<s>
Multivariate	B-General_Concept
statistics	I-General_Concept
is	O
a	O
subdivision	O
of	O
statistics	O
encompassing	O
the	O
simultaneous	O
observation	O
and	O
analysis	O
of	O
more	O
than	O
one	O
outcome	O
variable	O
,	O
i.e.	O
,	O
multivariate	B-General_Concept
random	I-General_Concept
variables	I-General_Concept
.	O
</s>
<s>
Multivariate	B-General_Concept
statistics	I-General_Concept
concerns	O
understanding	O
the	O
different	O
aims	O
and	O
background	O
of	O
each	O
of	O
the	O
different	O
forms	O
of	O
multivariate	O
analysis	O
,	O
and	O
how	O
they	O
relate	O
to	O
each	O
other	O
.	O
</s>
<s>
The	O
practical	O
application	O
of	O
multivariate	B-General_Concept
statistics	I-General_Concept
to	O
a	O
particular	O
problem	O
may	O
involve	O
several	O
types	O
of	O
univariate	O
and	O
multivariate	O
analyses	O
in	O
order	O
to	O
understand	O
the	O
relationships	O
between	O
variables	O
and	O
their	O
relevance	O
to	O
the	O
problem	O
being	O
studied	O
.	O
</s>
<s>
Certain	O
types	O
of	O
problems	O
involving	O
multivariate	B-General_Concept
data	I-General_Concept
,	O
for	O
example	O
simple	B-General_Concept
linear	I-General_Concept
regression	I-General_Concept
and	O
multiple	O
regression	O
,	O
are	O
not	O
usually	O
considered	O
to	O
be	O
special	O
cases	O
of	O
multivariate	B-General_Concept
statistics	I-General_Concept
because	O
the	O
analysis	O
is	O
dealt	O
with	O
by	O
considering	O
the	O
(	O
univariate	O
)	O
conditional	O
distribution	O
of	O
a	O
single	O
outcome	O
variable	O
given	O
the	O
other	O
variables	O
.	O
</s>
<s>
Multivariate	O
analysis	O
(	O
MVA	O
)	O
is	O
based	O
on	O
the	O
principles	O
of	O
multivariate	B-General_Concept
statistics	I-General_Concept
.	O
</s>
<s>
These	O
concerns	O
are	O
often	O
eased	O
through	O
the	O
use	O
of	O
surrogate	B-Algorithm
models	I-Algorithm
,	O
highly	O
accurate	O
approximations	O
of	O
the	O
physics-based	O
code	O
.	O
</s>
<s>
Since	O
surrogate	B-Algorithm
models	I-Algorithm
take	O
the	O
form	O
of	O
an	O
equation	O
,	O
they	O
can	O
be	O
evaluated	O
very	O
quickly	O
.	O
</s>
<s>
This	O
becomes	O
an	O
enabler	O
for	O
large-scale	O
MVA	O
studies	O
:	O
while	O
a	O
Monte	B-Algorithm
Carlo	I-Algorithm
simulation	I-Algorithm
across	O
the	O
design	O
space	O
is	O
difficult	O
with	O
physics-based	O
codes	O
,	O
it	O
becomes	O
trivial	O
when	O
evaluating	O
surrogate	B-Algorithm
models	I-Algorithm
,	O
which	O
often	O
take	O
the	O
form	O
of	O
response-surface	O
equations	O
.	O
</s>
<s>
Multivariate	B-General_Concept
analysis	I-General_Concept
of	I-General_Concept
variance	I-General_Concept
(	O
MANOVA	B-General_Concept
)	O
extends	O
the	O
analysis	B-General_Concept
of	I-General_Concept
variance	I-General_Concept
to	O
cover	O
cases	O
where	O
there	O
is	O
more	O
than	O
one	O
dependent	O
variable	O
to	O
be	O
analyzed	O
simultaneously	O
;	O
see	O
also	O
Multivariate	B-General_Concept
analysis	I-General_Concept
of	I-General_Concept
covariance	I-General_Concept
(	O
MANCOVA	B-General_Concept
)	O
.	O
</s>
<s>
Principal	B-Application
components	I-Application
analysis	I-Application
(	O
PCA	O
)	O
creates	O
a	O
new	O
set	O
of	O
orthogonal	O
variables	O
that	O
contain	O
the	O
same	O
information	O
as	O
the	O
original	O
set	O
.	O
</s>
<s>
Correspondence	B-Algorithm
analysis	I-Algorithm
(	O
CA	O
)	O
,	O
or	O
reciprocal	B-Algorithm
averaging	I-Algorithm
,	O
finds	O
(	O
like	O
PCA	O
)	O
a	O
set	O
of	O
synthetic	O
variables	O
that	O
summarise	O
the	O
original	O
set	O
.	O
</s>
<s>
Canonical	O
(	O
or	O
"	O
constrained	O
"	O
)	O
correspondence	B-Algorithm
analysis	I-Algorithm
(	O
CCA	O
)	O
for	O
summarising	O
the	O
joint	O
variation	O
in	O
two	O
sets	O
of	O
variables	O
(	O
like	O
redundancy	O
analysis	O
)	O
;	O
combination	O
of	O
correspondence	B-Algorithm
analysis	I-Algorithm
and	O
multivariate	O
regression	O
analysis	O
.	O
</s>
<s>
Discriminant	B-General_Concept
analysis	I-General_Concept
,	O
or	O
canonical	O
variate	O
analysis	O
,	O
attempts	O
to	O
establish	O
whether	O
a	O
set	O
of	O
variables	O
can	O
be	O
used	O
to	O
distinguish	O
between	O
two	O
or	O
more	O
groups	O
of	O
cases	O
.	O
</s>
<s>
Linear	B-General_Concept
discriminant	I-General_Concept
analysis	I-General_Concept
(	O
LDA	O
)	O
computes	O
a	O
linear	O
predictor	O
from	O
two	O
sets	O
of	O
normally	O
distributed	O
data	O
to	O
allow	O
for	O
classification	O
of	O
new	O
observations	O
.	O
</s>
<s>
Clustering	B-Algorithm
systems	I-Algorithm
assign	O
objects	O
into	O
groups	O
(	O
called	O
clusters	O
)	O
so	O
that	O
objects	O
(	O
cases	O
)	O
from	O
the	O
same	O
cluster	O
are	O
more	O
similar	O
to	O
each	O
other	O
than	O
objects	O
from	O
different	O
clusters	O
.	O
</s>
<s>
Recursive	B-General_Concept
partitioning	I-General_Concept
creates	O
a	O
decision	O
tree	O
that	O
attempts	O
to	O
correctly	O
classify	O
members	O
of	O
the	O
population	O
based	O
on	O
a	O
dichotomous	O
dependent	O
variable	O
.	O
</s>
<s>
Artificial	B-Architecture
neural	I-Architecture
networks	I-Architecture
extend	O
regression	O
and	O
clustering	B-Algorithm
methods	O
to	O
non-linear	O
multivariate	O
models	O
.	O
</s>
<s>
Statistical	B-Application
graphics	I-Application
such	O
as	O
tours	O
,	O
parallel	B-Application
coordinate	I-Application
plots	I-Application
,	O
scatterplot	O
matrices	O
can	O
be	O
used	O
to	O
explore	O
multivariate	B-General_Concept
data	I-General_Concept
.	O
</s>
<s>
Principal	B-General_Concept
response	I-General_Concept
curves	I-General_Concept
analysis	I-General_Concept
(	O
PRC	O
)	O
is	O
a	O
method	O
based	O
on	O
RDA	O
that	O
allows	O
the	O
user	O
to	O
focus	O
on	O
treatment	O
effects	O
over	O
time	O
by	O
correcting	O
for	O
changes	O
in	O
control	O
treatments	O
over	O
time	O
.	O
</s>
<s>
Iconography	B-General_Concept
of	I-General_Concept
correlations	I-General_Concept
consists	O
in	O
replacing	O
a	O
correlation	O
matrix	O
by	O
a	O
diagram	O
where	O
the	O
“	O
remarkable	O
”	O
correlations	O
are	O
represented	O
by	O
a	O
solid	O
line	O
(	O
positive	O
correlation	O
)	O
,	O
or	O
a	O
dotted	O
line	O
(	O
negative	O
correlation	O
)	O
.	O
</s>
<s>
There	O
is	O
a	O
set	O
of	O
probability	O
distributions	O
used	O
in	O
multivariate	O
analyses	O
that	O
play	O
a	O
similar	O
role	O
to	O
the	O
corresponding	O
set	O
of	O
distributions	O
that	O
are	O
used	O
in	O
univariate	B-General_Concept
analysis	I-General_Concept
when	O
the	O
normal	O
distribution	O
is	O
appropriate	O
to	O
a	O
dataset	O
.	O
</s>
<s>
Multivariate	B-General_Concept
Student-t	I-General_Concept
distribution	I-General_Concept
.	O
</s>
<s>
The	O
Inverse-Wishart	O
distribution	O
is	O
important	O
in	O
Bayesian	O
inference	O
,	O
for	O
example	O
in	O
Bayesian	B-General_Concept
multivariate	I-General_Concept
linear	I-General_Concept
regression	I-General_Concept
.	O
</s>
<s>
Anderson	O
's	O
1958	O
textbook	O
,	O
An	O
Introduction	O
to	O
Multivariate	O
Statistical	O
Analysis	O
,	O
educated	O
a	O
generation	O
of	O
theorists	O
and	O
applied	O
statisticians	O
;	O
Anderson	O
's	O
book	O
emphasizes	O
hypothesis	O
testing	O
via	O
likelihood	B-General_Concept
ratio	I-General_Concept
tests	I-General_Concept
and	O
the	O
properties	O
of	O
power	O
functions	O
:	O
admissibility	O
,	O
unbiasedness	O
and	O
monotonicity	O
.	O
</s>
