Geoscience Reference
In-Depth Information
sets from natural and artificial aquatic environments have found the greater part (>80%) of
fluorescence variability to be explained by just one or two principal components.
10.6 Parallel Factor Analysis
Parallel factor analysis (PARAFAC, less commonly termed canonical decomposition or
CANDECOMP), is a multiway decomposition method originating from the field of psy-
chometrics (Carroll and Chang,
1970
; Harshman and Lundy,
1994
). Although its utility
for analysing fluorescence data has been known for some years (Appellof and Davidson,
1981
), its first applications to DOM fluorescence occurred relatively recently (Søndergaard
et al.,
2003
; Stedmon et al.,
2003
). PARAFAC is now widely applied in the study of DOM
fluorescence, with more than 90 PARAFAC studies of DOM in soils and natural water bod-
ies published in 2005-2011, and a rapidly increasing number appearing in the engineering
literature related to monitoring of wastewaters and water treatment systems.
10.7 PARAFAC and Its Properties
The popularity of PARAFAC stems from its ability to mathematically separate the spectra
of overlapping fluorescence components. Hence, PARAFAC is similar to MCR mentioned
earlier (actually PARAFAC can be considered the three-way version of the two-way MCR)
with the important difference being that PARAFAC does not have rotational ambiguity
as does MCR. This means that if the model is correct, it
will
give chemically meaningful
results. For a mathematical explanation of PARAFAC including tutorials on its application
we refer the reader to other references (Bro,
1997
; Andersen and Bro,
2003
). Briefly stated,
PARAFAC of a three-way data set decomposes the data signal into a set of trilinear terms
and a residual array:
i = 1, …, I ; j = 1, …, J ; k = 1, …, K
where
x
ijk
is the intensity of the
i
th
sample at the
j
th
emission value and at the
k
th
excitation
value. When successfully modeling an EEM data set, the PARAFAC model allows for a
direct chemical interpretation as opposed to the abstract orthogonal components stemming
from PCA. For example, the parameter
a
if
is directly proportional to the concentration of
the
f
th
analyte of sample
i
; the vector
b
f
with elements
b
jf
is a scaled estimate of the emission
spectrum of the
f
th
analyte. Likewise, the vector
c
f
with elements
c
kf
is linearly proportional
to the specific absorption coefficient (i.e., molar absorptivity) of the
f
th
analyte. Finally,
e
ijk
is the residual representing the variability not accounted for by the model. Although
the simple so-called trilinear model is shown in the above equation, it can be extended to
higher-order data sets by increasing the number of terms following the summation sign.
For example, a four-way data set consisting of EEMs
x
ijkl
replicated in time would require
a further multiplicative term
d
lf
to model the fourth (time) dimension.
Three important assumptions for successful PARAFAC models are (1) variability:
no two chemical components can have identical excitation or emission spectra or have
