multivariate data frames - ucla statisticsdeleeuw/janspubs/2006/reports/deleeuw_r_06e.pdf · the...
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StatisticsUcla
Jan de Leeuw
Geometric Representation
of
Multivariate Data Frames
2
We discuss two classes of drawing methods for multivariate categoricaldata. Both are inspired by multidimensional scaling, and are intimatelylinked to the notion that similarity in the data is naturally represented asdistance in a low-dimensional Euclidean space. The objects that are mea-sured, or categorized, by our variables are represented as points. Eachvariable defines a partition of the points into subsets corresponding withthe values of the variable.
The first class of methods are the clumping methods, that try to representthe objects with the same values on a variable by small compact sub-sets of space. Since there are many ways to measure the size of a pointset, there are many clumping methods. The second class are separationmethods, which try to construct smooth surfaces from some parametricfamily to separate points having di!erent values on the variable.
Clumping and separation methods can be implemented using either leastsquares or likelihood based algorithms, which define the two main waysto measure and minimize badness-of-fit.
1. Data
Data are n observations on m categorical
variables. Variable j takes kj different values,
where 1 ! kj ! n. Numerical variables are just a
special case.
There are two basic ideas in the techniques we
discuss. The n objects are represented as points in
low-dimensional Euclidean space in such a way
that objects with similar profiles (values on the
variables) are relatively close (in some sense).
3
Somewhat more specifically, if two objects have
the same value on a variable, then that should
make them more close, if they have the same
values on all variables, then they should map into
the same point.
One easy way to portray the results of an analysis
is to make m copies of the plot of the n objects
and label them by the values (categories) of the
variable.
4
2. Clumping
In this class of techniques we use a measure of
the size of a cloud of points in Suppose X are
the coordinates of the points representing the n
objects. Variable j defines a partition into kj
subsets, which have measures
We also define a measure which measures
the size of the cloud of all n points.
5
Rp.
! j1(X), · · · ,! j,k j (X).
!(X),
It is now easy to define a loss function by the
simple rule
although other ways of combining the size
measures into a single numerical loss value are
certainly possible.
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!(X) =!m
j=1!k j
"=1 # j"(X)#(X)
,
A clumping technique finds n points xi ! Rp
such that !(X) is minimized.
Many measures of size have been proposed, and
some have actually been studied from a
computational points of view.
We mention the edge-length of the minimal spanning
tree, the circumference of the convex hull, the radius
of the circular or elliptical hull, the sum of the
distances to the Weber point, the maximum distance
between two points.
Anybody is free to suggest their own measures of
homogeneity. That's the easy part. However,
implementating good algorithms is quite another
matter.7
We discuss one particular choice of size in more
detail. Each point set with k points has a star,
which is the set of lines connecting the k points
with their centroid (average). The size of the star
is the sum of squares of the k lines.
The loss function associated with this definition
of size defines multiple correspondence analysis
(or homogeneity analysis). It is closely related
(but mathematically far simpler) than the Weber
correspondence analysis we mentioned before.
8
This can be expressed in matrix notation, which
already suggests why this is a fortunate choice.
Let G be the n x k binary indicator matrix (a.k.a.
dummy) indicating which objects belong to which
categories. Also D = G'G. Then Y=D-1G'X are
the k centroids, and
is the sum of the sizes (squared line lengths) of
the k stars.
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!(X) = tr (X !GY)"(X !GY)
The clumping technique we derive from measuring
star size (with squared distances) minimizes
over all centered configurations
X such that This clumping technique is
due to Louis Guttman in the early forties.
The optimization problem turns out to be an
eigenvalue problem for the matrix
where
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!1(X) + · · ·!m(X)tr X!X = 1.
P! =1m
m!
j=1
Pj,
Pj = G jD!1j G"j.
Alternatively we can define
And solve the generalized eigenvalue problem
where C=G'G and D=diag(C).
In fact, what I just said is not quite correct. We
have to require X'X=I instead of tr(X'X)=1. Then
X corresponds to the eigenvectors corresponding
to the p largest eigenvalues (except for the largest
one, which is always 1 and corresponds to a
column of ones).
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G = (G1| · · · |Gm).
Cy = m!Dy,
The resulting technique (MCA or homals) is
computationally very efficient. In the Gifi
(1990) system MCA is generalized to minimizing
over both X and Y with various restrictions on the Y.
Some of the other clumping techniques (size
measures) have been tried, and they generally have
failed.12
!(X; Y1, · · · ,Ym) =
=1m
m!
j=1
tr (X !G jYj)"Mj(X !G jYj)
Two problems with MCA is that it is somewhat
inelegant that we have to use a normalization to
force multidimensional solutions.
Moreover requiring orthogonality produces
horseshoes (for well-understood reasons).
Subsequent dimensions are quadratic, cubic, ...
functions of the first dimension.
To some extent we can get around this by
restricting the Yj, for instance requiring them to
be of rank one. This encourages the centroids to
be on straight lines through the origin.13
The GALO Data
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2
3
!0.02 0.00 0.02 0.04
!0.0
6!
0.0
4!
0.0
20.0
00.0
2
Voronoi plot for galo : advice
dimension 1
dim
ensio
n 2
ManMan
Man
Man
ManMan
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Ext
Man
Gen
Man
ManManManMan
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ManManManMan
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Man
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ManMan
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Man
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Man
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Man
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Man
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ManMan
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Man
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Gen
Gen
Man
Gen
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Ext
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ManMan
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Man
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Gen
Man
Gen
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Uni
Man
Gen
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Man
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GenGen
Gen
Man
ManManMan
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Ext
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Man
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ManMan
Ext
Man
Man
Man
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UniMan
Gen
Ext
Gen
Man
Man
Uni
Man Man
Gen
Gen
ManManManMan
Man
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Man
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Man
Man
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ExtNone
Ext
!0.02 0.00 0.02 0.04
!0.0
6!
0.0
4!
0.0
20.0
00.0
2
Voronoi plot for galo : SES
dimension 1
dim
ensio
n 2
ShopSkil
Unsk
Shop
SkilProf
Prof
Skil
Skil
MidWC
Shop
SkilSkilSkilSkil
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ShopShopMidWCShop
MidWCShopShop
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Shop
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Shop
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Shop
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MidWC
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ShopShopShop
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Prof
Skil
Skil
ShopShop
Shop
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ShopShop
Shop
Prof
Shop
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SkilSkil
Skil
Skil
LoWC
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Skil
Skil
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Skil
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ShopShop
MidWC
MidWC
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Shop
ProfShop
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SkilSkil
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MidWC
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ShopSkilShop
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16
17
!0.02 0.00 0.02 0.04
!0.0
4!
0.0
20.0
00.0
20.0
4
Voronoi plot for galo : gender
dimension 1
dim
ensio
n 2
M
F
M M
F
M
FM
M
MM
FFFF
F
MM
M
M
F
MM
M
M
F
F
M FF
M
M
M
M
MM
M
M
F
M
M
F
F
F
MMF
M
F
F
M
F
MM
MM
M
F
M
M
M
M
M
M
F
M
M
MM
MMM M
M
F
M
F
F
M
F
F
FF
F
F
MM
F
F
F
M
F
M
F
M
M
M
M
F
F
M
M
F
F
F
F
M
F F
F
FMF
M
F
F
M
F
F
M
F
F
F
M
M
M
F
M
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4
Voronoi plot for galo : IQ
dimension 1
dim
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n 2
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!0.02 0.00 0.02 0.04
!0.0
4!
0.0
20.0
00.0
20.0
4
Voronoi plot for galo : advice
dimension 1
dim
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n 2
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4
Voronoi plot for galo : SES
dimension 1
dim
ensio
n 2
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The Senate Data
18
19
!0.03 !0.02 !0.01 0.00 0.01 0.02 0.03
!0
.06
!0
.04
!0
.02
0.0
00
.02
0.0
4
Object score plot for senate
dimension 1
dim
en
sio
n 2
Sessions
ShelbyMurkowski
Stevens
Kyl
McCain
Hutchinson
Lincoln
Boxer
Feinstein
Allard
Campbell
Dodd
Lieberman
Biden
Carper
GrahamNelson
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Nelson1
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Thurmond
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3. Separation
A different starting point is not to concentrate on
making certain sets of object points small in size,
but on locating the points in space in such a way
that they can be separated by simple parametric
curves or surfaces.
Suppose, for instance, we have a variable with k
ordered categories (values). We want to locate the
n object points in, say, the plane such that the k
subsets can be separated by parallel lines, i.e. the
subsets must be in parallel strips.20
Or, alternatively, suppose we have a binary
variable taking values ''aye'' and ''nay''. We want
to locate the n points xi in the plane such that
there is a circle with all the "aye" points in the
circle and all the "nay" points outside the circle.
Separation techniques have been around since the
mid-sixties, and they are direct descendants of
the Shepard-Kruskal approach to non-metric
multidimensional scaling (1965). Although there
is Guttman (1940) and Coombs (1950). Because
of this ancestry the original separation techniques
mostly use least squares loss functions.21
We will go another way here, because we want to
measure fit (loss) on the probability scale. This
dispenses with the need for normalization, and it
may get rid of the horseshoes.
Consider the loss function
22
!(X,Y) =
= !m!
i=1
m!
j=1
k j!
!=1
gi j! logexp(!"xi ! y j!"2)
"k j
"=1 exp(!"xi ! y j""2).
Let's interpret optimizing this loss as a geometric
problem. If we can find X and Y such that
or, in words, such that each object is closest to
the category the object is in, then we can get
arbitrarily close to a solution with loss equal to
zero.
So one way to think of the method is to find an
approximate solution to a system of nonlinear
inequalities (Motzkin, Agmon, 1950)23
gi j! = 1! "xi # y j!" < "xi # y j"" $" ! !,
Where are the separations ? Well, if we let the
categories of each variable define Voronoi cells
in then the inequalities say that objects
should be in the correct Voronoi cells (for all
variables). Voronoi cells provide a non-
parametric system of separations.
As with MCA we can make this parametric by
constraining the Yj. Requiring them to be of rank
one, for instance makes the Voronoi cells into
strips bounded by parallel lines or planes.
24
Rp,
GALO Data
25 26
!10 !5 0 5
!10
!5
05
10
Voronoi plot for galo : gender
dimension 1
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gender_M
!10 !5 0 5
!10
!5
05
10
Voronoi plot for galo : IQ
dimension 1
dim
ensio
n 2 4
7
5
6
54
625
65
4444
6
55
4
4
5
55
33 5
5
5
7
4
55
6
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6
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6
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63
5
4
7
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7
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55 55
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5
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4
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5
36
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3
2
3IQ_1
IQ_2IQ_3
IQ_4 IQ_5IQ_6
IQ_7IQ_8IQ_9
!10 !5 0 5
!10
!5
05
10
Voronoi plot for galo : advice
dimension 1
dim
ensio
n 2 Man
Man
Man
Man
ManMan
GenExtMan
GenMan
ManManManMan
Gen
ManMan
Man
Man
Man
ManMan
ExtMan Man
Man
Man
Gen
Man
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GenGrls
GenGrls
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Man
Man
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Gen
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Man
Man
Man
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Man
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Man
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Gen
Gen
Gen
Man
Man
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ManExt
Gen
Gen
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Man
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Man Grls
Man
Gen
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Man
Gen
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Man
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Man
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Man
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Gen
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Man
Man
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Man
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Man
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Man
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Man
Man
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Man
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Man
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None GenMan
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Gen
Man
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Gen
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Gen
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Gen
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Man
Gen
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Man
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Man
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Gen
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Ext
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Man
Man
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Gen
Man
Gen
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Gen
Gen
Gen
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Gen
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Gen
ExtUni
Man
Man
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Man
ManUni
Man
Gen
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Gen
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Gen
Man
Ext
ExtGenGenGen
Man
Man
Man
Man
Gen
Gen
GenAgr
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Ext
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Ext
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Gen
Ext
Gen
Man
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Man
Man
Gen
Gen
Man
Man
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Man Gen
Gen
ManGen
Man
Man
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Man
Man
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Ext
None
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advice_Agr
advice_Ext
advice_Gen
advice_Grls
advice_Manadvice_None
advice_Uni
!10 !5 0 5
!10
!5
05
10
Voronoi plot for galo : SES
dimension 1
dim
ensio
n 2 Shop
Skil
Unsk
Shop
SkilProf
ProfSkilSkil
MidWCShop
SkilSkilSkilSkil
Skil
ShopShop
MidWC
Shop
MidWC
ShopShop
MidWCSkil Skil
LoWC
Skil
Prof
Shop
MidWCSkil
Shop
SkilMidWC
Shop
Shop
SkilMidWC
MidWC
Shop
ShopMidWC
Prof
ShopShop
ShopMidWC
LoWC
Shop
Shop
Prof
Shop MidWCShopShopShopShop
SkilShopShop
MidWC
Skil
ShopShop
ShopShopShopShop
Prof
MidWCMidWCSkil
Shop
Skil
Shop
SkilSkil
Shop
LoWC
ShopShopShopMidWC
Skil
ShopShop
ProfShopShop
Shop
Skil
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The Senate Data
27 28
!4 !3 !2 !1 0 1 2 3
!8
!6
!4
!2
02
4
dimension 1
dim
en
sio
n 2
Row Objects senate
Sessions
ShelbyMurkowski Stevens
Kyl
McCain
Hutchinson Lincoln
Boxer
Feinstein
Allard
Campbell
Dodd
Lieberman
Biden
Carper
GrahamNelson
ClelandMiller
Akaka
InouyeCraig
Crapo
Durbin
FitzgeraldBayh
LugarGrassley
Harkin
BrownbackRobertsBunning McConnell
Breaux
LandrieuCollinsSnowe
MikulskiSarbanes
KennedyKerryLevinStabenowDaytonWellstoneCochranLott
Bond
CarnahanBaucus
Burns
Hagel
Nelson1
Ensign
ReidGregg Smith1
Corzine
Torricelli
BingamanDomenici
ClintonSchumerEdwardsHelms ConradDorgan
DeWineVoinovichInhofe
Nickles
Smith WydenSantorum
Specter
Chafee
Reed
HollingsThurmond
DaschleJohnsonFrist
ThompsonGramm
HutchisonBennettHatchJeffords
LeahyAllenWarner Cantwell
Murray
Byrd
RockefellerFeingold
KohlEnzi
Thomas
Conclusions
Separation techniques became computationally
feasible only very recently. They use complicated
majorization algorithms.
They provide an interesting alternative to the
much more familiar (much less expensive, and
much better understood) least squares techniques.
http://www.cuddyvalley.org/psychoR
29