Extreme value statistics in records with long-term persistence

Many natural records exhibit long-term correlations characterized by a power-law decay of the autocorrelation function, $C\left(s\right)\sim s^{-\gamma }$, with time lag $s$ and correlation exponent $0<\gamma <1$. We study how the presence of such correlations affects the statistics of the extreme events, i.e., the maximum values of the signal within time segments of the fixed duration $R$. We find numerically that (i) the integrated distribution function of the maxima converges to a Gumbel distribution for large $R$ similar to uncorrelated signals, (ii) the deviations for finite $R$ depend on the initial distribution of the records and on their correlation properties, (iii) the maxima series exhibit long-term correlations similar to those of the original data, and most notably (iv) the maxima distribution as well as the mean maxima significantly depend on the history, in particular on the previous maximum. The last item implies that conditional mean maxima and conditional maxima distributions (with the value of the previous maximum as condition) should be considered for an improved extreme event prediction. We provide indications that this dependence of the mean maxima on the previous maximum occurs also in observational long-term correlated records.

Figure 1

Definition of maxima: A time series $\left(x_i\right),i=1,\dots ,N$, of, e.g., daily data is separated into segments of length $R=365\text{days}$. The maximum values $m_j$ (◯) in each segment, e.g., annual maxima, define another time series $\left(m_j\right),j=1,\dots ,N∕R$.

Figure 2

Distributions of maxima in segments of length $R$ for uncorrelated data with (a), (c), and (e) Gaussian and (b), (d), and (f) exponential distribution density $P\left(x\right)$. Pannels (a), and (b) show the distribution density function $P_R\left(m\right)$ of the maximum values for four segment sizes $R=6$ (circles), 30 (squares), 365 (diamonds), and 1500 (triangles). Pannels (c), and (d) show that a collapse of all four curves to a single curve is achieved in both cases, when the $m$ axis is replaced by $(m-u)∕\alpha$ and $P_R\left(m\right)$ is multiplied by the scale parameter $\alpha$. The solid line is the Gumbel distribution density, Eq. (4). Pannels (e), and (f) show the corresponding integrated distribution $G_R\left(m\right)$ together with the Gumbel function Eq. (3). Note that exponential records converge to the Gumbel distribution much faster than Gaussian records.

Figure 3

Distributions of maxima in segments of length $R$ for long-term correlated data with $\gamma =0.4$; for explanations of the plots and the symbols see Fig. 2. The curves in (a), and (b) appear broader than in Fig. 2, because in correlated data more small $m$ values are considered in $P_R\left(m\right)$ than in uncorrelated data. In (d) the curve for $R=6$ (circles) differs most from the theoretical curve for uncorrelated data, an effect caused by the correlations together with a rather small $R$ value: the left tail of $P_R\left(m\right)$ is strongly affected by the abrupt left end of the exponential. For larger $R$ values this effect disappears and the Gumbel distribution is well approached. For the Gaussian data in (c), and (e), the Gumbel law (solid line) is again not well approached.

Figure 4

Comparison of the distribution density $P_R\left(m\right)$ of the maxima for uncorrelated (circles) and long-term correlated ($\gamma =0.4$, squares) data for fixed $R=365$. For both (a) Gaussian and (b) exponentially distributed $\left(x_i\right)$, the long-term correlations lead to a similar broadening of the left tail, while the right tail is hardly affected by correlations. In (a), the probability $E_R\left(m^{*}\right)$ of finding a maximum $m$ larger than an (arbitrary but sufficiently large) $m^{*}=3.3$ is 0.14 for correlated data (gray area) compared with 0.16 for uncorrelated data, and in (b) 0.18 for correlated data compared with 0.2 for uncorrelated data for $m^{*}=7.4$. The difference in the probabilities $E_R\left(m^{*}\right)$ for large $m^{*}$ is marginal.

Figure 5

Autocorrelation function $C_m\left(s\right)$ of the maxima $\left(m_j\right)$ of (a) Gaussian and (b) exponentially distributed $\left(x_i\right)$ for different $R$ values, $R=6$ (circles), $R=30$ (squares), $R=365$ (diamonds), and $R=1500$ (triangles). The autocorrelation function $C_x\left(s\right)$ of the original data $\left(x_i\right)$ (crosses) shows the slope $-\gamma =-0.4$.

Figure 6

Study of finite-size effects in the autocorrelation function $C_m\left(s\right)$ of sequences of maxima $\left(m_j\right)$ for Gaussian distributed $\left(x_i\right)$ with (a) $R=365$ and (b) $R=1500$. The set lengths are $N=2^{21}$ (circles), $2^{20}$ (squares), $2^{19}$ (diamonds), and $2^{18}$ (triangles). The descent of the slopes of $C_m\left(s\right)$ from the slope of the straight line $(-\gamma =-0.4)$ with decreasing set length seems to be a finite-size effect.

Figure 7

Definition of the conditional maxima. In the sequence of (annual) maxima only those $m$ values (indicated by circles) are considered, which directly follow a maximum of approximate size $m_0\approx 6$ (gray band). The new sequence of $m$ values is the sequence of the conditional maxima.

Figure 8

Mean conditional maxima $m_R\left(m_0\right)$ for $\gamma =0.4$ and $R=6$ (circles), $R=30$ (boxes), $R=365$ (diamonds), and $R=1500$ (triangles) versus $m_0$ for (a) Gaussian and (b) exponential data. The straight lines represent the unconditional means $m_R$ for a given $R$. The width $\Delta m_0$ for the condition $m_0$ was chosen such that approximately $700\mathrm{m}$ values were obtained for each $m_0$ in each of the 150 runs of $N=2^{21}$ data points. Both figures show the memory effect in the form of $m_R\left(m_0\right)>m_R$ for rather large $m_0$ (above $m_R$) and $m_R\left(m_0\right)<m_R$ for rather small $m_0$ (below $m_R$).

Figure 9

(a)–(c) Mean conditional minima $m_R\left(m_0\right)$ for the annual data of the Nile river water level minima (squares) [35] and (d)–(f) mean conditional maxima for the reconstructed northern hemisphere annual temperatures after Moberg (circles) [36], for (a) and (d) $R=1\text{year}$ (b) and (e) $R=6\text{years}$, and (c) and (f) $R=12\text{years}$. The filled symbols show the results for the real data and the open symbols correspond to surrogate data where all correlations have been destroyed by random shuffling. The shuffled data have a different $m_0$ range due to the broadening of the left tail of $P_R\left(m\right)$ in the correlated data (see Fig. 4). The unconditional mean minima (a)–(c) and maxima (d)–(f) are indicated by dashed lines; the long-dashed lines correspond to the shuffled data.

Figure 10

(a) Conditional distribution density $P_R(m\mid m_0)$ of maximum values taken from correlated Gaussian data $(\gamma =0.4)$ with $R=365$ and $m_0=2.06$ (circles) as well as $m_0=3.55$ (squares). (b) shows the same as (a) for exponentially distributed data with $m_0=4.10$ (circles) and $m_0=8.65$ (squares). The width $\Delta m_0$ around $m_0$ is set such that a sufficient number of approximately 700 conditional maxima is obtained for each of the 150 data sets considered here. The probability $E_R(m^{*}\mid m_0)$ to find a $m$ value larger than an arbitrarily given $m^{*}$ [see Eq. (2)] also depends on the history $m_0$. For example, in (a), $E_{365}(3.30\mid 2.06)=0.08$ (gray area) is significantly smaller than $E_{365}(3.30\mid 3.55)=0.20$ (black area plus gray area), and in (b) $E_{365}(7.35\mid 4.10)=0.14<E_{365}(7.35\mid 8.65)=0.24$.

Figure 11

Conditional exceedance probabilities $E_R(m\mid m_0)$ to find a maximum larger than a given $m$ for (a) Gaussian and (b) exponentially distributed data with $\gamma =0.4$ and $R=365$ versus the condition $m_0$. The straight lines indicate the corresponding unconditional exceedance probabilities $E_R\left(m\right)$. The six $m$ values were chosen such that $E_R\left(m\right)=0.9$ (circles, $m=2.15$ for Gaussian and 4.40 for exponential data), 0.5 (squares, $m=2.75$ and 5.95), 0.3 (triangles up, $m=2.95$ and 6.70), 0.1 (diamonds, $m=3.35$ and 8.00), 0.05 (triangles down, $m=3.55$ and 8.75), and 0.01 (crosses, $m=3.95$ and $10.40$). Each point in the graph is based on a statistics of 500 conditional $m$ values and averaged over 150 runs of $N=2^{21}$ data points.

Figure 12

Ratios of the conditional and unconditional exceedance probabilities $E_R(m\mid m_0)$ and $E_R\left(m\right)$ to find a maximum larger than $m$ for (a) Gaussian and (b) exponentially distributed data with $\gamma =0.4$ and $R=365$. The symbols and the statistics are the same as in Fig. 11 except for the data corresponding to $E_R\left(m\right)=0.05$ (triangles down), which are not shown here to avoid overlapping symbols. The effect of long-term correlations seems to be strongest for the largest $m$ (crosses): depending on $m_0,E_R(m\mid m_0)$ varies from 0.4 up to 1.7 for Gaussian data, i.e., by a factor greater than 4. For exponential data this factor is still greater than 2.

Figure 13

Definition of centennial events: (a) the tail of a Gaussian distribution $P\left(x\right)$ (solid black line) with the $1∕36500$ quantile (dashed line), (b) the Gaussian tail (solid black line) and the histogram $P_{36500}\left(m\right)$ of centennial maxima with the average $m_{36500}$ (dashed line), and (c) the histogram $P_{365}\left(m\right)$ of the annual maxima and the corresponding Gumbel fit (Eq. (4), solid line) with its $1∕100$ quantile $Q_{100}$ [dashed line, Eq. (7)]. The deviations of the fit from the histogram are caused by the underlying Gaussian distribution, for which $R=365$ is not sufficiently large to reach the Gumbel limit.

Figure 14

Comparison of the quantile $q_{36500}$ (dashed vertical lines), the distribution density $P_{36500}\left(m\right)$ of centennial maxima (histograms), and the distributions of the $Q_{100}$ values based on $30\text{years}$ (squares) and $100\text{years}$ (circles) for Gaussian distributed (a) uncorrelated and (b) long-term correlated $(\gamma =0.4)$ data; see Fig. 13 and Eq. (7) for the definitions. Each $Q_{100}$ value was calculated from a different segment of length $36500\text{days}$ ($=100\text{years}$, circles) and $10950\text{days}$ ($=30\text{years}$, squares). The histograms were obtained from 150 artificial records with $N=2^{21}$.

Figure 15

Dependence of actual centennial events $m_{36500}$ and estimated centennial events $Q_{100}$ on the correlation exponent $\gamma$ of the data; small values of $\gamma$ indicating strong long-term correlations. The average (actual) centennial maxima (diamonds) and the corresponding upper fluctuations (range of one standard deviation indicated by dashed lines) is compared with the average $Q_{100}$ value (squares and circles) and its lower fluctuations (solid lines). The $Q_{100}$ values were obtained from segments of size (a) $10950\text{days}$ ($30\text{years}$, squares) and (b) $36500\text{days}$ ($100\text{years}$, circles). The solid horizontal line in both figures is the $1∕36500$ quantile $q_{36500}=4.034$.

Figure 16

Conditional average $Q_{100}$ (filled circles) and conditional $m_{36500}$ (squares) for Gaussian long-term correlated $(\gamma =0.4)$ data. The $⟨Q_{100}\left(Q_{100}^{\left(0\right)}\right)⟩$ values show a correlation based memory effect indicated by the upward trend, while the $m_{36500}$ values seem to have no memory due to the large $R$ value. The horizontal lines correspond to the unconditional average values 4.23 and 4.12 for $⟨Q_{100}⟩$ and $m_{36500}$, respectively. The $Q_{100}$ values were calculated from segments of a length of $36500\text{days}$.