Abstract
When studying air pollution measurements at different sites in a spatial area, we may search for a typical pattern, common to all curves, describing the underlying air pollution process in a preโspecified period. Another area of interest to support local authorities in air quality management may be the classification of the different sites in homogeneous clusters and the group ranking that follows. Yet, there is variation in both amplitude and dynamics among the air pollutant concentrations measured at the different monitoring stations. Analyzing such measurements, where the basic unit of information is the entire observed process rather than a string of numbers, involves finding the time shifts or the warping functions among curves. The analysis is much more complicated if we consider a multivariate process, that is, vectorโvalued air pollutant measurements. Following our previous work where an improved dynamic timeโwarping algorithm has been developed, especially in the multivariate case, and used both for classifying functional data and estimating the structural mean of a sample of curves, we analyzed the measurements of some air pollutants in Emilia Romagna (northern Italy). In addition, for the univariate analyses, we applied the selfโmodeling warping function approach, which is also convenient for these data. Indeed, this method was found to be modelโfree and enough flexible to capture very complex and highly nonโlinear patterns. Copyright ยฉ 2007 John Wiley & Sons, Ltd.