Biomedical Engineering Reference
In-Depth Information
= X
TP T +
X
+
E,
= Y
UQ T +
Y
+
F,
where T and U are the matrices of the PLS modes, P and Q are the loading matrices
which describe the weight of each variable in X
− Y respectively and
E and F are the residual terms which are the same size as X and Y respectively.
Additionally the following regression condition is imposed for the PLS modes:
− X and Y
U
=
TD
+
G,
(5.9)
where D is a diagonal matrix of weights and G a matrix of residuals. Due to this
added condition, the PLS loadings P and Q are not necessarily orthogonal as is the
case for PCA modes.
Algorithm 9 Partial least squares regression (PLS)
Input: Variables X and Y, number of components p ≤ N − 1 .
1: X 1 = X − X , Y 1 = Y − Y
2: for n =1 to p do
2: r n first eigenvector of X n T Y n Y n T X n
2: t n ← X n r n /r n n th PLS component of X
2: s n ← Y n t n / ( t n T t n )
2: u n ← Y n s n /s n n th PLS component of Y
2: p n ← X n t n / ( t n T t n ) n th loading of X
2: q n ← Y n u n / ( u n T u n ) n th loading of Y
2: X n +1 ← X n − t n p n T deflation of X
2: Y n +1 ← Y n − t n [ t n T Y n /t n T t n ] deflation of X
3: end for
4: return T =( t n ) n =1 ...p , P =( p n ) n =1 ...p , U =( u n ) n =1 ...p , Q =( q n ) n =1 ...p
Several algorithms have been proposed to compute the PLS modes. In this work,
we use the PLS1 method, an efficient iterative algorithm that does not require matrix
inversion as summarized in Algorithm 9 . X is the matrix of the initial velocity field
moments for all patients and Y is the vector of the BSA values for all patients. The
first five PLS modes are shown in Fig. 5.9 and the explained variance and correlation
of modes is shown in Fig. 5.10 . These modes account for 99 % of the covariance
between shape and BSA in the population and
% of the shape variability and
are oriented along increasing BSA. Visually we can see that Modes 1 and 3 display
an overall dilation in both the left and right ventricles. The second mode shows a
narrowing in the right ventricular outflow tract with a noticeable dilation in the left
ventricle which can be seen in the top view of the mode. Modes 4 and 5 show an
elongation at the right ventricular outflow tract. Mode 5 also shows an aneurysm in
the right ventricle outflow tract which is characteristic of rToF patients, although it is
still not clear if this is due to pulmonary regurgitations (although both are correlated)
or the initial surgical patch.
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