An efficient algorithm for the computation of the First Passage Time maps of a given signal

The First Passage Time (FPT) of a given signal is the time required to reach a level Xf for the first time, when starting from a level X0. For a given set (X0,Xf), this time can be obtained by moving along the given signal and clocking the time on when passing through level X0, then off when passing through level Xf for the first time. For random signals, the FPT is a random variable. Several samples of the first passage time are collected while moving along the given signal to compute the corresponding statistical moments and probability density function. The FPT map associates each statistical moment with every couple (X0,Xf). A map is constructed for every desired moment. A naive implementation of the signal processing method to compute the FPT map would require to move along the signal as many times as the desired pairs (X0,Xf). Instead, this paper introduces a new and optimized algorithm to establish the FPT maps of any statistical moment and, optionally, of the whole probability distribution. This algorithm combines different features to produce a computationally effective and memory-efficient method. The algorithm estimates the entire map by iterating only once across the signal. The accuracy of the algorithm has been assessed with particular cases where theoretical solutions exist as well as for lab data. Furthermore the algorithm's performance has also been evaluated in terms of computational burden. A specific attention is given to the computation time scaling for the evaluation of an FPT map, with respect to the signal size.