Applications of Filters
Running Average of length M
f=
-1
M
[1, 1, 1, …
T
1]
f
M-1
0
t0
tM-1
time, t
Note that average is “delayed” …
… since no future values of x are available
1
x
0
t0
tM-1
Could be ‘recentered’ during
plotting’
M-1
f*x
0
t0
tM-1
center
time, t
MatLab Example
M = 30; % length of filter
f = (1/M) * ones(M,1);
y = conv(f,T);
Note MatLab has an built-in
convolution function
If x has length N and y has length M
then x*y has length N+M-1
Laguardia Airport Mean Temperature
2 years of data
Blue: daily mean temperature data
Red: 14 day running average
Laguardia Airport Mean Temperature
2 years of data
Blue: daily mean temperature data
Red: 30 day running average
Laguardia Airport Mean Temperature
20 years of data
Blue: daily mean temperature data
Red: 365 day running average
Note “edge effect”
Criticism of Running Average: it
has sharp edges
sharp edge
f
M-1
0
t0
tM-1
time, t
The filter can produce rough results, because, as the filter advances in
time, an outlier suddenly goes from being inside to outside
Gaussian running average
f
M-1
0
t0
tM-1
time, t
f
0
67% of area
between t0
and tM-1
t0
tM-1
time, t
Need to discard a bit of future here
MatLab Example
width = 14;
% 67 percent width
sigma = 14/2;
% standard deviation
M = 6*round(sigma/dt); % truncate filter at +/- 3 sigma
delay = 3*sigma; % center of filter
f = exp( -(t(1:M)-delay).^2 / (2*sigma*sigma) );
amp = sum(f); % normalize to unit amplitude
f = f/amp;
14 day Gaussian filter
14 days
Laguardia Airport Mean Temperature
2 years of data
Blue: daily mean temperature data
Red: 14 day Gaussian running average
Laguardia Airport Mean Temperature
“Box car”
“Gaussian”
Blue: daily mean temperature data
Red: 14 day Gaussian running average
Exponential was easy to implement
f
recursively
M-1
0
t0
tM-1
f
M-1
0
time, t
67% of area
between t0
and tM-1
t0
tM-1
time, t
Some calculations
If f(t) = c-1 exp( -t/c )
What value of c puts A1 fraction of area between 0 and t1?
t1
A = 0 c-1 exp( -t/c ) dt = exp(-t/c)|
Note A(t=)=1
So A1 = 1- exp(-t1/c)
(1-A1) = exp(-t1/c)
ln(1-A1) = (-t1/c)
c = -t1 / ln(1-A1)
and t1 = -c / ln(1-A1)
t1
0 =
1- exp(-t1/c)
MatLab Code
width = 14; % 67% width
c = -width/log(1.0-0.67); % constant in exponential
M = 6*round(width/dt); % truncate filter at 3 widths
delay = -width/log(1.0-0.50); % delay at 50% of area
f = exp( -t(1:M)/c );
amp = sum(f); % normalize to sum to unity
f = f/amp;
Laguardia Airport Mean Temperature
“Gaussian”
“exponential”
Blue: daily mean temperature data
Red: 14 day Gaussian running average
first derivative filter
f = Dt [1, -1]T
But note delayed by ½Dt
M=2;
f = dt*[1,-1]';
delay = 0.5*dt;
second derivative filter
f=
2
(Dt)
[1, -2,
T
1]
But note delayed by Dt
First-derivative
note more-variable in winter
second-derivative
note more-variable in winter
prediction error filter
x = [x1, x2, x3, x4, … xN-1]T
f = [-1, f1, f2, f3, f4, … fN-1]T
Choose f such that f*x  0
f5, f4, f3, f2, f1, -1
… xM-2, xM-2, xM-1, xM, xM+1, xM+2, xM+3, …
xM = f1xM-1 + f2xM-2 + f3xM-3 …
xM predicted from past values
MatLab Code
M=10;
G = zeros(N+1,M);
d = zeros(N+1,1);
% filter of length M, data of length N
% solve by least-squares
% implement condition f0=1
% as if its prior information
for p = [1:M]
G(p:N,p) = T(1:N-p+1);
d(p)=0;
end
% usual G matrix for filter
G(N+1,1)=1e6;
d(N+1)=1e6;
% prior info, with epsilon=1e6
f = inv(G'*G)*G'*d;
% least-squares solution
y = conv(f,T);
% try out fileter, y is prediction error
% d vector is all zero
filter length M = 10 days
filter, f
filter length M = 10 days
x
f*x
error =
6.4105
filter length M = 100 days
filter, f
filter length M = 10 days
x
f*x
error =
6.2105
f*x is the unpredictable part of x
Let’s try it with the Neuse River
Hydrograph Dataset
What’s that?
Filter length
M=100
x
f*x
Close up of first year of data
Note that the prediction error, f*x, is
spikier than the hydrograph data, x.
I think that this means that some of
the dynamics of the river flow is
being captured by the filter, f, and
that the unpredictable part is mostly
the forcing, that is, precipitation
x
f*x