Environmental Data Analysis with - PowerPoint Presentation

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Environmental Data Analysis with MatLab

Lecture 9:

Fourier Series

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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 Transform Lecture 12 Power SpectraLecture 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

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purpose of the lecture

detect and quantify periodicities in data

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importance of periodicities

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Stream FlowNeuse River

discharge,

cfstime, days365 days1 year

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Air temperature

Black Rock Forest

365 days1 year

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Air temperature

Black Rock Forest

time, days1 day

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temporal periodicitiesand their periods

astronomical

rotationdailyrevolutionyearlyother naturalocean wavesa few secondsanthropogenicelectric power60 Hz

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delay,

t

0amplitude, Cperiod, Td(t)time, tsinusoidal oscillationf(t) = C cos{ 2π (t-t0

) / T }

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amplitude, C

lingo

temporalf(t) = C cos{ 2π t / T }spatialf(x) = C cos{ 2π x / λ }

amplitude,

C

period,

T

wavelength,

λ

frequency,

f=1/T

angular frequency,

ω=2

π

/T

wavenumber

,

k=2

π

/

λ

-

f(t)

= C

cos

(

ω

t)

f(x)

= C

cos

(

kx

)

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spatial periodicitiesand their wavelengths

natural

sand duneshundreds of meterstree ringsa few millimetersanthropogenicfurrows plowedin a fieldfew tens of cm

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pairing sines and cosines

to avoid using time delays

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derived using trig identity

A

B

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A

B

A=C cos(ωt0)B=C sin(ωt0)A2=C cos2 (ωt0)B2=C sin2 (ωt0)

A

2

+B

2

=C

2

[cos

2

(

ω

t

0

)

+sin

2

(

ω

t

0

)]

= C

2

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Fourier Serieslinear model containing nothing but

sines

and cosines

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A’s and

B

’s aremodel parametersω’s are auxiliary variables

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two choices

values of frequencies?

total number of frequencies?

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surprising fact about time series with evenly sampled data

Nyquist

frequency

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values of frequencies? evenly spaced,

ω

n = (n-1)Δ ω minimum frequency of zero maximum frequency of fnytotal number of frequencies? N/2+1 number of model parameters, M = number of data, N

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implies

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Number of Frequencieswhy

N/2+1

and not N/2 ?first and last sine are omitted from the Fourier Series since they are identically zero:

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cos

(

Δω t)cos(0t)sin(Δωt)cos(2Δω t)sin(2Δω t)cos(½NΔω t)

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Nyquist

Sampling Theorem

when m=n+Nanother way of stating itnote evenly sampled times

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ω

n

= (n-1)Δ ω and tk = (k-1) ΔtStep 1: Insert discrete frequencies and times into l.h.s. of equations.

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ω

n

= (n-1+N)Δ ω and tk = (k-1) ΔtStep 2: Insert discrete frequencies and times into r.h.s. of equations.

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same as

l.h.s

.same as l.h.s.Step 3: Note that l.h.s equals r.h.s.

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when

m=

n+Nor when ωm=ωn+2ωnyonly a 2ωny interval of the ω -axis is uniquesay from-ωny to +ωny Step 4: Identify unique region of ω-axis

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cos

(

ω t) has same shape as cos(-ω t) andsin(ω t) has same shape as sin(-ω t) so really only the0 to ωnypart of the ω-axis is unique Step 5: Apply symmetry of sines and cosines

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w

-

wnywny2wny3wny0

equivalent points on the

ω

-axis

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d

2

(t)d1(t)time, ttime, td1 (t) = cos(w1t), with w1=2Dwd2(t) = cos{w2t}, with w2=(2+N)Dw,

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problem of aliasing

high frequencies

masquerading as low frequenciessolution:pre-process data to remove high frequenciesbefore digitizing it

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Discrete Fourier Series

d

= Gm

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Least

S

quares Solution mest = [GTG]-1 GTdhas substantial simplification… since it can be shown that …

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% N = number of data, presumed even% Dt

is time sampling interval

t = Dt*[0:N-1]’;Df = 1 / (N * Dt );Dw = 2 * pi / (N * Dt);Nf = N/2+1;Nw = N/2+1;f = Df*[0:N/2];w = Dw*[0:N/2];frequency and time setupin MatLab

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% set up G G=zeros(N,M); % zero frequency column

G(:,1)=1;

% interior M/2-1 columns for i = [1:M/2-1] j = 2*i; k = j+1; G(:,j)=cos(w(i+1).*t); G(:,k)=sin(w(i+1).*t); end % nyquist column G(:,M)=cos(w(Nw).*t);Building G in MatLab

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gtgi = 2* ones(M,1)/N; gtgi

(1)=1/N;

gtgi(M)=1/N; mest = gtgi .* (G'*d);solving for model parameters in MatLab

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how to plot the model parameters?

A

’s and B ’splotagainst frequency

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power spectral density

big at frequency

ω whenwhen sine or cosine at the frequencyhas a large coefficient

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alternatively, plot

amplitude spectral density

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365.2

days

period, daysfrequency, cycles per dayamplitude spectral density182.6 days60.0 daysamplitude spectral densityStream FlowNeuse River

all interesting frequencies

near origin,

so plot period,

T=1/f

instead