5.4 Functional Principal Components
Principal Components Analysis yields dimension reduction in the multivariate framework.
The idea is to find normalized weight vectors
for which the linear transformations
of a
-dimensional random vector
, with
:
 |
(5.4) |
have maximal variance subject to:
Where
denotes the identificator function. The solution is the Jordan spectral decomposition of the covariance matrix, Härdle and Simar (2003).
In the Functional Principal Components Analysis (FPCA) the dimension reduction can be achieved via the same route,
i.e. by finding orthonormal weight functions
,
such that the variance of the linear transformation is maximal. In order to keep notation simple we assume
. The weight functions satisfy:
The linear transformation is:
and the desired weight functions solve:
 |
|
|
(5.5) |
or equivalently:
The solution is obtained by solving the Fredholm functional eigenequation
 |
(5.6) |
The eigenfunctions
sorted with respect to the corresponding
eigenvalues
solve the FPCA problem (5.5).
The following link between eigenvalues and eigenfunctions holds:
In the sampling problem, the unknown covariance function
needs to be replaced by the sample covariance
function
. Dauxois, Pousse, and Romain (1982) show that the eigenfunctions and eigenvalues
are consistent estimators for
and
and derive some asymptotic results for these estimators.
5.4.1 Basis Expansion
Suppose that the weight function
has expansion
Using this notation we can rewrite the left hand side of eigenequation (5.6):
so that:
The functional scalar product
corresponds to
in the truncated basis framework, in the sense
that if two functions
and
are orthogonal, the corresponding coefficient vectors
satisfy
. Matrix
is symmetric by definition.
Thus, defining
, one needs
to solve finally a symmetric eigenvalue problem:
and to compute the inverse transformation
.
For the orthonormal functional basis (i.e. also for the Fourier basis)
, i.e.
the problem of FPCA is reduced to the multivariate PCA performed on the matrix
.
Using the FPCA method on the IV-strings for 1M and 2M maturities we obtain the
eigenfunctions plotted in Figure 5.4. It can be seen, that the eigenfunctions
are too rough. Intuitively, this roughness is caused by the flexibility of the functional basis. In the next section we present
a way of incorporating the smoothing directly into the PCA problem.
Figure 5.4:
Weight functions for
and
maturity groups. Blue solid lines,
and
, are the first eigenfunctions,
green finely dashed lines,
and
, are the second
eigenfunctions, and cyan dashed lines,
and
, are the third eigenfunctions.
|