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The lowest drop of log-return in Fig. In September , Lehman Brothers collapsed as the financial crisis spread. Distribution of R ln values Fig. Curves are given for reference only. The larger Hurst exponent is visually seen as more slow evolving variations i. On the contrary, values 0. The q -order Hurst exponent H q is only one of several types of scaling exponents used to parameterize the multifractal structure of time series  , .
The q -order weights the influence of segments with large and small fluctuations. In our work, we use the standard multifractal detrended fluctuation analysis MFDFA algorithm  ,  for estimating the q -order Hurst exponents and the multifractal spectra directly from the time series:.
The local fluctuations F q with large and small magnitudes is graded by the magnitude of the negative or positive q -order, respectively;. The fractal structures of the positive and negative log-return time series and its deviations within time periods with large and small fluctuations are assessed by the q -order Hurst exponents see Fig. The q -order Hurst exponents H q for the time series of positive the dashed line and negative the bold line log-returns.
The slopes H q of the regression lines are q -dependent for the multifractal time series of positive the dashed line and negative the bold line log-returns. Decreasing H q with the q order indicates that the segments with small fluctuations have a random walk like structure whereas segments with large fluctuations have a noise like structure.
There are two primary approaches to analyzing extreme values the extreme deviations from the median of the probability distributions in data:. The GEV distribution has the following pdf  :. The Gumbel distribution , also named as the Extreme Value Type I distribution, has the following pdf and cdf :.
Generalizations of the Gumbel distribution, which are of flexible skewness and kurtosis due to the addition of one more shape parameter are widely used for extreme value data as they better fit data . The distribution in 4. The distribution has a light right tail, which declines exponentially, since its skewness and kurtosis coefficients are constant. The pdf and cdf of a Weibull random variable are shown as follows, respectively:. Further we show the application of the GEV model to the stock market close price using the weekly-return data that was calculated by.
The results of fitting the GEV distribution to weekly block maxima data is presented in Fig. Parameter estimates for the GEV fitted model with maximum likelihood estimator. Based on the statistical analysis presented above Fig. The density plot Fig. The classical approach for modeling extreme events is based on the GPD. It was proved  that if a threshold u is chosen and X 1 , X 2 , … , X n are observations above u , then the limiting distribution for excess over threshold is indeed GPD.
This case is the most relevant for financial time series, since it is heavy-tailed. Therefore, a Poisson process of exceedance times with generalized Pareto excess implies the classical extreme value distributions . The above suggests that generalized Pareto distribution is a practical tool for statistical estimation of the extreme values, given a sufficiently high threshold.
The rest of this chapter is devoted to a question about how high a threshold should be. One of the most common ways to determine a suitable threshold is to graphically inspect data. This approach  requires substantial expertise, that can be subjective and time consuming. In some cases, when dealing with several data sets, a uniform threshold may be proposed and kept fixed making the entire evaluation even more subjective.
The most common graphical tools are: mean excess plot  , threshold stability plot  , QQ-plot  , Hill plot  , return level plots  , etc. The mean excess plot is a tool widely used in the study of risk, insurance and extreme values. One use is in validating a generalized Pareto model for the excess distribution. The distribution of the excess over a threshold u for a random variable X with distribution function F is defined as.
This excess distribution is the foundation for peaks over threshold modeling which fits appropriate distributions to data on excesses and widespread with many application in hydrology  ,  , actuarial science  ,  , survival analysis . This modeling is based on the GPD that is suitable for describing properties of excesses.
The mean excess ME function is one of the most common tools to determine a suitable threshold u. The ME function of a random variable X is defined as. Davison and Smith  developed a simple graphical tool that checks data against a GPD model. If the ME plot is close to linear for sufficiently large values of the threshold then there is no evidence against use of a GPD model.
Recently, new graphical diagnostic tools have been introduced: a new multiple-threshold GP model with a piece-wise constant shape parameter  ; plots measuring a surprise at different candidates to select threshold, using results of Bayesian statistics  ,  ; structure of maximum likelihood estimators have been studied to develop diagnostic plots with more direct interpretability .
Similarly, we find a threshold for negative returns. In this case, all computations were repeated for absolute values of negative returns Fig. The threshold u is estimated at 0. A vertical dashed line marks this threshold. Such random limits lead to wrong impressions. Distribution of exceedances over respective thresholds are shown in the Fig. The curves are given for reference only. The QQ-plots shown in the Fig.
As was mentioned above, the graphical approaches as well as rules of thumb can be highly objective, time consuming, and require certain professional background. Thus some authors have proposed automatic selection methods that can treat chunks of Big Data: a pragmatic automated, simple and computationally inexpensive threshold selection method based on the distribution of the difference of parameter estimates when the threshold is changed  : it was shown that better performance is demonstrated by graphical methods and Goodness of Fit metrics that rely on pre-asymptotic properties of the GPD  using weighted least squares to fit linear models in the traditional mean residue life plot; the recently developed stopping rule ForwardStop  , which transforms the results of ordered, sequentially tested hypotheses to control the false discovery rate  that provides reasonable error control .
A particular interest has a method that suggests a way to determine threshold automatically without time consuming and rather subjective visual approaches based on L-moments of GPD that summarize probability distributions, perform estimation of parameters and hypothesis testing . Probability weighted moments, defined by Greenwood  , are precursors of L-moments.
Sample probability weighted moments computed from data values X 1 , X 2 , … , X n arranged in increasing order, are given by. L-moments are certain linear combinations of probability weighted moments that have simple interpretations as measure of location, dispersion and shape of the data sample. The first few L-moments are defined by. The first L-moment is the sample mean, a measure of location.
These are dimensionless quantities, independent of the units of measurement of the data. The Fig. Once again, we cannot choose one value of the threshold that can be absolutely accurate. Value of the thresholds for positive and negative log-return based on L-moments. The solid black and blue lines correspond to negative and positive log return thresholds, respectively, and based on a window of trading days.
The dotted black and blue lines correspond to negative and positive log return thresholds, respectively, and based on a window of trading days. For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article. As earlier noted, the threshold sequence is a function of the properties of the GPD provided that a population is in the domain of attraction of the GPD.
In case a distribution function F is known, derivation of the threshold selection is possible, however, in practice if F is unknown then there is no general form for the threshold sequence . There is a procedure that tries to find a region of stability among the estimates of the extreme value index .
This method depends on a tuning parameter, whose choice is further analysed in Neves and Alves . Unfortunately, no theoretical analysis exists for this approach . More comprehensive reviews of the threshold selection methods can be found in Scarrott and MacDonald . The process repeated until we exhausted the entire data set R.
We chose a window of days that was moving over the entire dataset producing a domain for threshold existence as shown in the Fig. It is clear from the Fig. A green strip represents positive log-return and an orange strip shows the threshold domain for negative log-return values. Similarly, the window of trading days, the Fig.
Once again, some values of thresholds can exist for almost entire period, while other can exist a few months and then must be replaced with an updated value. A green strip represents positive log-returns and an orange strip shows the threshold domain for negative log-return values. Based on Figs. Let n be the window of observation. In a similar fashion, we compute a set of threshold candidates for negative log-returns,.
The more data we use to estimate a value of a threshold, the more likely the threshold will stay unchanged, however, as we add data, the threshold should be reconsidered. It also brings another issue: in many cases, statistical analysis is performed on a historical dataset that does not reflect a phenomenon we study at the present time. The probability that a threshold of the log-return values will be changed on any given day calculated over the different data windows ranging from 25 to trading days.
The statistics of extreme events under threshold uncertainty can be described with the help of a simple model, in which the log-return of the index and the value of threshold are treated as random variables that yet can change inconsistently. The model that we are going to adopt and modify had been put forward by us for the first time in Floriani et al. We assume that the rate of daily variations of the log-return values is greater than or equal to that of the threshold values, ultimately determining whether the current log-return value is extreme or not.
The probability that a threshold of the log-return values will stay unchanged on any given day calculated over the different data windows ranging from 25 to trading days is shown in Fig. For example, in the special case of uniformly random updates of the threshold and log-return values, the probability function 4. For a general family of invariant measures of a map of the interval [0,1] with a fixed neutral point defined by the probability distributions F and G , absolutely continuous with respect to the Lebesgue measure, i.
The asymptotic decay of 4. The straight line shown in Fig. Once we assume statistics for log- returns, or ranges for threshold values are defined using different methods shown in Section 4. We measure uncertainty to justify an emergency scale to represent the extreme events. Using the rule of thumb Section 4. The case with the negative values of the threshold is analogous. We observed the Red Queen State and three emergency scales can be readily interpreted.
Location of extrema points of the uncertainty curves from Fig. With the analysis presented above, we define an emergency scale of three levels based on the three regions of the threshold values corresponding to three peaks of the uncertainty curve. This emergency scale is not sensitive to the size of the window of sufficient amount of data considered.
Moreover, the segments with small fluctuations have a random walk like structure whereas segments with large fluctuations have a noise like structure. We have reviewed different methods of threshold selection and studied the extreme events presented in the time series using different statistical approaches. We found that the distribution of the weekly-return data can be described by a combination of different distributions.
Based on a graphical approach for threshold selection, we chose separate thresholds for the positive and negative values of the log return, 0. With this choice, we registered instances of extreme events corresponding to raise of market and extreme events related to market declines.
With a few exceptions, exceedances over under for negative log return values the threshold follow the GPD. The rule of thumb showed that a threshold value depends on the width of observation window, and the threshold can change at any moment, once new data become available. Uncertainty of the threshold values can be determined by the probability of changing the threshold on any given day.
The moment we assume statistics of distributions or the dataset is fixed, it leads to uncertainty of the threshold value which can be resolved by the emergency scales rigid to variation on the size of the dataset. We suggested a statistical model that describes registration frequency of extreme events under threshold uncertainty.
We are grateful to Dr. Toda for her support. Commun Nonlinear Sci Numer Simul. Published online May Author information Article notes Copyright and License information Disclaimer. Copyright Published by Elsevier B. Elsevier hereby grants permission to make all its COVIDrelated research that is available on the COVID resource centre - including this research content - immediately available in PubMed Central and other publicly funded repositories, such as the WHO COVID database with rights for unrestricted research re-use and analyses in any form or by any means with acknowledgement of the original source.
This article has been cited by other articles in PMC. Abstract An event is extreme if its magnitude exceeds the threshold. Keywords: Extreme events, Emergency scales, Uncertainty of threshold. Introduction Not a single day passes by without hearing about extreme events which surround us almost everywhere. Open in a separate window. If the amount of data is not sufficient a solid line corresponding to an day window of obervation in Fig.
Our understanding of the events whether they are extreme or not is very limited and uncertainty is blurry. As events become more severe, our uncertainty that the events are extreme decreases. The observer realizes that the events are extreme, but a precise point at which the events turn to be severe cannot be determined. This case is called the Red Queen State. As the window of observation becomes larger, the uncertainty curve exhibits two maxima indicating that the amount of data is sufficient.
As we further extend the window the curve torrents into sharp peaks Fig. The latter ones clearly separate the threshold values into three regions: three levels of emergency. For each segment of length s and for each positive or negative values of the moment order q , the q -order fluctuation function,.
Tails, thresholds, and extreme events There are two primary approaches to analyzing extreme values the extreme deviations from the median of the probability distributions in data: i. The first and more classical approach reduces the data considerably by taking maxima of long blocks of data, e. The GEVD function has theoretical justification for fitting to block maxima of data .
The second approach is to analyze excesses over a high threshold. For this second approach the generalized Pareto distribution GPD function has similar justification for fitting to excesses over a high threshold . Table 1 Parameter estimates for the GEV fitted model with maximum likelihood estimator. How to choose a threshold The classical approach for modeling extreme events is based on the GPD.
Graphical approaches to estimate threshold One of the most common ways to determine a suitable threshold is to graphically inspect data. Automatic methods to estimate thresholds As was mentioned above, the graphical approaches as well as rules of thumb can be highly objective, time consuming, and require certain professional background. Rules of thumb to choose a threshold As earlier noted, the threshold sequence is a function of the properties of the GPD provided that a population is in the domain of attraction of the GPD.
These values will be a lower bound for a positive threshold and an upper bound for a negative threshold, respectively. First, we compute the k threshold candidates u i for positive log-return as. For each threshold candidate u i found in the previous step we compute its validity duration, i.
Statistics of extreme events under threshold uncertainty The statistics of extreme events under threshold uncertainty can be described with the help of a simple model, in which the log-return of the index and the value of threshold are treated as random variables that yet can change inconsistently.
Defining emergency scales by thresholds uncertainty. If amount of data is not sufficient a solid line corresponding to the day window of obervation in Fig. As the window of observation becomes larger, 25 days, for instance, the uncertainty curve exhibits two maxima indicating that the amount of data is sufficient. The locations of peaks of the curves are summarized in Table 2. The location of two peaks is not sensitive to the window of observation. Table 2 Location of extrema points of the uncertainty curves from Fig.
Declaration of Competing Interest The authors declare no conflict of interests. References 1. Kaplan S. Lexis Nexis; UK: Scale R.. Mercalli scale. Wind scale. Hurricane wind scale. Tornado F.. Damage scale. Security H.. Advisory system. Climate U. Extremes index CEI. Rohn E. A unified localizable emergency events scale.
Pisarchik A. Rogue waves in a multistable system. Phys Rev Lett. Extreme events in epileptic EEG of rodents after ischemic stroke. Eur Phys J Spec Top. Plotnick L. Brown K. Risk aversion, uncertain information and market aptitude. J Financ Econ. Corrado C. Risk aversion, uncertain information and market aptitude reexamining the evidence. Rev Quan Finance Account. Halt T.. Lee J. Bayesian threshold selection for extremal models using measures of surprise.
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