<s>
Vecchia	B-Algorithm
approximation	I-Algorithm
is	O
a	O
Gaussian	B-General_Concept
processes	I-General_Concept
approximation	B-Algorithm
technique	O
originally	O
developed	O
by	O
Aldo	O
Vecchia	O
,	O
a	O
statistician	O
at	O
United	O
States	O
Geological	O
Survey	O
.	O
</s>
<s>
It	O
is	O
one	O
of	O
the	O
earliest	O
attempts	O
to	O
use	O
Gaussian	B-General_Concept
processes	I-General_Concept
in	O
high-dimensional	O
settings	O
.	O
</s>
<s>
Vecchia	O
's	O
approximation	B-Algorithm
takes	O
the	O
form	O
,	O
for	O
example	O
,	O
</s>
<s>
and	O
so	O
use	O
of	O
the	O
approximation	B-Algorithm
requires	O
some	O
knowledge	O
of	O
which	O
events	O
are	O
close	O
to	O
conditionally	O
independent	O
given	O
others	O
.	O
</s>
<s>
Fortunately	O
,	O
in	O
many	O
cases	O
there	O
are	O
good	O
heuristics	O
making	O
decisions	O
about	O
how	O
to	O
construct	O
the	O
approximation	B-Algorithm
.	O
</s>
<s>
More	O
technically	O
,	O
general	O
versions	O
of	O
the	O
approximation	B-Algorithm
lead	O
to	O
a	O
sparse	O
Cholesky	O
factor	O
of	O
the	O
precision	O
matrix	O
.	O
</s>
<s>
Let	O
be	O
a	O
Gaussian	B-General_Concept
process	I-General_Concept
indexed	O
by	O
with	O
mean	O
function	O
and	O
covariance	O
function	O
.	O
</s>
<s>
Vecchia	O
also	O
suggested	O
that	O
the	O
above	O
approximation	B-Algorithm
be	O
applied	O
to	O
observations	O
that	O
are	O
reordered	O
lexicographically	O
using	O
their	O
spatial	O
coordinates	O
.	O
</s>
<s>
While	O
conceptually	O
simple	O
,	O
the	O
assumption	O
of	O
the	O
Vecchia	B-Algorithm
approximation	I-Algorithm
often	O
proves	O
to	O
be	O
fairly	O
restrictive	O
and	O
inaccurate	O
.	O
</s>
<s>
Different	O
special	O
cases	O
of	O
the	O
general	O
Vecchia	B-Algorithm
approximation	I-Algorithm
can	O
be	O
described	O
in	O
terms	O
of	O
how	O
these	O
three	O
elements	O
are	O
selected	O
.	O
</s>
<s>
Multiple	O
methods	O
of	O
choosing	O
and	O
have	O
been	O
proposed	O
,	O
most	O
notably	O
the	O
nearest-neighbour	O
Gaussian	B-General_Concept
process	I-General_Concept
(	O
NNGP	O
)	O
,	O
meshed	O
Gaussian	B-General_Concept
process	I-General_Concept
and	O
multi-resolution	O
approximation	B-Algorithm
(	O
MRA	O
)	O
approaches	O
using	O
,	O
standard	O
Vecchia	O
using	O
and	O
Sparse	O
General	O
Vecchia	O
where	O
both	O
and	O
are	O
non-empty	O
.	O
</s>
<s>
Several	O
packages	O
have	O
been	O
developed	O
which	O
implement	O
some	O
variants	O
of	O
the	O
Vecchia	B-Algorithm
approximation	I-Algorithm
.	O
</s>
