A novel semi-distance for measuring dissimilarities of curves with sharp local patterns

• An algorithm for comparison of curves with sharp local patterns is proposed.
• It is based on a new dissimilarity measure between curves.
• It takes not only vertical but also horizontal variation between two curves into account.
• The algorithm uses a representation of the curves in a signal-specific (unbalanced Haar) wavelet basis.
• Applications are (functional) classification and prediction.

A functional wavelet-based semi-distance is defined for comparing curves with potentially misaligned sharp local patterns. It is data-driven and highly adaptive to the curves. A main originality is that each curve is expanded in its own wavelet basis, which hierarchically encodes the patterns of the curve. The key to success is that shifts of the patterns along the abscissa and the ordinate axes are taken into account in a unified framework. We investigate how the use of the new semi-distance improves the performance of some common statistical tools for detecting and localizing differences between groups of curves. Further we apply our methodology to a set of 11H-NMR spectrometric curves.