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StatisticsUcla
Jan de Leeuw
Gifi Goes Logistic
2
Gifi goes Logistic
Abstract
The techniques in the Gifi system fordescriptive multivariate analysis arebased on least squares loss functions,alternating least squares algorithms,and star plots.
We develop an alternative system hereusing logistic likelihood functions,majorization algorithms, and Voronoiplots.
The Data
Suppose H={hij} is a data frame, i.e. a set of n
observations on m categorical variables. Variable
j has kj categories (values). There is no restriction
that n ! m.
The variables can have numerical, ordered, or
nominal categories, and they can be grouped into
sets of variables that have different roles in the
analysis (such as input-output, background,
predictors, confounders, outcomes, and so on).
3
Coding
Data are coded as m indicator matrices Gj, where
Gj is an n x kj binary matrix whose rows add up
to one.
There are extensions possible to missing data, in
which some rows add up to zero, and to fuzzy
indicators in which rows are non-negative and
add up to one but are not necessarily binary.
4
Star Plots
Suppose we have a representation (map) of the n
objects as n points in low-dimensional space
(usually the plane).
Make m copies of the map, one for each variable.
For each variable j connect all points in category
one to the centroid of category one, ... , all points
in category kj to the centroid of category kj. Thus
each map gets n lines, connecting the objects to
the centroid of the category they are in.
5
This plot is the star plot for variable j. We want
the kj stars in the plot to be small, relative to the
total size of the plot. Or: we want the within-
category variance to be small relative to the total
variance.
Thus the basic problem of homogeneity analysis:
make a map of the n objects such that the stars
are small (for each variable, if possible). In the
usual case we actually measure size by using
squared line length.
6
Homogeneity
A solution X for the objects is normalized if
X'X=I.
Heterogeneity of a normalized solution is
measured by
where
7
!(X) =1m
m!
j=1
tr (X !G jYj)"(X !G jYj),
Yj = (G!jG j)"1G!jX = D"1
j G!jX.
Alternating Least Squares
To minimize heterogeneity over normalized maps
we use alternating least squares or reciprocal
averaging.
8
Y (k)j = D!1
j G"jX(k),
Z(k)j =
1m
m!
j=1
G jY (k)j ,
X(k+1) = orth(Z(k)j ).
This is the Bauer-Rutishauser simultaneous
iteration method to compute the eigenvectors
corresponding to the p dominant eigenvalues of
The method capitalizes nicely on the sparseness
of the indicators and converges quickly and
reliable to the global minimum of the loss
function.
9
P! =1m
m!
j=1
Pj with Pj = G jD!1j G"j.
GALO Example
10
x
y
!0.02 0.00 0.02 0.04
!0.04
!0.02
0.00
0.02
11
!0.02 !0.01 0.00 0.01 0.02 0.03 0.04
!0.0
4!
0.0
20.0
00.0
2
Starplot for galo : gender
dimension 1
dim
ensio
n 2
FM
!0.02 !0.01 0.00 0.01 0.02 0.03 0.04
!0.0
4!
0.0
20.0
00.0
2
Starplot for galo : IQ
dimension 1
dim
ensio
n 2 1
2
3
4
5
6
7
8
9
!0.02 !0.01 0.00 0.01 0.02 0.03 0.04
!0.0
4!
0.0
20.0
00.0
2
Starplot for galo : advice
dimension 1
dim
ensio
n 2
Agr
Ext
GenGrls
Man
None
Uni
!0.02 !0.01 0.00 0.01 0.02 0.03 0.04
!0.0
4!
0.0
20.0
00.0
2
Starplot for galo : SES
dimension 1
dim
ensio
n 2 LoWC
MidWC
Prof
Shop
Skil
Unsk
12
Senate Example
13
!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
Cleland
Miller
Akaka
Inouye
Craig
Crapo
DurbinFitzgerald
BayhLugar
Grassley
Harkin
BrownbackRobertsBunning
McConnell
Breaux
Landrieu
CollinsSnowe
MikulskiSarbanes
KennedyKerry
LevinStabenowDaytonWellstone
Cochran
Lott
Bond Carnahan
Baucus
Burns
Hagel
Nelson1
EnsignReid
Gregg
Smith1
Corzine
TorricelliBingaman
Domenici
ClintonSchumerEdwards
Helms
ConradDorgan
DeWine
Voinovich
Inhofe
Nickles
Smith
Wyden
Santorum
Specter
Chafee
ReedHollings
Thurmond
Daschle
Johnson
Frist
Thompson
Gramm
Hutchison
BennettHatch
Jeffords
LeahyAllen
Warner
Cantwell
Murray
Byrd
RockefellerFeingold
Kohl
Enzi
Thomas
14
Cars Data
15
!0.2 !0.1 0.0 0.1 0.2
!0.2
!0.1
0.0
0.1
0.2
0.3
Object score plot for cars
dimension 1
dim
ensio
n 2
acura integra
daihatsu charade
dodge colt
eagle summit
ford escort
ford festiva
honda civic
hyundai excel!2hyundai excel!4
isuzu i!mark
mazda 323
mazda rx!7
mitsubishi mirage
mitsubishi starion
nissan pulsar nx
nissan sentra!4
nissan sentra!w
plymouth colt
pontiac lemans
subaru justy
toyota celica
toyota tercel
volkswagon golf
yugo gv
16
!0.15 !0.10 !0.05 0.00 0.05 0.10 0.15!
0.1
0.0
0.1
0.2
Hullplot for cars : body.style
dimension 1
dim
ensio
n 2
2!door
2!door
2!door
2!door
2!door
2!door
2!door
2!door"
4!door
4!door
4!door
wagon
wagon
!0.15 !0.10 !0.05 0.00 0.05 0.10 0.15
!0.1
0.0
0.1
0.2
Hullplot for cars : driver.protection
dimension 1
dim
ensio
n 2
certain injury
certain injury
certain injury
certain injury
fatal injury
fatal injury
fatal injury
fatal injurymoderate injury
moderate injury
moderate injury
no injury
no injuryno injury
no injury
severe injury
severe injury severe injury
!0.15 !0.10 !0.05 0.00 0.05 0.10 0.15
!0.1
0.0
0.1
0.2
Hullplot for cars : passenger.protection
dimension 1
dim
ensio
n 2
certain injury
certain injury
certain injury
moderate injury
moderate injury
moderate injury
moderate injury
no injury
no injury
no injury
no injury
no injury
no injury
!0.15 !0.10 !0.05 0.00 0.05 0.10 0.15
!0.1
0.0
0.1
0.2
Hullplot for cars : structural.integrity
dimension 1
dim
ensio
n 2
average
average
average
better
better
better better
better
better
better
much better
much better
much better
much better
much better
Two variables
If there are only two variables, then homogeneity
analysis becomes correspondence analysis. One
way of thinking about CA is making the
approximation
using a least squares loss function. Homogeneity
analysis with m > 2 makes a similar approximation
to the Burt table.
17
fi j ! !i" j(1 +p!
s=1
xisy js)
Rank and Level Restrictions
We see from the examples that order relations
between categories are not always respected, and
that in some cases maps are bend into somewhat
redundant horseshoes.
In the Gifi system these problems are resolved by
restricting the Yj in the loss function by the rank
restrictions
which requires the Yj to be on a line through the
origin.18
Yj = q ja!j,
Now for computation we use
and for interpretation we use
where R has the m columns Gjqj.
19
!(X,Y) =1m
tr (X !G jY j)"(X !G jY j)+
+1m
tr (Y j ! q ja"j)Dj(Yj ! q ja
"j).
!(X,Y) =1m
tr (X!G jq ja"j)"(X!G jq ja
"j) =
=1m
tr (R ! XA")"(R ! XA") + (p ! 1).
GALO Again
20
x
y
!0.02 0.00 0.02 0.04
!0.04
!0.02
0.00
0.02
0.04
21
!0.02 0.00 0.02 0.04
!0.0
20.0
00.0
20.0
4
Projection Plot for galo : gender
dimension 1
dim
ensio
n 2
F
M
!0.02 0.00 0.02 0.04
!0.0
20.0
00.0
20.0
4
Projection Plot for galo : IQ
dimension 1
dim
ensio
n 2
123 45
6
7
89
!0.02 0.00 0.02 0.04
!0.0
20.0
00.0
20.0
4
Projection Plot for galo : advice
dimension 1
dim
ensio
n 2
AgrExt
GenGrls
ManNone
Uni
!0.02 0.00 0.02 0.04
!0.0
20.0
00.0
20.0
4
Projection Plot for galo : SES
dimension 1
dim
ensio
n 2
LoWC
MidWC
Prof
Shop
Skil
Unsk
Interactive Coding and Additivity
If we have r variables with k1,...,kr categories
these can be coded as one variable with k1 x ... x
kr categories. We can then restrict the Y to be on a
grid
or on a rank-one grid
22
y j1,··· , jr ;s = y1j1 s + · · · + yr
jr s
y j1,··· , jr ;s = q1j1 a1
s + · · · + qrjr a
rs.
These additivity restrictions allow one to
efficiently incorporate sets of variables. If we
combine the predictors into a single set, for
instance, we have regression analysis; in the
same way we can have canonical analysis in its
various forms.
This allows one to incorporate much of classical
descriptive multivariate analysis, coupled with
the notion of optimal scaling or transformation.
23
GALO Again
24
x
y
!0.02 0.00 0.02 0.04 0.06
0.00
0.05
0.10
25
1.0 1.2 1.4 1.6 1.8 2.0
!0.006
!0.004
!0.002
0.000
0.002
0.004
0.006
Transformation plot for galo : gender
original
transformed
F
M
F
M
2 4 6 8
0.00
0.02
0.04
0.06
Transformation plot for galo : IQ
original
transformed
1 2 3
45
6
7
8
9
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7
!0.01
0.00
0.01
0.02
0.03
0.04
Transformation plot for galo : advice
original
transformed
Agr Ext
Gen Grls Man None
Uni
Agr Ext Gen Grls Man None Uni
1 2 3 4 5 6
0.00
0.02
0.04
0.06
0.08
0.10
Transformation plot for galo : SES
original
transformed
LoWCMidWC Prof Shop Skil Unsk
LoWC
MidWC Prof Shop
SkilUnsk
Summary
We have: flexible system with fast least squares
algorithms.
We have but may not want: we have to impose
orthogonality constraints to get multidimensional
solutions and they bring us horseshoes.
We do not have: interpretation on a natural
probability scale, notion of fitting a model, notion
of separation.
26
Gifi Goes Logistic
Instead of starting from
start from
with choices
27
fi j ! !i" j(1 +p!
s=1
xisy js)
fi j ! !i" j exp(#(xi, y j))
!(xi, y j) = x!i y j,
!(xi, y j) = "#xi " y j#,!(xi, y j) = "#xi " y j#2.
Instead of starting from least squares loss, start
from the negative Poisson log-likelihood
with
And instead of alternating least squares uses
majorization.
28
!(!, ", X,Y) =n!
i=1
m!
j=1
{#i j ! fi j log #i j},
!i j = "i# j exp($(xi, y j)).
In this context we use Uniform Quadratic
Majorization, which amounts to (lots of technical
detail omitted) minimizing (or continuously
decreasing) loss functions of the form
Here the target Z changes from one iteration to
the next, but all subproblems are standard least
squares multidimensional scaling problems.
29
!(", #, X,Y) =n!
i=1
m!
j=1
($(xi, y j) ! z(k)i j )2.
Rothkopf Morse Code Data
30
Now this just seems to generalize ordinary CA,
not homogeneity analysis. But now make the step
of applying the same loss function to indicator
matrices
where
31
!(!, ", X,Y) =n!
i=1
m!
j=1
k j!
#=1
{$i j# ! gi j# log $i j#},
!i j" = #i j$ j" exp(%(xi, y j")).
This brings us close to the basic Gifi setup, but
we need one final step.
except for some constants. And this is what we
hit with majorization, potentially using all the
Gifi restrictions on the Yj.
32
min!!(!, ", X,Y) =
=
n!
i=1
m!
j=1
k j!
#=1
gi j# log" j# exp($(xi, y j#))
"k j
%=1 " j% exp($(xi, y j%)),
GALO Again
33 34
!10 !5 0 5
!10
!5
05
10
Voronoi plot for galo : gender
dimension 1
dim
ensio
n 2 M
F
M
M
FM
FMM
MM
FFFF
F
MM
M
M
F
MM
MM F
F
M
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gender_F
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
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33 5
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!10 !5 0 5
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50
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Voronoi plot for galo : advice
dimension 1
dim
en
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n 2 Man
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!10 !5 0 5
!10
!5
05
10
Voronoi plot for galo : SES
dimension 1
dim
ensio
n 2 Shop
Skil
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SkilSkilSkil
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SES_LoWC
SES_MidWC
SES_Prof
SES_Shop
SES_Skil
SES_Unsk
35 36
!4 !2 0 2 4!
6!
4!
20
24
Voronoi plot for galo : gender
dimension 1
dim
ensio
n 2
M
F
MM
F
M
FMM
MM
FFFF
F
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M
F
MMM
M
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FM
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M
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M
M
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F
M
F
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MM
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FF
M
M
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MM
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F
MM
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F
M
F
M
MM
F
F
M
M
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M
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F
FM
M
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M
F
M
F
M
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M
M
M
F
M
F
F
M
F
MM
F
M
gender_F
gender_M
!4 !2 0 2 4
!6
!4
!2
02
4
Voronoi plot for galo : IQ
dimension 1
dim
ensio
n 2
4
7
56
5
4
6
25
65
4444
6
554
4
5
553
3
5
55
7
4
5
5
6
6
7
5
6
65
6
6
6 8
8
66
7
7
6
6
5
8
66
65
7
8
6
7
5 3
5
7
6
76
65 6
4 3 6
4
5
1
7
6
5
7
777
4
7
65
55
8
4
4
7
6
8
86
4
79
4
7
7
97
9
6
5
75
8
5 64
6
7
8
56
6
65
6
3
5
4
7
97
6
7
7
7
5
4
55 55
5
7
6
7
7
6
7
65
7
71
67
4
6
6
6
7
4
3 5
55
4
5
524
4
5
5
3
3
4
4
3
4
4
5
5
5
5
33
53 6
5
5
4
5
6
6
3
4
33
43 3
6
5
4
5
4
4
4
5
5
7
4
5
3
4
3
4
56
3
7
5
7
7
5
5
4
5
5
5
5
1 6
3
4
5
3
5
7
5
54
5
5
8
7
3
5
55 4
8
4
7
7
77
5
4
4
5
7
6
73
4
3
763
5
4
5
5
7
7
4
3
4
2
5
7
6
4
6
545 7
64
6
3
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4
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4
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3
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3
4
3
8
24
55
65 54
7
9
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5
6
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56
7
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4
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5
5
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4
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3
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4
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5
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5
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4
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5
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4 36
3
4
4
6
3
3
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3
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6
6
6
3
3
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4
6
57
4
5
6
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3
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5
64
5
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34
76
4 2
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6
3
5
5
2
4
44
56
7
7
53
4
6
5
6
8
85
55
9
7
6
5
5
65
35
6
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3
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5
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3
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4
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35
66
45
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4
73 6
8
3
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4
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37
5
5 6
5
8
3
3
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6
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3
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6
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4
7
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4
6
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6
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6
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6
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4
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5
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4
4
3
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63
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7
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4
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555
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4
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4
4
5
63
4
5
6
3
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5
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4
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4
65
4
3
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5
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6 3
4
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24
5
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5
65
6
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3
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2
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6
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3
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5
4
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4
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8
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5
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3
6
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5
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6
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7
4
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9
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3
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5
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7
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4
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6
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4
5
5
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4
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6
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2
5
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3
7
55
55
6
4
5
6
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6
4
5
5
6
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6
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4
4
53
6
6
4
7
5
3
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5
65
6
444
4
4
4
5
4
5
4
3
5
6
4
8
5
3
6
73
2
3
IQ_1IQ_2IQ_3IQ_4IQ_5
IQ_6IQ_7IQ_8IQ_9
!4 !2 0 2 4
!6
!4
!2
02
4
Voronoi plot for galo : IQ
dimension 1
dim
ensio
n 2
4
7
56
5
4
6
25
65
4444
6
554
4
5
553
3
5
55
7
4
5
5
6
6
7
5
6
65
6
6
6 8
8
66
7
7
6
6
5
8
66
65
7
8
6
7
5 3
5
7
6
76
65 6
4 3 6
4
5
1
7
6
5
7
777
4
7
65
55
8
4
4
7
6
8
86
4
79
4
7
7
97
9
6
5
75
8
5 64
6
7
8
56
6
65
6
3
5
4
7
97
6
7
7
7
5
4
55 55
5
7
6
7
7
6
7
65
7
71
67
4
6
6
6
7
4
3 5
55
4
5
524
4
5
5
3
3
4
4
3
4
4
5
5
5
5
33
53 6
5
5
4
5
6
6
3
4
33
43 3
6
5
4
5
4
4
4
5
5
7
4
5
3
4
3
4
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3
7
5
7
7
5
5
4
5
5
5
5
1 6
3
4
5
3
5
7
5
54
5
5
8
7
3
5
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8
4
7
7
77
5
4
4
5
7
6
73
4
3
763
5
4
5
5
7
7
4
3
4
2
5
7
6
4
6
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6
3
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6
4
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4
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4
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4
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IQ_1IQ_2IQ_3IQ_4IQ_5
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02
4
Voronoi plot for galo : SES
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Senate Again
37 38
!4 !3 !2 !1 0 1 2 3
!8
!6
!4
!2
02
4
dimension 1
dim
ensio
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
I did say earlier that the star plots are replaced by
Voronoi plots. This can be made more clear. Suppose
there are no weights and we use one of the distance
combination rules.
Or: if and only if each object can be placed closest to
the category point the object is in if and only if each
object is in the correct category Voronoi cell.
39
Loss can be made equal to zero if and only if there is asolution to the system of strict inequalities
!xi " y j!! < !xi " y j"! #i, j, ! $ gi j! = 1.
Summary
We have: an alternative system, which inherits
(and extends) the flexibility of the Gifi system. A
new geometrical interpretation, new convergent
algorithms, a likelihood interpretation.
We have but may not want: very heavy
computation, slow convergence, more clumping.
We do not have: experience with the new system.
40
