Shi, Y. and Nusholtz, G., "Assessment of Similarity of a Set of Impact Response Time Histories," SAE Int. J. Trans. Safety 3(1):28-38, 2015, doi:10.4271/2015-01-1441.
Two methods of assessing the similarity of a set of impact test signals have been proposed and used in the literature, which are cumulative variance-based and cross correlation-based. In this study, a normalized formulation unites these two approaches by establishing a relationship between the normalized cumulative variance metric (v), an overall similarity metric, and the normalized magnitude similarity metric (m) and shape similarity metric (s): v=1 − m · s. Each of these ranges between 0 and 1 (for the practical case of signals acquired with the same polarity), and they are independent of the physical unit of measurement. Under generally satisfied conditions, the magnitude similarity m is independent of the relative time shifts among the signals in the set; while the shape similarity s is a function of these. An optimal alignment is defined as the relative shifts corresponding to the minimum of the cumulative variance metric, or equivalently, to the maximum of the shape similarity metric. This system therefore quantifies the similarity of a given set of signals with an “as given” relative time alignment with the following: v = 1− m(p + sa), i.e., its overall similarity is partitioned into the shift-invariant normalized magnitude metric m, and the inherent (optimally aligned) normalized shape metric sa, and a normalized phase metric p which reflects the alignment.An algorithm is provided for automatic search of the optimal shifts for a given set of signals using the cumulative variance as the objective function. The shifts are treated as real numbers instead of integers so that standard continuous variable optimization functions can be used to treat the discrete signals. This is facilitated by an interpolation procedure for shifting a signal with a real-valued time shift. Numerical examples are presented to demonstrate the general effectiveness of the algorithm, and also specific limitations on consistent convergence to the global optimal. Practical measures that would mitigate the limitations and alternative numerical methods that would fundamentally remove these are suggested.