PARAFAC with splines: a case study

The PARAFAC model has been used in several applications in chemistry, e.g. for overlapped spectra resolution and second‐order calibration. In general, the PARAFAC method uses a vector space approach by considering the matrices resulting from the decomposition as a collection of vectors. This paper presents a PARAFAC application where the factors resulting from the decomposition are considered as functions. The functional objects used for this are spline functions. The methodology used performs the Spline‐PARAFAC decomposition based on the Bro–Sidiropoulos approach for the unimodality constraint. One of the advantages of using splines is the possibility of achieving a controlled degree of smoothing on the decomposed components. The amount of smoothing applied on the components in the presented methodology is controlled by a penalty parameter or by the number of basis functions. Thus Spline‐PARAFAC requires the calculation of the parameter λ and the number of basis functions, which were determined in this work by using ordinary cross‐validation (OCV). Spline‐PARAFAC was applied to a carbon monoxide data set comprising concentrations measured every hour during the years 1997 and 1999 in the city of São Paulo, Brazil. Each data set was arranged in a three‐way array of dimension (24 hours × 5 days × 52 weeks). Spline‐PARAFAC showed a good performance, producing smoothed profiles describing the daily variations in emitted gas and the seasonal effects during the year. Copyright © 2002 John Wiley & Sons, Ltd.