GeMCA EXIN THEORY - Neural-based Orthogonal Data Fitting: The EXIN Neural Networks - page 210

an MCA algorithm can replace the GeTLS EXIN algorithm by using matrix K
as autocorrelation matrix. The estimated minor component must be scaled by
using eq. (6.22) and normalized by constraining the last component equal to 1.

Choosing K as a autocorrelation matrix implies that the MCA EXIN neuron is
fed by input vectors m i ,definedas

−

a i
√ 2

T

b i

m i = , √ 2 (6.23)

ζ ( 1 − ζ )

a i being the column vector representing the i th row of matrix A .TheMCAEXIN
neuron whose input is preprocessed by means of eq. (6.23) is called the GeMCA
EXIN neuron .

6.2 ANALYSIS OF MATRIX K

Matrix K can be decomposed [9, Prop. 2.8.3, p. 43] as

A T A
ζ

I n 0 ζ

1 − ζ

0

A + b

I n

A + b T

ζ
1 − ζ

2 K =

2
2
1 − ζ

b ⊥

1

0 T

0 T

1

A T A
ζ

I n 0 0 ζ

1 − ζ

I n x OLS

= ζ

2
2
1 − ζ

x OLS

1 b ⊥

0 T

0 T

1

1 − ζ

S T K 1 S = (6.24)

where

I − A A T A − 1 A T b

b ⊥ = P A b =

(6.25)

represents the component of b orthogonal to the column space of A .Matrix P A
is the corresponding orthogonal projection matrix. Then , as seen before, b ⊥

2
2
is the sum of squares of the OLS residuals. It can be deduced that matrix K is
congruent to matrix K 1 and then, for Sylvester's theorem, it inherits the same
inertia [i.e., K is positive semidefinite, which is also evident from eq. (6.17)].
This analysis also yields the value of the determinant of matrix K :

det A T A
2 ζ

2
2
2 ( 1 − ζ )

2
2

det A T A

b ⊥ b ⊥

det K = = (6.26)

2 n + 1 ζ

n

( 1 − ζ )

By looking at matrix K [see eq. (6.17)] and identifying the principal submatrix
A T A / 2 ζ by means of the interlacing theorem, it follows ∀ i = 1, ... , n that
α i + 1 ≤ σ 2

i
2 ζ

(6.27)