<s>
Biplots	B-Application
are	O
a	O
type	O
of	O
exploratory	O
graph	O
used	O
in	O
statistics	O
,	O
a	O
generalization	O
of	O
the	O
simple	O
two-variable	O
scatterplot	B-Application
.	O
</s>
<s>
A	O
biplot	B-Application
overlays	O
a	O
score	O
plot	O
with	O
a	O
loading	O
plot	O
.	O
</s>
<s>
A	O
biplot	B-Application
allows	O
information	O
on	O
both	O
samples	O
and	O
variables	O
of	O
a	O
data	B-Algorithm
matrix	I-Algorithm
to	O
be	O
displayed	O
graphically	O
.	O
</s>
<s>
A	O
generalised	O
biplot	B-Application
displays	O
information	O
on	O
both	O
continuous	O
and	O
categorical	O
variables	O
.	O
</s>
<s>
The	O
biplot	B-Application
was	O
introduced	O
by	O
K	O
.	O
Ruben	O
Gabriel	O
(	O
1971	O
)	O
.	O
</s>
<s>
Gower	O
and	O
Hand	O
(	O
1996	O
)	O
wrote	O
a	O
monograph	O
on	O
biplots	B-Application
.	O
</s>
<s>
Yan	O
and	O
Kang	O
(	O
2003	O
)	O
described	O
various	O
methods	O
which	O
can	O
be	O
used	O
in	O
order	O
to	O
visualize	O
and	O
interpret	O
a	O
biplot	B-Application
.	O
</s>
<s>
The	O
book	O
by	O
Greenacre	O
(	O
2010	O
)	O
is	O
a	O
practical	O
user-oriented	O
guide	O
to	O
biplots	B-Application
,	O
along	O
with	O
scripts	O
in	O
the	O
open-source	O
R	B-Language
programming	I-Language
language	I-Language
,	O
to	O
generate	O
biplots	B-Application
associated	O
with	O
principal	B-Application
component	I-Application
analysis	I-Application
(	O
PCA	O
)	O
,	O
multidimensional	O
scaling	O
(	O
MDS	O
)	O
,	O
log-ratio	O
analysis	O
(	O
LRA	O
)	O
—	O
also	O
known	O
as	O
spectral	O
mapping	O
—	O
discriminant	B-General_Concept
analysis	I-General_Concept
(	O
DA	O
)	O
and	O
various	O
forms	O
of	O
correspondence	B-Algorithm
analysis	I-Algorithm
:	O
simple	O
correspondence	B-Algorithm
analysis	I-Algorithm
(	O
CA	O
)	O
,	O
multiple	O
correspondence	B-Algorithm
analysis	I-Algorithm
(	O
MCA	O
)	O
and	O
canonical	O
correspondence	B-Algorithm
analysis	I-Algorithm
(	O
CCA	O
)	O
(	O
Greenacre	O
2016	O
)	O
.	O
</s>
<s>
The	O
book	O
by	O
Gower	O
,	O
Lubbe	O
and	O
le	O
Roux	O
(	O
2011	O
)	O
aims	O
to	O
popularize	O
biplots	B-Application
as	O
a	O
useful	O
and	O
reliable	O
method	O
for	O
the	O
visualization	O
of	O
multivariate	O
data	O
when	O
researchers	O
want	O
to	O
consider	O
,	O
for	O
example	O
,	O
principal	B-Application
component	I-Application
analysis	I-Application
(	O
PCA	O
)	O
,	O
canonical	O
variates	O
analysis	O
(	O
CVA	O
)	O
or	O
various	O
types	O
of	O
correspondence	B-Algorithm
analysis	I-Algorithm
.	O
</s>
<s>
A	O
biplot	B-Application
is	O
constructed	O
by	O
using	O
the	O
singular	O
value	O
decomposition	O
(	O
SVD	O
)	O
to	O
obtain	O
a	O
low-rank	O
approximation	O
to	O
a	O
transformed	O
version	O
of	O
the	O
data	B-Algorithm
matrix	I-Algorithm
X	O
,	O
whose	O
n	O
rows	O
are	O
the	O
samples	O
(	O
also	O
called	O
the	O
cases	O
,	O
or	O
objects	O
)	O
,	O
and	O
whose	O
p	O
columns	O
are	O
the	O
variables	O
.	O
</s>
<s>
The	O
transformed	O
data	B-Algorithm
matrix	I-Algorithm
Y	O
is	O
obtained	O
from	O
the	O
original	O
matrix	O
X	O
by	O
centering	O
and	O
optionally	O
standardizing	O
the	O
columns	O
(	O
the	O
variables	O
)	O
.	O
</s>
<s>
The	O
biplot	B-Application
is	O
formed	O
from	O
two	O
scatterplots	B-Application
that	O
share	O
a	O
common	O
set	O
of	O
axes	O
and	O
have	O
a	O
between-set	O
scalar	O
product	O
interpretation	O
.	O
</s>
<s>
The	O
first	O
scatterplot	B-Application
is	O
formed	O
from	O
the	O
points	O
(	O
d1αu1i	O
,	O
d2αu2i	O
)	O
,	O
for	O
i	O
=	O
1	O
,...,	O
n	O
.	O
The	O
second	O
plot	O
is	O
formed	O
from	O
the	O
points	O
(	O
d11−αv1j	O
,	O
d21−αv2j	O
)	O
,	O
for	O
j	O
=	O
1	O
,...,	O
p	O
.	O
This	O
is	O
the	O
biplot	B-Application
formed	O
by	O
the	O
dominant	O
two	O
terms	O
of	O
the	O
SVD	O
,	O
which	O
can	O
then	O
be	O
represented	O
in	O
a	O
two-dimensional	O
display	O
.	O
</s>
<s>
Typical	O
choices	O
of	O
α	O
are	O
1	O
(	O
to	O
give	O
a	O
distance	O
interpretation	O
to	O
the	O
row	O
display	O
)	O
and	O
0	O
(	O
to	O
give	O
a	O
distance	O
interpretation	O
to	O
the	O
column	O
display	O
)	O
,	O
and	O
in	O
some	O
rare	O
cases	O
α	O
=	O
1/2	O
to	O
obtain	O
a	O
symmetrically	O
scaled	O
biplot	B-Application
(	O
which	O
gives	O
no	O
distance	O
interpretation	O
to	O
the	O
rows	O
or	O
the	O
columns	O
,	O
but	O
only	O
the	O
scalar	O
product	O
interpretation	O
)	O
.	O
</s>
<s>
The	O
set	O
of	O
points	O
depicting	O
the	O
variables	O
can	O
be	O
drawn	O
as	O
arrows	O
from	O
the	O
origin	O
to	O
reinforce	O
the	O
idea	O
that	O
they	O
represent	O
biplot	B-Application
axes	O
onto	O
which	O
the	O
samples	O
can	O
be	O
projected	O
to	O
approximate	O
the	O
original	O
data	O
.	O
</s>
