Mathematical Models of Spectral Density Function for Two ...
Environmental Data Analysis with MatLab Lecture 12: Power Spectral Density.
-
Upload
frank-simmons -
Category
Documents
-
view
232 -
download
3
Transcript of Environmental Data Analysis with MatLab Lecture 12: Power Spectral Density.
![Page 1: Environmental Data Analysis with MatLab Lecture 12: Power Spectral Density.](https://reader035.fdocuments.net/reader035/viewer/2022081506/56649d205503460f949f52f0/html5/thumbnails/1.jpg)
Environmental Data Analysis with MatLab
Lecture 12:
Power Spectral Density
![Page 2: Environmental Data Analysis with MatLab Lecture 12: Power Spectral Density.](https://reader035.fdocuments.net/reader035/viewer/2022081506/56649d205503460f949f52f0/html5/thumbnails/2.jpg)
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
![Page 3: Environmental Data Analysis with MatLab Lecture 12: Power Spectral Density.](https://reader035.fdocuments.net/reader035/viewer/2022081506/56649d205503460f949f52f0/html5/thumbnails/3.jpg)
purpose of the lecture
compute and understand
Power Spectral Density
of indefinitely-long time series
![Page 4: Environmental Data Analysis with MatLab Lecture 12: Power Spectral Density.](https://reader035.fdocuments.net/reader035/viewer/2022081506/56649d205503460f949f52f0/html5/thumbnails/4.jpg)
Nov 27, 2000
Jan 4, 2011
ground vibrations at the Palisades NY seismographic station
similar appearance of measurements separated by 10+ years apart
time, minutes
time, minutes
![Page 5: Environmental Data Analysis with MatLab Lecture 12: Power Spectral Density.](https://reader035.fdocuments.net/reader035/viewer/2022081506/56649d205503460f949f52f0/html5/thumbnails/5.jpg)
stationary time series
indefinitely long
but
statistical properties don’t vary with time
![Page 6: Environmental Data Analysis with MatLab Lecture 12: Power Spectral Density.](https://reader035.fdocuments.net/reader035/viewer/2022081506/56649d205503460f949f52f0/html5/thumbnails/6.jpg)
time, minutes
assume that we are dealing with a fragment of an indefinitely long time series
time series, dduration, Tlength, N
![Page 7: Environmental Data Analysis with MatLab Lecture 12: Power Spectral Density.](https://reader035.fdocuments.net/reader035/viewer/2022081506/56649d205503460f949f52f0/html5/thumbnails/7.jpg)
one quantity that might be stationary is …
![Page 8: Environmental Data Analysis with MatLab Lecture 12: Power Spectral Density.](https://reader035.fdocuments.net/reader035/viewer/2022081506/56649d205503460f949f52f0/html5/thumbnails/8.jpg)
“Power”
0T
![Page 9: Environmental Data Analysis with MatLab Lecture 12: Power Spectral Density.](https://reader035.fdocuments.net/reader035/viewer/2022081506/56649d205503460f949f52f0/html5/thumbnails/9.jpg)
0T
Power
mean-squared amplitude of time series
![Page 10: Environmental Data Analysis with MatLab Lecture 12: Power Spectral Density.](https://reader035.fdocuments.net/reader035/viewer/2022081506/56649d205503460f949f52f0/html5/thumbnails/10.jpg)
How is power related topower spectral density ?
![Page 11: Environmental Data Analysis with MatLab Lecture 12: Power Spectral Density.](https://reader035.fdocuments.net/reader035/viewer/2022081506/56649d205503460f949f52f0/html5/thumbnails/11.jpg)
write Fourier Series asd = Gmwere m are the Fourier coefficients
![Page 12: Environmental Data Analysis with MatLab Lecture 12: Power Spectral Density.](https://reader035.fdocuments.net/reader035/viewer/2022081506/56649d205503460f949f52f0/html5/thumbnails/12.jpg)
now use
![Page 13: Environmental Data Analysis with MatLab Lecture 12: Power Spectral Density.](https://reader035.fdocuments.net/reader035/viewer/2022081506/56649d205503460f949f52f0/html5/thumbnails/13.jpg)
now use
coefficients of sines and cosines
coefficients of complex exponentials
Fourier Transform
equals 2/T
![Page 14: Environmental Data Analysis with MatLab Lecture 12: Power Spectral Density.](https://reader035.fdocuments.net/reader035/viewer/2022081506/56649d205503460f949f52f0/html5/thumbnails/14.jpg)
so, if we define the power spectral density of a stationary time series as
the integral of the p.s.d. is the power in the time series
![Page 15: Environmental Data Analysis with MatLab Lecture 12: Power Spectral Density.](https://reader035.fdocuments.net/reader035/viewer/2022081506/56649d205503460f949f52f0/html5/thumbnails/15.jpg)
unitsif time series d has units of u
coefficients C also have units of u
Fourier Transform has units of u×time
power spectral density has units of u2×time2/time
e.g. u2-s or equivalently u2/Hz
![Page 16: Environmental Data Analysis with MatLab Lecture 12: Power Spectral Density.](https://reader035.fdocuments.net/reader035/viewer/2022081506/56649d205503460f949f52f0/html5/thumbnails/16.jpg)
we will assume that thepower spectral density
is a stationary quantity
![Page 17: Environmental Data Analysis with MatLab Lecture 12: Power Spectral Density.](https://reader035.fdocuments.net/reader035/viewer/2022081506/56649d205503460f949f52f0/html5/thumbnails/17.jpg)
when we measure the power spectral density of a finite-length time series,
we are making an estimate of the power spectral density of the indefinitely long time series
the two are not the samebecause of statistical fluctuation
![Page 18: Environmental Data Analysis with MatLab Lecture 12: Power Spectral Density.](https://reader035.fdocuments.net/reader035/viewer/2022081506/56649d205503460f949f52f0/html5/thumbnails/18.jpg)
finally
we will normally subtract out the mean of the time series
so that power spectral densityrepresents fluctuations about the
mean value
![Page 19: Environmental Data Analysis with MatLab Lecture 12: Power Spectral Density.](https://reader035.fdocuments.net/reader035/viewer/2022081506/56649d205503460f949f52f0/html5/thumbnails/19.jpg)
Example 1Ground vibration at Palisades NY
0 200 400 600 800 1000 1200 1400 1600
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
velo
city,
mic
rons/s
time, seconds
![Page 20: Environmental Data Analysis with MatLab Lecture 12: Power Spectral Density.](https://reader035.fdocuments.net/reader035/viewer/2022081506/56649d205503460f949f52f0/html5/thumbnails/20.jpg)
enlargement
0 5 10 15 20 25 30 35 40 45-0.4
-0.2
0
0.2
0.4
velo
city,
mic
rons/s
time, seconds
![Page 21: Environmental Data Analysis with MatLab Lecture 12: Power Spectral Density.](https://reader035.fdocuments.net/reader035/viewer/2022081506/56649d205503460f949f52f0/html5/thumbnails/21.jpg)
enlargement
0 5 10 15 20 25 30 35 40 45-0.4
-0.2
0
0.2
0.4
velo
city,
mic
rons/s
time, seconds
periods of a few seconds
![Page 22: Environmental Data Analysis with MatLab Lecture 12: Power Spectral Density.](https://reader035.fdocuments.net/reader035/viewer/2022081506/56649d205503460f949f52f0/html5/thumbnails/22.jpg)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.2
0.4
0.6
0.8
1
p.s
.d,
um2 /s
2 per
Hz
frequency, Hz
power spectral density
![Page 23: Environmental Data Analysis with MatLab Lecture 12: Power Spectral Density.](https://reader035.fdocuments.net/reader035/viewer/2022081506/56649d205503460f949f52f0/html5/thumbnails/23.jpg)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.2
0.4
0.6
0.8
1
p.s
.d,
um2 /s
2 per
Hz
frequency, Hz
power spectral density
frequencies of a few tenths of a Hzperiods of a few seconds
![Page 24: Environmental Data Analysis with MatLab Lecture 12: Power Spectral Density.](https://reader035.fdocuments.net/reader035/viewer/2022081506/56649d205503460f949f52f0/html5/thumbnails/24.jpg)
cumulative power
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.005
0.01
0.015
0.02
0.025
pow
er
frequency, Hz
power in time series
![Page 25: Environmental Data Analysis with MatLab Lecture 12: Power Spectral Density.](https://reader035.fdocuments.net/reader035/viewer/2022081506/56649d205503460f949f52f0/html5/thumbnails/25.jpg)
0 500 1000 1500 2000 2500 3000 3500 40000
1
2
x 104
time, days
disc
harg
e, c
fs
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.050
2
4
6
8
x 109
frequency, cycles per dayPS
D,
(cfs
)2 p
er c
ycle
/day
Example 2Neuse River Stream Flow
![Page 26: Environmental Data Analysis with MatLab Lecture 12: Power Spectral Density.](https://reader035.fdocuments.net/reader035/viewer/2022081506/56649d205503460f949f52f0/html5/thumbnails/26.jpg)
0 500 1000 1500 2000 2500 3000 3500 40000
1
2
x 104
time, days
disc
harg
e, c
fs
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.050
2
4
6
8
x 109
frequency, cycles per dayPS
D,
(cfs
)2 p
er c
ycle
/day
Example 2Neuse River Stream Flow
period of 1 year
![Page 27: Environmental Data Analysis with MatLab Lecture 12: Power Spectral Density.](https://reader035.fdocuments.net/reader035/viewer/2022081506/56649d205503460f949f52f0/html5/thumbnails/27.jpg)
0 500 1000 1500 2000 2500 3000 3500 40000
1
2
x 104
time, days
disc
harg
e, c
fs
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.050
2
4
6
8
x 109
frequency, cycles per dayPS
D,
(cfs
)2 p
er c
ycle
/day
power spectral density, s2(f)
frequency f, cycles/day
pow
er s
pect
ra d
ensi
tys2 (
f), (
cfs)
2 per
cyc
le/d
ay
![Page 28: Environmental Data Analysis with MatLab Lecture 12: Power Spectral Density.](https://reader035.fdocuments.net/reader035/viewer/2022081506/56649d205503460f949f52f0/html5/thumbnails/28.jpg)
0 500 1000 1500 2000 2500 3000 3500 40000
1
2
x 104
time, days
disc
harg
e, c
fs
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.050
2
4
6
8
x 109
frequency, cycles per dayPS
D,
(cfs
)2 p
er c
ycle
/day
power spectral density, s2(f)
frequency f, cycles/day
pow
er s
pect
ra d
ensi
tys2 (
f), (
cfs)
2 per
cyc
le/d
ay
period of 1 year
![Page 29: Environmental Data Analysis with MatLab Lecture 12: Power Spectral Density.](https://reader035.fdocuments.net/reader035/viewer/2022081506/56649d205503460f949f52f0/html5/thumbnails/29.jpg)
Example 3Atmospheric CO2
(after removing anthropogenic trend)
0 5 10 15 20 25 30 35 40 45 50
-4
-2
0
2
4
time, years
CO
2, p
pm
0 1 2 3 4 50
1
2
3
frequency, cycles per year
log1
0 ps
d of
CO
2
![Page 30: Environmental Data Analysis with MatLab Lecture 12: Power Spectral Density.](https://reader035.fdocuments.net/reader035/viewer/2022081506/56649d205503460f949f52f0/html5/thumbnails/30.jpg)
0 0.5 1 1.5 2 2.5 3-3
-2
-1
0
1
2
3
4
time, years
CO
2, p
pmenlargement
![Page 31: Environmental Data Analysis with MatLab Lecture 12: Power Spectral Density.](https://reader035.fdocuments.net/reader035/viewer/2022081506/56649d205503460f949f52f0/html5/thumbnails/31.jpg)
0 0.5 1 1.5 2 2.5 3-3
-2
-1
0
1
2
3
4
time, years
CO
2, p
pmenlargement
period of 1 year
![Page 32: Environmental Data Analysis with MatLab Lecture 12: Power Spectral Density.](https://reader035.fdocuments.net/reader035/viewer/2022081506/56649d205503460f949f52f0/html5/thumbnails/32.jpg)
0 5 10 15 20 25 30 35 40 45 50
-4
-2
0
2
4
time, years
CO
2, p
pm
0 1 2 3 4 50
1
2
3
frequency, cycles per year
log1
0 ps
d of
CO
2
power spectral density
frequency, cycles per year
![Page 33: Environmental Data Analysis with MatLab Lecture 12: Power Spectral Density.](https://reader035.fdocuments.net/reader035/viewer/2022081506/56649d205503460f949f52f0/html5/thumbnails/33.jpg)
0 5 10 15 20 25 30 35 40 45 50
-4
-2
0
2
4
time, years
CO
2, p
pm
0 1 2 3 4 50
1
2
3
frequency, cycles per year
log1
0 ps
d of
CO
2
power spectral density
frequency, cycles per year
1 year period ½ year
period
![Page 34: Environmental Data Analysis with MatLab Lecture 12: Power Spectral Density.](https://reader035.fdocuments.net/reader035/viewer/2022081506/56649d205503460f949f52f0/html5/thumbnails/34.jpg)
0 0.5 1 1.5 2 2.5 3-3
-2
-1
0
1
2
3
4
time, years
CO
2, p
pm
![Page 35: Environmental Data Analysis with MatLab Lecture 12: Power Spectral Density.](https://reader035.fdocuments.net/reader035/viewer/2022081506/56649d205503460f949f52f0/html5/thumbnails/35.jpg)
0 0.5 1 1.5 2 2.5 3-3
-2
-1
0
1
2
3
4
time, years
CO
2, p
pm
shallow side: 1 year and year½ out of phase steep side: 1 year and ½year in phase
![Page 36: Environmental Data Analysis with MatLab Lecture 12: Power Spectral Density.](https://reader035.fdocuments.net/reader035/viewer/2022081506/56649d205503460f949f52f0/html5/thumbnails/36.jpg)
0 1 2 3 4 5 60
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
frequency, cycles per year
pow
er
cumulative power
power in time series
![Page 37: Environmental Data Analysis with MatLab Lecture 12: Power Spectral Density.](https://reader035.fdocuments.net/reader035/viewer/2022081506/56649d205503460f949f52f0/html5/thumbnails/37.jpg)
Example 3:Tides
0 20 40 60 80 100 120-3
-2
-1
0
1
2
3
4
5
Ele
vation,
ft
time, days
90 days of data
![Page 38: Environmental Data Analysis with MatLab Lecture 12: Power Spectral Density.](https://reader035.fdocuments.net/reader035/viewer/2022081506/56649d205503460f949f52f0/html5/thumbnails/38.jpg)
enlargement
0 1 2 3 4 5 6 7-2
-1
0
1
2
3
4
Ele
vation,
ft
time, days
7 days of data
![Page 39: Environmental Data Analysis with MatLab Lecture 12: Power Spectral Density.](https://reader035.fdocuments.net/reader035/viewer/2022081506/56649d205503460f949f52f0/html5/thumbnails/39.jpg)
enlargement
0 1 2 3 4 5 6 7-2
-1
0
1
2
3
4
Ele
vation,
ft
time, days
7 days of data
period of day½
![Page 40: Environmental Data Analysis with MatLab Lecture 12: Power Spectral Density.](https://reader035.fdocuments.net/reader035/viewer/2022081506/56649d205503460f949f52f0/html5/thumbnails/40.jpg)
0 0.5 1 1.5 2 2.5 3-1
0
1
2
3
frequency, cycles per day
log1
0 ps
d
0 0.5 1 1.5 2 2.5 3-1
-0.5
0
0.5
1
frequency, cycles per day
log1
0 ps
d
power spectral density
cumulative power
power in time series
![Page 41: Environmental Data Analysis with MatLab Lecture 12: Power Spectral Density.](https://reader035.fdocuments.net/reader035/viewer/2022081506/56649d205503460f949f52f0/html5/thumbnails/41.jpg)
0 0.5 1 1.5 2 2.5 3-1
0
1
2
3
frequency, cycles per day
log1
0 ps
d
0 0.5 1 1.5 2 2.5 3-1
-0.5
0
0.5
1
frequency, cycles per day
log1
0 ps
d
power spectral density
cumulative power
power in time series
about ½ day period
about1 day period
fortnighly(2 wk) tide
![Page 42: Environmental Data Analysis with MatLab Lecture 12: Power Spectral Density.](https://reader035.fdocuments.net/reader035/viewer/2022081506/56649d205503460f949f52f0/html5/thumbnails/42.jpg)
MatLab
dtilde= Dt*fft(d-mean(d));
dtilde = dtilde(1:Nf);
psd = (2/T)*abs(dtilde).^2;
Fourier Transform
delete negative frequencies
power spectral density
![Page 43: Environmental Data Analysis with MatLab Lecture 12: Power Spectral Density.](https://reader035.fdocuments.net/reader035/viewer/2022081506/56649d205503460f949f52f0/html5/thumbnails/43.jpg)
MatLab
pwr=df*cumsum(psd);
Pf=df*sum(psd);
Pt=sum(d.^2)/N;
power as a function of frequency
total power
total power
should be the same!