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*Astron. Astrophys. 332, 459-478 (1998)*
## 3. PCA and ^{2} test: the formalism
A detailed description of the PCA technique can be found in Murtagh
& Heck (1987), and in Kendall (1980). Here we summarize its main
characteristics. The PCA is applied to a data set of *N* vectors
with *M* coordinates. In this *M* -dimensional space, each
object is a point and the sample forms a cloud of points. The central
problem which is solved by the PCA is the description of the cloud of
points by a set of P vectors of a new orthonormal base, with P
, and with a minimal
Euclidean distance from *each* point to the axes defined by the
new base. The eigenvectors of this new base are called principal
components (PC). Minimizing the sum of distances between spectra and
axes is equivalent to maximizing the sum of squared projections onto
axes, *i.e.,* maximizing the variance of the spectra when
projected onto these new axes.
The input for the PCA is a matrix of *N* spectra
*M* variables, which in our case are
spectral elements with 2 to 10 Å/pix, depending on the
resolution of the data. Each spectrum is
normalized by its norm (the square root of its scalar product with
itself), yielding the *N* normalized spectra
which serve as input to the PCA:
Other normalizations can be used (for example a flux
normalization), but it was shown by Connolly et al.
(1995) that the details of the normalization
applied to the input matrix do not have a
strong influence onto the PCA results. However, the interpretation of
the principal components does depend on the technique used to reduce
the input matrix. Because our input vectors are normalized by their
norm, we can apply the PCA onto the sum of squares and cross product
(intermediate) matrix (SSCP method), which does not rescale the data
nor center the data cloud. The normalized spectra then lie on the
surface of a *M* -dimensional hyper-sphere of radius 1, and the
first PC has the same direction as the average spectrum, but with norm
equal to 1. Two other procedures are based on the variance-covariance
matrix (VC method) and the correlation matrix (C method),
respectively. The VC method places the new origin onto the centroid of
the sample and the C method also re-scales the data in such a way that
the distance between variables is directly proportional to the
correlation between them. For the VC method the average spectrum has
to be used in order to reconstruct individual spectra. We emphasize
that neither the PC's nor the projections given by the SSCP method,
used in this paper, are the same as those given by the VC method.
However, if the normalized cloud of points is concentrated in a small
portion of the hyper-sphere, then the first PC of the VC method will
have almost the same direction as the second PC given by the SSCP
method (see Francis et al. 1992, Folkes et al. 1996). Although these
different methods give different PC's, if we take into account the
underlying transformations explained above, the physical
interpretation of the PC's and the projections does not change, and
the final result always satisfies the maximization conditions and the
orthonormality among the different principal components.
After application of the PCA using the SSCP method, we can write
each spectrum as
where is the reconstructed spectrum of
, is the projection of
spectrum onto the eigenspectrum
and is the number of
PC's taken into account for the reconstruction. In Eq. (2), the
PC's are in decreasing order of their contribution to the total
variance.
We show in Sect. 5 below, that if the S/N is high enough
(*i.e.,* 8), then we can take
= 3 or 4 to reconstruct
97 to 98% of the signal, respectively. If the S/N
8, it requires a higher number of PC's to
reproduce the initial spectrum to such high accuracy because of the
noise pattern. Therefore, the first 2 or 3 components carry most of
the signal in each spectrum, which leads us to use
, and
to describe the spectral sequence. We choose to
reduce these 3 parameters to the radius *r* and the angles
and defined by the
spectrum (as in Connolly et al. 1995) in
spherical coordinates ( the azimuth and
the polar angle taken from the equator),
(3*a*)
(3*b*)
(3*c*)
We express the values of and
*independently* of the value of
*r*:
(4*a*)
(4*b*)
Note that we prefer the use of and
(rather than the ratios
/ and /
) for defining the spectral sequence because
they have a geometrical meaning. In the next section, we show that the
physical meaning of is the relative
contribution of the red (or early) and the blue (or late) stellar
populations within a galaxy. Note that if , then
from Eq. (3c), Eq. (4b) approximates to
arcsin .
For comparison with the PCA, we have implemented a simple
test between the galaxies of the ESS sample and
a set of templates derived from the Kennicutt sample (Kennicutt 1992a,
see Sect. 5 and Sect. 6). In contrast to the PCA, the
test is dependent on the set of templates used
and can only provide a constrained classification procedure. The
between an observed spectrum and a template can
be written as
where and are the
values in the spectral element or bin *j* of the flux-calibrated
spectrum and the template, respectively. is the
total number of wavelength bins for both the spectrum and the template
(we take the largest wavelength interval common to the spectrum and
template, and rebin both to a common wavelength step of 5
Å/pix). The denominator measures the variance of the spectrum
and the template, assuming that the noise is Poissonian. Because for a
given observed spectrum is the same for all the
comparison templates, the value does not need
to be normalized. Therefore, if we have a set of *P* templates,
then the closest template *k* to the spectrum **S** is the one
which satisfies
Note that in the PCA treatment, the wavelength interval of all
input spectra must be identical. For the test,
the wavelength interval can be larger than the one used for the PCA
and varies from spectrum to spectrum. This difference will allow us to
check the dependence of the PCA classification on the wavelength
interval (cf. Sect. 6).
© European Southern Observatory (ESO) 1998
Online publication: March 23, 1998
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