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Environmental Data Analysis with MatLab Lecture 13: Filter Theory
Environmental Data Analysis with MatLab
Lecture 13:
Filter Theory
Lecture 01 Using MatLabLecture 02 Looking At DataLecture 03 Probability and Measurement Error Lecture 04 Multivariate DistributionsLecture 05 Linear ModelsLecture 06 The Principle of Least SquaresLecture 07 Prior InformationLecture 08 Solving Generalized Least Squares ProblemsLecture 09 Fourier SeriesLecture 10 Complex Fourier SeriesLecture 11 Lessons Learned from the Fourier TransformLecture 12 Power Spectral DensityLecture 13 Filter Theory Lecture 14 Applications of Filters Lecture 15 Factor Analysis Lecture 16 Orthogonal functions Lecture 17 Covariance and AutocorrelationLecture 18 Cross-correlationLecture 19 Smoothing, Correlation and SpectraLecture 20 Coherence; Tapering and Spectral Analysis Lecture 21 InterpolationLecture 22 Hypothesis testing Lecture 23 Hypothesis Testing continued; F-TestsLecture 24 Confidence Limits of Spectra, Bootstraps
SYLLABUS
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
Flame with time-varying heat h(t)Thermometer measuring temperature θ(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 10θ11= g 1 h11θ12= g 1 h 12. . .θN= g 1 h N where g 1 is a constant
But sometimes,past history is needed
Flame with time-varying heat h(t)
Thermometer measuring temperature θ(t)
Heats takes time to seep through plate
Plate retains heatθ(t=t’) history of h(t) for time t<t’
Steel plate
But sometimes,past history is needed
Flame with time-varying heat h(t)
Thermometer measuring temperature q(t)
Heats takes time to seep through plate
Plate retains heat
q(t=t’) history of h(t) for time t<t’
Steel plate “causal”: only past effects present (not future)
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
same shape
q(t)
q(t)
experiment is “time-shift 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 + …. . .where g 1, g 2, g 3, … are constants called a filter
coefficients all the same …filter is time-shift invariant …
this idea written as a summation
this idea written as a summation
inputoutput
filter
“convolution”, not multiplication
this idea written as matrix equation
output inputfilter
we’ve heard the word “convolution” beforein 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 doingis
“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 spikeh = [1, 0, 0, 0 … 0]T
then the output is the filteroutput
input filter
so the filter represents the
“impulse response”
of the experiment
time , t, after impulse
q(t),
K
0
time , t, after impulse
h(t),
W
0
A) Input is spike
B) Output is impulse response
spike
h(t0)g(t-t0)
h(t),
W
time, t
t0
q(t).
K
time, tt0
spike of amplitude, h(t0)
A)
B)
example: heat-generating layer
soil
soil heat-generating layer
observation point
z=0
z-axis
z
known impulse response
based on known physics of heat transfer
known impulse response
soil density1500 kg/m3 soil heat capacity,800 J/kg-K soil thermal diffusivity.1.25×10-6 m2/s
distance from center of layer
0 50 100 150 200 250 300 350 400 450 5000
5
x 10-3
time, days
g(t)
0 50 100 150 200 250 300 350 400 450 500-10
0
10
time, days
htru
e(t)
0 50 100 150 200 250 300 350 400 450 5000
0.5
time, days
qtru
e1(t)
Dt g
(t)htr
ue(t)
, W/m
2q
true
(t), K
time t, days
time t, days
A)
B)
C) time t, days
The Method
solve
using least-squares… well, use damped least square, just in case
0 50 100 150 200 250 300 350 400 450 5000
0.5
time, days
qobs
(t)
0 50 100 150 200 250 300 350 400 450 500-10
0
10
time, days
htru
e(t)
0 50 100 150 200 250 300 350 400 450 500-10
0
10
time, days
hest
(t)q
obs (t
), K
htrue
(t), W
/m2 B)
C)
hest (t
) ),
W/m
2
time t, days
time t, days
time t, days
A)
0 50 100 150 200 250 300 350 400 450 5000
0.5
time, days
qobs
(t)
0 50 100 150 200 250 300 350 400 450 500-10
0
10
time, days
htru
e(t)
0 50 100 150 200 250 300 350 400 450 500-10
0
10
time, days
hest
(t)q
obs (t
), K
htrue
(t), W
/m2 B)
C)
hest (t
) ),
W/m
2
time t, days
time t, days
time t, days
A)
noise in data …
… is amplified
Try adding
prior information of smoothness
(minimize second derivative)
0 50 100 150 200 250 300 350 400 450 5000
0.5
time, days
qobs
(t)
0 50 100 150 200 250 300 350 400 450 500-10
0
10
time, days
htru
e(t)
0 50 100 150 200 250 300 350 400 450 500-10
0
10
time, days
hest
(t)q
obs (t
), K
htrue
(t), W
/m2 B)
C)
hest (t
) ), W
/m2
time t, days
time t, days
time t, days
A)
much less noise
Tips onfilter calculations with MatLab
you should avoid constructing the matrix
because it contains so many redundant elements
G =
Tip #1
use the conv() function to calculate the convolution q = g * h
not the matrix multiplication q=Ghtmp = 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
