Environmental Engineering Reference
In-Depth Information
p
1
X
Xp
1
p
1
p
1
max
p
1
subject to
=
1
.
0
(3.1)
A first summary variable or score,
t
1
, is obtained simply by projecting
X
in the direc-
tion of
p
1
,
t
1
Xp
1
. This high variance direction is then removed from
X
, leaving a
residual matrix
E
1
=
t
1
p
1
, containing the variance of
X
that is not explained by
the first component. The construction of the PCA model can continue with the com-
putation of a second linear combination
p
2
, explaining the second highest amount
of variance in
X
. The objective in this case is the same as shown in Equation 3.1,
but replacing
p
1
by
p
2
and
X
by
E
1
and imposing the additional constraint that the
second component be orthogonal to the first one (
e.g.
,
p
1
=
X
−
0). This procedure
is repeated until the desired number of components is computed. The final structure
of the model is
X
p
2
=
TP
+
E
, which can be seen as an eigenvector or singular
value decomposition (SVD) of
X
X
. In fact, the
p
vectors are just the eigenvectors
of
X
X
and the
t
vectors are the eigenvectors of
XX
. When as many components
are computed as there is variables (
e.g.
,
A
=
J
), the decomposition of
X
is perfect and
E
=
=
0.
An alternative approach for computing the
p
and
t
vectors sequentially is to use
the Nonlinear iterative partial least squares (NIPALS) algorithm. The starting point
of this algorithm typically (but not always required) consists of mean-centering and
scaling of matrix
X
(discussed later in this section). The following steps are outlined
below:
1. set
t
to be one column of
X
;
2.
p
X
t
t
t
;
=
/
p
p
3.
p
=
p
/(
)
;
p
p
4.
t
;
5. continue iterating between 2. and 4. until convergence on
t
or
p
;
6. residual matrix:
E
=
Xp
/(
)
tp
;
7. store
p
and
t
in
P
and
T
, respectively;
8. calculate next dimensions by returning to 1, using
E
as the new
X
.
=
X
−
After computing each latent variable, one needs to decide whether another dimen-
sion should be added to the PCA model. Cross-validation [16] is often a typically
used criterion for selecting the number of components to keep in the model.
3.2.2 Projection to Latent Structures (PLS)
Projection to latent structures, or alternatively, partial least squares is a truly multi-
variate latent variable regression method. PLS is used to model relationships both
within and between two blocks of data,
X
and
Y
. A tutorial on PLS is found in [17]
and a review of PLS history is available in [18]. Some mathematical and statistical
properties of PLS were also addressed in [19, 20].
In PLS, the covariance structures of
X
and
Y
are modeled via a set of
A
latent
variables,
t
and
u
respectively, as shown in Figure 3.2(b). However, these latent
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