Alignment of Noisy and Uniformly Scaled Time Series

The alignment of noisy and uniformly scaled time series is an important but difficult task. Given two time series, one of which is a uniformly stretched subsequence of the other, we want to determine the stretching factor and the offset of the second time series within the first one. We adapted and enhanced different methods to address this problem: classical FFT-based approaches to determine the offset combined with a naïve search for the stretching factor or its direct computation in the frequency domain, bounded dynamic time warping and a new approach called shotgun analysis, which is inspired by sequencing and reassembling of genomes in bioinformatics. We thoroughly examined the strengths and weaknesses of the different methods on synthetic and real data sets. The FFT-based approaches are very accurate on high quality data, the shotgun approach is especially suitable for data with outliers. Dynamic time warping is a candidate for non-linear stretching or compression. We successfully applied the presented methods to identify steel coils via their thickness profiles.