Statistical Tests for Evaluating Earthquake Prediction Methods

The impact of including postcursors in the null hypothesis test is discussed. Unequal prediction probabilities can be included in the null hypothesis test using a generalization of the central limit theorem. A test for determining the enhancement factor over random chance is given. The seismic earthquake signal may preferentially precede earthquakes even if the VAN methodology fails to forecast the earthquakes. We formulate a statistical test for this possibility. Postcursors and High Significance Level The present paper by Varotsos et al. [1996, hereafter cited as VEVL] is devoted primarily to criticizing the criticism of Mulargia and Gasperini [1992, herafter cited as MG], although this paper does contain some clarification of the methodology of Varotsos et al. [1981, hereafter cited as VAN]. The central argument is whether the VEVL method predicts better than random chance. If the null hypothesis is true, the significance level is a random variable with a mean value of 1/2 and is uniformly distributed on [0,1]. Normally, one requires significance levels as low as 0.05 before one is confident that the random chance hypothesis is false. Many of Mulargia and Gasperini’s tests achieve a significance level of .999, which means that under the MG test, VAN do much worse than the hypothesis of random chance would expect. There are three possible explanations for the significance level of .999: 1) VAN are extremely unlucky; 2) VAN predictions are anticorrrelated with earthquakes; 3) the MG hypothesis that earthquakes are independent and uniformly distributed in time is wrong. Point 5 of the VEVL criticism suggests the likely culprit, i.e., many of MG’s earthquakes are probably aftershocks which VAN did not try to predict. Note that the significance level of the test will converge to 1.0 as the number of predictions tends to infinity if the null hypothesis of random chance is true and a small percentage of the spatial distribution of earthquakes are postcursors which VAN do not try to predict. Thus the MG test should be repreated after excluding these postcursors. It is unfortunate that VEVL did not formulate a criterion for excluding postcursors.