4.4 Principal Component Analysis (PCA)
101
obtained. The main diagonal of the matrix S is composed of eigenvalues, whereas the
remaining matrix elements are equal to zero. The values of the principal components k
are chosen in a way to keep the eigenvalues of the matrix S larger than 1. Equivalently,
the dependence of dispersion on the number of principal components k T RV (k) and
the dependence of the dispersion on the number of principal components k E RV (k)
should rapidly change their behaviour, i.e. the truncated singular series are employed.
After definition of the required number of principal components, the matrix T =
[t i j ]; t i j = (x i , a j ) has a dimension m × k, and its each row presents a projection of
the data vector onto k principal components (the number of columns k corresponds
to the number of vectors of the principal components chosen for the purpose of
projection). The loading matrix P = {a 1 , ..., a k } and each of its columns correspond
to the vector of principal components, whereas the number of rows n corresponds
to the dimension of the data space chosen for the purpose of projection. Each row
of computation matrix stands for the projection of the data vector onto k principal
components. The matrices U and V are orthogonal. The characteristics of dispersion
T RV and of dispersion ERV show the percentage of noise remaining after projection
onto the multidimensional space PC1-PCk. In other words, the whole characteristics
show what number of principal components is necessary for the purification of the
signal from noise.
4.5 Numerical Experiment
It is assumed that two beams satisfy the hypotheses of the first-order Euler-Bernoulli
approximation. The system (4.1) is solved taking into account the boundary (4.4)
and initial conditions (4.5).
Both ends of the beams are rigidly clamped:
w i (0, t) = w i (1, t) = u i (0, t) = u i (1, t) =
=
∂w i (x, t)
∂ x
x=0
=
∂w i (x, t)
∂ x
x=1
= 0, i = 1, 2,
(4.4)
and the initial conditions are taken in the following form:
w i (x, 0) = 0, u i (x, 0) = 0,
∂w i (x, t)
∂t
t=0
= 0,
∂u i (x, t)
∂t
t=0
= 0, i = 1, 2.
(4.5)
The beam 1 is subjected to the uniformly distributed transverse harmonic excitation of the following form
q = q 0 sin(ω p t),
(4.6)
where q 0 stands for the amplitude and ω p for the frequency of excitation.
101
obtained. The main diagonal of the matrix S is composed of eigenvalues, whereas the
remaining matrix elements are equal to zero. The values of the principal components k
are chosen in a way to keep the eigenvalues of the matrix S larger than 1. Equivalently,
the dependence of dispersion on the number of principal components k T RV (k) and
the dependence of the dispersion on the number of principal components k E RV (k)
should rapidly change their behaviour, i.e. the truncated singular series are employed.
After definition of the required number of principal components, the matrix T =
[t i j ]; t i j = (x i , a j ) has a dimension m × k, and its each row presents a projection of
the data vector onto k principal components (the number of columns k corresponds
to the number of vectors of the principal components chosen for the purpose of
projection). The loading matrix P = {a 1 , ..., a k } and each of its columns correspond
to the vector of principal components, whereas the number of rows n corresponds
to the dimension of the data space chosen for the purpose of projection. Each row
of computation matrix stands for the projection of the data vector onto k principal
components. The matrices U and V are orthogonal. The characteristics of dispersion
T RV and of dispersion ERV show the percentage of noise remaining after projection
onto the multidimensional space PC1-PCk. In other words, the whole characteristics
show what number of principal components is necessary for the purification of the
signal from noise.
4.5 Numerical Experiment
It is assumed that two beams satisfy the hypotheses of the first-order Euler-Bernoulli
approximation. The system (4.1) is solved taking into account the boundary (4.4)
and initial conditions (4.5).
Both ends of the beams are rigidly clamped:
w i (0, t) = w i (1, t) = u i (0, t) = u i (1, t) =
=
∂w i (x, t)
∂ x
x=0
=
∂w i (x, t)
∂ x
x=1
= 0, i = 1, 2,
(4.4)
and the initial conditions are taken in the following form:
w i (x, 0) = 0, u i (x, 0) = 0,
∂w i (x, t)
∂t
t=0
= 0,
∂u i (x, t)
∂t
t=0
= 0, i = 1, 2.
(4.5)
The beam 1 is subjected to the uniformly distributed transverse harmonic excitation of the following form
q = q 0 sin(ω p t),
(4.6)
where q 0 stands for the amplitude and ω p for the frequency of excitation.