Retrospective correlation

One of the main tools in phenomenological studies is the identification of correlations among different processes. This is essentially effected in retrospective with the specific aim of finding a positive result, and that leads to a parameter optimization which introduces a bias, so far only marginally considered, in the significance level of the results. If the correlation can be validated in a forward study in which parameters are kept fixed, such a bias is irrelevant. Unfortunately, forward studies are often infeasible for either cost or intrinsic reasons. This is the case of geophysics, due to the comparatively long time scale of recurrence of the phenomena. Unbiased estimates can be obtained in retrospective if each of the optimal choices is properly identified and accounted for. An estimate of the bias is made in a specific case, which can be written in closed form. While simulation confirms its good performance, the latter shows that apparently highly significant retrospective correlations may be insignificant.