1 Introduction
Space weather effects such as solar flares, geomagnetic storms and solar energetic particles can be hazardous to human activities, both on earth and in nearearth orbit, and the scientific community is faced with a number of such phenomena of great variability (Riley et al., 2017). A probabilistic assessment of the likelihood of such events – and their strength – is a top priority in space weather forecasting (Jonas et al., 2017). In particular, the prediction of occurrence of solar flares is crucial as even small intensity events can adversely affect geopositional systems and telecommunications (Lanzerotti, 2007). The immediate effects of solar flares are mainly related to technology i.e. telecommunications, radio transmissions, and satellites. Xray emissions can even affect and disturb the operation of radars and other devices. For example, during the Halloween storm of 2003 a set of perturbations related to the arrival of Coronal Mass Ejections at Earth were recorded: satellite communications, auroras from Texas to the Mediterranean, and several disruptions in the transport of high voltage energy (GICs) that produced power cuts (Watson et al., 2004). At long term and middle latitutes, highvoltage transmission networks are also affected by Space Weather events (Duro et al., 2012; Kappenman, 2003).
Solar flares are classified with letters: A, B, C, M, and X according to the peak flux measured in Wm
of Xrays as measured by the Geostationary Operational Environmental Satellite (GOES) spacecrafts. Each class of events has subclasses ranging from 1 to 9. Each letter specifies a flare of one order of magnitude larger than the previous one. For example an M2 is Wm whereas an X2 is Wm. Furthermore, an X2 is twice as strong as an X1 whilst four times more powerful than an M5 flare (Chamberlin et al., 2009).To forecast events that are of interest to human activity it is useful to study the distribution of the strengths of events recorded by the satellites mentioned above. Of particular interest is the study of the tail of the distribution of solar flares which describes the occurrence of extreme events such as the Carrington event of 1859 (Carrington, 1859). Up to today it has been assumed that the tail of the distribution of flare strengths follows a power law (Riley, 2012; Lu and Hamilton, 1991) i.e. , with values of ranging between 1.7 and 2 (Boffetta et al., 1999; Aschwanden and Freeland, 2012) or a lognormal distribution (Riley et al., 2017); however it is believed that a power law distribution overpredicts extreme events and their occurrence. In addition, in (Riley, 2012) as mathematical and numerical discrepancies have been noted (Parrott, a, b; Roodman, )
, the confidence intervals of the predictions are not reported
(Riley et al., 2017) and fits of straight lines on curved distributions overestimate the predicted risks as straight lines can be good approximations locally but cannot be accurately used for extrapolating to extremes.In this paper, we suggest more appropriate tools for the study of extreme space weather events for Xray flares of Carrington’s magnitude or more. We suggest the study of extreme solar flare events with the employment of Extreme Value Theory (EVT) for the estimation of the probability of extreme solar flares. The main result of this paper is that a Carringtonlike event (X45 ) (Cliver and Dietrich, 2013) is expected approximately once every 110 years whereas a Halloweenlike event (X35 5) (Cliver and Dietrich, 2013) is expected approximately once every 38 years. Furthermore, the frequencies of these events happening in the next solar cycle are in very close agreement with the NOAA predictions.
The paper is structured as follows: in Section 2 we introduce the methodology and the motivation behind EVT as well as its different approaches; in Section 3 we describe the data set used throughout this paper and discuss some modelling issues relevant to the dataset. In Section 4, we implement EVT and present the results and in Section 5 we evaluate the effect of the seasonality inherited from the sun in our predictions. In Section 6 we conclude with a discussion about our findings.
2 Extreme value theory
Extreme value theory (EVT) is a branch of statistics that offers tools and techniques for the study and estimation of probabilities of events outside the range of those that have already been observed. Since these extreme events are sparse their estimation requires extrapolation beyond the observed levels. EVT is designed precisely for such extrapolation and utilises asymptotic analyses for the foundation of extreme value models. Hence, EVT can model the stochastic nature of process of unusually large or small intensity events (Coles, 2001)
. What is remarkable about EVT is that it avoids any assumptions about the underlying probability distributions: the possible asymptotic distributions for extreme events are universal, independent of parent (or true distributions) describing the full process and thus mitigate the necessity of making any a priori assumptions. Hence, EVT is the most rigorous general approach allowing us to extrapolate from the past.
EVT has been previously used to model a range of natural phenomena such as extreme wind speeds (Fawcett and Walshaw, 2006), extreme draughts and flooding (Canfield et al., 1980; Katz et al., 2002) and is has even been used for public health and the prediction of extreme outbreaks of pneumonia and influenza (Thomas et al., 2016). In the area of space weather, EVT has been used to estimate the disturbance storm time (Dst) index (Tsubouchi and Omura, 2007; Silrergleit, 1996), the daily index (Silbergleit, 1999), speeds of fast Coronal Mass Ejections (CMEs) (Ruzmaikin et al., 2011), the magnetic index (Koons, 2001), energetic electron fluxes (Koons, 2001; O’Brien et al., 2007), relativistic electron fluxes (Meredith et al., 2015) as well as the millennial sunspot number series (Acero et al., 2018).
EVT is based on two main approaches: 1) the block maxima approach (also known as the FisherTippettGnedenko theorem) and 2) the peaksoverthreshold approach (also known as the PickandsBalkemade Haan theorem (Balkema and de Haan, 1974; III, 1975)) (Coles, 2001). In this paper we suggest the use of the latter in our analysis for reasons explained in Section 4.
2.1 Block maxima
The block maxima approach consists of grouping the observed data into blocks of equal length and extracting the maximum of each block. Then, according to EVT (Coles, 2001) the distribution of maxima in the large limit is a renormalised Generalised Extreme Value (GEV) distribution. Let
be a sequence of independent and identically distributed (i.i.d) random variables with common distribution
. Then, it can be proven that the cumulative distribution function (CDF) of
is:(1) 
if constants and can be found for the limit to exist^{1}^{1}1The constants and are chosen in order to allow a linear renormalisation of the variable so that the distribution of does not degenerate to a single point in the large limit. Appropriate choices of standardise and stabilise the location and scale of as increases. Explicit expressions for both and can be found in (Castillo et al., 2004), page 204., where is a nondegenerate distribution function which is a member of the GEV family:
(2) 
defined on {} and where , and .
The GEV family of distributions has three parameters, the location parameter , the scale parameter and the shape parameter . Estimators for the three parameters can be found by the Maximum Likelihood Estimate (MLE) method. Maximum likelihood estimators maximise the probability of observing the recorded data given a parameter dependent model. MLEs for the three parameters must be found numerically as there is no closed form solution. The GEV can be categorised according to the value of the parameter as follows: if then the distribution is Frechét (“heavy” tail), if the distribution is Weibull with an upper bound at (bounded tail) and in the limiting case of the distribution is Gumbel (“light” tail).
However, in this approach there are many drawbacks that cannot be easily overcome. For example, the choice of the block length is crucial. Although it is often convenient is to choose a block length of one year i.e. yearly maxima, this is not always the optimum choice for all datasets, and is not the most natural length for solar flares. The choice is usually the block length that balances the variance and the bias. Furthermore, since extreme events are scarce this method is not ideal as it wastes a lot of good data when the block size is large. This motivates the peaksoverthreshold approach.
2.2 Peaksoverthreshold (POT)
A different yet related way to determine which observations are extreme is the peaksoverthreshold (POT) approach where all observations greater than some high threshold are deemed extreme. The POT approach is used to model the tail of the distribution above the chosen threshold and the theory states that for a sufficiently high threshold the conditional distribution of exceedances is a Generalised Pareto Distribution (GPD) (Coles, 2001; Leadbetter et al., 1983).
To make this more concrete, let be a sequence of i.i.d random variable with . If in the large limit, with following a GEV then for large enough the distribution of conditional on , is approximately
(3) 
defined on and where . The parameters of the distribution are similar to the GEV with the shape, location, scale and threshold respectively. Equation 3 is the Generalised Pareto family and it implies that if block maxima are approximately GEV distributed then the threshold excesses are approximately GP distributed.
The issue of choosing a threshold is similar to the objectivity of the choice of block size in the GEV approach. Similarly, the choice should be one that assures balance between bias and variance. There are two ways to assess the suitability of the threshold: the mean residual life plot as well as the the stability of the parameters’ MLE estimates across a variety of different thresholds. For the former, if the threshold excesses follow a Pareto distribution then , the mean of the excesses of the threshold , should be a linear function of
at levels for which a Pareto model is appropriate. A good and unbiased estimator of the mean is the sample mean thus one should expect it to change linearly with
(within confidence intervals). For the latter, a complementary technique is to fit the GPD at a range of and look for stability of the GPD’s parameters, again within confidence intervals. If a GPD is appropriate for excesses of a threshold then excesses of a higher threshold should also follow a GPD with exactly the same shape and scale parameters. However, sampling variability does not guarantee exact stability but the parameters should be stable within their sampling errors. Plotting both and against with their confidence intervals, should be selected as the lowest value for which the estimates remain nearconstant.As with the GEV, and due to the duality between them, the shape parameter of the GPD determines its tail’s qualitative behaviour. Thus, yields the Pareto CDF (“heavy” tail), yields the Beta CDF (bounded tail) with upper bound at and the limiting case yields the exponential CDF (“light” tail).
Although there are several choices to estimate the parameters of the fitted GPD, in this paper we use the Maximum Likelihood Estimation (MLE). The estimated values of and , can be found by maximising the likelihood function; in fact, it is standard practice to maximise instead the loglikelihood. Assuming that are the exceedances above , with conditional on then the loglikelihood function to be maximised provided that is:
(4) 
provided that for otherwise
. However, as in the GEV case, there is no analytic solution for the maximisation of the likelihood so numerical iterative, optimisation methods are used. Along the estimates, we can obtain their standard errors and confidence intervals by using standard techniques (see
(Coles, 2001; Castillo et al., 2004)).2.3 Discussion between approaches
What is really remarkable about both aforementioned methods, is the fact that the three GEV distributions, and hence the three types of the GPD family, are the only possible limits for the distribution of maxima (Coles, 2001), no matter what the parent distribution
of the population is. This result can be seen as an extreme value theory analogue of the central limit theorem. This in turn shows that both the GEV and GP distributions are the appropriate limits for the two approaches when the number of samples becomes increasingly large
(Reiss and Thomas, 1997).The two mentioned methodologies are strongly connected: if block maxima obey the GEV distribution then exceedances over some high threshold obey an associated GPD. Furthermore, the shape parameters of the two distributions are expected to be identical asymptotically (Leadbetter et al., 1983; Coles, 2001). Although this argument stands in theory and the two approaches are equivalent in the limit of infinitely long time series, the GP distribution is more robust and provides more realistic results when finite time series are considered (Ding et al., 2008). Furthermore, as an approach it does not waste so many valuable extreme data as the block maxima, as it is known that extreme data are scarce and very hard to observe or collect. Thus, the GPD approach is the one that we will employ in this paper.
3 Data
The data used in this study were extracted from the SWPC/NOAA website and represent solar flares normalised from H spectral observations and Xray fluxes spanning a period of 43 years from November 1975 to July 2017 (https://www.ngdc.noaa.gov/stp/spaceweather/solardata/solarfeatures/solarflares/xrays/goes/xrs/). In order to take into account the last energetic events of the current solar cycle, we also included the records from July 2017 to October 2017.
The analysis was run on the peak Xray flux of each solar flare event converted in . The data have been scaled in order to yield consistency and long term continuity across all different GOES satellites that have been recording data since 1975. To get the true Xray fluxes for the GOES satellites the data were divided by 0.7 which ensures that the classification level of the solar flares is consistent across all GOES (Machol and Viereck, 2016).
The POT method is valid for independent and identically distributed (iid) data which we verify with post analysis diagnostics. Since the analysis is done on the peak Xray fluxes only, the data are assumed to be independent; any background noise or any consecutive events that are usually recorded multiple times in the GOES data, as most events last for a long time, have been removed by choosing the peak of each event only. This statement is further supported by the estimated extremal index of the data: for this we state the following theorem.
Extremal index
Theorem.
Let be a stationary series that satisfies Leadbetter’s condition (Coles, 2001; Leadbetter et al., 1983), i.e. showcasing sufficiently weak longrange independence, and let . Now let be an independent series with having the same distribution as and let . Then if has a nondegenerate limit law given by it follows that:
(5) 
for some . The parameter is known as the extremal index and quantifies the extremal dependence with a completely independent process and demonstrates increasing levels of dependence.
For the dataset used in this study and to reinforce the statement of independence above, the estimated extremal index of the data used, as estimated by the runs declustering method (Coles, 2001; Gilleland and Katz, 2016) , was found to be . We conclude that our data are completely independent.
Seasonality is an important aspect to consider when analysing time series. We analyse possible seasonality that could be inherited from the solar cycle in Section 6.
4 Results
The first step is to estimate the threshold that determines the extreme values of our data. To determine that, as mentioned before, one needs to examine the mean residual life plot looking for linearity as well as the plot of the GPD parameters against a series of thresholds looking for stability. Figure 1 shows these plots where the mean residual life plot becomes linear and the threshold range plot becomes stable for a threshold value of i.e. for X5 flares. For the rest of the statistical analysis, the value of is used as the threshold value as suggested by the diagnostics. According to NOAA (NOAA, ) a flare of intensity X5 is classified between strong and severe so using this value as a threshold balances the “extremity” of events and having enough data points so that the estimated results are valid. The chosen threshold value gives a set of 93 exceedances i.e. 93 flares are greater than X5 in our given dataset. The Maximum Likelihood Estimates (MLE) and standard errors for the distribution parameters are: and which were found by using R and the package “extRemes2” (Gilleland and Katz, 2016). In general, the normal approximation to the distribution of the maximum likelihood estimator may be poor and more accurate confidence intervals are extraced from the socalled profile likelihod (Coles, 2001). With this approach, the estimate of suggests an unbounded distribution () and the evidence is reasonably strong as the 95% profile likelihood confidence interval (C.I.) is almost exclusively in the positive domain with .
The validity of the model can be tested by the following diagnostic plots. From Figure 2 we can confirm the goodness of fit of the GPD for all solar flares above the chosen threshold  both diagnostic plots appear reasonable fits to the observed data, especially the loglog plot indicating that our model predicts extreme events accurately. Furthermore, the PPplot and the QQplot shown in Figure 3 look approximately linear, as expected when empirical predictions agree with the theoretical ones.
Since the model used is a good fit, it can be further used to estimate the extreme expected flares in a given period  this can be given by socalled return levels. The Nyear return level is expected to be exceeded on average once every years. More precisely, is exceeded by the annual maximum in any particular year with probability . This means that the year return level is given by:
(6) 
where is the number of years, the number of observations per year, and the estimated shape parameter of the GPD. Using the return levels we can estimate Carringtonlike events and their 95% confidence intervals. These can be found in Table 1, and a plot of these levels is provided in Figure 4. Caution is required in the interpretation of return level inferences, especially for return levels corresponding to long return periods since this corresponds to extreme extrapolation. Again better approximations are generally obtained from the appropriate profile likelihood function which is used in the results of Table 1. Though the EVT model is supported by mathematical arguments, its use in extrapolation is based on unverifiable assumptions, and measures of uncertainty on return levels should properly be regarded as lower bounds that could be much greater if uncertainty due to model correctness were taken into account.
According to the estimated return levels, based upon the data analysed here, a Carringtonlike event (X45) is expected as a one in 110year event whereas a “Halloween”like event (X35) is a one in 38years event. These have relatively large uncertainties as indicated by the confidence intervals (C.I). A Carringtonlike event is expected in the next decade with probability 9% whereas the probability of a Halloweenline event happening in the next decade is 23.8%. Comparing these results to those carried out by (Riley et al., 2017), we can say that they are comparable although a Carringtonlike event is predicted in the next decade with probability 10.3%. This could be an indication of the overprediction caused by the use of power law models.
Return level  Estimate  C.I. 

11year  X24.5  (X20.5, X35.5) 
20year  X29.5  (X23.5, X47.5) 
38year  Halloween (X35)  (X26.5, X65) 
50year  X37.5  (X27.5, X74) 
100year  X44  (X30.5, X103) 
110year  Carrington (X45)  (X31, X108) 
150year  X48  (X32.5, X125) 
In Figure 5, the first plot demonstrates the empirical and the fitted probability of at least one flare of a certain intensity happening in the next solar cycle. One can see that this is a relatively good fit to the data and it has been estimated as follows. Let be the unconditional probability of a single flare being greater than . The average number of flares in the solar cycle is , with the average number of observations per year. If is the number of flares in a cycle whose intensities are greater than then
is a random variable with a binomial distribution with parameters
and . The probability that in a cycle we observe at least one flare greater than is:(7) 
To lend further weight to the argument, the second plot of Figure 5 demonstrates the empirical and the fitted expected frequencies of a flare of a certain intensity in a solar cycle. These were estimated as the expected value from the binomial distribution described above and thus is equal to . One can see from this plot that the tail is predicted very well by the model. Furthermore, it is shown that our results are in close agreement with NOAA’s frequency predictions.
5 Seasonality
It has been suggested that seasonality inherited from the solar cycle affects the frequency as well as the severity of the solar flares emitted. To investigate this the solar seasonality has been subtracted from the data in the following way: a periodogram was used to identify the dominant frequencies of the discrete Fourier transform of the data,
and , which were found to correspond to periods of 11 and 14 years respectively. Sinusoidal waves of the form , were fitted to the data using a nonlinear leastsquares estimation procedure (“nls” in R). The estimated function was subtracted from the data and the analysis we used in Section 5 was repeated. We found the effects of seasonality to be negligible with regard to extreme events: the estimates of the distribution parameters of the deseasonalised data were almost equivalent to those reported earlier and were perfectly compatible taking their confidence interval into account. This suggests that solar seasonality does not significantly impact on extreme events and thus their prediction.6 Conclusions
EVT is a rigorous method to estimate the intensity of solar flares happening years into the future and acknowledges the errors in extrapolation in the estimates in the big limit. We have shown that the use of EVT is consistent with the data provided by SWPC.
The fit to the extremes of the sample distribution function is excellent and the results show that the extreme events observed are well fit by the EVT models. EVT also provides us with results that can be interpreted as the worst case scenario i.e. as the most extreme events in a given period of time. For example the return period of a Carringtonlike event (X45) is 110years and a “Halloween”like event (X35) is 38years. Moreover, for the purposes of anticipatory planning to cope with potential extreme events, it would be recommendable to take the prediction of the mode pessimistic value within confidence intervals for the most secure preparation. Moreover, the probability of a Carringtonlike event happening in the next decade is 9% whereas a Halloweenlike event is expected in the next decade with probability 23.8%. These two results are compatible with previous predictions (Riley et al., 2017) and could be an indication of the overprediction caused by the use of power law models.
Furthermore, the model used seems perfectly valid for the frequency distribution of the empirical data; the predictions conform to those provided by NOAA (NOAA, ). EVT is a great tool for the estimation of values that have not been observed and as more data are collected, the estimates of the parameters approach more closely the true parameters of the extreme value distribution function. Since the model seems appropriate for the whole tail of the data it could suggest that one mechanism of energy release is sufficient to produce solar flares across the whole range of strengths which might not be the case were we to have seen significant deviation from the tail of our fitter model for large observed flares.
The results show that taking into account the data from the last four solar cycles an X24 flare is expected once every 11 years. However in this solar cycle (24) the most energetic flare recorded was X9.3 on September 6, 2017. The amplitude of the solar cycle 24 was lower than previous cycles and as we observed it produced fewer energetic events. The influence of the amplitude of each solar cycle in the confidence to forecast in the long term requires a better understanding of the particularities of each sub set of data (by cycle) and their relation with global statistics.
Acknowledgements
Both authors acknowledge access to solar flare’s flux strength data provided by NOAA’s National Centers for Environmental Information website found at https://www.ngdc.noaa.gov/stp/spaceweather/solardata/solarfeatures/solarflares/xrays/goes/xrs/. TT would like to thank J.P. Edwards for fruitful discussions whilst writing this paper.
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