Stat 315c: Transposable Data Correspondence Analysis

Stat 315c: Transposable DataCorrespondence Analysis

Art B. Owen

Stanford Statistics

Art B. Owen (Stanford Statistics) Correspondence Analysis 1 / 17

Correspondence Analysis

It plots both variables and cases in the same plane.

Clearest motivation is for contingency table data. It gets usedelsewhere too.

Emphasis is on presenting the data themselves as opposed toilluminating an underlying model.

This is an old and classical statistical technique pioneered byJean-Paul Benzecri in the 1960s.

The treatment by Greenacre is particularly clear.

Contingency tables

I × J table of counts

n11 n12 · · · n1J

n21 n22 · · · n2J...

.... . .

...nI1 nI2 · · · nIJ

Nomenclature

Correspondence matrix P pij = nij/n••

Row masses ri = pi• = ni•/n••

Column masses cj = pi• = ni•/n••

Row profiles ri = (pi1/ri, . . . , piJ/ri)′ ∈ RJ

Column profiles cj = (p1j/cj , . . . , pIj/cj)′ ∈ RI

These are conditional and marginal distributions

Contingency tables

n11 n12 · · · n1J

n21 n22 · · · n2J...

.... . .

...nI1 nI2 · · · nIJ

Nomenclature

Contingency tables

n11 n12 · · · n1J

n21 n22 · · · n2J...

.... . .

...nI1 nI2 · · · nIJ

Nomenclature

First moments: centroids

Row centroid

I∑i=1

riri =I∑

ri, . . . ,

)′= (c1, . . . , cJ)′ ≡ c

Column centroid

J∑j=1

cjcj = (r1, . . . , rI)′ ≡ r

Upshot

‘Mass’ weighted average of row profiles is marginal distribution overcolumns

Row centroid

I∑i=1

riri =I∑

ri, . . . ,

)′= (c1, . . . , cJ)′ ≡ c

Column centroid

J∑j=1

cjcj = (r1, . . . , rI)′ ≡ r

Upshot

Row centroid

I∑i=1

riri =I∑

ri, . . . ,

)′= (c1, . . . , cJ)′ ≡ c

Column centroid

J∑j=1

cjcj = (r1, . . . , rI)′ ≡ r

Upshot

Second moments: inertias

Chisquare for independence as weighted Euclidean distance

X2 =∑

(nij − ni•n•j/n••)2

ni•n•j/n••

(nij/ni• − n•j/n••)2

n•j/n••

= n••

n••

n•j/n••

= n••

ri(ri − c)′diag(c)−1(ri − c)

= n•• × Inertia

This is the total inertia of the row profiles. It equals total inertia ofcolumn profiles.

Second moments: inertias

Chisquare for independence as weighted Euclidean distance

X2 =∑

(nij − ni•n•j/n••)2

ni•n•j/n••

n•j/n••

= n••

n••

n•j/n••

= n••

ri(ri − c)′diag(c)−1(ri − c)

= n•• × Inertia

This is the total inertia of the row profiles. It equals total inertia ofcolumn profiles.

Geometry

For J = 3

12 8 87 7 68 8 106 9 89 8 7We can plot profiles in R3

Low inertia

Geometry

This example has higher inertia

Geometry

Still higher inertia.

χ2 statistics describes variationof row profiles

Similarly for col profiles

Rescale

Distances

Euclidean distance in plot ignores column values

Replace ri by ri with rij =rij√cj

Euclidean dist between ri and ri′ is “χ2 dist” between ri and ri′ .

Rescale

Distances

Euclidean distance in plot ignores column values

Replace ri by ri with rij =rij√cj

Euclidean dist between ri and ri′ is “χ2 dist” between ri and ri′ .

Reason for χ2

Invariance

Suppose rows i and i′ are proportional

nij/ni′j = α all j = 1, . . . , J

Suppose also that we pool these rows

New ni∗j = nij + ni′j

and delete originals

New χ2 distance between cols j and j′ equals old dist

Principle of distributional equivalence

Common profile, summed mass

Role of χ2 in statistical significance is not considered important in thisliterature

Reason for χ2

Invariance

nij/ni′j = α all j = 1, . . . , J

Reason for χ2

Invariance

nij/ni′j = α all j = 1, . . . , J

Dimension reduction

Now we have a plot

With rows and cols both in RJ−1

If J is too big

reduce dimension

by principal components of

rij − cj√cj

plot in reduced dimension

along with images of corners

Dimension reduction

Now we have a plot

With rows and cols both in RJ−1

If J is too big

reduce dimension

by principal components of

rij − cj√cj

plot in reduced dimension

along with images of corners

Duality

Rows lie in min(I − 1, J − 1) dimensional space

So do columns

In PC of row profiles . . . columns are outside

In PC of column profiles . . . rows are outside

Symmetric correspondence analysis overlap the points after rescaling

More notation

Dr = diag(r) = diag(r1, . . . , rI)

Dc = diag(c) = diag(c1, . . . , cJ)

Duality

Rows lie in min(I − 1, J − 1) dimensional space

So do columns

In PC of row profiles . . . columns are outside

In PC of column profiles . . . rows are outside

Symmetric correspondence analysis overlap the points after rescaling

More notation

Dr = diag(r) = diag(r1, . . . , rI)

Dc = diag(c) = diag(c1, . . . , cJ)

Symmetric analysis

Uses SVD S = UΣV ′ where

sij =pij − ricj√

Total inertia is ‖S‖2F

’principal inertias’ are λ2j

Coordinates

Rows (1st k cols of)

F = (D−1r P − 1c′)D−1

c V1:k = D−1r UΣ

Columns (1st k cols of)

G = (D−1c P − 1r′)D−1

r U1:k = D−1c V Σ′

Symmetric analysis

Uses SVD S = UΣV ′ where

sij =pij − ricj√

Total inertia is ‖S‖2F

’principal inertias’ are λ2j

Coordinates

Rows (1st k cols of)

F = (D−1r P − 1c′)D−1

c V1:k = D−1r UΣ

Columns (1st k cols of)

G = (D−1c P − 1r′)D−1

r U1:k = D−1c V Σ′

Symmetric analysis

Interpretation is tricky/controversial

ri near ri′√

cj near cj′√

ri near cj ??Rows and columns are not in the same space

Biplots

Due to Gabriel (1971) Biometrika

For matrix Xij

plot rows as ui ∈ R2

cols as vj ∈ R2

with u′ivj.= Xij

A biplot interpretation applies to asymmetric plots

Some finer points

Ghost points

Apply projection to point not in table

E.G. hypothetical row entity,1 impute a president’s ’senate voting record’2 compare a state’s economy to those of countries

Treat as fixed profile with mass ↓ 0

Merged points

Add linear combination or sum of rows, E.G.1 pool columns for math and statistics into “math sciences”2 pool rows for EU countries into an EU point

Some finer points

Ghost points

Apply projection to point not in table

E.G. hypothetical row entity,1 impute a president’s ’senate voting record’2 compare a state’s economy to those of countries

Treat as fixed profile with mass ↓ 0

Merged points

Add linear combination or sum of rows, E.G.1 pool columns for math and statistics into “math sciences”2 pool rows for EU countries into an EU point

Data types

Counts are straightforward

Other ’near measures’ are reasonableI rainfalls, heights, volumes, temperatures KelvinI dollars spentI parts per million

Reweight cols to equalize inertia ≈ standardizing to equalize varianceRequires iteration

Fuzzy coding

x ∈ R becomes two columns

(1, 0) for small x say x < L

(0, 1) for large x say x > U

(1− t, t) for intermediate x t = (x− L)/(U − L)

Generalizations to > 2 columns

Data types

Counts are straightforward

Other ’near measures’ are reasonableI rainfalls, heights, volumes, temperatures KelvinI dollars spentI parts per million

Reweight cols to equalize inertia ≈ standardizing to equalize varianceRequires iteration

Fuzzy coding

x ∈ R becomes two columns

(1, 0) for small x say x < L

(0, 1) for large x say x > U

(1− t, t) for intermediate x t = (x− L)/(U − L)

Generalizations to > 2 columns

Puzzlers

Does it scale? (eg 108 points in the plane)

Is there a tensor version? (Beyond all pairs of two way versions)

Distributional equivalence vs Poisson models

Stat 315c: Transposable Data Correspondence Analysis

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