Misalignment of Spectral Data: Constrained Optimization in a Functional Data Analysis Framework

Across several branches of sciences, a large number of applications involves data represented as functions and curves, for which functional data analysis can play a central role in solving a variety of problem formulations. With some thecnologies, the obtained data are spectra containing a vast amount of information concerning the composition of a sample: in order to infer the chemical composition of the materials from spectra, functional data analysis offers a valuable mean for characterizing the spectral response through identification of peaks position and intensity. The collection of data from different measurement may exhibit similar peak pattern but display misalignment in their peaks. In general, the multiple alignment is crucial in the subsequent analysis; the method proposed faces with the challenge of random shifts in the peaks and implements constraints in a proper objective function to optimize the alignment. The constraints are based on a priori information that is formalized in the choice of a set of peaks across functions. Spectrum data from X-ray Fluorescence (XRF) and Total Reflectance-Fourier Transform Infra-Red (TR-FTIR) spectroscopies are considered to illustrate the approach and to provide useful comparison with other approaches.