On the progress of the 2015 – 2016 El Niño event

It has been recently reported that the current 2015–2016 El Niño could become “one of the 11 strongest on record”. To further explore this claim, we performed the new analysis described 12 in detail in Varotsos et al. (2015) that allows the detection of precursory signals of the strong 13 El Niño events by using a recently developed non-linear dynamics tool. In this context, the 14 analysis of the Southern Oscillation Index time series for the period 1876–2015 shows that the 15 running 2015–2016 El Niño would be rather a “moderate to strong” or even a “strong” event 16 and not “one of the strongest on record”, as that of 1997–1998. 17

Very recently, Klein (2015) reported that the running 2015-2016 El Niño could become "one of the strongest on record".Furthermore, the Australian Government Bureau of Meteorology In this study, we further explore these claims, by applying to the SOI time series the recently proposed analysis by Varotsos et al. (2015).The ability of accurate predictions of such severe natural events, like El Niño, is of crucial importance especially nowadays, where the global annual average temperature in 2015 reached the warmest on record values, which might be associated with the 2015 El Niño event (WMO, 2016).

Results and discussion
As mentioned in the previous section, we analyse the SOI time series (Troup, 1965;Power and Kociuba, 2011) for the period January 1876 -October 2015 by employing the method described in detail in Varotsos et al. (2015).More specifically, we conduct the analysis of the SOI monthly values by using the dataset, entitled " The method used by Varotsos et al. (2015) is based on the entropy change in natural time under time reversal ∆S i (e.g., see Varotsos et al., 2005Varotsos et al., , 2007Varotsos et al., , 2009b;;Sarlis et al., 2010Sarlis et al., , 2011) ) calculated for a window size of i events.Within this window, natural time k χ characterizing the k-th event is defined by i k k = χ (Varotsos et al., 2002).The analysis in natural time is based on the study of the pair ( ) where Q k (> 0) is proportional to the "energy" emitted during the k-th event.Thus, one can define the quantity which can be considered as a probability, since it is positive and satisfies the condition ∑ = = i n n p 1 1 (Varotsos et al., 2011).For the study of El Niño, Varotsos et al. (2015) suggested that k Q could be considered proportional to (SOI + c), where c is an appropriate constant, since k Q should be positive.Under these assumptions, the average values of quantities, which are functions of natural time χ, can be evaluated by and the entropy in natural time can be defined by Varotsos et al., 2005Varotsos et al., , 2011)).Τhe latter quantity changes to a value S -if, instead of the true sequence of events, one uses the timereversed process that is described by , where T ˆdenotes the time reversal operator in the window of i events.The quantity ∆S i (= S -S -) reveals the breaking of timesymmetry by capturing the difference in the dynamics as the system evolves from present to future and vice-versa.In short, it has been shown (e.g., see Varotsos et al., 2007Varotsos et al., , 2011) that positive values of ∆S i correspond to a decreasing time-series in natural time, and hence when ∆S i exceeds a certain threshold this reveals that SOI is approaching at small values indicating El Niño (Varotsos et al., 2015).Varotsos et al. (2015) have also shown (see their Fig. 4) that the most useful window size for this purpose is i = 20 events (months).In their prediction scheme, the monthly SOI values for the past 20 months are used for the calculation of ∆S 20 (see the red crosses in Figs. 1 and 3) and compared with a threshold ∆S thres , which can be determined on the basis of Receiver Operating Characteristics (ROC, see Fawcett, 2006).
ROC is a method for the visualization, evaluation, and selection of prediction schemes based on their performance, which is quantified by a plot of the hit rate vs. the false alarm rate obtained by the following procedure applied to the present case.When ∆S 20 ≥ ∆S thres , one issues an alarm that the value of SOI for the next month will be smaller than or equal to T (see the black broken line in Fig. 2).If this turns out to be true, then we have a true positive prediction.If ∆S 20 < ∆S thres and the next month's SOI is larger than T, then we have a true negative prediction.All other combinations lead to errors (which are inevitable in stochastic prediction), which can be either false positive or false negative predictions.Figure 2 depicts the ROC curve obtained, when using ∆S 20 as a predictor for the SOI value of the next month with T = -14 (which is the upper limit of the yellow area in Figs. 1 and 3 discussed below).This is a diagram of the hit rate (or True Positive rate, i.e., the number of true positive predictions over all cases with SOI ≤ T = −14) vs. the false alarm rate (or False Positive rate, i.e., the number of false positive predictions over all cases with SOI > -14) as we vary ∆S thres .
A method to estimate an appropriate value of ∆S thres is that of iso-performance lines suggested by Provost andFawcett (1998, 2001).In this scheme, a line of constant slope m (see the blue line in Fig. 2) is selected on the basis of the relative cost of false positive predictions over the cost of false negative predictions multiplied by the relative frequency of negatives over positives, i.e., see Eq. (1) of Fawcett (2006).As a typical selection we chose m = 1.We fitted ROC points with the red curve (having a simple analytical form ) and determined the point at which the slope was unity.This leads to the ROC point indicated by an arrow in Fig. 2 and corresponds to ∆S thres = 0.0035 (i.e., a value very close to that 0.00326 presented in Table 1 of Varotsos et al (2015) for T = -15).Thus, in Figs. 1 and 3 when ∆S 20 ≥ 0.0035 the alarm is set on for the SOI value of the next month.
The time progress of the SOI monthly values as well as the entropy change in natural time under time reversal (for the window length i = 20 months) ∆S 20 are depicted in Fig. 1 (as well as in Fig. 3).Beyond the information gained from the exploration of the ∆S 20 dynamics and in order to further identify if 2015-2016 El Niño could be characterized as a "very strong" one or even more as "one of the strongest on record", we followed the classification and characterization of the past El Niño events given by BOM (http://www.bom.gov.au/climate/enso/enlist/). The coloured areas in Figs. 1 and 3 represent the mean minimum negative values of SOI along with the 1σ standard deviation bands for the two cases of "weak, weak to moderate, moderate, moderate to strong" (green band) and "strong, very strong" (yellow band) El Niño events.
As can be clearly seen in Fig. 3, the SOI values during the last three months remain in the green band and in the limits of the yellow one, indicating that 2015 El Niño should be rather characterized as a "moderate to strong" or even "strong" event and not "one of the strongest on record", as also shown by comparing with the El Niño events of 1982-1983 and 1997-1998.Furthermore, the variation of ∆S 20 during the 2015 El Niño in comparison with 1982-1983 and 1997-1998 El Niño events is not as sharp, confirming that the undergoing El Niño event is not "one of the strongest on record".In order to estimate the extent of this variation, we plot with the black curve in Fig. 4 the probability density function (PDF) of ∆S 20 obtained from the estimator , where O i are the observed values of ∆S 20 since the beginning of our study, N is the total number of these observations, the kernel having the value ( ) as suggested by Mercik et al. (1999).We observe in Fig. 4 that only rarely ∆S 20 exceeds the value of 0.02, which can be also verified by the red histogram obtained for ∆S 20 using the TISEAN package (Hegger et al., 1999) (also plotted in Fig. 4).In the latter histogram, the minimum non-zero height is observed in the bar that includes the value ∆S 20 = 0.02 covering the range up to approximately 0.0205.To detect when ∆S 20 exceeds the latter value, we plot with blue crosses the time series of ∆S 20 vs. time, which can be read in the right axis of Fig. 4. We see (blue arrows in Fig. 4) that 0205 .0 20 > ∆S is observed only in the three strong El Niño events of 1905-1906, 1982-1983 and 1997-1998.This inequality, however, is not fulfilled in the current case (2015-2016 El Niño), since the currently observed values are close to 0.01, i.e., markedly smaller than the value of 0.0205.

Conclusions
Recent reports indicate that 2015-2016 El Niño event could become "one of the strongest on record" or could be already characterized as "the strongest El Niño since 1997-98".In order to investigate these assertions, we analyzed the SOI time series for the period January 1876 -October 2015 by using the method described in Varotsos et al. (2015) based on the entropy change in natural time under time reversal.The results obtained indicate that the undergoing 2015-2016 El Niño event should be rather characterized as a "moderate to strong" or even "strong" event and not "one of the strongest on record".and hence it corresponds to the m = 1 iso-performance line of the ROC space (e.g., see Fawcett, 2006;Provost andFawcett, 1998, 2001).and hence it corresponds to the m = 1 iso-performance line of the ROC space (e.g., see Fawcett, 2006;Provost andFawcett, 1998, 2001).
and b N is related with the standard deviation σ of the observed ∆S 20 values by 34

Figure captions Figure 1 .
Figure captions

Figure 2 .
Figure 2. The hit rate vs. false alarm rate when using ∆S 20 as a predictor for the SOI value of the next month.The ROC point indicated by the arrow has been selected so that the slope of the tangent of the analytical fitting of the ROC points indicated by the red curve has unit slope

Figure 3 .
Figure 3.As in Fig. 1, but only for the 1982-1983, 1997-1998 (the two strongest in the last century) and the current 2015-2016 El Niño events.

Figure 4 .
Figure 4.The PDF of ∆S 20 (black curve, left scale) together with the corresponding histogram (red bars, left scale) obtained from the time series of ∆S 20 , which is also plotted vs. time (blue crosses, right scale) along the vertical axis.The arrows indicate when ∆S 20 exceeds 0.0205 and are labeled by the corresponding ongoing strong El Niño events.

Figure 1 .
Figure 1.The entropy change ∆S 20 in natural time for the window length i = 20 months (red line, left scale) along with SOI monthly values (blue line, right scale) for the period January 1980 -October 2015.The alarm is set on (black line), when ∆S 20 exceeds the threshold value ∆S thres = 0.0035.

Figure 2 .
Figure 2. The hit rate vs. false alarm rate when using ∆S 20 as a predictor for the SOI value of the next month.The ROC point indicated by the arrow has been selected so that the slope of the tangent of the analytical fitting of the ROC points indicated by the red curve has unit slope

Figure 3 .
Figure 3.As in Fig. 1, but only for the 1982-1983, 1997-1998 (the two strongest in the last century) and the current 2015-2016 El Niño events.

Figure 4 .
Figure 4.The PDF of ∆S 20 (black curve, left scale) together with the corresponding histogram (red bars, left scale) obtained from the time series of ∆S 20 , which is also plotted vs. time (blue crosses, right scale) along the vertical axis.The arrows indicate when ∆S 20 exceeds 0.0205 and are labeled by the corresponding ongoing strong El Niño events.