# Natural Time Analysis: The Area under the Receiver Operating Characteristic Curve of the Order Parameter Fluctuations Minima Preceding Major Earthquakes

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## Abstract

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## 1. Introduction

**Figure 2.**(color online) Variability $\beta $ versus the conventional time depicted in consecutive 6 year periods in the upper graphs of the panels (

**a**–

**e**), respectively. The six minima preceded the M ≥ 7.6 EQs are marked by red circles at the ${\beta}_{200}$ curve while the nine minima followed by smaller EQs by green squares again at the ${\beta}_{200}$ curve; the values of ${\beta}_{300,min}/{\beta}_{200,min}$ are also written in red and green, respectively (see, e.g., Tables 1 and 2 of Reference [30]). No data are plotted in (e) after M9 Tohoku EQ. The horizontal red line corresponds to the shallowest ${\beta}_{200}$ minimum that preceded a M ≥ 7.6 EQ and the EQs are marked as black arrows whose magnitude can be read in the right scale. In addition, below the variability graph in each panel, we depict the time dependent seismicity rate $(\lambda \left(t\right)-\mu )/{K}_{0}={\sum}_{{t}_{i}<t}exp\left[\alpha ({M}_{i}-{M}_{c})\right]/{(t-{t}_{i}+c)}^{p}$ of the temporal epidemic-type aftershock sequence (ETAS) model [47,48,49] according to Equation (1) of Ogata et al. [50] together with the seismicity (black vertical lines ending at circles whose magnitude can be read in the right scale). The ETAS model parameters $(\alpha ,p,c)$ are the same as those presented in Figure 2a in Reference [50].

## 2. Receiver Operating Characteristics Technique

## 3. Data Analyzed

## 4. Results

## 5. Discussion

`VISROC.f`, provided in Reference [55], gives that the probability p to obtain this operation point (8.74%, 100%) by chance based on k-ellipses is of the order of ${10}^{-5}$. In other words, this result ($p={10}^{-5}$) demonstrates that it is incorrect to claim [78] that since there exists an abundance of false positives (i.e., 6 true positives while 9 false positives in the present case) the method of identifying ${\beta}_{W,min}$ that are simultaneous with SES activities is unusable without, however, paying attention to its statistical significance (cf. This can be also shown analytically by applying the formulation given in the Appendix of Reference [79]). Such a line of thinking (abundance of false positives) is in contrast with the proper use of ROC curves as discussed in detail in Section 6 of Reference [46], see Equation (1) there, according to which the selection of the operating point should be selected on the basis of the trade of the losses in case of false negative and the cost of a false positive.

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

AUC | Area under the ROC curve |

ECA | Event coincidence analysis |

EQ | Earthquake |

JMA | Japan Meteorological Agency |

ROC | Receiver operating characteristic |

SES | Seismic electric signals |

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**Figure 1.**(color online) Epicenters (red stars) of all six shallow EQs with M ≥ 7.6 within the area N${}_{25}^{46}$ E${}_{125}^{148}$ since 1 January 1984 until the M9 Tohoku EQ. The smaller green stars indicate the epicenters of all M ≥ 3.5 EQs.

**Figure 3.**ROC curves resulting from the study of the minima of the fluctuations of the order parameter of seismicity in Japan. For all the three ROCs, the corresponding AUCs have a mean value 0.95 reflecting an outstanding discrimination (see p. 162 of Reference [53]). The cyan and magenta lines are the k-ellipses that correspond [55] to the 95% and 99% confidence intervals. They indicate how far away from the diagonal the ROC curve of a random predictor may scatter with probability 1/20 or 1/100.

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**MDPI and ACS Style**

Sarlis, N.V.; Skordas, E.S.; Christopoulos, S.-R.G.; Varotsos, P.A.
Natural Time Analysis: The Area under the Receiver Operating Characteristic Curve of the Order Parameter Fluctuations Minima Preceding Major Earthquakes. *Entropy* **2020**, *22*, 583.
https://doi.org/10.3390/e22050583

**AMA Style**

Sarlis NV, Skordas ES, Christopoulos S-RG, Varotsos PA.
Natural Time Analysis: The Area under the Receiver Operating Characteristic Curve of the Order Parameter Fluctuations Minima Preceding Major Earthquakes. *Entropy*. 2020; 22(5):583.
https://doi.org/10.3390/e22050583

**Chicago/Turabian Style**

Sarlis, Nicholas V., Efthimios S. Skordas, Stavros-Richard G. Christopoulos, and Panayiotis A. Varotsos.
2020. "Natural Time Analysis: The Area under the Receiver Operating Characteristic Curve of the Order Parameter Fluctuations Minima Preceding Major Earthquakes" *Entropy* 22, no. 5: 583.
https://doi.org/10.3390/e22050583