3.4 The Elliptical Structure Theorem
For another picture of convergence, two variables were randomly selected from the fifty symptoms and their scatterplots were displayed at each iteration. The sequence of scatterplots for columns TH2 and NC2 are displayed in Figure 5. An interesting pattern of a gradually formed ellipse is identified in the sequence of the
scatterplots. The near perfect ellipse was formed at matrices
to and later these fifty points rapidly converge to the points (1, 1) and (1, 1). These two point clusters are then magnified at matrix
, one iteration from , and similar elliptical structure in these two epsilon boxes was identified. The pointclouds concentrate on the lower left and upper right corners indicating that these two columns (symptoms) converge to the same group.
Figure 5. Sequence of Scatterplots of Symptoms TH2 and NC2.
Actually, this formation of ellipse in the scatterplot for the correlation matrices exists in all C(50, 2) = 1225 possible pairs of columns. The following theorem and corollaries say that every correlation matrix
in the converging sequence has it's column vectors fall on the surface of the
dimensional ellipsoid generated by the kernel
or , the inverse or generalized inverse of
.
Theorem 3.1.
Given a full rank correlation matrix R, all p column (row) vectors of R,
, i=1,…,p, fall on the pdimensional ellipsoid generated by the kernel of
, the inverse of R.
Proof. Observe that the right hand side of the equality
is a correlation matrix, hence all diagonal elements equal one. That is
diag() = diag() =(1,1,…,1), which leads to
, i=1,…,p, thus completing the proof.
In general, only the original proximity matrix D could be of full rank, all the subsequent correlation matrices
have ranks smaller than p, the dimension of D. We can then substitute
in Theorem 3.1 with the generalized inverse
. When the correlation matrix is not orthogonal, usually the case, we have a rotated version of Theorem 3.1.
Corollary 3.1.
Given a pdimensional correlation matrix
with rank , and the spectral decomposition
, where
is a kdimensional diagonal matrix with a list of nonzero eigenvalues (,,…,) ( are not necessarily distinct) on the diagonal and
contains the
eigenvectors ,,…. All the p principal components of
, , i=1,…,p fall on the kdimensional ellipsoid generated by the kernel of
, the inverse of .
Usually when one deals with the ellipsoid generated from the quadratic form of a positive definite matrix like the correlation matrix, then it is Corollary 3.2 that is of interest and not Corollary 3.1.
Corollary 3.2. With the same setup as in Corollary 3.1, all the p rows of
, , i=1,…,p fall on the kdimensional ellipsoid generated by the kernel of
.
Figure 6 shows these two ellipse generated by
and
for the fifty columns at with all the fifty
’s and ’s falling on them. Besides, the halflengths of the principal axes for these two ellipsoids can be computed from the
eigenvalues. For the ellipsoid generated by , we have
[…
]diag[ …
][…
]^{T}=1, so =1.
The halflength for each principal axis of the ellipsoid generated by
is , i=1,…,k. For the ellipsoid generated by
, the halflengths are , i=1,…,k. In Figure 6, these halflengths are (6.401851, 3.002717) for
and (0.156205, 0.333032) for .
Figure 6. Two Ellipses Generated by
(outer) and
(inner) for the Fifty Symptom Example.
We have now shown that each of the correlation matrix in the sequence
has an exact dimensional ellipsoid structure embedded in it. In the converging process, each time the rank (numerical) decreases, the ellipsoid collapses down to a lower dimensional one. The sequence of scatterplots for symptoms TH2 and NC2 with elliptical structure in Figure 5 are the twodimensional images projected from the fiftydimensional ellipsoids. The plotting of leading eigenvectors with ellipses to study the structure of original proximity matrix D is illustrated in Section 5.1.
