K. Jagiełło et al.
20
E
(j)
matrix represents the residual spectra of j order. the residual spectra for j < k
have the character of the differences spectra, for j k
≥ the residual spectrum represents random noise. thus, further analysis of the residual spectra allows to specify
the correct value of k.
2.2.3 Numerical Spectrum Decomposition Technique and
Physicochemical Model of a Process
Applying PCA analysis let us know not only the number of spectral forms presented
in the samples, but also the first spectra approximation, however, does not allow to
designate individual forms, because the principal components are obtained with the
assumption of perfect orthogonality.
therefore, next step in chemometrical analysis of sets of spectra is to obtain the
spectra of indywidauals and estimation of its molar fractions. to reach this goal, Z
matrix should be expressed as a multiplication of the two other matrices: the matrix
of components B, in which the columns represent the spectrum of pure spectral
forms, and matrix X, which columns represent different forms of molar fractions
[3, 4].
Z BX
=
(2.6)
Knowing that the ends of the centered vectors of two components mixtures are on
a straight line (Fig. 2.2b), the position of the spectra of pure components on this
line is unknown. therefore, in order to solve the Eq. (2.6) there is need to use the
iterative procedure called Numerical Spectrum decomposition [1]. to determine
spectra of the complexes, as well as relative amount of each species, an iterative
self-consisting procedure—multivariate curve resolution-alternating least squares
(mCR-ALS) has to be used [11]. the most outer vectors w1 i wm participate the
most within the spectral forms according to the equation.
w
x f x f
x
x
21
1
1 1 1
2
1 1
2 1
1
0
=
+
≈
≈
,
(2.7)
w
x f x f
x
x
m
1 m 1
2m
1m
2m
=
+
≈
≈
2
1
0
,
(2.8)
In addition, in the two components mixtures, the molar ratios should not be negative, and the sum of the molar fractions of the two forms must be equal to 1:
x
x =1
11
21
+
(2.9)
x
x
1
m
2 m
1
+
=
(2.10)
20
E
(j)
matrix represents the residual spectra of j order. the residual spectra for j < k
have the character of the differences spectra, for j k
≥ the residual spectrum represents random noise. thus, further analysis of the residual spectra allows to specify
the correct value of k.
2.2.3 Numerical Spectrum Decomposition Technique and
Physicochemical Model of a Process
Applying PCA analysis let us know not only the number of spectral forms presented
in the samples, but also the first spectra approximation, however, does not allow to
designate individual forms, because the principal components are obtained with the
assumption of perfect orthogonality.
therefore, next step in chemometrical analysis of sets of spectra is to obtain the
spectra of indywidauals and estimation of its molar fractions. to reach this goal, Z
matrix should be expressed as a multiplication of the two other matrices: the matrix
of components B, in which the columns represent the spectrum of pure spectral
forms, and matrix X, which columns represent different forms of molar fractions
[3, 4].
Z BX
=
(2.6)
Knowing that the ends of the centered vectors of two components mixtures are on
a straight line (Fig. 2.2b), the position of the spectra of pure components on this
line is unknown. therefore, in order to solve the Eq. (2.6) there is need to use the
iterative procedure called Numerical Spectrum decomposition [1]. to determine
spectra of the complexes, as well as relative amount of each species, an iterative
self-consisting procedure—multivariate curve resolution-alternating least squares
(mCR-ALS) has to be used [11]. the most outer vectors w1 i wm participate the
most within the spectral forms according to the equation.
w
x f x f
x
x
21
1
1 1 1
2
1 1
2 1
1
0
=
+
≈
≈
,
(2.7)
w
x f x f
x
x
m
1 m 1
2m
1m
2m
=
+
≈
≈
2
1
0
,
(2.8)
In addition, in the two components mixtures, the molar ratios should not be negative, and the sum of the molar fractions of the two forms must be equal to 1:
x
x =1
11
21
+
(2.9)
x
x
1
m
2 m
1
+
=
(2.10)