Introduction
El Niño/La Niña Southern Oscillation (ENSO) is an oceanic-atmospheric
quasi-periodic phenomenon with several impacts on climate and weather not
only in the tropical Pacific, but in many regions all over the world
(Varotsos and Deligiorgi, 1991; Kondratyev and Varotsos, 1995a, b; Klein et
al., 1999; Xue et al., 2000; Eccles and Tziperman, 2004; Cracknell and
Varotsos, 2007, 2011; Lin, 2007; Chattopadhyay and Chattopadhyay, 2011;
Efstathiou et al., 1998, 2011; Varotsos, 2013; C. Varotsos et al., 2009,
2012, 2014a, b). The disastrous effects of the strong ENSO events necessitate
their reliable short- and long-term prediction (Latif et al., 1998; Stenseth
et al., 2003; Monks et al., 2009; Hsiang et al., 2011; Cheng et al., 2011;
Barnston et al., 2012; Krapivin and Shutko, 2012; Tippett et al., 2012). In
this context, Varotsos et al. (2015) presented a new method (see also
Varotsos and Tzanis, 2012) for the detection of precursory signals of the
strong El Niño events by using the entropy change in “natural time” (a
new time domain, see Varotsos et al., 2002) under time reversal. The analysis
of the Southern Oscillation Index (SOI) time series by using this modern
method provided significant precursory signals of two of the strongest El
Niño events (1982–1983 and 1997–1998).
The entropy change ΔS20 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 ΔS20 exceeds the threshold value
ΔSthres=0.0035.
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 (BOM) in their report
(http://www.bom.gov.au/climate/enso/archive/ensowrap_20150901.pdf) of
1 September 2015 stated that “The 2015 El Niño is now the strongest El
Niño since 1997–98” and moreover on 29 September 2015 they reported
that most international climate models indicate current El Niño
(http://www.bom.gov.au/climate/enso/archive/ensowrap_20150929.pdf) “is
likely to peak towards the end of 2015” as also reported on 8 October 2015
by the Climate Prediction Center, National Centers for Environmental
Prediction, National Oceanic and Atmospheric Administration (NOAA)/National
Weather Service
(http://www.cpc.ncep.noaa.gov/products/analysis_monitoring/enso_disc_oct2015/ensodisc.pdf).
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 analyze 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
data set, entitled “Monthly SOIPhase 1887–1989 Base”,
(https://www.longpaddock.qld.gov.au/seasonalclimateoutlook/southernoscillationindex/soidatafiles/index.php)
derived from the Long Paddock site. It should be clarified that we use the
monthly values of SOI, instead of the daily ones, as the latter introduce
significant noise due to daily weather patterns variability. It should be
noted here that El Niño and La Niña episodes are associated with
negative and positive values of the SOI, respectively, and
SOI = 10×PA(Tahiti)-PA(Darwin)/SDD, where the Pressure Anomaly
(PA) is the monthly mean minus long-term mean (1887–1989 base period) and
SDD is the standard deviation of the difference (1887–1989 base period) of
mean sea level pressure between Tahiti and Darwin.
The hit rate vs. false alarm rate when using ΔS20 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 and hence
it corresponds to the m=1 iso-performance line of the ROC space (e.g.,
see Fawcett, 2006; Provost and Fawcett, 1998, 2001).
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.
The method suggested by Varotsos et al. (2015) is based on the entropy change
in natural time under time reversal ΔSi (e.g., see P. A. Varotsos
et al., 2005, 2007, 2009; Sarlis et al., 2010, 2011) calculated for a window
size of i events (SOI monthly values). To this end, Varotsos et al. (2015)
converted the original SOI time series to a new one Qk=(SOIk+|min(SOI)|), where min(SOI) is the minimum value of SOI
during the whole study period, keeping the temporal sequence of the events
and not considering their time of occurrence. Hence, for each Qk value
we calculate the ratio (χk) of the order of its occurrence (k) and
the total number (i) of events within the window, i.e. χk=k/i. The
latter quantity, which replaces the conventional time (t), is natural time
χk characterizing the kth event (Varotsos et al., 2002). This way,
Varotsos et al. (2015) introduced a new series the members of which are the
pairs χk,Qk where Qk > 0. Thus, one
can define the quantity pk=Qk/Σn=1iQn which can
be considered as a probability, since it is positive and satisfies the
condition ∑n=1ipn=1 (Varotsos et al., 2011). Under
these assumptions, the average values of quantities, which are functions of
natural time χ, can be evaluated by f(χ)=∑n=1if(χn)pn and the entropy
in natural time can be defined by S=χlnχ-χlnχ (Varotsos et al., 2005, 2011). The latter quantity changes to
a value S- if, instead of the true sequence of events, one uses the
time-reversed process that is described by pk′=T^pk=pi-k+1, where T^ denotes the time reversal operator in the window of i
events. The quantity ΔSi (= S-S-) reveals the breaking
of time symmetry 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., 2007, 2011) that positive values of ΔSi correspond to a decreasing time series in natural time, and hence when
ΔSi 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 ΔS20 (see the red crosses in Figs. 1 and 3) and
compared with a threshold ΔSthres, 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 ΔS20≥ΔSthres, 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 ΔS20 < ΔSthres 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 ΔS20 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 ΔSthres. A method to estimate
an appropriate value of ΔSthres is that of iso-performance
lines suggested by Provost and Fawcett (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 a+bx+cxd) 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 ΔSthres=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 ΔS20≥0.0035 the alarm is
set on for the SOI value of the next month.
The PDF of ΔS20 (black curve, left scale) together with
the corresponding histogram (red bars, left scale) obtained from the time
series of ΔS20, which is also plotted vs. time (blue crosses,
right scale) along the vertical axis. The arrows indicate when ΔS20 exceeds 0.0205 and are labeled by the corresponding ongoing strong El
Niño events.
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)
ΔS20 are depicted in Fig. 1 (as well as in Fig. 3). Beyond the
information gained from the exploration of the ΔS20 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
colored 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 ΔS20 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
ΔS20 obtained from the estimator fN(ΔS20)=1N∑i=1N1bNKΔS20-OibN, where Oi are the observed values of
ΔS20 since the beginning of our study, N is the total number of
these observations, the kernel K(x) is non-zero only when x<1 having the value K(x)=341-x2 and
bN is related with the standard deviation σ of the observed
ΔS20 values by bN=10.25σ/N0.34 as suggested by
Mercik et al. (1999). We observe in Fig. 4 that only rarely ΔS20
exceeds the value of 0.02, which can be also verified by the red histogram
obtained for ΔS20 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 ΔS20=0.02
covering the range up to approximately 0.0205. To detect when ΔS20
exceeds the latter value, we plot with blue crosses the time series of
ΔS20 vs. time, which can be read in the right axis of Fig. 4. We
see (blue arrows in Fig. 4) that ΔS20>0.0205 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.