Introduction
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The following slides are intended to complement articles that I have been
publishing about dynamic graphics.
The “printed” articles have a selection of frames from the animations but on
this web site you can view the complete animation sequence.
All the frames have been calculated and drawn in R (www.r-project.org)
The frames have been put together into an animated GIF file using the
software Animation Shop.
Since the animations are fairly large files you might not see them
immediately. Also on the web they suffer minor problems such as “blank
flashes” – still to be resolved,
An alternative is to download the whole Powerpoint presentation file (an
even bigger file) and then watch the show offline, which is much better
quality and much faster, once everything is downloaded. The link to that file
is:
www.econ.upf.edu/~michael/animations/animations.pps
I hope you enjoy them!
Michael Greenacre
[email protected]
www.econ.upf.edu/~michael
IF YOU ARE USING A WEB
BROWSER TO VIEW THESE
ANIMATIONS TRY EXPLORER
RATHER THAN FIREFOX.
Statistical Animations
by Michael Greenacre
Greenacre, M.J. & Lewi, P. (2008).
Distributional Equivalence and Subcompositional Coherence in the
Analysis of Compositional Data, Contingency Tables and Ratio-Scale
Measurements
To appear in the Journal of Classification
In this paper we show that how unweighted log-ratio analysis (LRA) can
be improved by introducing differential weights for the “variables” of the
data matrix (these are components in a compositional data matrix).
When the weights are proportional to the marginal totals, this weighted
LRA is exactly the “spectral map” which Lewi defined almost 30 years
ago for analysing biological activity spectra.
In compositional data analysis this means that rare components will
receive less weight and the problems associated with low frequency
components (higher relative error, higher logratios) are downweighted.
We show one animation here, for a compositional data set from archeology
published by Baxter, Cool and Heyworth (1990, J. Appl. Statist.), where the
element Manganese (Mn) appears in very low concentration as an oxide.
Mn severely influences the unweighted LRA map – the weighted LRA map
is a great improvement. We also show the transition from correspondence
analysis (CA) to the weighted LRA map – there is a very small difference.
Unweighted to
weighted
logratio
analysis (LRA),
Baxter et al
compositional
data on Roman
glass cups
The map starts with
the huge influence of
Mn, which only takes
on three different
(and small) values in
the data set, but as
the weights are
introduced this effect
is phased out in
favour of seeing
other more
interesting aspects
of the data. Notice
that in reality the
data have much
lower variance – the
variance was
originally inflated by
the effect of the rare
components such as
manganese (Mn)
Correspondence
analysis (CA) to
weighted LRA,
Baxter et al
compositional
data on Roman
glass cups
Here there is only a
small difference – it
is known that when
the data have low
variance, as in this
example, the CA
weighted LRA
solutions will be
similar. Both in turn
resemble the leastsquares solution of
Goodman’s RC
association model, a
bilinear model
defined originally for
contingency tables.
Pardo, R. and Greenacre, M.J. (2008).
Positioning the "middle" categories in survey research: a
multidimensional
From keynote address at the European Association of Methodology’s
biennial conference in Oviedo, Spain, July 2008.
In this talk we looked at questionnaire data and the position of the “middle”
response categories (e.g., “neither agree nor disagree” on a 5-point
bipolar scale) across a number of questions.
To compare what we observe in real data with what we would expect in an
idealized situation where there was a single underlying response gradient,
with the middle categories perfectly “between” agree and disagree, we
show the multiple correspondence analysis (MCA) of simulated data.
Animation is used here to show the configuration in three dimensions,
where in the first two dimensions the category points form a parabola, the
well-known “arch effect” in CA, while with respect to axes one and three
the configuration becomes a cubic.
Rotating a threedimensional
MCA solution
Here we see the
quadratic
relationship of axis 2
with respect to axis 1
changing to a cubic
one as we rotate
around the first axis
to gradually bring
into view the third
dimension. These
data were generated
according to aperfect
underlying onedimensional
response gradient,
for which we can
prove the polynomial
relationships
illustrated here
between the
coordinates
Greenacre, M.J. (2008).
Power transformations in correspondence analysis.
To appear in the Special Issue of Correspondence Analysis and Related
methods, Computational Statistics and Data Analysis
In this paper I show how power transformations in correspondence
analysis (CA) have as a limiting case the method of logratio analysis
(LRA).
A straightforward powering of the original data to a power  followed by
the application of CA with the rescaling of the singular values by 1/
tends to unweighted LRA as  tends to 0. In this case the row and
column margins depend on  and tend to constants (hence the
“unweighted”...) .
A powering of the contingency ratios, keeping the row and column
margins fixed, and applying the usual CA algorithm, again with the final
rescaling by 1/, tends to weighted LRA as  tends to 0.
The transition from CA to LRA is illustrated with two data sets: the MN
population genetic data set, and the author data.
Figure 1: CA to weighted LRA, MN genetic data
Here we clearly see how the power transformation opens up the arch in the CA and at the limit the
configuration is almost one-dimensional. In fact, a perfectly linear configuration of the M, MN and N
genotypes in the logratio analysis would imply Hardy-Weinberg equilibrium.
Figure 2: CA to weighted LRA for the author data
This example has very little inertia. The difference between CA and LRA will be very small in
this case, as shown by Greenacre & Lewi (2005, to appear in Journal of Classification, 2008)
Figure 3: CA (chi-square distance) to Hellinger distance: MN data
Even though this is a high inertia example, the transition to Hellinger distance is very slight
(notice that the power drops to only ½ here, which is the Hellinger analysis)
Figure 4: CA to NSCA: author data
There is almost no
difference! (again this is
because of the very small
inertia in these data)
Greenacre, M.J. (2008).
Dynamic graphics of parametrically linked multivariate methods used in
compositional data analysis
Paper presented at the 3rd International Workshop on Compositional
Data Analysis, June 2008, Girona , Spain
You can get a PDF of this paper at:
http://www.econ.upf.es/en/research/onepaper.php?id=1082
where there are some dynamic graphics embedded in the file.
Notice, however, that this does not work on all platforms – we are trying
to ascertain exactly why this occurs..
In addition to this paper, I presented an animation of the logratio analysis
of a large compositional data set (known as “Darssil”) where the large
number of zeros in the data were replaced by 0.1 and then in decreasing
steps of 0.001, i.e. 0.099, 0.098, until 0.001. This shows graphically
where the zero-replacement strategy starts to break down. This is
shown on the next slides, first for unweighted LRA, then weighted LRA.
Zeroreplacements
in Darssil data
from 0.1 to
0.001 in
decreasing
steps of
0.001:
unweighted
LRA
breakdown
starts to occur
about 0.01 (we
could estimate
this point more
precisely by
studying the
rate of change
of the
configuration
according to the
Procrustes
statistics, for
example)
There is almost no
difference! (again this is
because of the very small
inertia in these data)
Zeroreplacements
in Darssil data
from 0.1 to
0.001 in
decreasing
steps of
0.001:
weighted LRA
notice that
weighted LRA is
more stable
than the
unweighted
form on the
previous slide
and breakdown
starts to occur
much later,
when the zeroreplacement
value is much
closer to 0