Predictability of fat-tailed extremes

We conjecture for a linear stochastic differential equation that the predictability of threshold exceedances (I) improves with the event magnitude when the noise is a so-called correlated additive-multiplicative noise, no matter the nature of the stochastic innovations, and also improves when (II) the noise is purely additive, obeying a distribution that decays fast, i.e., not by a power law, and (III) deteriorates only when the additive noise distribution follows a power law. The predictability is measured by a summary index of the receiver operating characteristic curve. We provide support to our conjecture—to compliment reports in the existing literature on (II)—by a set of case studies. Calculations for the prediction skill are conducted in some cases by a direct numerical time-series-data-driven approach and in other cases by an analytical or semianalytical approach developed here.

Figure 1

Schematic depiction of a conjecture on the predictability of threshold exceedances in the linear stochastic differential equation (SDE) (1). Round shapes represent sets of models of certain properties identified by capital letter codes, with matching colors or line type of borders, as follows: P, the noise distribution features a power-law tail; V, the variance of the noise distribution does not exist; and C, the noise is of the CAM-type ($c\ne 0$). The rectangular shape is a representation of all possibilities (except that $b=0$). In each region defined by a unique combination of these properties we put a $+$ or $-$ sign according to whether the asymptotic limit of the predictability with ever-increasing event magnitude is approached from above or below. (Note that a larger value of the measure of predictability measured by the $ROC-D$ (17) means poorer predictability.) In these regions a code stands for the section number where a particular scenario is examined. The regions in color represent fat-tailed extremes.

Figure 2

Illustration of the prediction problem: Given a discrete-time irregular time series $x_n$ that every so often overshoots a relatively high threshold $x_{*}$ (straight horizontal line), we want to predict these overshoots, i.e., extreme events, at the time, say, of the immediately preceding observation. The discrete data points are connected by straight lines to indicate their order in time.

Figure 3

Predictability of extremes in a P2-type process (Sec. 2a) using two different fat-tailed noise variables of power exponent 3/2 both (see main text for details). One noise variable is of a Lévy-type (blue and red curves with plus sing + and dot $•$ markers, respectively), and the other one is of a matching (see main text for details) Pareto-type (magenta and green curves with circle $\circ$ and dot $•$ markers, respectively). The calculations were done both by a direct numerical method (Sec. 2c1, red and green curves, $N={10}^6$, $B=2\times {10}^4$) and by a semianalytical method (Sec. 2c2 blue and magenta curves). A straight line of slope 1/2 is included for reference.

Figure 4

Predictability of extremes in a P1-type process (Sec. 2a) using two different noise variables of equal variance. One noise variable is a Gaussian one (blue and red curves with cross $\times$ and dot $•$ markers, respectively), and the other one is uniformly distributed (green curve with dot $•$ markers). The calculations were done both by a direct numerical method (Sec. 2c1, red and green curves, $N={10}^6$, $B=2\times {10}^4$, $\mathrm{\Delta }t={10}^{-2}$) and by a semianalytical method (Sec. 2c2, blue curve). Straight lines of slope 1/2 are included in both panels for reference.

Figure 5

Predictability of extremes in a P1-type process (Sec. 2a) using a uniformly distributed noise variable and a binary precursory variable (see main text for details). The calculations were done both by a direct numerical method (Sec. 2c1, red and blue curves with dot $•$ markers, $N={10}^6$, $B=2\times {10}^4$, $\mathrm{\Delta }t={10}^{-2}$) and by a fully analytical method (Sec. 2c2, thick solid green curve). The direct numerical results have been repeated $2\times 20$ times. For the red curves the median defining the binary precursor (32) is estimated from data, and for the blue curves the true value of it is used. Discrete data points are connected by straight lines.

Figure 6

Predictability of extremes in a P2-type process (Sec. 2a) using stable noise variables (see the main text for details). Sample values for $\alpha$ are $0.7+0.1k,k=0,\cdots ,13$. The matching curves of $D^{\mathrm{opt}}\left(x_{*}\right)$ and ${\mathcal{L}}_{*}^{\mathrm{opt}}\left(x_{*}\right)$ are color coded with matching colors. The ${\mathcal{L}}_{*}^{\mathrm{opt}}\left(x_{*}\right)$ curves follow an order from right to left for increasing values of $\alpha$. Purple $D^{\mathrm{opt}}\left(x_{*}\right)$ curves with circle $\circ$ markers are extended by evaluating $D^{\mathrm{opt}}$ using the same seminalytical technique (Sec. 2c2) for more sample values of $x_{*}$ (diamond $\diamond$ markers in magenta), and $D^{\mathrm{opt}}$ is evaluated also based on long time series data (Sec. 2c1, $N={10}^6$, $B=2\times {10}^3$) for a series of sample values of $x_{*}$ (cross $\times$ markers in magenta). The close match of results by the two independent methods gives confidence in these results.

Figure 7

Predictability of extremes in a P2-type process (Sec. 2a) using symmetrized Pareto noise variables (see main text for details). The calculations have been carried out by a direct numerical approach (Sec. 2c1, $N={10}^6$, $B=2\times {10}^3$). Sample values for $\alpha$ are $0.7+0.1k,k=0,\cdots ,23$. The curves follow an order from right to left for increasing values of $\alpha$. For two sample values, 1 (circle $\circ$ marker) and 1.7 (diamond $\diamond$ marker), similar results are generated applying our semianalytical approach (Sec. 2c2).

Figure 8

Predictability of extremes in a P1-type process (Sec. 2a) using a Gaussian noise variable (see main text for details). Calculations are done both by the semianalytical (Sec. 2c2, blue line with circle $\circ$ markers) and direct numerical approaches (Sec. 2c1, red line with dot $•$ markers, $N={10}^6$, $B=2\times {10}^4$, $\mathrm{\Delta }t={10}^{-2}$). Outliers in both diagrams indicate that in that regime the algorithm has difficulties dealing with fast-varying functions.

Figure 9

Predictability of extremes in the same P1-type process as in Fig. 8 but using stable noise variables (see main text for details). Calculations are done only by the direct numerical approach (Sec. 2c1, $N={10}^6$, $\mathrm{\Delta }t={10}^{-2}$). Sample values of the stability parameter values used are $\alpha =1+0.1k,k=0,\cdots ,10$, and also 1.95. These values of $\alpha$ positively correlate with the peak height of the corresponding curves. We chose $B=2\times {10}^4$, but for $\alpha =1.6$ and 2 we also used $B=2\times {10}^3$. With a time series of length $N={10}^6$ these give, respectively, $N/B=50$ and 500 number of data points in each bin. A blowup is needed to show features of some curves.