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6. DATA ANALYSIS

FACTOR-ANALYTIC METHODS FOR PHYSICAL SCIENCES

Pentti Paatero, Philip K. Hopke* and Sirkka Juntto**

Matrix factorization methods for physical sciences ("Factor Analysis")
are applicable to many problems where a number of "spectra" have been measured
in similar situations or of similar samples consisting of same (perhaps
unknown) constituents in different proportions. Examples: chromatographic
"spectra", aerosol size distributions, compositions of environmental samples,
MEG (magnetoencephalographic) measurements.

A newly developed method "PMF" or "Positive Matrix Factorization" is
evaluated and applied in the present work. The essential features of PMF
are:

- utilization of error information of the measured data matrix

- implementation of strict non-negativity constraints for the
factor matrices

- production of meaningful error estimates for the computed
factors.

The method has been developed both for two-dimensional and for three-dimensional
data arrays. The 3-way model is often called PARAFAC. The present 3-way
solution is more efficent than the customary solutions of the PARAFAC problem
and produces error estimates for the results.

In 1998, journal articles describing the application of the methods
to various measurements of pollution in the Arctic air have been submitted
for publication.

Generalization of the factor analytic methods has lead to a "multilinear"
program which was completed in 1998. This table-driven program allows that
different mathematical or physical models of data analysis may be formulated
and computed by individual scientists by using the same program. The individual
models are described by a large "structure table"; reprogramming of the
fitting algorithm is not needed when a new model is to be solved.

* Clarkson Univ., NY, USA

** Finnish Meteorological Institute