Retrospective validation of the time association of precursors

This paper examines the problems raised by the retrospective association of events, a process which underlies any search for precursors to geophysical phenomena. Association is analysed here with respect to time only, disregarding the spatial component, so the results are more readily applicable to volcanic eruptions than to earthquakes. Two sources of bias are examined, viz. the time lag between the precursor and event series, and the tolerance or time window accepted on that lag, which is also linked to the number of associated events. These parameters are typically chosen a posteriori to minimize the apparent probability of random association in the given series. In general, given different input data, optimal association of the two series will be achieved for different values of these parameters. Thus, any valid system of significance testing must include a step which estimates the effect of retrospectively choosing optimal parameters rather than fixing them in advance. This paper shows that, as long as at least one of the two time-series—either the precursors or the events—is Poissonian, the correction factors for the significance level can be written in a simple closed form. The correction scheme is applied to a proposed association between seismic clusters and flank eruptions of Mount Etna. Although the association appears to be highly significant when assessed using simple standard tests, this significance is not confirmed when the effect of retrospective parameter optimization is taken into account. The key point is that retrospective studies can be investigated using standard techniques, but where such techniques are employed, the threshold for significance of association must be greatly lowered in order to account for the bias due to the optimal choice of parameters.