<s>
Correspondence	B-Algorithm
analysis	I-Algorithm
(	O
CA	O
)	O
is	O
a	O
multivariate	O
statistical	O
technique	O
proposed	O
by	O
Herman	O
Otto	O
Hartley	O
(	O
Hirschfeld	O
)	O
and	O
later	O
developed	O
by	O
Jean-Paul	O
Benzécri	O
.	O
</s>
<s>
It	O
is	O
conceptually	O
similar	O
to	O
principal	B-Application
component	I-Application
analysis	I-Application
,	O
but	O
applies	O
to	O
categorical	O
rather	O
than	O
continuous	O
data	O
.	O
</s>
<s>
In	O
a	O
similar	O
manner	O
to	O
principal	B-Application
component	I-Application
analysis	I-Application
,	O
it	O
provides	O
a	O
means	O
of	O
displaying	O
or	O
summarising	O
a	O
set	O
of	O
data	O
in	O
two-dimensional	O
graphical	O
form	O
.	O
</s>
<s>
Its	O
aim	O
is	O
to	O
display	O
in	O
a	O
biplot	B-Application
any	O
structure	O
hidden	O
in	O
the	O
multivariate	O
setting	O
of	O
the	O
data	O
table	O
.	O
</s>
<s>
As	O
such	O
it	O
is	O
a	O
technique	O
from	O
the	O
field	O
of	O
multivariate	O
ordination	B-Algorithm
.	O
</s>
<s>
Since	O
the	O
variant	O
of	O
CA	O
described	O
here	O
can	O
be	O
applied	O
either	O
with	O
a	O
focus	O
on	O
the	O
rows	O
or	O
on	O
the	O
columns	O
it	O
should	O
in	O
fact	O
be	O
called	O
simple	O
(	O
symmetric	O
)	O
correspondence	B-Algorithm
analysis	I-Algorithm
.	O
</s>
<s>
It	O
is	O
traditionally	O
applied	O
to	O
the	O
contingency	B-Application
table	I-Application
of	O
a	O
pair	O
of	O
nominal	O
variables	O
where	O
each	O
cell	O
contains	O
either	O
a	O
count	O
or	O
a	O
zero	O
value	O
.	O
</s>
<s>
If	O
more	O
than	O
two	O
categorical	O
variables	O
are	O
to	O
be	O
summarized	O
,	O
a	O
variant	O
called	O
multiple	B-Algorithm
correspondence	I-Algorithm
analysis	I-Algorithm
should	O
be	O
chosen	O
instead	O
.	O
</s>
<s>
Depending	O
on	O
the	O
scores	O
used	O
CA	O
preserves	O
the	O
chi-square	B-General_Concept
distance	O
between	O
either	O
the	O
rows	O
or	O
the	O
columns	O
of	O
the	O
table	O
.	O
</s>
<s>
Because	O
CA	O
is	O
a	O
descriptive	O
technique	O
,	O
it	O
can	O
be	O
applied	O
to	O
tables	O
regardless	O
of	O
a	O
significant	O
chisquared	B-General_Concept
test	I-General_Concept
.	O
</s>
<s>
Although	O
the	O
statistic	O
used	O
in	O
inferential	O
statistics	O
and	O
the	O
chi-square	B-General_Concept
distance	O
are	O
computationally	O
related	O
they	O
should	O
not	O
be	O
confused	O
since	O
the	O
latter	O
works	O
as	O
a	O
multivariate	O
statistical	O
distance	O
measure	O
in	O
CA	O
while	O
the	O
statistic	O
is	O
in	O
fact	O
a	O
scalar	O
not	O
a	O
metric	O
.	O
</s>
<s>
Like	O
principal	B-Application
components	I-Application
analysis	I-Application
,	O
correspondence	B-Algorithm
analysis	I-Algorithm
creates	O
orthogonal	B-Application
components	O
(	O
or	O
axes	O
)	O
and	O
,	O
for	O
each	O
item	O
in	O
a	O
table	O
i.e.	O
</s>
<s>
Correspondence	B-Algorithm
analysis	I-Algorithm
is	O
performed	O
on	O
the	O
data	O
table	O
,	O
conceived	O
as	O
matrix	B-Architecture
C	O
of	O
size	O
m×n	O
where	O
m	O
is	O
the	O
number	O
of	O
rows	O
and	O
n	O
is	O
the	O
number	O
of	O
columns	O
.	O
</s>
<s>
In	O
the	O
following	O
mathematical	O
description	O
of	O
the	O
method	O
capital	O
letters	O
in	O
italics	O
refer	O
to	O
a	O
matrix	B-Architecture
while	O
letters	O
in	O
italics	O
refer	O
to	O
vectors	O
.	O
</s>
<s>
Understanding	O
the	O
following	O
computations	O
requires	O
knowledge	O
of	O
matrix	B-Architecture
algebra	O
.	O
</s>
<s>
Before	O
proceeding	O
to	O
the	O
central	O
computational	O
step	O
of	O
the	O
algorithm	O
,	O
the	O
values	O
in	O
matrix	B-Architecture
C	O
have	O
to	O
be	O
transformed	O
.	O
</s>
<s>
Here	O
is	O
the	O
sum	O
of	O
all	O
cell	O
values	O
in	O
matrix	B-Architecture
C	O
,	O
or	O
short	O
the	O
sum	O
of	O
C	O
,	O
and	O
is	O
a	O
column	O
vector	O
of	O
ones	O
with	O
the	O
appropriate	O
dimension	O
.	O
</s>
<s>
In	O
simple	O
words	O
,	O
Matrix	B-Architecture
is	O
just	O
the	O
data	O
matrix	B-Architecture
(	O
contingency	B-Application
table	I-Application
or	O
binary	O
table	O
)	O
transformed	O
into	O
portions	O
i.e.	O
</s>
<s>
Note	O
,	O
the	O
vectors	O
and	O
are	O
combined	O
in	O
an	O
outer	O
product	O
resulting	O
in	O
a	O
matrix	B-Architecture
of	O
the	O
same	O
dimensions	O
as	O
.	O
</s>
<s>
In	O
words	O
the	O
formula	O
reads	O
:	O
matrix	B-Architecture
is	O
subtracted	O
from	O
matrix	B-Architecture
and	O
the	O
resulting	O
matrix	B-Architecture
is	O
scaled	O
(	O
weighted	O
)	O
by	O
the	O
diagonal	B-Algorithm
matrices	I-Algorithm
and	O
.	O
</s>
<s>
Multiplying	O
the	O
resulting	O
matrix	B-Architecture
by	O
the	O
diagonal	B-Algorithm
matrices	I-Algorithm
is	O
equivalent	O
to	O
multiply	O
the	O
i-th	O
row	O
(	O
or	O
column	O
)	O
of	O
it	O
by	O
the	O
i-th	O
element	O
of	O
the	O
diagonal	O
of	O
or	O
,	O
respectively	O
.	O
</s>
<s>
Subtracting	O
matrix	B-Architecture
from	O
matrix	B-Architecture
is	O
the	O
matrix	B-Architecture
algebra	O
version	O
of	O
double	O
centering	B-Algorithm
the	O
data	O
.	O
</s>
<s>
Multiplying	O
this	O
difference	O
by	O
the	O
diagonal	O
weighting	O
matrices	O
results	O
in	O
a	O
matrix	B-Architecture
containing	O
weighted	O
deviations	O
from	O
the	O
origin	O
of	O
a	O
vector	O
space	O
.	O
</s>
<s>
This	O
origin	O
is	O
defined	O
by	O
matrix	B-Architecture
.	O
</s>
<s>
In	O
fact	O
matrix	B-Architecture
is	O
identical	O
with	O
the	O
matrix	B-Architecture
of	O
expected	O
frequencies	O
in	O
the	O
chi-squared	B-General_Concept
test	I-General_Concept
.	O
</s>
<s>
where	O
and	O
are	O
the	O
left	O
and	O
right	O
singular	O
vectors	O
of	O
and	O
is	O
a	O
square	O
diagonal	B-Algorithm
matrix	I-Algorithm
with	O
the	O
singular	O
values	O
of	O
on	O
the	O
diagonal	O
.	O
</s>
<s>
In	O
other	O
words	O
,	O
the	O
multivariate	O
information	O
that	O
is	O
contained	O
in	O
as	O
well	O
as	O
in	O
is	O
now	O
distributed	O
across	O
two	O
(	O
coordinate	O
)	O
matrices	O
and	O
and	O
a	O
diagonal	O
(	O
scaling	O
)	O
matrix	B-Architecture
.	O
</s>
<s>
While	O
a	O
principal	B-Application
component	I-Application
analysis	I-Application
may	O
be	O
said	O
to	O
decompose	O
the	O
(	O
co	O
)	O
variance	O
,	O
and	O
hence	O
its	O
measure	O
of	O
success	O
is	O
the	O
amount	O
of	O
(	O
co	O
-	O
)	O
variance	O
covered	O
by	O
the	O
first	O
few	O
PCA	O
axes	O
-	O
measured	O
in	O
eigenvalue	O
-	O
,	O
a	O
CA	O
works	O
with	O
a	O
weighted	O
(	O
co	O
-	O
)	O
variance	O
which	O
is	O
called	O
inertia	O
.	O
</s>
<s>
and	O
are	O
presented	O
in	O
the	O
form	O
of	O
a	O
scree	B-Application
plot	I-Application
.	O
</s>
<s>
In	O
fact	O
a	O
scree	B-Application
plot	I-Application
is	O
just	O
a	O
bar	B-Application
plot	I-Application
of	O
all	O
principal	O
inertia	O
portions	O
.	O
</s>
<s>
Since	O
the	O
weighting	O
includes	O
the	O
singular	O
values	O
of	O
the	O
matrix	B-Architecture
of	O
standardized	O
residuals	O
these	O
coordinates	O
are	O
sometimes	O
referred	O
to	O
as	O
singular	O
value	O
scaled	O
singular	O
vectors	O
,	O
or	O
,	O
a	O
little	O
bit	O
misleading	O
,	O
as	O
eigenvalue	O
scaled	O
eigenvectors	O
.	O
</s>
<s>
To	O
represent	O
the	O
result	O
of	O
CA	O
in	O
a	O
proper	O
biplot	B-Application
,	O
those	O
categories	O
which	O
are	O
not	O
plotted	O
in	O
principal	O
coordinates	O
,	O
i.e.	O
</s>
<s>
When	O
computing	O
standard	O
coordinates	O
the	O
singular	O
values	O
are	O
omitted	O
which	O
is	O
a	O
direct	O
result	O
of	O
applying	O
the	O
biplot	B-Application
rule	I-Application
by	O
which	O
one	O
of	O
the	O
two	O
sets	O
of	O
singular	O
vector	O
matrices	O
must	O
be	O
scaled	O
by	O
singular	O
values	O
raised	O
to	O
the	O
power	O
of	O
zero	O
i.e.	O
</s>
<s>
it	O
leads	O
to	O
meaningful	O
interpretations	O
of	O
their	O
spatial	O
relations	O
in	O
a	O
biplot	B-Application
.	O
</s>
<s>
Note	O
that	O
a	O
scaling	O
1	O
biplot	B-Application
in	O
ecology	O
implies	O
the	O
rows	O
to	O
be	O
in	O
principal	O
and	O
the	O
columns	O
to	O
be	O
in	O
standard	O
coordinates	O
while	O
scaling	O
2	O
implies	O
the	O
rows	O
to	O
be	O
in	O
standard	O
and	O
the	O
columns	O
to	O
be	O
in	O
principal	O
coordinates	O
.	O
</s>
<s>
scaling	O
1	O
implies	O
a	O
biplot	B-Application
of	O
together	O
with	O
while	O
scaling	O
2	O
implies	O
a	O
biplot	B-Application
of	O
together	O
with	O
.	O
</s>
<s>
The	O
visualization	O
of	O
a	O
CA	O
result	O
always	O
starts	O
with	O
displaying	O
the	O
scree	B-Application
plot	I-Application
of	O
the	O
principal	O
inertia	O
values	O
to	O
evaluate	O
the	O
success	O
of	O
summarizing	O
spread	O
by	O
the	O
first	O
few	O
singular	O
vectors	O
.	O
</s>
<s>
The	O
actual	O
ordination	B-Algorithm
is	O
presented	O
in	O
a	O
graph	O
which	O
could	O
-	O
at	O
first	O
look	O
-	O
be	O
confused	O
with	O
a	O
complicated	O
scatter	B-Application
plot	I-Application
.	O
</s>
<s>
In	O
fact	O
it	O
consists	O
of	O
two	O
scatter	B-Application
plots	I-Application
printed	O
one	O
upon	O
the	O
other	O
,	O
one	O
set	O
of	O
points	O
for	O
the	O
rows	O
and	O
one	O
for	O
the	O
columns	O
.	O
</s>
<s>
But	O
being	O
a	O
biplot	B-Application
a	O
clear	O
interpretation	O
rule	O
relates	O
the	O
two	O
coordinate	O
matrices	O
used	O
.	O
</s>
<s>
Usually	O
the	O
first	O
two	O
dimensions	O
of	O
the	O
CA	O
solution	O
are	O
plotted	O
because	O
they	O
encompass	O
the	O
maximum	O
of	O
information	O
about	O
the	O
data	O
table	O
that	O
can	O
be	O
displayed	O
in	O
2D	O
although	O
other	O
combinations	O
of	O
dimensions	O
may	O
be	O
investigated	O
by	O
a	O
biplot	B-Application
.	O
</s>
<s>
A	O
biplot	B-Application
is	O
in	O
fact	O
a	O
low	B-Algorithm
dimensional	I-Algorithm
mapping	B-Algorithm
of	O
a	O
part	O
of	O
the	O
information	O
contained	O
in	O
the	O
original	O
table	O
.	O
</s>
<s>
Traditionally	O
,	O
originating	O
from	O
the	O
french	O
tradition	O
in	O
CA	O
,	O
early	O
CA	O
biplots	B-Application
mapped	O
both	O
entities	O
in	O
the	O
same	O
coordinate	O
version	O
,	O
usually	O
principal	O
coordinates	O
,	O
but	O
this	O
kind	O
of	O
display	O
is	O
misleading	O
insofar	O
as	O
:	O
"	O
Although	O
this	O
is	O
called	O
a	O
biplot	B-Application
,	O
it	O
does	O
not	O
have	O
any	O
useful	O
inner	O
product	O
relationship	O
between	O
the	O
row	O
and	O
column	O
scores	O
"	O
as	O
Brian	O
Ripley	O
,	O
maintainer	O
of	O
R	B-Language
package	O
MASS	O
points	O
out	O
correctly	O
.	O
</s>
<s>
A	O
scaling	O
1	O
biplot	B-Application
(	O
rows	O
in	O
principal	O
coordinates	O
,	O
columns	O
in	O
standard	O
coordinates	O
)	O
is	O
interpreted	O
as	O
follows	O
:	O
</s>
<s>
The	O
distances	O
between	O
row	O
points	O
approximate	O
their	O
chi-square	B-General_Concept
distance	O
.	O
</s>
<s>
Several	O
variants	O
of	O
CA	O
are	O
available	O
,	O
including	O
detrended	B-Algorithm
correspondence	I-Algorithm
analysis	I-Algorithm
(	O
DCA	O
)	O
and	O
canonical	B-Algorithm
correspondence	I-Algorithm
analysis	I-Algorithm
(	O
CCA	O
)	O
.	O
</s>
<s>
The	O
extension	O
of	O
correspondence	B-Algorithm
analysis	I-Algorithm
to	O
many	O
categorical	O
variables	O
is	O
called	O
multiple	B-Algorithm
correspondence	I-Algorithm
analysis	I-Algorithm
.	O
</s>
<s>
An	O
adaptation	O
of	O
correspondence	B-Algorithm
analysis	I-Algorithm
to	O
the	O
problem	O
of	O
discrimination	O
based	O
upon	O
qualitative	O
variables	O
(	O
i.e.	O
,	O
the	O
equivalent	O
of	O
discriminant	B-General_Concept
analysis	I-General_Concept
for	O
qualitative	O
data	O
)	O
is	O
called	O
discriminant	O
correspondence	B-Algorithm
analysis	I-Algorithm
or	O
barycentric	O
discriminant	B-General_Concept
analysis	I-General_Concept
.	O
</s>
<s>
In	O
the	O
social	O
sciences	O
,	O
correspondence	B-Algorithm
analysis	I-Algorithm
,	O
and	O
particularly	O
its	O
extension	O
multiple	B-Algorithm
correspondence	I-Algorithm
analysis	I-Algorithm
,	O
was	O
made	O
known	O
outside	O
France	O
through	O
French	O
sociologist	O
Pierre	O
Bourdieu	O
's	O
application	O
of	O
it	O
.	O
</s>
<s>
The	O
data	O
visualization	O
system	O
Orange	B-Application
include	O
the	O
module	O
:	O
orngCA	O
.	O
</s>
<s>
The	O
statistical	O
programming	O
language	O
R	B-Language
includes	O
several	O
packages	O
,	O
which	O
offer	O
a	O
function	O
for	O
(	O
simple	O
symmetric	O
)	O
correspondence	B-Algorithm
analysis	I-Algorithm
.	O
</s>
<s>
Using	O
the	O
R	B-Language
notation	O
 [ package_name::function_name ] 	O
the	O
packages	O
and	O
respective	O
functions	O
are	O
:	O
ade4::dudi.coa( )	O
,	O
ca::ca( )	O
,	O
ExPosition::epCA( )	O
,	O
FactoMineR::CA( )	O
,	O
MASS::corresp( )	O
,	O
vegan::cca( )	O
.	O
</s>
<s>
The	O
Freeware	O
PAST	O
(	O
PAleontological	O
STatistics	O
)	O
offers	O
(	O
simple	O
symmetric	O
)	O
correspondence	B-Algorithm
analysis	I-Algorithm
via	O
the	O
menu	O
"	O
Multivariate/Ordination/Correspondence	O
(	O
CA	O
)	O
"	O
.	O
</s>
