<s>
Smoothing	B-Algorithm
splines	I-Algorithm
are	O
function	O
estimates	O
,	O
,	O
obtained	O
from	O
a	O
set	O
of	O
noisy	O
observations	O
of	O
the	O
target	O
,	O
in	O
order	O
to	O
balance	O
a	O
measure	O
of	O
goodness	O
of	O
fit	O
of	O
to	O
with	O
a	O
derivative	O
based	O
measure	O
of	O
the	O
smoothness	O
of	O
.	O
</s>
<s>
The	O
most	O
familiar	O
example	O
is	O
the	O
cubic	O
smoothing	B-Algorithm
spline	I-Algorithm
,	O
but	O
there	O
are	O
many	O
other	O
possibilities	O
,	O
including	O
for	O
the	O
case	O
where	O
is	O
a	O
vector	O
quantity	O
.	O
</s>
<s>
This	O
is	O
often	O
estimated	O
by	O
generalized	O
cross-validation	O
,	O
or	O
by	O
restricted	O
marginal	O
likelihood	O
(	O
REML	O
)	O
which	O
exploits	O
the	O
link	O
between	O
spline	B-Algorithm
smoothing	I-Algorithm
and	O
Bayesian	O
estimation	O
(	O
the	O
smoothing	O
penalty	O
can	O
be	O
viewed	O
as	O
being	O
induced	O
by	O
a	O
prior	O
on	O
the	O
)	O
.	O
</s>
<s>
As	O
(	O
no	O
smoothing	O
)	O
,	O
the	O
smoothing	B-Algorithm
spline	I-Algorithm
converges	O
to	O
the	O
interpolating	B-Algorithm
spline	I-Algorithm
.	O
</s>
<s>
As	O
(	O
infinite	O
smoothing	O
)	O
,	O
the	O
roughness	O
penalty	O
becomes	O
paramount	O
and	O
the	O
estimate	O
converges	O
to	O
a	O
linear	B-General_Concept
least	I-General_Concept
squares	I-General_Concept
estimate	O
.	O
</s>
<s>
The	O
roughness	O
penalty	O
based	O
on	O
the	O
second	B-Algorithm
derivative	I-Algorithm
is	O
the	O
most	O
common	O
in	O
modern	O
statistics	O
literature	O
,	O
although	O
the	O
method	O
can	O
easily	O
be	O
adapted	O
to	O
penalties	O
based	O
on	O
other	O
derivatives	O
.	O
</s>
<s>
It	O
is	O
useful	O
to	O
think	O
of	O
fitting	O
a	O
smoothing	B-Algorithm
spline	I-Algorithm
in	O
two	O
steps	O
:	O
</s>
<s>
Given	O
the	O
vector	O
of	O
fitted	O
values	O
,	O
the	O
sum-of-squares	O
part	O
of	O
the	O
spline	B-Algorithm
criterion	O
is	O
fixed	O
.	O
</s>
<s>
It	O
remains	O
only	O
to	O
minimize	O
,	O
and	O
the	O
minimizer	O
is	O
a	O
natural	B-Algorithm
cubic	I-Algorithm
spline	I-Algorithm
that	O
interpolates	O
the	O
points	O
.	O
</s>
<s>
where	O
are	O
a	O
set	O
of	O
spline	B-Algorithm
basis	O
functions	O
.	O
</s>
<s>
,	O
and	O
,	O
the	O
distances	O
between	O
successive	O
knots	B-Algorithm
(	O
or	O
x	O
values	O
)	O
.	O
</s>
<s>
In	O
practice	O
,	O
since	O
cubic	B-Algorithm
splines	I-Algorithm
are	O
mostly	O
used	O
,	O
is	O
usually	O
.	O
</s>
<s>
For	O
,	O
when	O
approaches	O
,	O
converges	O
to	O
the	O
"	O
natural	O
"	O
spline	B-Algorithm
interpolant	O
to	O
the	O
given	O
data	O
.	O
</s>
<s>
Having	O
means	O
the	O
solution	O
is	O
the	O
"	O
natural	O
"	O
spline	B-Algorithm
interpolant	O
.	O
</s>
<s>
The	O
first	O
approach	O
simply	O
generalizes	O
the	O
spline	B-Algorithm
smoothing	I-Algorithm
penalty	O
to	O
the	O
multidimensional	O
setting	O
.	O
</s>
<s>
The	O
thin	B-Algorithm
plate	I-Algorithm
spline	I-Algorithm
approach	O
can	O
be	O
generalized	O
to	O
smoothing	O
with	O
respect	O
to	O
more	O
than	O
two	O
dimensions	O
and	O
to	O
other	O
orders	O
of	O
differentiation	O
in	O
the	O
penalty	O
.	O
</s>
<s>
The	O
thin	B-Algorithm
plate	I-Algorithm
splines	I-Algorithm
are	O
isotropic	O
,	O
meaning	O
that	O
if	O
we	O
rotate	O
the	O
co-ordinate	O
system	O
the	O
estimate	O
will	O
not	O
change	O
,	O
but	O
also	O
that	O
we	O
are	O
assuming	O
that	O
the	O
same	O
level	O
of	O
smoothing	O
is	O
appropriate	O
in	O
all	O
directions	O
.	O
</s>
<s>
The	O
second	O
class	O
of	O
generalizations	O
to	O
multi-dimensional	O
smoothing	O
deals	O
directly	O
with	O
this	O
scale	O
invariance	O
issue	O
using	O
tensor	O
product	O
spline	B-Algorithm
constructions	O
.	O
</s>
<s>
Such	O
splines	B-Algorithm
have	O
smoothing	O
penalties	O
with	O
multiple	O
smoothing	O
parameters	O
,	O
which	O
is	O
the	O
price	O
that	O
must	O
be	O
paid	O
for	O
not	O
assuming	O
that	O
the	O
same	O
degree	O
of	O
smoothness	O
is	O
appropriate	O
in	O
all	O
directions	O
.	O
</s>
<s>
Smoothing	B-Algorithm
splines	I-Algorithm
are	O
related	O
to	O
,	O
but	O
distinct	O
from	O
:	O
</s>
<s>
Regression	O
splines	B-Algorithm
.	O
</s>
<s>
In	O
this	O
method	O
,	O
the	O
data	O
is	O
fitted	O
to	O
a	O
set	O
of	O
spline	B-Algorithm
basis	O
functions	O
with	O
a	O
reduced	O
set	O
of	O
knots	B-Algorithm
,	O
typically	O
by	O
least	B-Algorithm
squares	I-Algorithm
.	O
</s>
<s>
(	O
See	O
also	O
multivariate	B-General_Concept
adaptive	I-General_Concept
regression	I-General_Concept
splines	I-General_Concept
.	O
)	O
</s>
<s>
Penalized	O
splines	B-Algorithm
.	O
</s>
<s>
This	O
combines	O
the	O
reduced	O
knots	B-Algorithm
of	O
regression	O
splines	B-Algorithm
,	O
with	O
the	O
roughness	O
penalty	O
of	O
smoothing	B-Algorithm
splines	I-Algorithm
.	O
</s>
<s>
Thin	B-Algorithm
plate	I-Algorithm
splines	I-Algorithm
and	O
Elastic	B-Algorithm
maps	I-Algorithm
method	O
for	O
manifold	O
learning	O
.	O
</s>
<s>
This	O
method	O
combines	O
the	O
least	B-Algorithm
squares	I-Algorithm
penalty	O
for	O
approximation	O
error	O
with	O
the	O
bending	O
and	O
stretching	O
penalty	O
of	O
the	O
approximating	O
manifold	O
and	O
uses	O
the	O
coarse	O
discretization	O
of	O
the	O
optimization	O
problem	O
.	O
</s>
<s>
Source	O
code	O
for	O
spline	B-Algorithm
smoothing	I-Algorithm
can	O
be	O
found	O
in	O
the	O
examples	O
from	O
Carl	O
de	O
Boor	O
's	O
book	O
A	O
Practical	O
Guide	O
to	O
Splines	B-Algorithm
.	O
</s>
<s>
The	O
examples	O
are	O
in	O
the	O
Fortran	B-Application
programming	I-Application
language	I-Application
.	O
</s>
