dos Reis M. M., Ferreira M. M. C., “PARAFAC with splines: A case study”. Copenhagen, Denmark, 19-23/08/2001: 7th Scandinavian Symposium on Chemometrics SSC7, Book of Abstracts, A22 (2001). Oral A22 in section: Multi-way Methods.
PARAFAC with splines: A case study
Marlon M. Reis, marlon@iqm.unicamp.br, Chemistry
Institute - UNICAMP. Cidade Universitária Zeferino
Vaz, s/n, Campinas, Brazil.
Márcia M. C. Ferreira, marcia@iqm.unicamp.br,
Universidade Estadual de Campinas, Instituto de Química.
Campinas, SP, 13083-970, Brazil
Keywords: PARAFAC, smoothing splines constraint, carbon monoxide
The PARAFAC model has been used in several applications
in chemistry, e.g. for overlapped
spectra curve resolution, second order calibration and
others. In general, PARAFAC approach
considers the decomposed multilinear components as being
vectors. This work presents a
PARAFAC approach where the decomposed multilinear components
are considered as
functions.
The functional objects used to constrain the PARAFAC decomposition
are Splines. The
methodology used to promote the Spline-PARAFAC decomposition
is based on Bro-
Sidiropoulos' approach (1) for the unimodality constraint.
The spline-PARAFAC requires and additional calculation
of a penalty parameter or the number
basis functions, which were found in this work by using
an Ordinary Cross Validation
OCV(2). The Spline-PARAFAC was applied on a data set,
which corresponds to the
concentration of carbon monoxide measured every hour
during one year in the São Paulo city
in Brazil. The data was arranged as a Three Way Array
having the modes: (Hours of the Day-
Days of Week-Weeks of the Year). The Spline-PARAFAC presented
a good performance.
The authors acknowledge the financial support from FAPESP
for carrying out this work and
CETESB for kindly supplying the data set.
References:
1. Bro, R.; Sidiropoulos, N. D. "Least Squares Algorithms
Under Unimodality and Non-
Negativity Constraints", Journal of Chemometrics 1998,
12, 223-247.
2. Silverman, B. W. "Some Aspect of the Spline Smoothing
Approach to Non-parametric
Regression Curve Fitting", Journal of Royal Statistical
Society, 1985, 47, (1), 1-52.
A22