The paper is concerned with data representing a sample of smooth curves which can be considered as independent realizations of an underlying biological (chemical, $\ldots$) process. Such samples of curves often possess the following features: There is a typical structural pattern common to all curves of the sample. On the other hand, individual realizations of the typical shape show different dynamics and intensity. In particular, typical peaks are shifted from individual to individual. Differences in dynamics complicate the analysis of samples of curves. For example, the cross-sectional average usually does not reflect an average pattern. Due to shifts, structure is smeared or might even disappear. Our approach consists in synchronizing the individual curves before determining the average or any further statistics. Pointwise averaging of the synchronized curves then leads to an average curve which represents the common structure with average dynamics and average intensity. The method requires the introduction of new statistical objects. They are defined mathematically, their properties are discussed, and possible estimators are proposed. The asymptotic bias and variance of the estimators are derived. An application to visually evoked brain potentials illustrates the approach.
"Statistical Tools to Analyze Data Representing a Sample of Curves." Ann. Statist. 20 (3) 1266 - 1305, September, 1992. https://doi.org/10.1214/aos/1176348769