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Introduction to Principal Components Analysis 1
Paul J. Francis
Department of Physics & Theoretical Physics, Australian National
University, Canberra, Australia; pfrancis@mso.anu.edu.au
Beverley J. Wills
McDonald Observatory & Astronomy Department, University of Texas
at Austin, TX, USA 78712; bev@astro.as.utexas.edu
Abstract.
Understanding the inverse equivalent width { luminosity relationship
(Baldwin Eect), the topic of this meeting, requires extracting informa-
tion on continuum and emission line parameters from samples of AGN.
We wish to discover whether, and how, dierent subsets of measured
parameters may correlate with each other. This general problem is the
domain of Principal Components Analysis (PCA). We discuss the pur-
pose, principles, and the interpretation of PCA, using some examples
from QSO spectroscopy. The hope is that identication of relationships
among subsets of correlated variables may lead to new physical insight.
1. Introduction
The point of all statistics is simplication. The amount of data the world can
throw at us would swamp Einstein: we have to simplify to survive. Statistics is
the art of extracting simple comprehensible facts that tell us what we want to
know for practical reasons, from the oods of data washing over us.
Consider the fuel consumption of cars, for example. Every car will be
dierent, depending on its model, year, maintenance state, and the aggression
level of its driver. To fully characterize the fuel economy of cars in the USA would
require a dierent number for every car/driver combination: that is, more than
10 8 numbers. For most purposes, however, such as working out the nation's
likely oil usage, these 10 8 numbers can be replaced with one: the average fuel
consumption. An enormous simplication!
Principal Components Analysis (PCA) is a tool for simplifying one partic-
ular class of data. Imagine that you have n objects (where n is large), and you
can measure p parameters for each of them (where p is also large). For example,
the objects could be the n QSO researchers attending a meeting in La Serena,
and the parameters could be the p things you know about each of them: e.g.,
1 This paper is for our friends Leah Cutter and Mike Brotherton, who announced their engage-
ment during the La Serena meeting.
1

their heights, weights, number of publications, frequent ier miles and the fuel
consumption of their cars. Let us imagine that you want to investigate how
these p parameters are related to each other. For example, do astronomers who
spend most of their lives in airports publish more? Does this depend on how fat
they are? Do people with ineôcient cars y more, or is it only the smart ones
(with lots of publications) that do so? Do these correlations represent real causal
connections, or it it just that once you get tenure you buy a new car, become
fat, stop publishing and give lots of invited talks in exotic foreign locations?
The traditional way of dealing with this type of problem is to plot everything
against everything else and look for correlations. Unfortunately, as the number
of parameters increases, this rapidly becomes impossibly complicated. It is easy
to get lost in the web of parameters, each of which correlates more or less well
with some combination of the other parameters. The human brain can cope with
two or three parameters easily. By plotting all the dierent variables against
each other separately, we can just about learn something about 5 { 7 variables.
But once we are beyond this, the human brain needs help.
PCA is specically designed to help in situations like this: when you know
lots of things about lots of objects, and want to see how all these properties are
inter-related. Basically, PCA looks for sets of parameters that always correlate
together. By grouping these into one new parameter, an enormous saving in
complexity can be achieved with minimal loss of information.
PCA is one of a family of algorithms (known as multivariate statistics)
designed to handle complex problems of this sort. It was rst widely applied
in the social sciences. The most infamous early application of PCA was to
intelligence testing. You can test the intellectual ability of people in many
ways. For example, you could give a sample of n people a set of p exams, with
questions testing their creativity, memory, math skills, verbal skills etc. Do
people who score well on one test score well on all? Or do the scores break
up into sub-groups, such as verbal or logical scores, which correlate well with
the scores on other similar tests? PCA was applied to these exercises, and it
was found that nearly all the scores correlate well with each other. Thus, it was
claimed, a single underlying variable (known as IQ) can be used to replace all the
individual scores, and once you know someone's IQ, you can accurately predict
their performance on all the tests. (See Steven Jay Gould's `The Mismeasure of
Man' for a hilarious account of the misuse of this application of PCA.)
2. Overview
The task of PCA is then, given a sample of n objects with p measured quantities
for each, i.e. p variables, x j (j = 1, . . . , p), nd a set of p new, orthogonal (i.e.
independent) variables,  1 ; : : : ;  i ; : : : ;  p , each one a linear combination of the
original variables, x j :
 i = a i1 x 1 +


+ a ij x j +


+ a ip x p
Determine the constants a ij such that the smallest number of new variables
account for as much of the variance of the sample as possible. The  i are called
principal components.
2

If most of the variance in the original data can be accounted for by just a few
of the p new variables, we will have found a simpler description of the original
dataset. A smaller number of variables may point to a way of classifying the
data. More interesting, beyond the realm of statistical description, the PCA, by
showing which original variables correlate together, may lead to new physical
insight. Of course it will sometimes happen that the observed variables are
uncorrelated, or at least, lead to no dominant principal components. That may
be useful to know, but not very interesting.
The concept of PCA is usually introduced either algebraically, through co-
variance matrices, or geometrically. We will rst give a geometrical overview,
then illustrate with examples and interpretation. Many textbooks on multivari-
ate statistics give rigorous mathematical treatments (e.g., Kendall 1980).
3. A Geometrical Approach to Principal Components Analysis
Consider the case of p variables. The data of n QSOs are represented by a
large cloud in p-dimensional space. If two or more parameters are correlated,
the cloud will be elongated along some direction in hyper-space dened by their
axes. Large extensions can arise when a few parameters are correlated, or when
smaller correlated variations occur for a substantial number of variables.
PCA identies these extended directions and uses them as a set of axes
for the parameterization of the multidimemsional space. Following the analysis,
each QSO can be represented by its coordinates in the new space. The new axes
are identied sequentially: PCA rst nds the most extended direction in the
original p-dimensional space by minimizing the sums of squares of the deviations
from that direction. This direction forms the rst principal component (often
called eigenvector 1), and accounts for the largest single linear variation among
measured QSO properties. Next we consider the (p 1)-dimensional hyper-plane
orthogonal to the rst principal component. We then search for the direction
that represents the greatest variance in (p 1)-space, thus dening the second
principal component. This process is continued, dening a total of p orthogonal
directions.
4. Examples using Real Data
4.1. PCA with Two Variables
Consider the case of 22 QSOs each with measured values of X-ray spectral
index  x (dened by F  /  x between 0.15 keV and 2keV), and FWHM
H (full width at half maximum for the broad H emission line). The data
points are distributed in an elongated cloud in 2 dimensions, as shown in Fig.
1. It is standard practice to subtract the mean value from each variable, and
normalize by dividing by the standard deviation. One can nd the direction of
the rst principal component axis by rotating an axis to align with the direction
of maximum elongation, actually maximum variance, of the data. The result
of this is shown by the dashed line labeled PC1 in Fig. 2. Because the points
remain the same distance from the origin, by Pythagoras' theorem, maximizing
the variance along PC1 is equivalent to minimizing the sums of squares of the
3

Figure 1. An important optical correlation, soft X-ray spectral in-
dex vs. width of the broad H emission line. Left: In natural units.
Right: In normalized units, with mean subtracted, then divided by the
standard deviation. The dashed line shows the direction of the rst
principal component (PC1), representing the maximum deviation of
the cloud of data points. Dotted lines project the data points onto this
direction. PC1 represents the direction that minimizes the sums of the
squares of the lengths of the dotted lines. The value (score) of PC1
for a given point is the distance of the point from the origin, projected
onto PC1. Similarly the lengths of the dotted lines represent the values
of PC2 for each data point.
distances of the points from this line through the origin; these distances are
shown as dotted lines. The distance of a point from the origin, projected onto
the direction PC1 represents the value (score) of the rst principal component
for that data point. Clearly, PC1 is a linear combination of the original input
variables. The variance of PC1 is 1.764 { rather more than the unit variance
of the original variables. The total variance of the sample is the sum of the
variances for each variable, in this case, 2. Because the new coordinates are
found simply by rotation, the distance of the points from the origin remains
unchanged, so the total variance remains the same. Thus the rst principal
component accounts for 1.764/2 = 88.2% of the total sample variance. The
remaining variance of the sample can be accounted for by the projection of the
data points onto the axis PC2, perpendicular to PC1 { or 0.236/2 = 11.8%.
These projections (lengths of the dotted lines) are the values or scores of the
second principal component.
We have succeeded in dening a new variable, a linear combination of  x
and log FWHM H, that accounts for most of the variation within the sample
(PC1). The interpretation of this parameter is a hotly contended topic (e.g.,
Pounds, Done & Osborne 1995, Laor et al. 1997, Brandt & Boller 1998). Is
PC2 of any signicance? The astronomer, with knowledge of the measurement
uncertainties, may have more hope of addressing this. If the original variables
4

had been uncorrelated, we could still dene PC1 and PC2 mathematically, but
we would be no better o as a result of the analysis.
4.2. PCA with More Variables
Table 1. Input Data
PG Name log  x log FWHM FeII/ log EW log FWHM
L 1216
a H H [OIII] CIII]
0947+396 45.66 1.51 3.684 0.23 1.18 3.520
0953+414 45.83 1.57 3.496 0.25 1.26 3.432
1001+054 44.93 . . . 3.241 0.82 0.85 3.424
1114+445 44.99 0.88 3.660 0.20 1.23 3.654
1115+407 45.41 1.89 3.236 0.54 0.78 3.403
1116+215 46.00 1.73 3.465 0.47 1.00 3.446
1202+281 44.77 1.22 3.703 0.29 1.56 3.434
1216+069 46.03 1.36 3.715 0.20 1.00 3.514
1226+023 46.74 0.94 3.547 0.57 0.70 3.477
1309+355 45.55 1.51 3.468 0.28 1.28 3.406
1322+659 45.42 1.69 3.446 0.59 0.90 3.351
1352+183 45.34 1.52 3.556 0.46 1.00 3.548
1402+261 45.74 1.93 3.281 1.23 0.30 3.229
1411+442 44.93 1.97 3.427 0.49 1.18 3.275
1415+451 45.08 1.74 3.418 1.25 0.30 3.434
1425+267 45.72 0.94 3.974 0.11 1.56 3.666
1427+480 45.54 1.41 3.405 0.36 1.76 3.300
1440+356 45.23 2.08 3.161 1.19 1.00 3.192
1444+407 45.92 1.91 3.394 1.45 0.30 3.479
1512+370 46.04 1.21 3.833 0.16 1.76 3.546
1543+489 46.02 2.11 3.193 0.85 0.00 . . .
1626+554 45.48 1.94 3.652 0.32 0.95 3.631
Number 22 21 22 22 22 21
Mean 45.56 1.57 3.498 0.56 0.99 3.446
Std dev'n 0.47 0.38 0.212 0.40 0.47 0.129
a Log of continuum luminosity at 1216  A in units of erg s 1
(Ho = 50 km s 1 Mpc 1 , qo = 0:5.)
FWHM are in km s 1 ; rest-frame equivalent widths (EW) are in  A.
PCA achieves its real usefulness in multivariate problems. We perform a
PCA 1 on the small sample of 22 QSOs discussed by Wills et al. (1998a,b),
using a subset of the available measured properties shown in Tables 1 and 2.
Unavoidably, there are missing data, so the number of objects available depends
1 Several widely available statistical packages include a task for Principal Components Analyses
(Statistical Package for the Social Sciences { SSPS, Statistical Analysis System { SAS, Minitab
{ Minitab Reference Manual 1992).
5

on the variables chosen for the PCA. Many correlations are plotted, and the
Pearson correlation coeôcients tabulated, by Wills et al. (this volume).
Table 2. Input Data, continued
PG Name log EW log EW CIV/ log EW SiIII/ NV/ 1400/
Ly CIV Ly CIII] CIII] Ly Ly
0947+396 2.08 1.78 0.45 1.24 0.306 0.179 0.143
0953+414 2.19 1.78 0.40 1.24 0.164 0.189 0.093
1001+054 2.25 1.76 0.40 1.43 0.443 0.462 0.174
1114+445 2.27 1.85 0.42 1.48 0.222 0.175 0.092
1115+407 1.90 1.51 0.33 1.14 0.385 0.228 0.134
1116+215 2.14 1.71 0.34 1.20 0.440 0.254 0.126
1202+281 2.72 2.41 0.69 1.87 0.164 0.154 0.098
1216+069 2.12 1.95 0.54 1.20 0.037 0.121 0.056
1226+023 1.64 1.44 0.45 1.00 0.280 0.174 0.018
1309+355 2.01 1.68 0.41 1.15 0.303 0.131 0.064
1322+659 2.19 1.85 0.41 1.30 0.291 0.135 0.097
1352+183 2.14 1.80 0.41 1.29 0.357 0.203 0.116
1402+261 1.91 1.59 0.39 1.09 0.568 0.227 0.161
1411+442 . . . 1.88 . . . 1,42 0.314 . . . 0.093
1415+451 2.32 1.78 0.29 1.40 0.688 0.210 0.142
1425+267 . . . 2.17 . . . 1.43 0.398 . . . 0.055
1427+480 2.03 1.82 0.49 1.21 0.265 0.126 0.117
1440+356 2.14 1.54 0.21 1.05 0.747 0.141 0.092
1444+407 1.99 1.34 0.21 1.06 0.809 0.335 0.164
1512+370 2.02 2.05 0.75 1.28 0.228 0.182 0.050
1543+489 1.93 1.60 0.44 . . . . . . 0.398 0.174
1626+554 2.14 1.80 0.39 1.36 0.197 0.217 0.118
Number 20 22 20 21 21 20 22
Mean 2.11 1.78 0.421 1.279 0.362 0.212 0.108
Std dev'n 0.21 0.24 0.131 0.194 0.199 0.091 0.043
Tables 1 and 2 also present, for each variable, the number of data points,
the mean and the standard deviation. Notice the completely dierent units
for dierent measured parameters. Clearly, from our two dimensional example,
one can see in Fig. 1 or 2 that the deviations from PC1, hence PC1 itself,
will depend on the units chosen (the weighting of the variables). In order to
weight the variables more or less equally, after subtracting the mean values, we
normalize by the variance. The choice of weights is a diôcult issue, and depends
on the user's knowledge of the data, and preferences, as well as the use to which
the results will be put. The results of performing a PCA on these normalized
variables are shown in Table 3. Columns (2){(6) show the rst 5 out of a total of
13 principal components. The rst row gives the variances (eigenvalues) of the
data along the direction of the corresponding principal component. The sums
of all the variances add up to the sums of the variances of the input variables,
6

in this case, 13. By convention, the principal components are given in order
of their contribution to the total variance. This is given as `Proportion' in the
second line, and the `Cumulative' proportion on the third line. Thus, among
the parameters we have chosen to use, the rst principal component contributes
50% of the spectrum-to-spectrum variance, the second 22%, the third, 12%. The
rst two principal components together contribute 71% of the variance, the rst
3, 84%, and the rst 4, nearly 90%.
Table 3. Results of Eigenanalysis { The Principal Components a
PC1 PC2 PC3 PC4 PC5
Eigenvalue 6.4505 2.8157 1.5879 0.6257 0.5698
Proportion 0.496 0.217 0.122 0.048 0.044
Cumulative 0.496 0.713 0.835 0.883 0.927
Variable PC1 PC2 PC3 PC4 PC5
log L 1216 0.053 0.535 0.123 0.029 0.405
 x 0.295 0.198 0.079 0.485 0.155
FWHM H 0.330 0.077 0.357 0.082 0.141
Fe II/H 0.341 0.140 0.003 0.487 0.212
log EW [O III] 0.310 0.016 0.255 0.394 0.095
log FWHM C III] 0.198 0.077 0.623 0.054 0.402
log EW Ly 0.177 0.502 0.006 0.143 0.033
log EW C IV 0.336 0.262 0.048 0.050 0.303
C IV/Ly 0.342 0.062 0.025 0.074 0.584
log EW C III] 0.262 0.413 0.124 0.176 0.008
Si III]/C III] 0.342 0.149 0.018 0.311 0.116
NV/Ly 0.231 0.050 0.573 0.107 0.288
1400/Ly 0.223 0.351 0.225 0.441 0.216
a 18 of 22 QSO spectra used; 4 cases contain missing values.
The columns of numbers for each principal component represent the weights
assigned to each input variable. Thus PC1 = 0:053x 1 +0:295x 2 0:330x 3 +
. . . , where x 1
, x 2
, and x 3
are the values of the normalized variables corresponding
to log L 1216 ,  x , FWMH H, etc. By convention these weights are chosen so
that the sum of their squares = 1. This arbitrarily xes the scale of the new
variable. The sign of the new variable is therefore arbitrary.
4.3. Interpretation
The rst principal component is elongated with variance 6.5 times that of any
individual measurements, and accounts for about half the total variance. This
is therefore likely to be highly signicant. If all measured, normalized quantities
contributed equally to PC1, they would all have weight 0.277 (1=
p
13 for 13
variables), but each variable contributes more or less than this. One way to test
the signicance of the contribution of any one measured variable, is to perform
7

the PCA without that variable, then check the signicance of the correlation
between that variable and the scores of the new principal component. This
procedure shows that all measured variables except L 1216
, log FWHM CIII],
and log EW Ly, correlate with PC1, but correlations involving NV/Ly and
1400/Ly are not very strong. PC2, accounting for 22% of the variance in this
dataset, appears to link the EW Ly, EW CIV, and EW CIII] with L 1216 , so
EW CIV and EW CIII] appear to contribute to both PC1 and PC2, but EW
Ly contributes predominantly to PC2. Is PC2 a signicant component? A
similar correlation test shows that individually the EWs do anti-correlate with
L 1216
, but this result depends on the lowest EWs for the highest luminosity QSO
PG1226+023 and the highest EWs for the low luminosity QSO PG1202+281.
However L 1216 correlates signicantly (Pearson's ordinary correlation coeôcient
= 0:77) with PC2 formed when L 1216 is excluded. Thus there is a signicant
overall correlation between EW and L 1216
, although a larger sample is clearly
needed to investigate the individual EW correlations. Another test may be to
check correlations between observed measurements for those measurements that
contribute to only one signicant principal component { for example, CIV/Ly
vs. Fe II/H (see Fig. and Table of Wills et al. in this volume).
As a rule-of-thumb, any principal component with variance greater than
1, should be considered seriously. It is also worth investigating any principal
component with variance rather greater than that of the remaining principal
components. In our example, this could mean the rst three principal compo-
nents.
PCA is a linear analysis. Tests should be performed to check on the lin-
earity of the principal components. If a linear analysis is valid, plotting the
scores of PC1 vs. PC2 should show a normal distribution consistent with no
correlation between the two. Mathematically, there cannot be a correlation, but
a non-random distribution of points, or individual outlying points, may indicate
non-linearity of the relationships { or some other problem with the uniformity
of the data-set. Outliers could be rejected and the analysis repeated, or a trans-
form of co-ordinates, for example to logarithmic co-ordinates, may reduce the
problem to a linear analysis. A PCA performed using the ranks rather than
the actual (normalized) measurements may be more robust to both non-random
distributions and outliers. (Compare the present results with those from the
anaylsis of the ranks, in Table 2 of Ch. n, this volume.) These tests are an
important tool for examining non-linearities in the data, and for discovering
individual unusual objects.
5. Some Examples from the Literature
Increasing awareness of statistical methods has led to the establishment of the
Statistical Consulting Center For Astronomy at Penn State University (Akritas
et al. 1997, Feigelson et al. 1995, see also http://www.stat.psu.edu/scca/ and
www.astro.psu.edu/statcodes), and a series of conference and other volumes
devoted to statistics in astronomy (Murtagh & Heck 1987; Feigelson, Babu, &
Jogesh 1992), including PCA.
PCA is being increasingly applied in astrophysics. Investigations of low and
high redshift galaxies depend on their classication (by morphology, photome-
8

try, kinematics, etc.), in terms of the purely observational \fundamental plane",
a subspace of the p-dimensional parameter space (Djorgovski & Davis 1987, see
also an interesting PCA paper by M. Han 1995). The same area of astronomy
has also extensively applied `neural network' techniques for the automated clas-
sication of galaxy images, and more (Odewahn 1998; Rawson, Bailey & Francis
1996.)
An example similar to that presented here, using a subset of the param-
eters we consider, but for a much larger sample, is provided in the paper by
Boroson & Green (1992). Other examples of PCA analyses are given and dis-
cussed by Whitney (1983a, b), and Murtagh and Heck (1987). Some PCAs
have a larger number of variables than input observables, p > n. This results
in a singular matrix and therefore requires modications to the techniques to
solve the eigenvector equations. These techniques are discussed, for example,
by Wilkinson (1978), and mentioned by Mittaz, Penston, & Snijders (1990).
This situation occurs in `Spectral PCA'. The principles are identical, but the
number of variables is larger than the number of QSO spectra. Here the QSO
spectra are divided into many discrete bins, by wavelength or log (wavelength)
(or velocity), and the p variables are the uxes in these p bins. An excellent
example and discussion of interpretation is given by Francis et al. (1992) 2 . For
another example, see Wills, Brotherton, Wills & Thompson (1997). Spectral
PCA also nds application to spectral time variability. For example, Mittaz et
al. (1990) analyze the spectra of NGC 4151 at 59 epochs, binning each spectrum
in wavelength space (1375 bins). A more recent example is given by Turler &
Courvoisier (1998).
Recommended for further reading, is chapter 6 from Manly's `Multivariate
Statistical Methods' (1994), which gives a good brief discussion of the method,
with useful insights into interpretation. A more rigorous mathematical treat-
ment, together with discussion, is given by the great researcher and expositor of
statistics M. Kendall (Chapters 1 and 2 of `Multivariate Analysis'.)
Acknowledgments. We gratefully acknowledge many discussions with M.
S. Brotherton (Lawrence Livermore National Laboratory, Livermore, CA). B.J.W.
thanks Kris Berg, Dirk Grupe, and Sprout Berg for friendship and a great deal
of patience. We also thank M. Cornell and R. Wilhelm who provide computer
support in the Astronomy Department of the University of Texas. B.J.W. is
supported by NASA through LTSA grant number NAG5-3431 and grant num-
ber GO-06781 from the Space Telescope Science Institute, which is operated by
the Association of Universities for Research in Astronomy, Inc., under NASA
contract NAS5-26555. We used MINITAB.
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