<s>
Partial	B-Algorithm
least	I-Algorithm
squares	I-Algorithm
regression	I-Algorithm
(	O
PLS	O
regression	O
)	O
is	O
a	O
statistical	O
method	O
that	O
bears	O
some	O
relation	O
to	O
principal	B-Algorithm
components	I-Algorithm
regression	I-Algorithm
;	O
instead	O
of	O
finding	O
hyperplanes	O
of	O
maximum	O
variance	O
between	O
the	O
response	O
and	O
independent	O
variables	O
,	O
it	O
finds	O
a	O
linear	B-General_Concept
regression	I-General_Concept
model	I-General_Concept
by	O
projecting	O
the	O
predicted	O
variables	O
and	O
the	O
observable	O
variables	O
to	O
a	O
new	O
space	O
.	O
</s>
<s>
Partial	B-Algorithm
least	I-Algorithm
squares	I-Algorithm
discriminant	O
analysis	O
(	O
PLS-DA	O
)	O
is	O
a	O
variant	O
used	O
when	O
the	O
Y	O
is	O
categorical	O
.	O
</s>
<s>
PLS	O
is	O
used	O
to	O
find	O
the	O
fundamental	O
relations	O
between	O
2	O
matrices	B-Architecture
(	O
X	O
and	O
Y	O
)	O
,	O
i.e.	O
</s>
<s>
Partial	B-Algorithm
least	I-Algorithm
squares	I-Algorithm
was	O
introduced	O
by	O
the	O
Swedish	O
statistician	O
Herman	O
O	O
.	O
</s>
<s>
An	O
alternative	O
term	O
for	O
PLS	O
is	O
projection	B-Algorithm
to	I-Algorithm
latent	I-Algorithm
structures	I-Algorithm
,	O
but	O
the	O
term	O
partial	B-Algorithm
least	I-Algorithm
squares	I-Algorithm
is	O
still	O
dominant	O
in	O
many	O
areas	O
.	O
</s>
<s>
where	O
is	O
an	O
matrix	O
of	O
predictors	O
,	O
is	O
an	O
matrix	O
of	O
responses	O
;	O
and	O
are	O
matrices	B-Architecture
that	O
are	O
,	O
respectively	O
,	O
projections	O
of	O
(	O
the	O
X	O
score	O
,	O
component	O
or	O
factor	O
matrix	O
)	O
and	O
projections	O
of	O
(	O
the	O
Y	O
scores	O
)	O
;	O
and	O
are	O
,	O
respectively	O
,	O
and	O
orthogonal	O
loading	O
matrices	B-Architecture
;	O
and	O
matrices	B-Architecture
and	O
are	O
the	O
error	O
terms	O
,	O
assumed	O
to	O
be	O
independent	O
and	O
identically	O
distributed	O
random	O
normal	O
variables	O
.	O
</s>
<s>
A	O
number	O
of	O
variants	O
of	O
PLS	O
exist	O
for	O
estimating	O
the	O
factor	O
and	O
loading	O
matrices	B-Architecture
and	O
.	O
</s>
<s>
Most	O
of	O
them	O
construct	O
estimates	O
of	O
the	O
linear	B-General_Concept
regression	I-General_Concept
between	O
and	O
as	O
.	O
</s>
<s>
Algorithms	O
also	O
differ	O
on	O
whether	O
they	O
estimate	O
the	O
factor	O
matrix	O
as	O
an	O
orthogonal	O
(	O
that	O
is	O
,	O
orthonormal	B-Algorithm
)	O
matrix	O
or	O
not	O
.	O
</s>
<s>
It	O
estimates	O
as	O
an	O
orthonormal	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
In	O
pseudocode	O
it	O
is	O
expressed	O
below	O
(	O
capital	O
letters	O
are	O
matrices	B-Architecture
,	O
lower	O
case	O
letters	O
are	O
vectors	O
if	O
they	O
are	O
superscripted	O
and	O
scalars	O
if	O
they	O
are	O
subscripted	O
)	O
.	O
</s>
<s>
Another	O
extension	O
of	O
PLS	O
regression	O
,	O
named	O
L-PLS	O
for	O
its	O
L-shaped	O
matrices	B-Architecture
,	O
connects	O
3	O
related	O
data	O
blocks	O
to	O
improve	O
predictability	O
.	O
</s>
<s>
In	O
2015	O
partial	B-Algorithm
least	I-Algorithm
squares	I-Algorithm
was	O
related	O
to	O
a	O
procedure	O
called	O
the	O
three-pass	O
regression	O
filter	O
(	O
3PRF	O
)	O
.	O
</s>
