<s>
Kernel	B-General_Concept
density	I-General_Concept
estimation	I-General_Concept
is	O
a	O
nonparametric	B-General_Concept
technique	O
for	O
density	B-General_Concept
estimation	I-General_Concept
i.e.	O
,	O
estimation	O
of	O
probability	O
density	O
functions	O
,	O
which	O
is	O
one	O
of	O
the	O
fundamental	O
questions	O
in	O
statistics	O
.	O
</s>
<s>
It	O
can	O
be	O
viewed	O
as	O
a	O
generalisation	O
of	O
histogram	B-Algorithm
density	B-General_Concept
estimation	I-General_Concept
with	O
improved	O
statistical	O
properties	O
.	O
</s>
<s>
Apart	O
from	O
histograms	B-Algorithm
,	O
other	O
types	O
of	O
density	O
estimators	O
include	O
parametric	B-General_Concept
,	O
spline	B-Algorithm
,	O
wavelet	O
and	O
Fourier	O
series	O
.	O
</s>
<s>
Kernel	B-General_Concept
density	I-General_Concept
estimators	I-General_Concept
were	O
first	O
introduced	O
in	O
the	O
scientific	O
literature	O
for	O
univariate	B-General_Concept
data	O
in	O
the	O
1950s	O
and	O
1960s	O
and	O
subsequently	O
have	O
been	O
widely	O
adopted	O
.	O
</s>
<s>
It	O
was	O
soon	O
recognised	O
that	O
analogous	O
estimators	O
for	O
multivariate	B-General_Concept
data	I-General_Concept
would	O
be	O
an	O
important	O
addition	O
to	O
multivariate	B-General_Concept
statistics	I-General_Concept
.	O
</s>
<s>
Based	O
on	O
research	O
carried	O
out	O
in	O
the	O
1990s	O
and	O
2000s	O
,	O
multivariate	B-General_Concept
kernel	I-General_Concept
density	I-General_Concept
estimation	I-General_Concept
has	O
reached	O
a	O
level	O
of	O
maturity	O
comparable	O
to	O
its	O
univariate	B-General_Concept
counterparts	O
.	O
</s>
<s>
We	O
take	O
an	O
illustrative	O
synthetic	B-General_Concept
bivariate	O
data	O
set	O
of	O
50	O
points	O
to	O
illustrate	O
the	O
construction	O
of	O
histograms	B-Algorithm
.	O
</s>
<s>
This	O
requires	O
the	O
choice	O
of	O
an	O
anchor	O
point	O
(	O
the	O
lower	O
left	O
corner	O
of	O
the	O
histogram	B-Algorithm
grid	O
)	O
.	O
</s>
<s>
For	O
the	O
histogram	B-Algorithm
on	O
the	O
left	O
,	O
we	O
choose	O
( −	O
1.5	O
,	O
−	O
1.5	O
)	O
:	O
for	O
the	O
one	O
on	O
the	O
right	O
,	O
we	O
shift	O
the	O
anchor	O
point	O
by	O
0.125	O
in	O
both	O
directions	O
to	O
( −	O
1.625	O
,	O
−	O
1.625	O
)	O
.	O
</s>
<s>
Both	O
histograms	B-Algorithm
have	O
a	O
binwidth	O
of	O
0.5	O
,	O
so	O
any	O
differences	O
are	O
due	O
to	O
the	O
change	O
in	O
the	O
anchor	O
point	O
only	O
.	O
</s>
<s>
The	O
left	O
histogram	B-Algorithm
appears	O
to	O
indicate	O
that	O
the	O
upper	O
half	O
has	O
a	O
higher	O
density	O
than	O
the	O
lower	O
half	O
,	O
whereas	O
the	O
reverse	O
is	O
the	O
case	O
for	O
the	O
right-hand	O
histogram	B-Algorithm
,	O
confirming	O
that	O
histograms	B-Algorithm
are	O
highly	O
sensitive	O
to	O
the	O
placement	O
of	O
the	O
anchor	O
point	O
.	O
</s>
<s>
One	O
possible	O
solution	O
to	O
this	O
anchor	O
point	O
placement	O
problem	O
is	O
to	O
remove	O
the	O
histogram	B-Algorithm
binning	O
grid	O
completely	O
.	O
</s>
<s>
The	O
result	O
of	O
summing	O
these	O
kernels	O
is	O
given	O
on	O
the	O
right	O
figure	O
,	O
which	O
is	O
a	O
kernel	B-General_Concept
density	I-General_Concept
estimate	I-General_Concept
.	O
</s>
<s>
The	O
most	O
striking	O
difference	O
between	O
kernel	B-General_Concept
density	I-General_Concept
estimates	I-General_Concept
and	O
histograms	B-Algorithm
is	O
that	O
the	O
former	O
are	O
easier	O
to	O
interpret	O
since	O
they	O
do	O
not	O
contain	O
artifices	O
induced	O
by	O
a	O
binning	O
grid	O
.	O
</s>
<s>
The	O
goal	O
of	O
density	B-General_Concept
estimation	I-General_Concept
is	O
to	O
take	O
a	O
finite	O
sample	O
of	O
data	O
and	O
to	O
make	O
inferences	O
about	O
the	O
underlying	O
probability	O
density	O
function	O
everywhere	O
,	O
including	O
where	O
no	O
data	O
are	O
observed	O
.	O
</s>
<s>
In	O
kernel	B-General_Concept
density	I-General_Concept
estimation	I-General_Concept
,	O
the	O
contribution	O
of	O
each	O
data	O
point	O
is	O
smoothed	O
out	O
from	O
a	O
single	O
point	O
into	O
a	O
region	O
of	O
space	O
surrounding	O
it	O
.	O
</s>
<s>
The	O
previous	O
figure	O
is	O
a	O
graphical	O
representation	O
of	O
kernel	B-General_Concept
density	I-General_Concept
estimate	I-General_Concept
,	O
which	O
we	O
now	O
define	O
in	O
an	O
exact	O
manner	O
.	O
</s>
<s>
Let	O
x1	O
,	O
x2	O
,	O
...	O
,	O
xn	O
be	O
a	O
sample	O
of	O
d-variate	O
random	B-General_Concept
vectors	I-General_Concept
drawn	O
from	O
a	O
common	O
distribution	O
described	O
by	O
the	O
density	O
function	O
ƒ	O
.	O
</s>
<s>
H	O
is	O
the	O
bandwidth	O
(	O
or	O
smoothing	O
)	O
d×d	O
matrix	O
which	O
is	O
symmetric	B-Algorithm
and	O
positive	B-Algorithm
definite	I-Algorithm
;	O
</s>
<s>
K	O
is	O
the	O
kernel	O
function	O
which	O
is	O
a	O
symmetric	B-Algorithm
multivariate	O
density	O
;	O
</s>
<s>
The	O
choice	O
of	O
the	O
kernel	O
function	O
K	O
is	O
not	O
crucial	O
to	O
the	O
accuracy	O
of	O
kernel	B-General_Concept
density	I-General_Concept
estimators	I-General_Concept
,	O
so	O
we	O
use	O
the	O
standard	O
multivariate	O
normal	O
kernel	O
throughout	O
:	O
,	O
where	O
H	O
plays	O
the	O
role	O
of	O
the	O
covariance	O
matrix	O
.	O
</s>
<s>
That	O
the	O
bandwidth	O
matrix	O
also	O
induces	O
an	O
orientation	O
is	O
a	O
basic	O
difference	O
between	O
multivariate	B-General_Concept
kernel	I-General_Concept
density	I-General_Concept
estimation	I-General_Concept
from	O
its	O
univariate	B-General_Concept
analogue	O
since	O
orientation	O
is	O
not	O
defined	O
for	O
1D	O
kernels	O
.	O
</s>
<s>
The	O
three	O
main	O
parametrisation	O
classes	O
(	O
in	O
increasing	O
order	O
of	O
complexity	O
)	O
are	O
S	O
,	O
the	O
class	O
of	O
positive	O
scalars	O
times	O
the	O
identity	B-Algorithm
matrix	I-Algorithm
;	O
D	O
,	O
diagonal	O
matrices	O
with	O
positive	O
entries	O
on	O
the	O
main	O
diagonal	O
;	O
and	O
F	O
,	O
symmetric	B-Algorithm
positive	I-Algorithm
definite	I-Algorithm
matrices	O
.	O
</s>
<s>
Heuristically	O
this	O
statement	O
implies	O
that	O
the	O
AMISE	O
is	O
a	O
'	O
good	O
 '	O
approximation	O
of	O
the	O
MISE	B-General_Concept
as	O
the	O
sample	O
size	O
n	O
→	O
∞	O
.	O
</s>
<s>
Substituting	O
this	O
into	O
the	O
MISE	B-General_Concept
formula	O
yields	O
that	O
the	O
optimal	O
MISE	B-General_Concept
is	O
O( n−	O
4/	O
( d+4	O
)	O
)	O
.	O
</s>
<s>
Thus	O
as	O
n	O
→	O
∞	O
,	O
the	O
MISE	B-General_Concept
→	O
0	O
,	O
i.e.	O
</s>
<s>
the	O
kernel	B-General_Concept
density	I-General_Concept
estimate	I-General_Concept
converges	O
in	O
mean	O
square	O
and	O
thus	O
also	O
in	O
probability	O
to	O
the	O
true	O
density	O
f	O
.	O
These	O
modes	O
of	O
convergence	O
are	O
confirmation	O
of	O
the	O
statement	O
in	O
the	O
motivation	O
section	O
that	O
kernel	O
methods	O
lead	O
to	O
reasonable	O
density	O
estimators	O
.	O
</s>
<s>
We	O
concentrate	O
on	O
two	O
classes	O
of	O
selectors	O
which	O
have	O
been	O
shown	O
to	O
be	O
the	O
most	O
widely	O
applicable	O
in	O
practice	O
:	O
smoothed	O
cross	B-Application
validation	I-Application
and	O
plug-in	O
selectors	O
.	O
</s>
<s>
Smoothed	O
cross	B-Application
validation	I-Application
(	O
SCV	O
)	O
is	O
a	O
subset	O
of	O
a	O
larger	O
class	O
of	O
cross	B-Application
validation	I-Application
techniques	O
.	O
</s>
<s>
In	O
the	O
optimal	O
bandwidth	O
selection	O
section	O
,	O
we	O
introduced	O
the	O
MISE	B-General_Concept
.	O
</s>
<s>
the	O
kernel	B-General_Concept
density	I-General_Concept
estimator	I-General_Concept
is	O
asymptotically	O
unbiased	O
;	O
and	O
that	O
the	O
variance	O
tends	O
to	O
zero	O
.	O
</s>
<s>
we	O
have	O
that	O
the	O
MSE	O
tends	O
to	O
0	O
,	O
implying	O
that	O
the	O
kernel	B-General_Concept
density	I-General_Concept
estimator	I-General_Concept
is	O
(	O
mean	O
square	O
)	O
consistent	O
and	O
hence	O
converges	O
in	O
probability	O
to	O
the	O
true	O
density	O
f	O
.	O
The	O
rate	O
of	O
convergence	O
of	O
the	O
MSE	O
to	O
0	O
is	O
the	O
necessarily	O
the	O
same	O
as	O
the	O
MISE	B-General_Concept
rate	O
noted	O
previously	O
O( n−	O
4/	O
( d+4	O
)	O
)	O
,	O
hence	O
the	O
covergence	O
rate	O
of	O
the	O
density	O
estimator	O
to	O
f	O
is	O
Op( n−	O
2/	O
( d+4	O
)	O
)	O
where	O
Op	O
denotes	O
order	O
in	O
probability	O
.	O
</s>
<s>
The	O
functional	O
covergence	O
is	O
established	O
similarly	O
by	O
considering	O
the	O
behaviour	O
of	O
the	O
MISE	B-General_Concept
,	O
and	O
noting	O
that	O
under	O
sufficient	O
regularity	O
,	O
integration	O
does	O
not	O
affect	O
the	O
convergence	O
rates	O
.	O
</s>
<s>
It	O
has	O
been	O
established	O
that	O
the	O
plug-in	O
and	O
smoothed	O
cross	B-Application
validation	I-Application
selectors	O
(	O
given	O
a	O
single	O
pilot	O
bandwidth	O
G	O
)	O
both	O
converge	O
at	O
a	O
relative	O
rate	O
of	O
Op( n−	O
2/	O
( d+6	O
)	O
)	O
i.e.	O
,	O
both	O
these	O
data-based	O
selectors	O
are	O
consistent	O
estimators	O
.	O
</s>
<s>
The	O
in	O
R	B-Language
implements	O
the	O
plug-in	O
and	O
smoothed	O
cross	B-Application
validation	I-Application
selectors	O
(	O
amongst	O
others	O
)	O
.	O
</s>
<s>
The	O
code	O
fragment	O
computes	O
the	O
kernel	B-General_Concept
density	I-General_Concept
estimate	I-General_Concept
with	O
the	O
plug-in	O
bandwidth	O
matrix	O
Again	O
,	O
the	O
coloured	O
contours	O
correspond	O
to	O
the	O
smallest	O
region	O
which	O
contains	O
the	O
respective	O
probability	O
mass	O
:	O
red	O
=	O
25%	O
,	O
orange	O
+	O
red	O
=	O
50%	O
,	O
yellow	O
+	O
orange	O
+	O
red	O
=	O
75%	O
.	O
</s>
<s>
The	O
MISE	B-General_Concept
is	O
the	O
expected	O
integrated	O
L2	O
distance	O
between	O
the	O
density	O
estimate	O
and	O
the	O
true	O
density	O
function	O
f	O
.	O
It	O
is	O
the	O
most	O
widely	O
used	O
,	O
mostly	O
due	O
to	O
its	O
tractability	O
and	O
most	O
software	O
implement	O
MISE-based	O
bandwidth	O
selectors	O
.	O
</s>
<s>
There	O
are	O
alternative	O
optimality	O
criteria	O
,	O
which	O
attempt	O
to	O
cover	O
cases	O
where	O
MISE	B-General_Concept
is	O
not	O
an	O
appropriate	O
measure	O
.	O
</s>
<s>
Its	O
mathematical	O
analysis	O
is	O
considerably	O
more	O
difficult	O
than	O
the	O
MISE	B-General_Concept
ones	O
.	O
</s>
<s>
The	O
KL	O
can	O
be	O
estimated	O
using	O
a	O
cross-validation	B-Application
method	O
,	O
although	O
KL	O
cross-validation	B-Application
selectors	O
can	O
be	O
sub-optimal	O
even	O
if	O
it	O
remains	O
consistent	O
for	O
bounded	O
density	O
functions	O
.	O
</s>
<s>
The	O
resulting	O
kernel	B-General_Concept
density	I-General_Concept
estimate	I-General_Concept
converges	O
rapidly	O
to	O
the	O
true	O
probability	O
distribution	O
as	O
samples	O
are	O
added	O
:	O
at	O
a	O
rate	O
close	O
to	O
the	O
expected	O
for	O
parametric	B-General_Concept
estimators	O
.	O
</s>
<s>
This	O
kernel	O
estimator	O
works	O
for	O
univariate	B-General_Concept
and	O
multivariate	O
samples	O
alike	O
.	O
</s>
<s>
The	O
optimal	O
kernel	O
is	O
defined	O
in	O
Fourier	O
space	O
—	O
as	O
the	O
optimal	O
damping	O
function	O
(	O
the	O
Fourier	O
transform	O
of	O
the	O
kernel	O
)	O
--	O
in	O
terms	O
of	O
the	O
Fourier	O
transform	O
of	O
the	O
data	O
,	O
the	O
empirical	O
characteristic	O
function	O
(	O
see	O
Kernel	B-General_Concept
density	I-General_Concept
estimation	I-General_Concept
)	O
:	O
</s>
<s>
There	O
are	O
various	O
ways	O
to	O
define	O
this	O
filter	O
function	O
,	O
and	O
a	O
simple	O
one	O
that	O
works	O
for	O
univariate	B-General_Concept
or	O
multivariate	O
samples	O
is	O
called	O
the	O
'	O
lowest	O
contiguous	O
hypervolume	O
filter	O
 '	O
;	O
is	O
chosen	O
such	O
that	O
the	O
only	O
accepted	O
frequencies	O
are	O
a	O
contiguous	O
subset	O
of	O
frequencies	O
surrounding	O
the	O
origin	O
for	O
which	O
(	O
see	O
for	O
a	O
discussion	O
of	O
this	O
and	O
other	O
filter	O
functions	O
)	O
.	O
</s>
