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THE (GRAPHICAL) REPRESENTATION OF PROXIMITY INFORMATION: DESCRIPTIVE
DATA ANALYSIS RUN AMUCK

\bigskip

Lawrence Hubert

The University of Illinois

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The MATLAB M-files used in these examples (other than the one taken
from the Statistics Toolbox for multidimensional scaling) are
available at:

http://cda.psych.uiuc.edu/srpm\_mfiles

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\begin{slide}
I might begin with a little story.  I was at a conference recently
and was listening to a very opaque and rather confused paper
presentation by one of my quantitative colleagues.  After the paper
was finished, an obviously more substantively-oriented listener
sitting next to me, commented that he and his colleagues generally
had the ability to share their research interests in presentations.
He wondered why quantitative people often appeared only able to
inflict theirs on others.

Well, hopefully I'll share more than I inflict today --- but if you like, we could take a vote when I'm done.

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To begin, it seems that for quite a long time now, most of my
research interests have centered around data analysis strategies for
what we might generically call proximity data or proximity
information.  Proximity merely refers to a collection of numerical
information we have about the relationship between whatever objects
or things we are interested in.

Also, I have pursued a path of making \mbox{MATLAB} M-files
(open-source and freely available) to implement everything being
developed.  So, all the analyses we will discuss here are repeatable
easily by the listener (with access to MATLAB or to a free clone
like OCTAVE; plus in one case, the MATLAB Statistics Toolbox).   The
place to get all these M-files was noted on the title slide.


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OUTLINE

A) A very brief review of what proximities are and where they may
come from ---

B) The general problem of attempting to represent (at least in an
approximate manner) the information contained in proximity data by
some formal structure intended to help explain the patterning of the
original data ---

C) Review five such formal structures in the context of a specific
example (that deals with proximities between facial expressions of
emotion):

1) Multidimensional scaling (representation by distances from
placements in both Euclidean and city-block spaces)

2) Hierarchical clustering (representation through a sequence of nested partitions)

3) Additive-tree analysis (representation by distances from a placement in a specific tree-like graphical structure)

4) Unidimensional scaling, both Linear (representation by distances
from a placement along a line) and Circular (representation by
distances from a placement around a closed continuum)

5) Imposition of an anti-Robinson pattern (representation by fitting
a certain gradient structure to the data matrix)

D) Possible extensions (and for which analysis routines are presently
available in the way of M-files; for the present, however, we will not review these
extensions):


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1) The use of multiple (additive) structures for representation

2) Alternative formal representation structures, including those
that incorporate a double-centering operation (through a centroid
component)

3) Proximity data defined between two distinct sets of objects (two-mode as opposed to one-mode)

4) The inclusion of proximity data transformations (e.g., monotonic,
or more constrained forms, such as convex, concave, or some of both)

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Given some set of objects of interest --- people, stimuli,
variables, situations, and the like --- the phrase ``proximity data'' merely
refers to numerically specified information about the relation
between each pair of objects.

Usually organized as an $n \times n$ matrix, say, $\mathbf{P} = \{p_{ij}\}$, where $p_{ij}$ denotes the relationship between objects
$i$ and $j$ (which is symmetric, so, $p_{ij} = p_{ji}$).

Where do proximities come from?

A) Direct measures (or judgements) of proximity ---


pair comparisons: subjects judge the similarity of object pairs (stimuli), possibly according to some specific type of similarity

same/different judgements: percentage of ``same'' judgements for pairs of stimuli, or latencies for ``same'' judgements

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\begin{slide}
sorting: subjects partition objects into groups according to
similarity, with proximity then defined by object-pair co-occurrence
frequencies

joint occurrence frequencies: proportion of times that object pairs
co-occur together (over sites, conditions, and so on)

interaction measures: degrees of communication or flow between objects

confusion measures: frequencies with which objects are confused with one another

B) Indirect (or derived) measures ---

Given initial object by variable (attribute) data:

profile dissimilarity (distance measures) between objects

correlation or other measures of association between variables



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GENERAL REPRESENTATION TASK

Given a proximity matrix $\mathbf{P}$, find some matrix, say
$\mathbf{P}^*$, that is:

(a) ``close'' to $\mathbf{P}$ (captures [a major amount of] the information present in $\mathbf{P}$

(b) the entries in  $\mathbf{P}^*$ have some particularly convenient
structure that can be represented formally by some (graphical)
mechanism

Thus, to explain what is going on in $\mathbf{P}$, we use
$\mathbf{P}^*$, and the graphical mechanism it induces.

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The criterion of ``closeness'' we use is consistently least-squares;
thus we seek to minimize \[ \sum_{i,j} (p_{ij} - p_{ij}^{*})^2 \ ,
\] by the choice of  $\mathbf{P}^{*}$.  Also, as a measure of ``fit'' adequacy we use the
variance-accounted-for (vaf) criterion of \[ \mathrm{vaf} = 1 -
\frac{\sum_{i,j} (p_{ij} - p_{ij}^{*})^2 }{\sum_{i,j} (p_{ij} -
\bar{p})^2 } \ , \] where $\bar{p}$ is the mean of the off-diagonal
entries in $\mathbf{P}$.

Analogies abound --- e.g., to explain some single dependent measure,
choose a multiple regression equation based on a collection of
independent variables; interpret what is ``going on'' from the
weights on the independent variables. ``Closeness'' here is also
least-squares.



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An Example for Illustration:

To make all of this a little more concrete, it might be useful if I
introduce an example at this point, that I can carry through during
the rest of the talk.


Some very early research dating back to the 1930s on the psychology
of expression was occupied with the question of whether subjects
could correctly identify an intended emotional message from a
person's facial expression.  It was found that the errors in the
rather extensive number of misinterpretations were not random, and
the emotion perceived was usually ``psychologically similar'' to the
emotion expressed by the sender.  A number of individuals,
particularly Schlosberg, attempted to develop a theory of the
differentiability of facial expressions, concluded that three
perceptual ``dimensions''  were needed for a meaningful
classification: pleasant/unpleasant, attention/rejection,
tension/sleep.

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Over a variety of different studies, subjects could fairly reliably
classify facial expressions within this system.  Although subjects
may be able to classify facial expressions according to the
Schlosberg scales when made explicit, it is still uncertain whether
judges that are uninstructed use these particular ``dimensions'' in
making judgements about facial expression --- or possibly would use
others.

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\begin{flushleft}

\begin{tabular}{llll}



 Scene (Lightfoot series) &   PU  &  AR &  TS  \\ [2ex]

   1) [7] Grief at death of mother &  3.8 & 4.2  &  4.1  \\

  2) [13] Savoring a coke & 5.9   & 5.4   &  4.8  \\

  3) [15] Very pleasant surprise & 8.8   & 7.8   & 7.1  \\

  4) [16] Maternal love --- & 7.0  & 5.9  & 4.0  \\
baby in arms & & & \\
 5)[20] Physical exhaustion  & 3.3  & 2.5  & 3.1  \\

 6) [28] Something wrong   &  3.5 & 6.1  & 6.8  \\

 with plane & & & \\

 7)  [29] Anger seeing dog beaten  & 2.1  &  8.0 & 8.2 \\

 8) [30]  Pulling hard on seat  & 6.7  & 4.2 & 6.6  \\
 of chair & & & \\
 9) [32] Unexpectedly meets  & 7.4  & 6.8  & 5.9  \\
old boyfriend & & & \\
 10)[36]  Revulsion  & 2.9  & 3.0  & 5.1  \\

 11) [37] Extreme pain  & 2.2  & 2.2  & 6.4  \\

 12) [51] Knows plane will crash  & 1.1  & 8.6  & 8.9  \\

13) [56]  Light sleep  & 4.1  & 1.3  & 1.0  \\



    \end{tabular}

    \end{flushleft}

    \bigskip

    Note: the entries in brackets, [ ], indicate the original scene
    number in the Lightfoot series.


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    PU: pleasant/unpleasant;

    AR: attention/rejection;

    TS: tension/sleep

    (Schlosberg Scale Values are from Engen, Levy, \& Schlosberg,
    1958, \emph{JEP}, 454--458 --- empirical averages on a 9-point
    scale over a group of subjects)

    Correlations:

    PU vs. AR: .18

    PU vs. TS: $-$.15

    AR vs. TS: .75




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MULTIDIMENSIONAL SCALING

Given the proximity matrix $\mathbf{P}$, find a new matrix
$\mathbf{P}^*$ that is ``close'' to $\mathbf{P}$ and the entries in
$\mathbf{P}^*$ are (a linear transformation of) Euclidean or
city-block distances (between the objects [faces] placed in some
$K$-dimensional space).

For now, we use the MATLAB Statistical Toolbox routine,
\verb+mdscale.m+, with \verb+Criterion+ set to \verb+metricstress+
for the Euclidean scaling; for the city-block alternative, we use my
routine, \verb+biscalqa.m+.

Formally, if $x_{1}, \ldots, x_{n}$ and $y_{1}, \ldots, y_{n}$ denote the coordinates in two dimensions, then the distances between objects $i$ and $j$ are

city-block: $d_{ij} = |x_{j} - x_{i}| + |y_{j} - y_{i}|$

euclidean:  $d_{ij} = \sqrt{(x_{j} - x_{i})^2 + (y_{j} - y_{i})^2}$



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HIERARCHICAL CLUSTERING

Given the proximity matrix $\mathbf{P}$, find a new matrix
$\mathbf{P}^*$ that is ``close'' to $\mathbf{P}$ (in a least squares
sense), \emph{and} the entries in $\mathbf{P}^*$ satisfy the
ultrametric property:  for any three objects [faces] $i$, $j$, and
$k$, among the three corresponding entries in $\mathbf{P}^*$,
$p_{ij}^*$,  $p_{ik}^*$, and $p_{jk}^*$, the largest two must be
equal.

Or less intuitively, we have the trio inequality: $p_{ij} \le
\max\{p_{ik},p_{jk}\}$.

The MATLAB routine used is called \verb+ultrafnd.m+

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ADDITIVE TREE ANALYSIS

Given the proximity matrix $\mathbf{P}$, find a new matrix
$\mathbf{P}^*$ that is ``close'' to $\mathbf{P}$ \emph{and} the
entries in $\mathbf{P}^*$ satisfy the additive-tree property: for
any four objects [faces], $i$, $j$, $k$, and $h$, among the three sums,
$p_{ij}^* + p_{kh}^*$, \ $p_{ik}^* + p_{jh}^*$, and $p_{ih}^* +
p_{jk}^*$, the two largest must be equal.

Or, to keep the musical motif going, we have the less intuitive
quartet inequality: $p_{ij}^* + p_{kh}^* \le \max\{p_{ij}^* +
p_{kh}^*, \  p_{ij}^* + p_{kh}^*\}$.

Generally, in representing an additive tree graphically, each branch
represents the common feature of those ``below''  (the progeny); the
branch length indicates the importance of this group of common
features (all within a Tverskian notion of common and distinctive
features).

The MATLAB routine used is called \verb+atreefnd.m+

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An alternative view of additive trees represents them (very
non-uniquely) as an ultrametric plus a centroid ``metric'' matrix. A
(symmetric ) centroid matrix, say, $\mathbf{C} = \{c_{ij}\}$, has
main-diagonal entries, $c_{ii} = 0$, for $ 1 \le i \le n$, and
off-diagonal entries ($i \ne j$), $c_{ij} = g_{i} + g_{j}$, for some
set of values, $g_{1}, \ldots, g_{n}$.  Because some  $g_{1},
\ldots, g_{n}$ may be negative (and lead to negative entries in
$\mathbf{C}$), we put the word ``metric'' in quotes.

A (closed-form) least-squares approximation to $\mathbf{P}$ by a
centroid ``metric'', in effect double-centers the residual matrix
(so row and column sums are zero).

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UNIDIMENSIONAL SCALING

Linear (LUS):

Find a set of coordinates, $x_{1}, \ldots, x_{n}$, to minimize \[
\sum_{i < j} (p_{ij} - \{ |x_{j} - x_{i}| - c \})^2 \ ,\] where $c$
is an additional additive constant to be estimated; here $c$ could be considered part of the model bing fitted, as we suggest above; or alternatively, we could interpret the proximities as being translated, i.e., $p_{ij} + c$ is fit by $|x_{j} - x_{i}|$.

Once an appropriate order is obtained, the coordinate estimation is
immediate.  (The MATLAB routines we use are called \verb+order.m+
(to generate an appropriate object order), and \verb+linfitac.m+ (to estimate $c$
and the coordinates based on the found order.)

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Circular (CUS):

Find a set of coordinates, $x_{1}, \ldots, x_{n}$, and an $(n +
1)$st value, $x_{0}$ (the circumference of a circular structure), $x_{0} \ge
|x_{j} - x_{i}|$ for all $ 1 \le i \ne j \le n$, minimizing \[
\sum_{i < j} (p_{ij} - [ \min \{|x_{j} - x_{i}|, x_{0} - |x_{j} -
x_{i}| \} - c ])^2 \ , \] where $c$ is again an additive constant to
be estimated. (The MATLAB routine we use is called
\verb+unicirac.m+.)

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SCALING A MATRIX TO BE IN ANTI-ROBINSON FORM

Given the proximity matrix $\mathbf{P}$, find a new matrix
$\mathbf{P}^*$ that is ``close'' to $\mathbf{P}$ \emph{and} the
entries in $\mathbf{P}^*$ satisfy  the anti-Robinson property: there
is some reordering of the rows and columns of $\mathbf{P}^*$ so that
the entries within each row and within each column never decrease in
moving away from the main diagonal.

In other words, we have a regular gradient present both within the
rows and within the columns.

The $n(n-1)/2$ subsets defined by the choice of endpoints for an
interval in the given order used to demonstrate the anti-Robinson form,
along with their subset diameters (as measures of salience), can be
used to ``explain'' the gradient (at least hopefully).

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