with the cumulant function – Part 2: An application to El Niño, Nonlin. Processes Geophys., 15, 169–177, doi:10.5194/npg-15-169-2008, 2008.

Biau, G., Zorita, E., von Storch, H., and Wackernagel, H.: Estimation of precipitation by kriging in the EOF space of the sea level pressure field, J. Climate, 12, 1070–1085, 1999.

Blender, R., Fraedrich, K., and Sienz, F.: Extreme event return times in long-term memory processes near 1/f, Nonlin. Processes Geophys., 15, 557–565, doi:10.5194/npg-15-557-2008, 2008.

Bliefernicht, J. and Bárdossy, A.: Probabilistic forecast of daily areal precipitation focusing on extreme events, Nat. Hazards Earth Syst. Sci., 7, 263–269, doi:10.5194/nhess-7-263-2007, 2007.

Boldi, M. and Davison, A. C.: Maximum empirical likelihood estimation of the spectral measure of an extreme-value distribution, J. Roy. Stat. Soc.: Series B, 69, 217–229, 2007.

Bowman, D. D., Ouillon, G., Sammis, C. G., Sornette, A., and Sornette, D.: An observational test of the critical earthquake concept, J. Geophys. Res. Solid Earth, 103, 24359–24372, 1998.

Box, G. E. P. and Jenkins, G. M.: Time Series Analysis, Forecasting and Control, Holden-Day, San Francisco, 575 pp., 1970.

Brankovic, C., Matiacic, B., Ivatek-Sahdan, S., and Buizza, R.: Dynamical downscaling of ECMWF ensemble forecasts for cases of severe weather: ensemble statistics and cluster analysis, Mon. Weather Rev., 136, 3323–3342, 2008.

Brémaud, P.: Point Processes and Queues, Springer-Verlag, New York, 373 pp., 1981.

Bremnes, J. B.: Probabilistic forecasts of precipitation in terms of quantiles using NWP model output, Mon. Weather Rev., 132,
338–347, 2004.

Brillinger, D. R.: Time Series: Data Analysis and Theory, Holden-Day, San Francisco, 540 pp., 1981.

Brillinger, D. R., Guttorp, P. M., and Schoenberg, F. P.: Point processes, temporal, in: Encyclopedia of Environmetrics, 3, edited by: El-Shaarawi, A. H. and Piegorsch, W. W., 1577–1581, J. Wiley & Sons, Chichester, 2002.

Brockwell, P. J. and Davis, R. A.: Time Series: Theory and Methods, Springer Series in Statistics, Springer, Berlin, 584 pp., 1991.

Broomhead, D. S. and King, G. P.: Extracting qualitative dynamics from experimental data, Physica, D20, 217–236, 1986.

Bufe, C. G. and Varnes, D. J.: Predictive Modeling of the Seismic Cycle of the Greater San-Francisco Bay-Region, J. Geophys. Res. Solid Earth, 98, 9871–9883, 1993.

Bui Trong, L.: Les racines de la violence, De l'émeute au communautarisme, Louis Audibert, 221 pp., 2003.

Bunde, A., Eichner, J. F., Havlin, S., and Kantelhart, J. W.: The effect of long-term correlations on the return periods of rare events, Physica A, 330, 1–7, 2003.

Bunde, A., Eichner, J. F., Havlin, S., Koscielny-Bunde, E., Schellnhuber, H. J., and Vyushin, D.: Comment on “Scaling of Atmosphere and Ocean Temperature Correlations in Observations and Climate Models”, Phys. Rev. Lett., 92, 039801, doi:10.1103/PhysRevLett.92.039801, 2004.

Burridge, R. and Knopoff, L.: Model and Theoretical Seismicity, Bull. Seism. Soc. Am., 57, 341–371, 1967.

Busuioc, A. and von Storch, H.: Conditional stochastic model for generating daily precipitation time series, Clim. Res., 24, 181–195, 2003.

Busuioc, A., Tomozeiu, R., and Cacciamani, C.: Statistical downscaling model based on canonical correlation analysis for winter extreme precipitation events in the Emilia-Romagna region, Int. J. Climatol., 28, 449–464, 2008.

Caballero, R., Jewson, S., and Brix, A.: Long memory in surface air temperature: detection, modeling, and application to weather derivative valuation, Clim. Res., 21, 127–140, 2002.

Caires, S., Swail, V., and Wang, X.: Projection and analysis of extreme wave climate, J. Climate, 19, 5581–5605, 2006.

Cannon, A. J. and Whitfield, P. H.: Downscaling recent streamflow conditions in British Columbia, Canada using ensemble neural network models, J. Hydrol., 259, 136–151, 2002.

Carreau, J. and Bengio, Y.: A Hybrid Pareto Model for Asymmetric Fat-Tail, Tech. rep., Dept. IRO, Université de Montréal, 2006.

Chang, P., Wang, B., Li, T., and Ji, L.: Interactions between the seasonal cycle and the Southern Oscillation: Frequency entrainment and chaos in intermediate coupled ocean-atmosphere model, Geophys. Res. Lett., 21, 2817–2820, 1994.

Chang, P., Ji, L., Wang, B., and Li, T.: Interactions between the seasonal cycle and El Niño – Southern Oscillation in an intermediate coupled ocean-atmosphere model., J. Atmos. Sci., 52, 2353–2372, 1995.

Chavez-Demoulin, V. and Davison, A. C.: Generalized additive models for sample extremes, Appl. Stat., 54, 207–222, 2005.

Chen, Y. D., Chen, X., Xu, C.-Y., and Shao, Q.: Downscaling of daily precipitation with a stochastic weather generator for the subtropical region in South China, Hydrol. Earth Syst. Sci. Discuss., 3, 1145–1183, doi:10.5194/hessd-3-1145-2006, 2006.

Chen, Z., Ivanov, P. C., Hu, K., and Stanley, H. E.: Effects of non-stationarities on detrended fluctuation analysis, Phys. Rev. E, 65,

041107, doi:10.1103/PhysRevE.65.041107, 2002.

Chiarella, C. and Flaschel: The Dynamics of Keynesian Monetary Growth, Cambridge University Press, 434 pp., 2000.

Chiarella, C., Flaschel, P., and Franke, R.: Foundations for a Disequilibrium Theory of the Business Cycle, Cambridge University Press, 550 pp., 2005.

Chiarella, C., Franke, R., Flaschel, P., and Semmler, W.: Quantitative and Empirical Analysis of Nonlinear Dynamic Macromodels, Elsevier, 564 pp., 2006.

Clauset, A., Shalizi, C., and Newman, M.: Power-law distributions in empirical data, SIAM Rev., 49, doi:10.1137/070710111, 2009.

Coles, S.: An Introduction to Statistical Modeling of Extreme Values, Springer series in statistics, Springer, London, New York, 224 pp., 2001.

Coles, S. and Casson, E.: Spatial regression for extremes, Extremes, 1, 339–365, 1999.

Coles, S. and Dixon, M.: Likelihood-based inference of extreme value models, Extremes, 2, 2–53, 1999.

Coles, S., Heffernan, J., and Tawn, J.: Dependence measures for extreme value analysis, Extremes, 2, 339–365, 1999.

Coles, S. G. and Powell, E. A.: Bayesian methods in extreme value modelling: A review and new developments, Intl. Statist. Rev., 64, 119–136, 1996.

Collet, P. and Eckmann, J.-P.: Iterated Maps on the Interval as Dynamical Systems, Birkhauser Verlag, Boston, 248 pp., 1980.

Cooley, D., Naveau, P., and Poncet, P.: Variograms for spatial max-stable random fields, in: Statistics for dependent data, Lect Notes Stat., Springer, 373–390, 2006.

Cooley, D., Nychka, D., and Naveau, P.: Bayesian spatial modeling of extreme precipitation return levels, J. Amer. Stat. Assoc., 102, 824–840, 2007.

Cowan, G. A., Pines, D., and Melzer, D.: Complexity: Metaphors, Models and Reality, Addison-Wesley, Reading, Mass, 731 pp., 1994.

Cox, D. R. and Hinkley, D. V.: Theoretical Statistics, Chapman & Hall, London, 528 pp., 1994.

Dahlhaus, R.: Efficient parameter estimation for self-similar processes, Ann. Stat., 17, 1749–1766, 1989.

Davis, R. and Mikosch, T.: Extreme Value Theory for Space-Time Processes with Heavy-Tailed Distributions, Stoch. Proc. Appl., 118, 560–584, 2008.

Davis, R. and Mikosch, T.: Extreme Value Theory for GARCH Processes, in: Handbook of Financial Time Series, Springer, New York, 187–200, 2009.

Davis, R. and Resnick, S. I.: Prediction of stationary max-stable processes, Ann. Applied Probab., 3, 497–525, 1993.

Davis, R., Dolan, R., and Demme, G.: Synoptic climatology of Atlantic coast northeasters, Int. J. Climatol., 13, 171–189, 1993.

Davison, A. C. and Ramesh, N. I.: Local likelihood smoothing of sample extremes, J. Roy. Statist. Soc. B, 62, 191–208, 2000.

Day, R.: Irregular growth cycles, Amer. Econ. Rev., 72, 406–414, 1982.

De Haan, L. and Ferreira, A.: Extreme Value Theory: An Introduction, Springer, 436 pp., New York, 2006.

De Haan, L. and Pereira, T. T.: Spatial extremes: Models for the stationary case, Ann. Statist, 34, 146–168, 2006.

De Haan, L. and Resnick, S. I.: Limit theory for multivariate sample extremes, Z. Wahrscheinlichkeitstheorie u. verwandte Gebiete, 40, 317–337, 1977.
De Putter, T., Loutre, M. F., and Wansard, G.: Decadal periodicities of Nile River historical discharge (AD 622–1470) and climatic implications, Geophys. Res. Lett., 25, 3193–3196, 1998.

Dee, D. and Ghil, M.: Boolean difference equations, I: Formulation and dynamic behavior, SIAM J. Appl. Math., 1984.

Deidda, R.: Rainfall downscaling in a space-time multifractal framework, Water Resour. Res., 36, 1779–1794, 2000.

Dettinger, M. D., Ghil, M., Strong, C. M., Weibel, W., and Yiou, P.: Software expedites singular-spectrum analysis of noisy time series, Eos Trans. AGU, 76, 12, 20, 21, 1995.

Diaz, H. and Markgraf, V.: El Niño: Historical and Paleoclimatic Aspects of the Southern Oscillation, Academic Press, Cambridge University Press, New York, 476 pp., 1993.

Diebolt, J., Guillou, A., Naveau, P., and Ribereau, P.: Improving probability-weighted moment methods for the generalized extreme value distribution, REVSTAT – Statistical Journal, 6, 33–50, 2008.

Dijkstra, H. A.: Nonlinear Physical Oceanography: A Dynamical Systems Approach to the Large-Scale Ocean Circulation and El Niño, Springer, Dordrecht, The Netherlands, 534 pp., 2005.

Dijkstra, H. A. and Ghil, M.: Low-frequency variability of the large-scale ocean circulation: A dynamical systems approach, Rev. Geophys., 43, RG3002, doi:10.1029/2002RG000122, 2005.

Drazin, P. G. and King, G. P., eds.: Interpretation of Time Series from Nonlinear Systems (Proc. of the IUTAM Symposium and NATO Advanced Research Workshop on the Interpretation of Time Series from Nonlinear Mechanical Systems), University of Warwick, England, North-Holland, 1992.

Dumas, P., Ghil, M., Groth, A., and Hallegatte, S.: Dynamic coupling of the climate and macroeconomic systems, Math. Social Sci., in press, 2011.

Dupuis, D.: Exceedances over high thresholds: A guide to threshold selection, Extremes, 1, 251–261, 1999.

Eckmann, J. and Ruelle, D.: Ergodic theory of chaos and strange attractors, Rev. Mod. Phys., 57, 617–656, and 57, 1115, 1985.

Eichner, J. F., Koscielny-Bunde, E., Bunde, A., Havlin, S., and Schellnhuber, H.-J.: Power law persistence in the atmosphere: A detailed study of long temperature records, Phys. Rev. E, 68, 2003.

Einmahl, J. and Segers, J.: Maximum empirical likelihood estimation of the spectral measure of an extreme-value distribution, Ann. Statist., 37, 2953–2989, 2009.

El Adlouni, S. and Ouarda, T. B. M. J.: Comparaison des méthodes d'estimation des paramètres du modèle GEV non-stationnaire, Revue des Sciences de l'Eau, 21, 35–50, 2008.

El Adlouni, S., Ouarda, T. B. M. J., Zhang, X., Roy, R., and Bobee, B.: Generalized maximum likelihood estimators for the nonstationary generalized extreme value model, Water Resour. Res., 43, doi:10.1029/2005WR004545, 2007.

Embrechts, P., Klüppelberger, C., and Mikosch, T.: Modelling Extremal Events for Insurance and Finance, Springer, Berlin/New York, 648 pp., 1997.

European Commission: Economic Portrait of the European Union 2001, European Commission, Brussels, Panorama of the European Union, 161 pp., 2002.

Evans, M., Hastings, N., and Peacock, B.: Statistical Distributions, Wiley-Interscience, New York, 170 pp., 2000.

Feliks, Y., Ghil, M., and Robertson, A. W.: Oscillatory climate modes in the Eastern Mediterranean and their synchronization

with the North Atlantic Oscillation, J. Clim., 23, 4060–4079, 2010.

Feller, W.: An Introduction to Probability Theory and its Applications (3rd ed.), I, Wiley, New York, 509 pp., 1968.

Ferro, C. A. T.: Inference for clusters of extreme values, J. Roy. Statist. Soc. B, 65, 545–556, 2003.

Flaschel, P., Franke, R., and Semmler, W.: Dynamic Macroeconomics: Instability, Fluctuations and Growth in Monetary Economies, MIT Press, 455 pp., 1997.

Fougères, A.: Multivariate extremes, in: Extreme Values in Finance, Telecommunication and the Environment, edited by: Finkenstadt, B. and Rootzen, B., Chapman and Hall CRC Press, London, 373–388, 2004.

Fraedrich, K. and Blender, R.: Scaling of atmosphere and ocean temperature correlations in observations and climate models, Phys. Rev. Lett., 90, 108501, doi:10.1103/PhysRevLett.90.108501, 2003.

Fraedrich, K. and Blender, R.: Reply to Comment on “Scaling of atmosphere and ocean temperature correlations in observations and climate models”, Phys. Rev. Lett., 92, 039802–1, 2004.

Fraedrich, K., Luksch, U., and Blender, R.: 1/f-model for long-time memory of the ocean surface temperature, Phys. Rev. E, 70, 037301(1–4), 2004.

Fraedrich, K., Blender, R., and Zhu, X.: Continuum climate variability: long-term memory, extremes, and predictability, Intl. J. Modern Phys. B, 23, 5403–5416, 2009.

Freitas, A. and Freitas, J.: Extreme values for Benedicks-Carleson quadratic maps, Ergod. Theor. Dyn. Syst., 28(4), 1117–1133, 2008b.

Freitas, A., Freitas, J., and Todd, M.: Extreme value laws in dynamical systems for non-smooth observations, J. Stat. Phys., 142(1), 108–126, 2011.

Freitas, A. C. M. and Freitas, J. M.: On the link between dependence and independence in extreme value theory for dynamical systems, Stat. Prob. Lett., 78, 1088–1093, 2008a.

Freitas, J., Freitas, A., and Todd, M.: Hitting time statistics and extreme value theory, Probab. Theory Related Fields, 147(3–4), 675–710, 2010.

Friederichs, P.: Statistical downscaling of extreme precipitation using extreme value theory, Extremes, 13, 109–132, 2010.

Friederichs, P. and Hense, A.: Statistical downscaling of extreme precipitation events using censored quantile regression, Mon. Weather Rev., 135, 2365–2378, 2007.

Friederichs, P. and Hense, A.: A Probabilistic forecast approach for daily precipitation totals, Weather Forecast., 23, 659–673, 2008.

Friederichs, P., Goeber, M., Bentzien, S., Lenz, A., and Krampitz, R.: A probabilistic analysis of wind gusts using extreme value statistics, Meteorol. Z., 18, 615–629, 2009.

Frigessi, A., Haug, O., and Rue, H.: A dynamic mixture model for unsupervised tail estimation without threshold selection, Extremes, 5, 219–235, 2002.

Frisch, R.: Propagation Problems and Impulse Problems in Dynamic Economics, in: Economic Essay in honor of Gustav Cassel, 171–206, George Allen and Unwin, London, 1933.

Furrer, E. and Katz, R. W.: Improving the simulation of extreme precipitation events by stochastic weather generators, Water Resour. Res., 44, W12439, doi:10.1029/2008WR007316, 2008.

Gabrielov, A., Keilis-Borok, V., Zaliapin, I., and Newman, W. I.: Critical transitions in colliding cascades, Phys. Rev. E, 62, 237–
249, 2000a.

Gabriellov, A., Zaliapin, I., Keilis-Borok, and Newman, W. I.: Col-
liding cascades model for earthquake prediction, J. Geophys.
Intl., 143, 427–437, 2000b.

Gagneur, J. and Casari, G.: From molecular networks to qualitative
cell behavior, Febs Letters, 579, 1867–1871, 2005.

Gale, D.: Pure exchange equilibrium of dynamic economic models,
J. Econ. Theory, 6, 12–36, 1973.

Gallant, J. C., Moore, I. D., Hutchinson, M. F., and Gessler, P.: Estimating fractal dimension of profiles, Math. Geol., 26, 455–481, 1994.

Garca, J. A., Gallego, M. C., Serrano, A., and Vaquero, J. M.: De-
pendence estimation and vizualisation in multivariate the Iberian
Peninsula in the second half of the Twentieth Century, J. Climate,
20, 113–130, 2007.

Geisel, T., Zacherl, A., and Radons, G.: Generic 1/f-noise in
chaotic Hamiltonian dynamics, Phys. Rev. Lett., 59, 2503–2506,
1987.

Gelfand, I. M., Guberman, S. A., Keilis-Borok, V. I., Knopoff, L.,
Press, F., Ranzman, E. Y., Rotwain, I. M., and Sadovsky, A. M.:
Pattern recognition applied to earthquake epicenters in Califor-
nia, Phys. Earth Planet. Inter., 11, 227–283, 1976.

Geweke, J. and Porter-Hudak, S.: The estimation and application of
long memory time series models, J. Time Ser. Anal., 4, 221–237,
1983.

Ghaleb, K. O.: Le Mikyas ou Nilomètre de lîle de Rodah, Mem.
Inst. Egypte, 54, 182 pp., 1951.

Ghil, M.: The SSA-MTM Toolkit: Applications to analysis and
prediction of time series, Proc. SPIE, 3165, 216–230, 1997.

Ghil, M. and Childress, S.: Topics in Geophysical Fluid Dynamics: Atmospheric Dynamics, Dynamo Theory, and Climate Dynamics, Springer-Verlag, New York, 485 pp., 1987.

Ghil, M. and Jiang, N.: Recent forecast skill for the El Niño/Southern Oscillation, Geophys. Res. Lett., 25, 171–174, 1998.

Ghil, M. and Mo, K. C.: Intraseasonal oscillations in the global
atmosphere. Part I: Northern hemisphere and tropics, J. Atmos.
Sci., 48, 752–779, 1991.

Ghil, M. and Mullhaupt, A.: Boolean delay equations. 2. Periodic
and aperiodic solutions, J. Stat. Phys., 41, 125–173, 1985.

Ghil, M. and Robertson, A.: General circulation models and their
role in the climate modeling hierarchy, in: General Circulation
Model Developments: Past, Present and Future, edited by: Ran-
dall, D., Academic Press, San Diego, 285–325, 2000.

Ghil, M. and Robertson, A. W.: “Waves” vs. “particles” in the at-
mosphere’s phase space: A pathway to long-range forecasting?,
Proc. Natl. Acad. Sci. USA, 99 (Suppl. 1), 2493–2500, 2002.

Ghil, M. and Vautard, R.: Interdecadal oscillations and the warming
trend in global temperature time series, Nature, 350, 324–327,
1991.

Ghil, M., Halem, M., and Atlas, R.: Time-continuous assimilation
of remote-sounding data and its effect on weather forecasting,
Mon. Weather Rev., 107, 140–171, 1979.

Ghil, M., Benzi, R., and Parisi, G.: Turbulence and Predictability
in Geophysical Fluid Dynamics and Climate Dynamics, North-
Holland, Amsterdam, 1985.

Ghil, M., Mullhaupt, A., and Pestiaux, P.: Deep water formation
and Quaternary glaciations, Clim. Dyn., 2, 1–10, 1987.

Ghil, M., Allen, M. R., Dettinger, M. D., Ide, K., Kondrashov, D.,

Mann, M. E., Robertson, A. W., Saunders, A., Tian, Y., Varadi,
F., and Yiou, P.: Advanced spectral methods for climatic time se-
ries, Rev. Geophys., 40, 3.1–3.41, doi:10.1029/2000RG000092,
2002.

Ghil, M., Zaliapin, I., and Coluzzi, B.: Boolean delay equations: A simple way of looking at complex systems, Physica D, 237, 2967–2986, 2008a.

Ghil, M., Zaliapin, I., and Thompson, S.: A delay differential model of ENSO variability: parametric instability and the distribution of extremes, Nonlin. Processes Geophys., 15, 417–433, doi:10.5194/npg-15-417-2008, 2008b.

Glahn, H. R. and Lowry, D. A.: The use of model output statistics (MOS) in objective weather forecasting, J. Appl. Meteor., 11, 1203–1211, 1972.

Gleick, J.: Chaos: Making a New Science, Viking, New York,
352 pp., 1987.

Goodwin, R.: The non-linear accelerator and the persistence of
business cycles, Econometrica, 19, 1–17, 1951.

Goodwin, R.: A growth cycle, in: Socialism, Capitalism and Eco-
nomic Growth, edited by Feinstein, C., Cambridge University
Press, Cambridge, 54–58, 1967.

Grandmont, J.-M.: On endogenous competitive business cycles,
Econometrica, 5, 995–1045, 1985.

Granger, C. W. and Joyeux, R.: An Introduction to Long-Memory
Time Series Models and Fractional Differences, J. Time Ser.
Anal., 1, 15–29, 1980.

Greenwood, J. A., Landwehr, J. M., Matalas, N. C., and Wallis,
J. R.: Probability Weighted Moments - Definition and Relation to
Parameters of Several Distributions Expressable in Inverse Form,
Water Resour. Res., 15, 1049–1054, 1979.

Guckenheimer, J. and Holmes, P.: Nonlinear Oscillations, Dynam-
ical Systems, and Bifurcations of Vector Fields, Springer, New
York, corr. 5th print. edn., 480 pp., 1997.

Gupta, C., Holland, M., and Nicol, M.: Extreme value theory and
return time statistics for dispersing billiard maps and flows, Lozi
maps and Lorenz-like maps, Ergodic Theory Dynam. Systems, in
press, http://sites.google.com/site/9p9emrak/B111785S, 2011.

Gutowitz, H.: Cellular Automata: Theory and Experiment, MIT
Press, Cambridge, MA, 1991.

Guttorp, P., Brillinger, D. R., and Schoenberg, F. P.: Point pro-
cesses, spatial, in: Encyclopedia of Environmetrics, vol. 3, edited
by: El-Shaarawi, A. H. and Piegorsch, W. W., 1571–1573, J. Wi-
ley & Sons, Chichester, 2002.

Hahn, F. and Solow, R.: A Critical Essay on Modern Macroecon-
nomic Theory, The MIT Press, Cambridge, Massachusetts and
London, England, 208 pp., 1995.

Hall, P. and Tajvidi, N.: Distribution and dependence-function es-
timation for bivariate extreme-value distributions, Bernoulli, 6,
835–844, 2000.

Hallegatte, S. and Ghil, M.: Natural disasters impacting a macroe-
conomic model with endogenous dynamics, Ecol. Econ., 68,
582–592, doi:10.1016/j.ecolecon.2008.05.022, 2008.

Hallegatte, S., Hourcade, J.-C., and Dumas, P.: Why economic
dynamics matter in assessing climate change damages: illustra-
tion on extreme events, Ecol. Econ., 62, 330–340, doi:10.1016/j.
ecolecon.2006.06.006, 2007.

Hallegatte, S., Ghil, M., Dumas, P., and Hourcade, J.-C.: Business
cycles, bifurcations and chaos in a neo-classical model with in-
vestment dynamics, J. Economic Behavior & Organization, 67,
57–77, doi:10.1016/j.jebo.2007.05.001, 2008.

Hamill, T. M. and Whitaker, J. S.: Probabilistic quantitative precipitation forecasts based on reforecast analogs: theory and application, Mon. Weather Rev., 134, 3209–3229, 2006.

Hamill, T. M., Whitaker, J. S., and Mullen, S. L.: Reforecasts: An important dataset for improving weather predictions, B. Am. Meteorol. Soc., 87, 33–46, 2006.

Hannan, E. J.: Time Series Analysis, Methuen, New York, 152 pp., 1960.

Harrod, R.: An essay on dynamic economic theory, Economic J., 49, 1433, 1939.

Hasselmann, K.: Stochastic climate models, Tellus, 6, 473–485, 1976.

Hastie, T. and Tibshirani, R.: Generalized Additive Models, Chapman and Hall, 335 pp., 1990.

Haylock, M., Cawley, G., Harpham, C., Wilby, R., and Goodess, C.: Downscaling heavy precipitation over the United Kingdom: a comparison of dynamical and statistical methods and their future scenarios, Int. J. Climatol., 26, 1397–1415, 2006.

Haylock, M. R. and Goodess, C. M.: Interannual variability of European extreme winter rainfall and links with mean large-scale circulation, Int. J. Climatol., 24, 759–776, 2004.

Heffernan, J. E. and Tawn, J.: A conditional approach for multivariate extreme values, J. Roy. Statist. Soc. B, 66, 497–546, 2004.

Hénon, M.: Sur La Topologie Des Lignes De Courant Dans Un Cas Particulier, C. R. Acad. Sci. Paris, 262, 312–414, 1966.

Hewitson, B. and Crane, R.: Self organizing maps: applications to synoptic climatology, Clim. Res., 26, 1315–1337, 2002.

Hicks, J.: The Cycle in Outline, in: A Contribution to the Theory of the Trade Cycle, Oxford University Press, Oxford, 8, 95–107, 1950.

Hillinger, C.: Cyclical Growth in Market and Planned Economies, Oxford University Press, 224 pp., 1992.

Holland, M. P., Nicol, M., and Török, A.: Extreme value theory for non-uniformly expanding dynamical systems, Trans. Amer. Math. Soc., in press, 2011.

Hosking, J. R. M.: Fractional differencing, Biometrika, 68, 165–176, 1981.

Hosking, J. R. M.: Maximum-likelihood estimation of the parameters of the generalized extreme-value distribution, Appl. Stat., 34, 301–310, 1985.

Hosking, J. R. M.: L-moment: Analysis and estimation of distributions using linear combinations of order statistics, J. Roy. Statist. Soc. B, 52, 105–124, 1990.

Hosking, J. R. M., Wallis, J. R., and Wood, E.: Estimation of the generalized extreme-value distribution by the method of probability-weighted moments, Technometrics, 27, 251–261, 1985.

Hurst, H. E.: Long-term storage capacity of reservoirs, Trans. Am. Soc. Civil Eng., 116, 770–799, 1951.

Hurst, H. E.: The Nile, Constable, London, 326 pp., 1952.

Hüsler, J. and Reiss, R.-D.: Maxima of normal random vectors: Between independence and complete dependence, Stat. Probabil. Lett., 7, 283–286, 1989.

Huth, R.: Disaggregating climatic trends by classification of circulation patterns, Int. J. Climatol., 21, 135–153, 2001.

Huth, R.: Statistical downscaling of daily temperature in central Europe, J. Climate, 15, 1731–1742, 2002.

Jacob, F. and Monod, J.: Genetic regulatory mechanisms in synthe-

sis of proteins, J. Molecular Biol., 3, 318–356, 1961.

Jarsulic, M.: A nonlinear model of the pure growth cycle, J. Econ. Behav. Organ., 22, 133–151, 1993.

Jaume, S. C. and Sykes, L. R.: Evolving towards a critical point: A review of accelerating seismic moment/energy release prior to large and great earthquakes, Pure Appl. Geophys., 155, 279–305, 1999.

Jenkins, G. M. and Watts, D. G.: Spectral Analysis and its Applications, Holden-Day, San Francisco, 525 pp., 1968.

Jin, F.: Tropical ocean-atmosphere interaction, the Pacific cold tongue, and the El Niño-Southern Oscillation, Science, 274, 1996.

Jin, F., Neelin, J., and Ghil, M.: El Niño on the Devil's staircase: Annual subharmonic steps to chaos, Science, 264, 70–72, 1994.

Jin, F., Neelin, J., and Ghil, M.: El Nino/Southern Oscillation and the annual cycle: Subharmonic frequency locking and aperiodicity, Physica D, 98, 1996.

Jin, F. F. and Neelin, J. D.: Modes of Interannual Tropical Ocean-Atmosphere Interaction - a Unified View.1. Numerical Results, J. Atmos. Sci., 50, 3477–3503, 1993a.

Jin, F. F. and Neelin, J. D.: Modes of Interannual Tropical Ocean-Atmosphere Interaction – a Unified View.2. Analytical results in fully coupled cases, J. Atmos. Sci., 50, 3504–3522, 1993b.

Jin, F. F. and Neelin, J. D.: Modes of Interannual Tropical Ocean-Atmosphere Interaction – a Unified View.3. Analytical Results in Fully Coupled Cases, J. Atmos. Sci., 50, 3523–3540, 1993c.

Joe, H., Smith, R., and Weissmann, I.: Bivariate threshold methods for extremes, Royal Statisitcal Society, Series B, 54, 171–183, 1992.

Kabluchko, Z., Schlather, M., and de Haan, L.: Stationary max-stable fields associated to negative definite functions, 37, 2042–2065, 2009.

Kaldor, N.: A model of the trade cycle, Econ. J., 50, 78–92, 1940.

Kalecki, M.: A theory of the business cycle, Review of Economic Studies, 4, 77–97, 1937.

Kallache, M., Rust, H. W., and Kropp, J.: Trend assessment: applications for hydrology and climate research, Nonlin. Processes Geophys., 12, 201–210, doi:10.5194/npg-12-201-2005, 2005.

Kalnay, E., Kanamitsu, M., Kistler, R., Collins, W., Deaven, D., Gandin, L., Iredell, M., Saha, S., White, G., Woollen, J., Zhu, Y., Chelliah, M., Ebisuzaki, W., Higgins, W., Janowiak, J., Mo, K., Ropelewski, C., Wang, J., Leetmaa, A., Reynolds, R., Jenne, R., and Joseph, D.: The NCEP/NCAR 40-year reanalysis project, Bull. Amer. Meteorol. Soc., 77, 437–471, 1996.

Kantelhardt, J. W., Koscielny-Bunde, E., Rego, H. H. A., Havlin, S., and Bunde, A.: Detecting Long-range Correlations with Trended Fluctuation Analysis, Physica A, 295, 441–454, 2001.

Kantz, H. and Schreiber, T.: Nonlinear Time Series Analysis, 2nd Edition, Cambridge Univ. Press, Cambridge, 388 pp., 2004.

Kantz, H., Altman, E. G., Hallerberg, S., Holstein, D., and Riegert, A.: Dynamical interpretation of extreme events: Predictability and prediction, in: Extreme Events in Nature and Society, edited by: Albeverio, S., Jentsch, V., and Kantz, H., 69–94, Birkhäuser, Basel/Boston, 2005.

Katz, R.: Precipitation as a chain-dependent process, J. Appl. Meteorol., 16, 671–676, 1977.

Katz, R. W., Parlange, M. B., and Naveau, P.: Statistics of extremes in hydrology, Adv. Water Resour., 25, 1287–1304, 2002.

Kauffman, S.: The Origins of Order: Self-Organization and Selec-
tion in Evolution, Oxford University Press, 709 pp., 1993.

Kauffman, S. A.: Metabolic stability and epigenesis in randomly constructed genetic nets, J. Theor. Biol., 22, 437–467, 1969.

Kauffman, S. A.: At Home in the Universe: The Search for Laws of Self-Organization and Complexity, Oxford University Press, New York, 336 pp., 1995.

Keilis-Borok, V. I.: Symptoms of instability in a system of earthquake-prone faults, Physica D, 77, 193–199, 1994.

Keilis-Borok, V. I.: Intermediate-term earthquake prediction, Proc. Natl. Acad. Sci. USA, 93, 3748–3755, 1996.

Keilis-Borok, V. I.: Earthquake prediction: State-of-the-art and emerging possibilities, Ann. Rev. Earth Planet. Sci., 30, 1–33, 2002.

Keilis-Borok, V. I. and Kossobokov, V.: Premonitory activation of earthquake flow: algorithm M8, Phys. Earth Planet. Inter., 61, 73–83, 1990.

Keilis-Borok, V. I. and Malinovskaya, L.: One regularity in the occurrence of strong earthquakes, J. Geophys. Res., 69, 3019–3024, 1964.

Keilis-Borok, V. I. and Shebalin, P.: Dynamics of lithosphere and earthquake prediction, Phys. Earth Planet. Int., 111, 179–330, 1999.

Keilis-Borok, V. I. and Soloviev, A. A.: Nonlinear Dynamics of the Lithosphere and Earthquake Prediction, Springer-Verlag, Berlin-Heidelberg, 338 pp., 2003.

Keilis-Borok, V. I., Knopoff, L., and Rotwain, I. M.: Bursts of after-shocks, long-term precursors of strong earthquakes, Nature, 283, 258–263, 1980.

Keilis-Borok, V. I., Knopoff, L., Rotwain, I. M., and Allen, C. R.: Intermediate-term prediction of occurrence times of strong earthquakes, Nature, 335, 690–694, 1988.

Keilis-Borok, V. I., Stock, J. H., Soloviev, A., and Mikhalev, P.: Pre-recession pattern of six economic indicators in the USA, J. Forecast., 19, 65–80, 2000.

Keilis-Borok, V. I., Gascon, D. J., Soloviev, A. A., Intriligator, M. D., Pichardo, R., and Winberg, F. E.: On predictability of homicide surges in megacities, in: Risk Science and Sustainability, edited by: Beer, T. and Ismail-Zadeh, A., vol. 112 of NATO Science Series. II. Mathematics, Physics and Chemistry, 91–110, Kluwer Academic Publishers, Dordrecht-Boston-London, 2003.

Keilis-Borok, V. I., Soloviev, A. A., Allègre, C. B., Sobolevskii, A. N., and Intriligator, M. D.: Patterns of macroeconomic indicators preceding the unemployment rise in Western Europe and the USA, Pattern Recognition, 38, 423–435, 2005.

Keizer, K., Lindenberg, S., and Steg, L.: The spreading of disorder, Science, 322, 1681–1685, doi:10.1126/science.1161405, 2008.

Kelling, G. L. and Wilson, J. Q.: Broken windows, Atlantic Monthly, 1982, 29 pp., March 1982.

Keppenne, C. L. and Ghil, M.: Extreme weather events, Nature, 358, 547, 1992a.

Keppenne, C. L. and Ghil, M.: Adaptive filtering and prediction of the Southern Oscillation Index, J. Geophys. Res, 97, 20449–20454, 1992b.

Keppenne, C. L. and Ghil, M.: Adaptive filtering and prediction of noisy multivariate signals: An application to subannual variability in atmospheric angular momentum, Intl. J. Bifurcation & Chaos, 3, 625–634, 1993.

Kesten, H.: Random difference equations and renewal theory for products of random matrices, Acta Mathematica, 131, 207–248,

1973.

Kharin, V. V. and Zwiers, F.: Changes in the extremes in an ensemble of transient climate simulation with a coupled atmosphere-ocean GCM, J. Climate, 13, 3760–3788, 2000.

Kharin, V. V., Zwiers, F. W., Zhang, X., and Hegerl, G. C.: Changes in temperature and precipitation extremes in the IPCC ensemble of global coupled model simulations, J. Climate, 20, 1419–1444, 2007.

King, R. and Rebelo, S.: Resuscitating real business cycles, in: Handbook of Macroeconomics, edited by: Taylor, J. and Woodford, M., North-Holland, Amsterdam, 927–1007, 2000.

Király, A. and Jánosi, I. M.: Detrended fluctuation analysis of daily temperature records: Geographic dependence over Australia, Meteorol. Atmos. Phys., 88, 119–128, 2004.

Klein, W. H.: Computer Prediction of Precipitation Probability in the United States, J. Appl. Meteorol., 10, 903–915, 1971.

Knopoff, L., Levshina, T., Keilis-Borok, V. I., and Mattoni, C.: Increased long-range intermediate-magnitude earthquake activity prior to strong earthquakes in California, J. Geophys. Res. Solid Earth, 101, 5779–5796, 1996.

Koenker, R. and Bassett, B.: Regression quantiles, Econometrica, 46, 33–49, 1978.

Kolmogorov, A. N.: Local structures of turbulence in fluid for very large Reynolds numbers, in: Transl. in Turbulence, edited by: Friedländer, S. K. and Topper, L., Interscience Publisher, New York, 1961, 151–155, 1941.

Kondrashov, D. and Ghil, M.: Spatio-temporal filling of missing points in geophysical data sets, Nonlin. Processes Geophys., 13, 151–159, doi:10.5194/npg-13-151-2006, 2006.

Kondrashov, D., Feliks, Y., and Ghil, M.: Oscillatory modes of extended Nile River records (AD 622–1922), Geophys. Res. Lett., 32, L10702, doi:10.1029/2004GL022156, 2005.

Kondrashov, D., Kravtsov, S., Robertson, A. W., and Ghil, M.: A hierarchy of data-based ENSO models, J. Climate, 18, 4425–4444, 2005b.

Koscielny-Bunde, E., Bunde, A., Havlin, S., and Goldreich, Y.: Analysis of daily temperature fluctuations, Physica A, 231, 393–396, 1996.

Kossobokov, V.: User Manual for M8, in: Algorithms for Earthquake Statistics and Prediction, edited by: Healy, J. H., Keilis-Borok, V. I., and Lee, W. H. K., 6 of IASPEI Software Library, Seismological Society of America, El Cerrito, CA, 167–222, 1997.

Kossobokov, V.: Quantitative Earthquake Prediction on Global and Regional Scales, in: Recent Geodynamics, Georisk and Sustainable Development in the Black Sea to Caspian Sea Region, edited by: Ismail-Zadeh, A., American Institute of Physics Conference Proceedings, Melville, New York, 825, 32–50, 2006.

Kossobokov, V. and Shebalin, P.: Earthquake Prediction, in: Nonlinear Dynamics of the Lithosphere and Earthquake Prediction, edited by: Keilis-Borok, V. I. and Soloviev, A. A., Springer-Verlag, Berlin-Heidelberg, 141–207, 2003.

Kossobokov, V. G., Maeda, K., and Uyeda, S.: Precursory activation of seismicity in advance of Kobe, 1995, M =7.2 earthquake, Pure Appl. Geophys., 155, 409–423, 1999a.

Kossobokov, V. G., Romashkova, L. L., Keilis-Borok, V. I., and Healy, J. H.: Testing earthquake prediction algorithms: Statistically significant real-time prediction of the largest earthquakes in the Circum-Pacific, 1992–1997, Phys. Earth Planet. Inter., 111,
187–196, 1999b.

Kydland, F. and Prescott, E.: Time to build and aggregate fluctuations, Econometrica, 50, 1345–1370, 1982.

Lalaurette, F.: Early detection of abnormal weather using a probabilistic extreme forecast index, Q. J. R. Meteorol. Soc., 129, 3037–3057, 2003.

Landwehr, J. M., Matalas, N. C., and Wallis, J. R.: Probability Weighted Moments Compared with Some Traditional Techniques in Estimating Gumbel Parameters and Quantiles, Water Resour. Res., 15, 1055–1064, 1979.

Lasota, A. and Mackey, M. C.: Chaos, Fractals, and Noise: Stochastic Aspects of Dynamics, 97, Appl. Mathematical Sci., Springer-Verlag, New York, 2 edn., 472 pp., 1994.

Latif, M., Barnett, T., Cane, M., Flügel, M., Graham, N., Von Storch, H., Xu, J., and Zebiak, S.: A review of ENSO prediction studies, Clim. Dyn., 9, 167–179, 1994.

Leadbetter, M. R., Lindgren, G., and Rootzén, H.: Extremes and Related Properties of Random Sequences and Processes, Springer Series in Statistics, Springer-Verlag, New York, 336 pp., 1983.

Ledford, A. W. and Tawn, J. A.: Modelling dependence within joint tail regions, J. Roy. Statist. Soc. B, 59, 475–499, 1997.

Lichtman, A. J. and Keilis-Borok, V. I.: Aggregate-level analysis and prediction of midterm senatorial elections in the United States, Proc. Natl. Acad. Sci. USA, 86, 10176–10180, 1989.

Lo, A. W.: Long-term memory in stock market prices, Econometrica, 59, 1279–1313, 1991.

Lomb, N.: Least-squares frequency analysis of unequally spaced data, Astrophys. Space Sci, 39, 447–462, 1976.

Lorenz, E.: Deterministic nonperiodic flow, J. Atmos. Sci., 20, 130–141, 1963.

Lorenz, E. N.: Atmospheric predictability as revealed by naturally occurring analogues, J. Atmos. Sci., 26, 636–646, 1969.

Lye, L. M., Hapuarachchi, K. P., and Ryan, S.: Bayes estimation of the extreme-value reliability function, IEEE Trans. Reliab., 42, 641–644, 1993.

Malamud, B.: Tails of natural hazards, Physics World, 17, 31–35, 2004.

Malamud, B. and Turcotte, D.: Self-affine time series: measures of weak and strong persistence, J. Statistical Planning Inference, 80, 173–196, 1999.

Mandelbrot, B.: The Fractal Geometry of Nature, W. H. Freeman, New York, 460 pp., 1982.

Mandelbrot, B. B. and Wallis, J. R.: Noah, Joseph and operational hydrology, Water Resour. Res., 4, 909–918, 1968a.

Mandelbrot, B. B. and Wallis, J. R.: Robustness of the rescaled range R/S and measurement of non-cyclic long-run statistical dependence, Water Resour. Res., 5, 967–988, 1968b.

Mañé, R.: On the dimension of the compact invariant sets of certain non-linear maps, in: Dynamical Systems and Turbulence, edited by: Rand, D. A. and Young, L.-S., 898, Lect. Notes Math., Springer, Berlin, 230–242, 1981.

Mann, M. E. and Lees, J. M.: Robust estimation of background noise and signal detection in climatic time series, Clim. Change, 33, 409–445, 1996.

Manneville, P.: Intermittency, self-similarity and 1/f spectrum in dissipative dynamical systems, J. Physique, 41, 1235–1243, 1980.

Maraun, D., Rust, H. W., and Timmer, J.: Tempting long-memory-

on the interpretation of DFA results, Nonlin. Processes Geophys., 11, 495–503, doi:10.5194/npg-11-495-2004, 2004.

Maraun, D., Wetterhall, F., Ireson, A., Chandler, R., Kendon, E., Widmann, M., Brienen, S., Rust, H., Sauter, T., Themessl, M., Venema, V., Chun, K., Goodess, C., Jones, R., Onof, C., Vrac, M., and Thiele-Eich, I.: Precipitation Downscaling under climate change. Recent developments to bridge the gap between dynamical models and the end user, Rev. Geophys., 48, RG3003, doi: 10.1029/2009RG000314, 2010.

Markovic, D. and Koch, M.: Sensitivity of Hurst parameter estimation to periodic signals in time series and filtering approaches, Geophys. Res. Lett., 32, L17401, doi:10.1029/2005GL024069, 2005.

Marsigli, C., Boccanera, F., Montani, A., and Paccagnella, T.: The COSMO-LEPS mesoscale ensemble system: validation of the methodology and verification, Nonlin. Processes Geophys., 12, 527–536, doi:10.5194/npg-12-527-2005, 2005.

Marzban, C., Sandgathe, S., and Kalnay, E.: MOS, Perfect Prog, and Reanalysis Data, Mon. Weather Rev., 134, 657–663, 2005.

Mertins, A.: Signal Analysis: Wavelets, Filter Banks, Time-Frequency Transforms and Applications, 330 pp., 1999.

Mestre, O. and Halegatte, S.: Predictors of tropical cyclone numbers and extreme hurricane intensities over the North Atlantic using generalized additive and linear models, J. Climate, 22, 633–648, doi:10.1175/2008JCLI2318.1, 2009.

Metzler, R.: Comment on “Power-Law corelations in the southern-oscillation-index fluctuations characterizing El Niño”, Phys. Rev. E, 67, doi:10.1103/Physreve.67.018201, 2003.

Min, S.-K., Zhang, X., Zwiers, F. W., Friederichs, P., and Hense, A.: Signal detectability in extreme precipitation changes assessed from twentieth century climate simulations., Clim. Dynam., 32, 95–111, 2009.

Mitchell, J. M.: An overview of climatic variability and its causal mechanisms, Quatern. Res., 6, 481–493, 1976.

Mitzenmacher, M.: A Brief History of Generative Models for Power Law and Lognormal Distributions, Internet Mathematics, 1, 226–251, 2003.

Molchan, G. M.: Earthquake prediction as a decision-making problem, Pure Appl. Geophys., 149, 233–247, 1997.

Molchan, G. M., Dmitrieva, O., Rotwain, I., and Dewey, J.: Statistical analysis of the results of earthquake prediction, based on burst of aftershocks, Phys. Earth Planet. Int., 61, 128–139, 1990.

Monetti, R. A., Havlin, S., and Bunde, A.: Long term persistence in the sea surface temperature fluctuations, Physica A, 320, 581–589, 2001.

Montanari, A., Rosso, R., and Taqqu, M. S.: A seasonal fractional ARIMA model applied to the Nile River monthly flows at Aswan, Water Resour. Res., 36, 1249–1259, 2000.

Montani, A., Capaldo, M., Cesari, D., Marsigli, C., Modigliani, U., Nerozzi, F., Paccagnella, T., Patruno, P., and Tibaldi, S.: Operational limited-area ensemble forecasts based on the Lokal Modell, ECMWF Newsletter, 98, 2–7, 2003.

Mudelsee, M.: Long memory of rivers from spatial aggregation, Water Resour. Res., 43, doi:10.129/2006WR005721, 2007.

Mullhaupt, A. P.: Boolean Delay Equations: A Class of Semi-Discrete Dynamical Systems, Ph.D. thesis, New York University, 193 pp., 1984.

Nadarajah, S.: A polynomial model for bivariate extreme value distributions, Stat. Probabil. Lett., 42, 15–25, 1999.
Narteau, C., Shebalin, P., and Holschneider, M.: Loading rates in California inferred from aftershocks, Nonlin. Processes Geophys., 15, 245–263, doi:10.5194/npg-15-245-2008, 2008.

Naveau, P., Guillou, A., Cooley, D., and Diebolt, J.: Modelling pairwise dependence of maxima in space, Biometrika, 96, 1–17, 2009.

Nelder, J. A. and Wedderburn, R. W.: Generalized Linear Models, J. Roy. Statist. Soc. A, 135, 370–384, 1972.

Newman, W. I., Gabrielov, A., and Turcotte, D.: Nonlinear Dynamics and Predictability of Geophysical Phenomena, American Geophysical Union, Washington, DC, 107 pp., 1994.

Nicolis, C. and Nicolis, S. C.: Return time statistics of extreme events in deterministic dynamical systems, Europ. Phys. Lett., 80, doi:10.1209/0295-5075/80/40003, 2007.

Nicolis, C., Balakrishnan, V., and Nicolis, G.: Extreme events in deterministic dynamical systems, Phys. Rev. Lett., 97, doi:10.1103/PhysRevLett.97.210602, 2006.

Nicolis, G., Balakrishnan, V., and Nicolis, C.: Probabilistic aspects of extreme events generated by periodic and quasiperiodic deterministic dynamics, Stoch. Dynam., 8, 115–125, 2008.

Nikaido, H.: Prices, Cycles, and Growth, MIT Press, Cambridge, MA, 285 pp., 1996.

Nordhaus, W. D. and Boyer, J.: Roll the DICE Again: The Economics of Global Warming, Tech. rep., Yale University, http://www.econ.yale.edu/~nordhaus/homepage/dicemodels.htm, 1998.

Orlowsky, B. and Fraedrich, K.: Upscaling European surface temperatures to North Atlantic circulation-pattern statistics, Intl. J. Climatology, 29, 839–849, doi:10.1002/joc.1744, 2009.

Orlowsky, B., Bothe, O., Fraedrich, K., Gerstengarbe, F.-W., and Zhu, X.: Future climates from bias-bootstrapped weather analogues: an application to the Yangtze river basin, J. Climate, 23, 3509–3524, 2010.

Ott, E., Sauer, T., and Yorke, J. A., eds.: Coping with Chaos, Analysis of Chaotic Data and the Exploitation of Chaotic Systems, J. Wiley & Sons, New York, 418 pp., 1994.

Packard, N. H., Crutchfield, J. P., Farmer, J. D., and Shaw, R. S.: Geometry from a time series, Phys. Rev. Lett., 45, 712–716, 1980.

Padoan, S. A., Ribatet, M., and Sisson, S. A.: Likelihood-Based Inference for Max-Stable Processes, Journal of the American Statistical Association, 105, 263–277, doi:10.1198/jasa.2009.tm08577, 2010.

Penland, C., Ghil, M., and Weickmann, K. M.: Adaptive filtering and maximum entropy spectra, with application to changes in atmospheric angular momentum, J. Geophys. Res., 96, 22659–22671, 1991.

Pepke, G., Carlson, J., and Shaw, B.: Prediction of large events on a dynamical model of fault, J. Geophys. Res., 99, 6769–6788, 1994.

Percival, D. B. and Walden, A. T.: Spectral Analysis for Physical Applications, Cambridge University Press, Cambridge, UK, 583 pp., 1993.

Percival, D. B. and Walden, A. T.: Wavelet Methods for Time Series Analysis, Cambridge University Press, Cambridge, UK, 600 pp., 2000.

Perica, S. and Foufoula-Georgiou, E.: Linkage of scaling and thermodynamic parameters of rainfall: Results from midlatitude mesoscale convective systems, J. Geophys. Res., 101, 7431–7448, 1996.

Philander, S. G. H.: El Niño, La Niña, and the Southern Oscillation, Academic Press, San Diego, 289 pp., 1990.

Plaut, G. and Vautard, R.: Spells of low-frequency oscillations and weather regimes in the Northern Hemisphere, J. Atmos. Sci., 51, 210–236, 1994.

Plaut, G., Schuepbach, E., and Doctor, M.: Heavy precipitation events over a few Alpine sub-regions and the links with large-scale circulation, Clim. Res., 17, 285–302, 2001.

Popper, W.: The Cairo Nilometer, Univ. of Calif. Press, Berkeley, 269 pp., 1951.

Priestley, M. B.: Spectral Analysis and Time Series, Academic Press, London, 890 pp., 1992.

Procaccia, I. and Schuster, H.: Functional renormalization-group theory of universal 1/f noise in dynamical systems, Phys. Rev. A, 28, 1210–1212, 1983.

Pryor, S., Schoof, J. T., and Barthelmie, R.: Empirical downscaling of wind speed probability distributions, J. Geophys. Res., 110, doi:10.1029/2005JD005899, 2005.

Ramos, A. and Ledford, A.: A new class of models for bivariate joint tails, J. Roy. Stat. Soc. B, 71, 219–241, 2009.

Resnick, S. I.: Extreme Values, Regular Variation, and Point Processes, 4 of Applied Probability, Springer-Verlag, New York, 334 pp., 1987.

Resnick, S. I.: Heavy-Tail Phenomena: Probabilistic and Statistical Modeling, Operations Research and Financial Engineering, Springer, New York, 410 pp., 2007.

Ribatet, M.: SpatialExtremes: A R package for Modelling Spatial Extremes, http://cran.r-project.org/, 2008.

Ribatet, M.: POT: Generalized Pareto Distribution and Peaks Over Threshold, http://people.epfl.ch/mathieu.ribatet, http://r-forge.r-project.org/projects/pot/, r package version 1.0-9, 2009.

Risk Management Solutions: Hurricane Katrina: Profile of a Super Cat. Lessons and Implications for Catastrophe Risk Management., Tech. rep., Risk Management Solutions, Newark, CA, 30 pp., 2005.

Robertson, A. W. and Ghil, M.: Large-scale weather regimes and local climate over the Western United States, J. Climate, 12, 1796–1813, 1999.

Robinson, P. M.: Log-periodogram regression of time series with long-range dependence, Ann. Statist., 23, 1048–1072, 1995.

Robinson, P. M.: Time Series with Long Memory, Advanced Texts in Econometrics, Oxford University Press, 392 pp., 2003.

Rossi, M., Witt, A., Guzzetti, F., Malamud, B. D., and Peruccacci, S.: Analysis of historical landslide time series in the Emilia-Romagna region, northern Italy, Earth Surf. Proc. Land., 35, 1123–1137, 2010.

Roux, J. C., Rossi, A., Bachelart, S., and Vidal, C.: Representation of a strange attractor from an experimental study of chemical turbulence, Phys. Lett. A, 77, 391–393, 1980.

Rudin, W.: Real and Complex Analysis (3rd ed.), McGraw-Hill, New York, 483 pp., 1987.

Ruelle, D.: Small random perturbations of dynamical systems and the definition of attractors, Commun. Math. Phys., 82, 137–151, 1981.

Ruelle, D. and Takens, F.: On the nature of turbulence, Commun. Math. Phys., 20, 167–192, and 23, 343–344, 1971.

Rundle, J., Turcotte, D., and Klein, W., eds.: Geocomplexity and the
Physics of Earthquakes, American Geophysical Union, Washington, DC, 2000.

Rust, H. W.: Detection of Long-Range Dependence – Applications in Climatology and Hydrology, Ph.D. thesis, Potsdam University, Potsdam, http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-13347, 2007.

Rust, H. W.: The effect of long-range dependence on modelling extremes with the generalised extreme value distribution, Europ. Phys. J. Special Topics, 174, 1, 91–97, doi:10.1140/epjst/e2009-01092-8, 2009.

Rust, H. W., Mestre, O., and Venema, V. K. C.: Fewer jumps, less memory: Homogenized temperature records and long memory, J. Geophys. Res., 113, doi:10.1029/2008JD009919, 2008.

Rust, H. W., Kallache, M., Schellnhuber, H.-J., and Kropp, J.: Confidence intervals for flood return level estimates using a bootstrap approach, in: In Extremis, edited by: Kropp, J. and Schellnhuber, H.-J., Springer, 61–81, 2011.

Rybski, D., Bunde, A., Havlin, S., and von Storch, H.: Long-term persistence in the climate and the detection problem, Geophy. Res. Lett., 33, doi:10.1029/2005GL025591, 2006.

Salameh, T., Drobinski, P., Vrac, M., and Naveau, P.: Statistical downscaling of near-surface wind over complex terrain in southern France, Meteorol. Atmos. Phys., 103, 243–256, 2009.

Sanchez-Gomez, E. and Terray, L.: Large-scale atmospheric dynamics and local intense precipitation episodes, Geophys. Res. Lett., 32, L24711, doi:10.1029/2005GL023990, 2005.

Samuelson, P.: A Synthesis of the Principle of Acceleration and the Multiplier, J. Political Econ., 47, 786–797, 1939.

Sato, K.-I.: Lévy Processes and Infinitely Divisible Distributions, Cambridge University Press, London, 486 pp., 1999.

Sauer, T., Yorke, J. A., and Casdagli, M.: Embedology, J. Stat. Phys., 65, 579–616, 1991.

Saunders, A. and Ghil, M.: A Boolean delay equation model of ENSO variability, Physica D, 160, 54–78, 2001.

Schlather, M.: Models for stationary max-stable random fields, Extremes, 5, 33–44, 2002.

Schlather, M.: RandomFields: Simulations and Analysis of Random Fields, R package version 1.3.28, 2006.

Schlather, M. and Tawn, J.: Inequalities for the extremal coefficients of multivariate extreme value distributions, Extremes, 1, 87–102, 2009.

Schlather, M. and Tawn, J. A.: A dependence measure for multivariate and spatial extreme values: Properties and inference, Biometrika, 90, 139–156, 2003.

Schmidli, J., Goodess, C., Frei, C., Haylock, M., Hundecha, Y., Ribalaygua, J., and Schmith, T.: Statistical and dynamical downscaling of precipitation: An evaluation and comparison of scenarios for the European Alps, J. Geophys. Res.-Atmos., D04105, doi:10.1029/2005JD007026, 2007.

Schnur, R. and Lettenmaier, D.: A case study of statistical downscaling in Australia using weather classification by recursive partitioning, J. Hydrol., 212–213, 362–379, 1998.

Scholz, C. H.: The mechanics of earthquakes and faulting, Cambridge University Press, Cambridge, U.K.; New York, 2nd edn., 471 pp., 2002.

Schölzel, C. and Friederichs, P.: Multivariate non-normally distributed random variables in climate research – introduction to the copula approach, Nonlin. Processes Geophys., 15, 761–772, doi:10.5194/npg-15-761-2008, 2008.

Semenov, M. and Barrow, E.: Use of a stochastic weather generator in the development of climate change scenarios, Clim. Res., 35, 397–414, 1997.

Semenov, M., Brooks, R., Barrow, E., and Richardson, C.: Comparison of the WGEN and the LARS-WG stochastic weather generators in diverse climates, Clim. Res., 10, 95–107, 1998.

Serinaldi, F.: Comment on “Multivariate non-normally distributed random variables in climate research – introduction to the copula approach” by C. Schölzel and P. Friederichs, Nonlin. Processes Geophys., 15, 761–772, 2008, Nonlin. Processes Geophys., 16, 331–332, doi:10.5194/npg-16-331-2009, 2009.

Simonnet, E., Ghil, M., and Dijkstra, H. A.: Homoclinic bifurcations in the quasi-geostrophic double-gyre circulation, J. Mar. Res., 63, 931–956, 2005.

Slutsky, E.: The summation of random causes as a source of cyclic processes, III(1), Conjuncture Institute, Moscow, Econometrica, 5, 105–146, 1927.

Smale, S.: Differentiable dynamical systems, Bull. Amer. Math. Soc., 73, 747–817, 1967.

Smith, H. F.: An empirical law describing heterogeneity in the yields of acricultural crops, J. Agric. Sci., 28, 1–23, 1938.

Smith, R. L.: Maximum-likelihood estimation in a class of nonregular cases, Biometrika, 72, 67–90, 1985.

Smith, R. L.: Extreme value analysis of environmental time series: An application to trend detection in ground-level ozone, Stat. Sci., 4, 367–393, 1989.

Smith, R. L.: Max-stable processes and spatial extremes, http://www.stat.unc.edu/postscript/rs/spatex.pdf, 1990.

Smith, R. L.: Statistics of extremes, with applications in environment, insurance and finance., in: Extreme Values in Finance, Telecommunication and the Environment, edited by: B. Finkenstadt, and Rootzen, H., 1–78, Chapman and Hall CRC Press, London, 1–78, 2004.

Smith, R. L., Tawn, J., and Coles, S. G.: Markov chain models for threshold exceedances, Biometrika, 84, 249–268, doi:10.1093/biomet/84.2.249, 1997.

Snell, S., Gopal, S., and Kaufmann, R.: Spatial interpolation of surface air temperatures using artificial neural networks: Evaluating their use for downscaling GCMs, J. Climate, 13, 886–895, 2000.

Solomon, S., Qin, D., Manning, M., Chen, Z., Marquis, M., Averyt, K., Manning, M., and Miller, H., eds.: Climate Change 2007: The Physical Science Basis. Contribution of Working Group I to the Fourth Assessment Report of the Intergovernmental Panel on Climate Change, Cambridge University Press, Cambridge; New York, 2007.

Soloviev, A.: Transformation of frequency-magnitude relation prior to large events in the model of block structure dynamics, Nonlin. Processes Geophys., 15, 209–220, doi:10.5194/npg-15-209-2008, 2008.

Solow, R. M.: A contribution to the theory of economic growth, Q. J. Econ., 70, 65–94, 1956.

Sornette, D.: Multiplicative processes and power laws, Phys. Rev. E, 57, 4811–4813, 1998.

Sornette, D.: Critical Phenomena in Natural Sciences, Chaos, Fractals, Self-organization and Disorder: Concepts and Tools, 2nd edn., Springer, Heidelberg, 528 pp., 2004.

Soros, G.: The New Paradigm for Financial Markets: The Credit Crisis of 2008 and What It Means, BBS, PublicAffairs, New York, 162 pp., 2008.
Stedinger, J. R., Vogel, R. M., Foufoula-Georgiou, E., and Maidment, D. R.: Frequency analysis of extreme events, in: Handbook of Hydrology, McGraw-Hill, 18–23, 1993.

Stern, N.: The Economics of Climate Change. The Stern Review, Cambridge University Press, 712 pp., 2006.

Takayasu, H., Sato, A.-H., and Takayasu, M.: Stable infinite variance fluctuations in randomly amplified Langevin systems, Phys. Rev. Lett., 79, 966–969, 1997.

Takens, F.: Detecting strange attractors in turbulence, in: Dynamical Systems and Turbulence, edited by: Rand, D. A. and Young, L.-S., 898 Lec. Notes Math., 366–381, Springer, Berlin, 1981.

Taqqu, M. S., Teverovsky, V., and Willinger, W.: Estimators for long-range dependence: An empirical study, Fractals, 3, 785–798, 1995.

Taricco, C., Alessio, S., and Vivaldo, G.: Sequence of erupitive events in the Vesuvio area recorded in shallow-water Ionian Sea sediments, Nonlin. Processes Geophys., 15, 25–32, doi:10.5194/npg-15-25-2008, 2008.

Thomas, R.: Boolean formalization of genetic-control circuits, J. Theor. Biol., 42, 563–585, 1973.

Thomas, R.: Logical analysis of systems comprising feedback loops, J. Theor. Biol., 73, 631–656, 1978.

Thomas, R.: Kinetic Logic: A Boolean Approach to the Analysis of Complex Regulatory Systems, Springer-Verlag, Berlin, Heidelberg, New York, 507 pp., 1979.

Thomson, D. J.: Spectrum estimation and harmonic analysis, Proc. IEEE, 70, 1055–1096, 1982.

Timmer, J. and König, M.: On generating power law noise, Astron. Astrophys., 300, 707–710, 1995.

Toussoun, O.: Mémoire sur l'histoire du Nil, Mem. Inst. Egypte, 18, 366–404, 1925.

Tucker, W.: The Lorenz attractor exists, C. R. Acad. Sciences–Math., 328, 1197–1202, 1999.

Turcotte, D.: Fractals and Chaos in Geology and Geophysics, Cambridge University Press, Cambridge, 2nd edition edn., 231 pp., 1997.

Turcotte, D., Newman, W., and Gabrielov, A.: A statistical physics approach to earthquakes, in: Geocomplexity and the Physics of Earthquakes, edited by: Rundle, J., Turcotte, D., and Klein, W., American Geophysical Union, Washington, DC, 2000.

Tziperman, E., Stone, L., Cane, M. A., and Jarosh, H.: El Niño Chaos - Overlapping of Resonances between the Seasonal Cycle and the Pacific Ocean-Atmosphere Oscillator, Science, 264, 72–74, 1994.

Tziperman, E., Cane, M., and Zebiak, S.: Irregularity and locking to the seasonal cycle in an enso prediction model as explained by the quasi-periodicity route to chaos, J. Atmos. Sci., 50, 293–306, 1995.

Unal, Y. S. and Ghil, M.: Interannual and Interdecadal Oscillation Patterns in Sea-Level, Clim. Dyn., 11, 255–278, 1995.

Vannitsem, S. and Naveau, P.: Spatial dependences among precipitation maxima over Belgium, Nonlin. Processes Geophys., 14, 621–630, doi:10.5194/npg-14-621-2007, 2007.

Vautard, R. and Ghil, M.: Singular spectrum analysis in nonlinear dynamics, with applications to paleoclimatic time series, Physica, D35, 395–424, 1989.

Vautard, R., Yiou, P., and Ghil, M.: Singular spectrum analysis: a toolkit for short noisy chaotic signals, Physica D, 58, 95–126, 1992.

Vislocky, R. L. and Young, G. S.: The use of perfect prog forecasts to improve model output statistics forecasts of precipitation probability, Weather Forecast., 4, 202–209, 1989.

Von Neumann, J.: Theory of Self-Reproducing Automata, University of Illinois Press, Urbana, Illinois, 488 pp., 1966.

Voss, R. F.: Random fractal forgeries, in: Fundamental Algorithms for Computer Graphics, edited by: Earnshaw, R., Springer, Berlin, 805–835, 1985.

Vrac, M. and Naveau, P.: Stochastic downscaling of precipitation: From dry events to heavy rainfalls, Water Resour. Res., 43, doi: 10.1029/2006WR005308, 2007.

Vrac, M., Naveau, P., and Drobinski, P.: Modeling pairwise dependencies in precipitation intensities, Nonlin. Processes Geophys., 14, 789–797, doi:10.5194/npg-14-789-2007, 2007a.

Vrac, M., Marbaix, P., Paillard, D., and Naveau, P.: Non-linear statistical downscaling of present and LGM precipitation and temperatures over Europe, Clim. Past, 3, 669–682, doi:10.5194/cp-3-669-2007, 2007b.

Vrac, M., Stein, M., and Hayhoe, K.: Statistical downscaling of precipitation through nonhomogeneous stochastic weather typing, Clim. Res., 34, 169–184, doi:10.3354/cr00696, 2007c.

Vrac, M., Stein, M., Hayhoe, K., and Liang, X.-Z.: A general method for validating statistical downscaling methods under future climate change, Geophys. Res. Lett., 34, doi:10.1029/2007GL030295, 2007d.

Vyushin, D., Mayer, J., and Kushner, P.: PowerSpectrum: Spectral Analysis of Time Series, r-package version 0.3, http://www.atmosph.physics.utoronto.ca/people/vyushin/mysoftware.html, 2008.

Vyushin, D. I. and Kushner, P. J.: Power-law and long-memory characteristics of the atmospheric general circulation, J. Climate, 22, 2890–2904, doi:10.1175/2008JCLI2528.1, 2009.

Vyushin, D. I., Fioletov, V. E., and Shepherd, T. G.: Impact of long-range correlations on trend detection in total ozone, J. Geophys. Res., 112, D14307, doi:10.1029/2006JD008168, 2007.

Wetterhall, F., Halldin, S., and Xu, C.-Y.: Statistical precipitation downscaling in central Sweden with the analogue method, J. Hydrol., 306, 174–190, 2005.

Whitcher, B., Byers, S., Guttorp, P., and Percival, D.: Testing for homogeneity of variance in time series: Long memory, wavelets, and the Nile River, Water Resour. Res., 38, 5, 1054, doi:10.1029/2001WR000509, 2002.

White, E., Enquist, B., and Green, J.: On estimating the exponent of power-law frequency distributions, Ecology, 89, 905–912, 2008.

Whitney, B.: Differentiable manifolds, Ann. Math., 37, 645–680, 1936.

Widmann, M., Bretherton, C., and Salathé, E.: Statistical precipitation downscaling over the Northwestern United States using numerically simulated precipitation as a predictor, J. Climate, 16, 799–816, 2003.

Wigley, T. M., Jones, P., Briffa, K., and Smith, G.: Obtaining subgrid scale information from coarse resolution general circulation output, J. Geophys. Res., 95, 1943–1953, 1990.

Wilby, R. and Wigley, T.: Downscaling general circulation model output: a review of methods and limitations, Prog. Phys. Geogr., 21, 530–548, doi:10.1177/030913339702100403, 1997.

Wilby, R., Charles, S., Zorita, E., Timbal, B., Whetton, P., and Mearns, L.: Guidelines for use of Climate Scenarios Developed from Statistical Downscaling Methods, IPCC Task Group on
Data and Scenario Support for Impact and Climate Analysis (TG-ICA), http://ipcc-ddc.cru.uea.ac.uk/guidelines/StatDownGuide.pdf, 2004.

Wilks, D. S.: Multi-site generalization of a daily stochastic precipitation model, J. Hydrol., 210, 178–191, 1998.

Wilks, D. S.: Multi-site downscaling of daily precipitation with a stochastic weather generator, Clim. Res., 11, 125–136, 1999.

Wilks, D. S. and Wilby, R. L.: The weather generation game: a review of stochastic weather models, Prog. Phys. Geogr., 23, 329–357, 1999.

Witt, A., Malamud, B. D., Rossi, M., Guzzetti, F., and Peruccacci, S.: Temporal correlations and clustering of landslides, Earth Surf. Proc. Land., 35, 1138–1156, 2010.

Wolfram, S.: Statistical mechanics of cellular automata, Rev. Mod. Phys., 55, 601–644, 1983.

Wolfram, S., ed.: Cellular Automata and Complexity: Collected Papers, Addison-Wesley, Reading, Mass., 1994.

World Bank: Turkey: Marmara Earthquake Assessment, The World Bank, Washington, D.C., world Bank Working Paper, 1999.

Yee, T. W.: A User's Guide to the VGAM Package, Tech. rep., Department of Statistics, University of Auckland, http://www.stat.auckland.ac.nz/~yee/VGAM, 2006.

Yee, T. W. and Hastie, T. J.: Reduced-rank vector generalized linear models, Stat. Modell., 3, 15–41, 2003.

Yee, T. W. and Stephenson, A.: Vector Generalized Linear and Additive Extreme Value Models, Extremes, 10, 1386–1999, 2007.

Yee, T. W. and Wild, C. J.: Vector generalized additive models, J. Roy. Statist. Soc. B, 58, 481–493, 1996.

Yiou, P. and Nogaj, M.: Extreme climatic events and weather regimes over the North Atlantic: When and where?, Geophys. Res. Lett., 31, L07202, doi:10.1029/2003GL019119, 2004.

Yiou, P., Loutre, M. F., and Baert, E.: Spectral analysis of climate data, Surveys Geophys., 17, 619–663, 1996.

Yiou, P., Sornette, D., and Ghil, M.: Data-adaptive wavelets and multi-scale singular-spectrum analysis, Physica D, 142, 254–290, 2000.

Yiou, P., Goubanova, K., Li, Z. X., and Nogaj, M.: Weather regime dependence of extreme value statistics for summer temperature and precipitation, Nonlin. Processes Geophys., 15, 365–378, doi:10.5194/npg-15-365-2008, 2008.

Yosida, K.: Functional Analysis, Academic Press, San Diego, 513 pp., 1968.

Zaliapin, I., Keilis-Borok, and Ghil, M.: A Boolean delay equation model of colliding cascades, Part I: Multiple seismic regimes., J. Stat. Phys., 111, 2003a.

Zaliapin, I., Keilis-Borok, V., and Ghil, M.: A Boolean delay equation model of colliding cascades. Part II: Prediction of critical transitions., J. Stat. Phys., 111, 2003b.

Zhang, J.: Likelihood moment estimation for the generalized pareto distribution, Aust Nz J. Stat., 49, 69–77, 2007.

Zhang, X., Zwiers, F., and Li, G.: Monte Carlo experiments on the detection of trends in extreme values, J. Climate, 17, 1945–1952, 2004.

Zorita, E. and von Storch, H.: The analog method as a simple statistical downscaling technique: Comparison with more complicated methods, J. Climate, 12, 2474–2489, 1998.

Zorita, E., Hughes, J., Lettenmaier, D., and von Storch, H.: Stochastic characterization of regional circulation patterns for climate model diagnosis and estimation of local precipitation, Report of the Max-Planck-Institut für Meteorologie, Hamburg, 109, 1993.

Zwiers, F. and Kharin, V. V.: Changes in the extremes of the climate simulated by CCC GCM2 under CO₂ doubling, J. Climate, 11, 2200–2222, 1998.