Environmental Data Analysis with MatLab
Lecture 13:
Filter Theory
SYLLABUS
Lecture 01
Lecture 02
Lecture 03
Lecture 04
Lecture 05
Lecture 06
Lecture 07
Lecture 08
Lecture 09
Lecture 10
Lecture 11
Lecture 12
Lecture 13
Lecture 14
Lecture 15
Lecture 16
Lecture 17
Lecture 18
Lecture 19
Lecture 20
Lecture 21
Lecture 22
Lecture 23
Lecture 24
Using MatLab
Looking At Data
Probability and Measurement Error
Multivariate Distributions
Linear Models
The Principle of Least Squares
Prior Information
Solving Generalized Least Squares Problems
Fourier Series
Complex Fourier Series
Lessons Learned from the Fourier Transform
Power Spectral Density
Filter Theory
Applications of Filters
Factor Analysis
Orthogonal functions
Covariance and Autocorrelation
Cross-correlation
Smoothing, Correlation and Spectra
Coherence; Tapering and Spectral Analysis
Interpolation
Hypothesis testing
Hypothesis Testing continued; F-Tests
Confidence Limits of Spectra, Bootstraps
purpose of the lecture
develop the
Linear Filter
as a way to describe the way past events
influence present-time observations
sometimes, no past history is needed
Thermometer measuring temperature θ(t)
Flame with time-varying heat h(t)
Flame instantaneously heats the thermometer
Thermometer retains no heat
θ(t)  h(t)
this idea described as a linear model
θ1= g 1 h 1
...
θ10= g 1 h 10
θ11= g 1 h11
θ12= g 1 h 12
...
θN= g 1 h N
where g 1 is a constant
But sometimes,
past history is needed
Thermometer measuring temperature θ(t)
Steel plate
Flame with time-varying heat h(t)
Heats takes time to seep through plate
Plate retains heat
θ(t=t’)  history of h(t) for time t<t’
But sometimes,
past history is needed
Thermometer measuring temperature q(t)
Steel plate
“causal”: only past
effects present (not
future)
Flame with time-varying heat h(t)
Heats takes time to seep through plate
Plate retains heat
q(t=t’)  history of h(t) for time t<t’
this idea described as a linear model
...
θ10= g 1 h 10 + g 2 h 9 + g 3 h 8 + g 4 h 7 + …
θ11= g 1 h 11 + g 2 h 10 + g 3 h 9 + g 4 h 8 + …
θ12= g 1 h 12 + g 2 h 11 + g 3 h 10 + g 4 h 9 + …
...
where g 1, g 2, g 3, … are constants
called a “filter”
Special case: temperature depends only on the time elapsed
since the flame was turned on, and not on the experiment as
performed on Monday or Wednesday
q(t)
q(t)
same
shape
experiment is “timeshift invariant”
this idea described as a linear model
...
θ10= g 1 h 10 + g 2 h 9 + g 3 h 8 + g 4 h 7 + …
θ11= g 1 h 11 + g 2 h 10 + g 3 h 9 + g 4 h 8 + …
θ12= g 1 h 12 + g 2 h 11 + g 3 h 10 + g 4 h 9 + …
...
coefficients all the same …
filter is time-shift invariant …
where g 1, g 2, g 3, … are constants
called a filter
this idea written as a summation
this idea written as a summation
output
input
filter
“convolution”, not
multiplication
this idea written as matrix equation
filter
output
input
we’ve heard the word “convolution” before
in Lecture 11
it’s the name of this integral
but the integral can be approximated as the
summation we’ve just seen
so, mathematically, what we’re doing
is
“convolution”
convolutions can be written two ways
...
θ10= g 1 h 10 + g 2 h 9 + g 3 h 8 + g 4 h 7 + …
θ11= g 1 h 11 + g 2 h 10 + g 3 h 9 + g 4 h 8 + …
θ12= g 1 h 12 + g 2 h 11 + g 3 h 10 + g 4 h 9 + …
...
g inside
matrix
h inside
matrix
implying that the convolution operation
is symmetric
g*h = h*g
meaning of the filter g
suppose the input is a spike
h = [1, 0, 0, 0 … 0]T
output
then the output is the filter
input
filter
so the filter represents the
“impulse response”
of the experiment
A) Input is spike
h(t), W
spike
0
time , t, after impulse
q(t), K
B) Output is impulse response
0
time , t, after impulse
A)
h(t), W
spike of amplitude, h(t0)
time, t
h(t0)g(t-t0)
q(t). K
B)
t0
t0
time, t
example: heat-generating layer
z-axis
soil
z
observation
point
z=0
soil
heat-generating
layer
known impulse response
based on known physics of heat transfer
known impulse response
soil density
soil heat capacity,
1500 kg/m3
800 J/kg-K
distance
from
center of
layer
soil thermal
diffusivity.
1.25×10-6
m2/s
A)
-3
Dtg(t)g(t)
x 10
5
htruehtrue(t)
(t), W/m2
0
0
50
100
150
B)
250
300
time, days
350
400
450
500
250
300
time, days
350
400
450
500
250
300
time, days
350
400
450
500
time t, days
10
0
-10
0
50
100
150
200
time t, days
C)
q trueqtrue1(t)
(t), K
200
0.5
0
0
50
100
150
200
time t, days
The Method
solve
using least-squares
… well, use damped least square, just in case
qobs(t)
(t), K
htruehtrue(t)
(t), W/m2 q obs
A)
0.5
0
0
50
100
150
200
B)
250
300
time, days
350
400
450
500
250
300
time, days
350
400
450
500
250
300
time, days
350
400
450
500
time t, days
10
0
-10
0
50
100
150
200
time t, days
C)
),
hest(t)
hest(t)
W/m2
10
0
-10
0
50
100
150
200
time t, days
qobs(t)
(t), K
htruehtrue(t)
(t), W/m2 q obs
noise in data …
A)
0.5
0
0
50
100
150
200
B)
250
300
time, days
350
400
450
500
250
300
time, days
350
400
450
500
250
300
time, days
350
time t, days
10
0
-10
0
50
100
150
200
time t, days
C)
… is amplified
),
hest(t)
hest(t)
W/m2
10
0
-10
0
50
100
150
200
time t, days
400
450
500
Try adding
prior information of smoothness
(minimize second derivative)
obs(t), K
q qobs(t)
htruehtrue(t)
(t), W/m2
W/m2
hest(t) ),
hest(t)
A)
0.5
0
0
50
100
150
200
250
300
time, days
350
400
450
500
250
300
time, days
350
400
450
500
time t, days
B)
10
0
-10
0
50
100
150
200
time t, days
C)
10
much less noise
0
-10
0
50
100
150
200
250
300
time, days
time t, days
350
400
450
500
Tips on
filter calculations with MatLab
you should avoid constructing the
matrix
G=
because it contains so many redundant
elements
Tip #1
use the conv() function
to calculate the convolution q = g * h
not the matrix multiplication q=Gh
tmp = conv(g, h);
q=tmp(1:N);
note that we truncate the output of
conv(), so that is has the same length,
N, as the input time series
Tip #2
use the bicg() solver, together with the function
filterrfun() to solve Fq=h by generalized least squares
not matrix division hest=(FTF)\(FTq)
it implements the matrix multiplications, GT(Gv), using the
timeseries g, and does not ever construct G
filterrfun() is described in the text