<s>
In	O
statistics	O
and	O
image	B-Algorithm
processing	I-Algorithm
,	O
to	O
smooth	O
a	O
data	B-General_Concept
set	I-General_Concept
is	O
to	O
create	O
an	O
approximating	O
function	O
that	O
attempts	O
to	O
capture	O
important	O
patterns	O
in	O
the	O
data	O
,	O
while	O
leaving	O
out	O
noise	O
or	O
other	O
fine-scale	O
structures/rapid	O
phenomena	O
.	O
</s>
<s>
In	O
smoothing	B-Application
,	O
the	O
data	O
points	O
of	O
a	O
signal	O
are	O
modified	O
so	O
individual	O
points	O
higher	O
than	O
the	O
adjacent	O
points	O
(	O
presumably	O
because	O
of	O
noise	O
)	O
are	O
reduced	O
,	O
and	O
points	O
that	O
are	O
lower	O
than	O
the	O
adjacent	O
points	O
are	O
increased	O
leading	O
to	O
a	O
smoother	O
signal	O
.	O
</s>
<s>
Smoothing	B-Application
may	O
be	O
used	O
in	O
two	O
important	O
ways	O
that	O
can	O
aid	O
in	O
data	O
analysis	O
(	O
1	O
)	O
by	O
being	O
able	O
to	O
extract	O
more	O
information	O
from	O
the	O
data	O
as	O
long	O
as	O
the	O
assumption	O
of	O
smoothing	B-Application
is	O
reasonable	O
and	O
(	O
2	O
)	O
by	O
being	O
able	O
to	O
provide	O
analyses	O
that	O
are	O
both	O
flexible	O
and	O
robust	O
.	O
</s>
<s>
Many	O
different	O
algorithms	O
are	O
used	O
in	O
smoothing	B-Application
.	O
</s>
<s>
Smoothing	B-Application
may	O
be	O
distinguished	O
from	O
the	O
related	O
and	O
partially	O
overlapping	O
concept	O
of	O
curve	B-Algorithm
fitting	I-Algorithm
in	O
the	O
following	O
ways	O
:	O
</s>
<s>
curve	B-Algorithm
fitting	I-Algorithm
often	O
involves	O
the	O
use	O
of	O
an	O
explicit	O
function	O
form	O
for	O
the	O
result	O
,	O
whereas	O
the	O
immediate	O
results	O
from	O
smoothing	B-Application
are	O
the	O
"	O
smoothed	B-Application
"	O
values	O
with	O
no	O
later	O
use	O
made	O
of	O
a	O
functional	O
form	O
if	O
there	O
is	O
one	O
;	O
</s>
<s>
the	O
aim	O
of	O
smoothing	B-Application
is	O
to	O
give	O
a	O
general	O
idea	O
of	O
relatively	O
slow	O
changes	O
of	O
value	O
with	O
little	O
attention	O
paid	O
to	O
the	O
close	O
matching	O
of	O
data	O
values	O
,	O
while	O
curve	B-Algorithm
fitting	I-Algorithm
concentrates	O
on	O
achieving	O
as	O
close	O
a	O
match	O
as	O
possible	O
.	O
</s>
<s>
smoothing	B-Application
methods	O
often	O
have	O
an	O
associated	O
tuning	O
parameter	O
which	O
is	O
used	O
to	O
control	O
the	O
extent	O
of	O
smoothing	B-Application
.	O
</s>
<s>
Curve	B-Algorithm
fitting	I-Algorithm
will	O
adjust	O
any	O
number	O
of	O
parameters	O
of	O
the	O
function	O
to	O
obtain	O
the	O
'	O
best	O
 '	O
fit	O
.	O
</s>
<s>
In	O
the	O
case	O
that	O
the	O
smoothed	B-Application
values	O
can	O
be	O
written	O
as	O
a	O
linear	B-Architecture
transformation	I-Architecture
of	O
the	O
observed	O
values	O
,	O
the	O
smoothing	B-Application
operation	O
is	O
known	O
as	O
a	O
linear	O
smoother	O
;	O
the	O
matrix	O
representing	O
the	O
transformation	O
is	O
known	O
as	O
a	O
smoother	O
matrix	O
or	O
hat	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
The	O
operation	O
of	O
applying	O
such	O
a	O
matrix	O
transformation	O
is	O
called	O
convolution	B-Language
.	O
</s>
<s>
Thus	O
the	O
matrix	O
is	O
also	O
called	O
convolution	B-Language
matrix	O
or	O
a	O
convolution	B-Language
kernel	O
.	O
</s>
<s>
In	O
the	O
case	O
of	O
simple	O
series	O
of	O
data	O
points	O
(	O
rather	O
than	O
a	O
multi-dimensional	O
image	O
)	O
,	O
the	O
convolution	B-Language
kernel	O
is	O
a	O
one-dimensional	O
vector	O
.	O
</s>
<s>
In	O
image	B-Algorithm
processing	I-Algorithm
and	O
computer	B-Application
vision	I-Application
,	O
smoothing	B-Application
ideas	O
are	O
used	O
in	O
scale	B-Algorithm
space	I-Algorithm
representations	I-Algorithm
.	O
</s>
<s>
The	O
simplest	O
smoothing	B-Application
algorithm	O
is	O
the	O
"	O
rectangular	O
"	O
or	O
"	O
unweighted	O
sliding-average	O
smooth	O
"	O
.	O
</s>
<s>
The	O
triangular	O
smooth	O
is	O
like	O
the	O
rectangular	O
smooth	O
except	O
that	O
it	O
implements	O
a	O
weighted	O
smoothing	B-Application
function	O
.	O
</s>
<s>
Some	O
specific	O
smoothing	B-Application
and	O
filter	O
types	O
,	O
with	O
their	O
respective	O
uses	O
,	O
pros	O
and	O
cons	O
are	O
:	O
</s>
<s>
+AlgorithmOverview	O
and	O
usesProsConsAdditive	O
smoothingused	O
to	O
smooth	O
categorical	O
data.Butterworth	O
filterSlower	O
roll-off	B-Algorithm
than	O
a	O
Chebyshev	O
Type	O
I/Type	O
II	O
filter	O
or	O
an	O
elliptic	O
filter	O
More	O
linear	O
phase	O
response	O
in	O
the	O
passband	B-Algorithm
than	O
Chebyshev	O
Type	O
I/Type	O
II	O
and	O
elliptic	O
filters	O
can	O
achieve	O
.	O
</s>
<s>
Designed	O
to	O
have	O
a	O
frequency	O
response	O
as	O
flat	O
as	O
possible	O
in	O
the	O
passband	B-Algorithm
.	O
</s>
<s>
requires	O
a	O
higher	O
order	O
to	O
implement	O
a	O
particular	O
stopband	O
specificationChebyshev	O
filterHas	O
a	O
steeper	O
roll-off	B-Algorithm
and	O
more	O
passband	B-Algorithm
ripple	O
(	O
type	O
I	O
)	O
or	O
stopband	O
ripple	O
(	O
type	O
II	O
)	O
than	O
Butterworth	O
filters	O
.	O
</s>
<s>
Minimizes	O
the	O
error	O
between	O
the	O
idealized	O
and	O
the	O
actual	O
filter	O
characteristic	O
over	O
the	O
range	O
of	O
the	O
filter	O
Contains	O
ripples	O
in	O
the	O
passband.Digital	O
filterUsed	O
on	O
a	O
sampled	B-Algorithm
,	O
discrete-time	O
signal	O
to	O
reduce	O
or	O
enhance	O
certain	O
aspects	O
of	O
that	O
signalElliptic	O
filterExponential	O
smoothing	B-Application
Used	O
to	O
reduce	O
irregularities	O
(	O
random	O
fluctuations	O
)	O
in	O
time	O
series	O
data	O
,	O
thus	O
providing	O
a	O
clearer	O
view	O
of	O
the	O
true	O
underlying	O
behaviour	O
of	O
the	O
series	O
.	O
</s>
<s>
Also	O
,	O
provides	O
an	O
effective	O
means	O
of	O
predicting	O
future	O
values	O
of	O
the	O
time	O
series	O
(	O
forecasting	O
)	O
.Easton	O
,	O
V	O
.	O
J.	O
;	O
&	O
McColl	O
,	O
J	O
.	O
H	O
.	O
(	O
1997	O
)	O
"	O
Time	O
series	O
"	O
,	O
STEPS	O
Statistics	O
GlossaryKalman	O
filter	O
Uses	O
a	O
series	O
of	O
measurements	O
observed	O
over	O
time	O
,	O
containing	O
statistical	B-General_Concept
noise	I-General_Concept
and	O
other	O
inaccuracies	O
by	O
estimating	O
a	O
joint	O
probability	O
distribution	O
over	O
the	O
variables	O
for	O
each	O
timeframe.Estimates	O
of	O
unknown	O
variables	O
it	O
produces	O
tend	O
to	O
be	O
more	O
accurate	O
than	O
those	O
based	O
on	O
a	O
single	O
measurement	O
aloneKernel	O
smoother	O
used	O
to	O
estimate	O
a	O
real	O
valued	O
function	O
as	O
the	O
weighted	O
average	O
of	O
neighboring	O
observed	O
data	O
.	O
</s>
<s>
Uses	O
a	O
series	O
of	O
iterations	B-Algorithm
of	O
a	O
moving	O
average	O
filter	O
of	O
length	O
m	O
,	O
where	O
m	O
is	O
a	O
positive	O
,	O
odd	O
integer	O
.	O
</s>
<s>
Software	O
implementations	O
for	O
time	O
series	O
,	O
longitudinal	O
and	O
spatial	O
data	O
have	O
been	O
developed	O
in	O
the	O
popular	O
statistical	O
package	O
R	B-Language
,	O
which	O
facilitate	O
the	O
use	O
of	O
the	O
KZ	O
filter	O
and	O
its	O
extensions	O
in	O
different	O
areas.Laplacian	O
smoothingalgorithm	O
to	O
smooth	O
a	O
polygonal	B-Algorithm
mesh	I-Algorithm
..	O
Local	O
regression	O
also	O
known	O
as	O
"	O
loess	O
"	O
or	O
"	O
lowess	O
"	O
a	O
generalization	O
of	O
moving	O
average	O
and	O
polynomial	O
regression	O
.	O
</s>
<s>
Used	O
for	O
continuous	O
time	O
realization	O
and	O
discrete	O
time	O
realization.Moving	O
average	O
A	O
calculation	O
to	O
analyze	O
data	O
points	O
by	O
creating	O
a	O
series	O
of	O
averages	O
of	O
different	O
subsets	O
of	O
the	O
full	O
data	B-General_Concept
set	I-General_Concept
.	O
</s>
<s>
a	O
smoothing	B-Application
technique	O
used	O
to	O
make	O
the	O
long	O
term	O
trends	O
of	O
a	O
time	O
series	O
clearer	O
.	O
</s>
<s>
engineers	O
use	O
the	O
stretched	B-Algorithm
grid	I-Algorithm
method	I-Algorithm
to	O
design	O
tents	O
and	O
other	O
tensile	O
structures	O
.	O
</s>
