Reply to Takayama's comment

The goal of our paper (Mulargia & Gasperini 1992, which will be from now on referred as (I)) was to derive a solution to the following problem: given a set of predictions based on some hypothesized precursor, could a similar success have been obtained by issuing an equal number of predictions at chance, i.e. observing no precursor at all? A crucial point is how to define these ‘chance’ predictions. The definition is not easy since in general the analysis must be made in retrospect on a limited set and with laws of the game which are provided by the same proponent of the precursor. According to statistical decision theory, the ‘chance’ set must be statistically identical to the real set. In other words, the real and the ‘chance’ sets must have the same size and the same distribution in: (a) magnitude, (b) time and (c) space. It is infeasible to fully satisfy this constraint. In fact, if predictions are to be effective, their distribution should mimic that of seismic events, which is joint in magnitude, time and space, i.e. each of the three distributions cannot be fitted separately. Unfortunately, the joint magnitude, time and space distribution of seismic events is very complex (cf. Gasperini & Mulargia 1989), and appears inipractical to translate into a significance test on precursors. In (I) we, therefore, adopted an approximate solution aimed at preserving this joint character, which has a particularly important effect on precursor validation: in a finite set some phenomenon might appear as a seismic precursor, but in fact have a post-seismic nature of whatever, known or unknown, origin. Note how this would be particularly evident for aftershocks, since a post-seismic phenomenon relative to the principal event would appear as a precursor to its aftershocks. The solution we adopted in (I) was based on: (a) using for the ‘chance’ set the same magnitude distribution as the real one; (b) using for each prediction of the ‘chance’ set the global spatial distribution of the real one, at the same time allowing for alternative (double) predictions; this is equivalent to assuming that the right epicentre could have been predicted ‘by chance’, i.e. that predictions were mostly originated by a post-seismic effect; (c) using a totally random distribution in time, i.e. Poisson; (d) checking the validity of hypothesis (b) by verifying that the significance level of the time association in reverse time is much lower than in forward, i.e. that earthquakes lead to predictions rather than vice versa. As a main result of (I), hypothesis (b) was found to be entirely consistent. In his Comment, Takayama transcribes our reasoning with three crucial differences: (i) the main variables entering the significance level equation, i.e. the number of predictions and of the earthquakes that should have been predicted, have been reversed; (ii) ‘chance’ predictions are assumed to be distributed uniformly at random in space within the area S on which precursors are operational; this leads to weight the number of predictions with a factor As/S, where As = n Ar2 is the epicentral indetermination area of each prediction, (iii) no study of reverse-time association is performed. Two notes are in order. First, the number of predictions cannot be weighted, since the ‘chance’ predictions set must have the same size and distribution of the real one. It is only the number of events that should have been predicted that can be weighted to account for the fact that each prediction covers a subset of the global region on which predictions are operational. Fortunately, this interchange has little practical consequence on the significance level of time association, as can be immediately seen from the symmetry with respect to the number of predictions m and events n of the Poisson equation