Post on 16-Dec-2015
description
Emery & Thomson (2005) : Chapter 5: Time Series Analysis Methods Basic Concepts (375-378) Correlation Function (p. 378-384) Fourier Analysis (p.384-388)
Fourier Series
. any function could be written as an infinite sum of the trigonometric functions cosine and sine..
purpose of the lecture
detect and quantify periodicities in data
importance of periodicities
Time Series SSH & SST (2007-2012)
Annual Fundamental period Strength of amplitude ?
Time Series Zonal & Merid Current (2007-2012)
Fundamental period ? Strength of amplitude ?
Decomposition of SST time series
Periodicity: Inter-annual, Annual, Semi-annual, Intra-seasonal
Temporal Periodicities and their periods
astronomical
rotation daily
revolution
yearly
..
other natural
ocean waves a few seconds
ocean currents days months
.
anthropogenic
electric power 60 Hz
..
0 10 20 30 40 50 60 70 80 90-3
-2
-1
0
1
2
3
time, t
d(t)
cosine example
delay, t0
amplitude, C
period, T
d(t)
time, t
sinusoidal oscillation f(t) = C cos{ 2 (t-t0) / T }
amplitude, C
Periodicities
temporal f(t) = C cos{ 2 t / T } spatial f(x) = C cos{ 2 x / }
amplitude, Cperiod, T wavelength,
frequency, f=1/Tangular frequency, =2 /T wavenumber, k=2 / -
f(t) = C cos(t) f(x) = C cos(kx)
spatial periodicities and their wavelengths
natural
sand dunes hundreds of meters
tree rings a few millimeters
anthropogenic
furrows plowed in a field
few tens of cm
pairing sines and cosines to avoid using time delays
derived using trigonometri identity
A B
A BA=C cos(t0)B=C sin(t0)
A2=C cos2 (t0)B2=C sin2 (t0)
A2+B2=C2 [cos2 (t0)+sin2 (t0)]= C2
Fourier Series linear model containing nothing but
sines and cosines
As and Bs are model parameters
s are auxiliary variables
two choices
values of frequencies?
total number of frequencies?
surprising fact about time series with evenly sampled data
Nyquist frequency
values of frequencies? evenly spaced, n = (n-1) minimum frequency of zero maximum frequency of fny
total number of frequencies? N/2+1
number of model parameters, M = number of data, N
implies
Number of Frequencies why N/2+1 and not N/2 ?
first and last sine are omitted from the
Fourier Series since they are identically zero:
-2 0 20
5
10
15
20
25
30
col 1
time,
s
-2 0 20
5
10
15
20
25
30
col 2-2 0 20
5
10
15
20
25
30
col 3-2 0 20
5
10
15
20
25
30
col 4-2 0 20
5
10
15
20
25
30
col 5-2 0 20
5
10
15
20
25
30
col 32cos( t)cos(0t) sin(t) cos(2 t) sin(2 t)cos(N t)
Nyquist Sampling Theorem
when m=n+Nanother way of stating itnote evenly
sampled times
problem of aliasing
high frequencies
masquerading as low frequencies
solution: pre-process data to remove high
frequencies before digitizing it
Discrete Fourier Series
d = Gm
Least Squares Solution mest = [GTG]-1 GTd has substantial simplification
since it can be shown that
% 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 setup in MatLab
% 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
gtgi = 2* ones(M,1)/N; gtgi(1)=1/N; gtgi(M)=1/N; mest = gtgi .* (G'*d);
solving for model parameters in MatLab
how to plot the model parameters? As and B s plot
against frequency
power spectral density
big at frequency when when sine or cosine at the frequency
has a large coefficient
alternatively, plot amplitude spectral density
Tambahan . Transformasi Fourier
2. Fourier Transform
Transformasi Fourier adalah transformasi matematika dengan banyak aplikasi dalam fisika dan teknik.
Mengubah fungsi matematika waktu f(t) menjadi fungsi baru F atau dengan argumen frekuensi dg satuan siklus/detik (hertz).
Fungsi baru ini dikenal sebagai Transformasi Fourier atau spektrum frekuensi dari fungsi f.
f disebut domain waktu, dan F adalah domain frekuensi. Transformasi Fourier dinyatakan sebagai :
Variabel bebas x mewakili waktu (detik) dan variabel transformasi merupakan frekuensi (hertz).
untuk setiap bilangan riil (zeta)
Transformasi Fourier menghubungkan fungsi domain waktu f (ditampilkan dalam warna merah)
Visualisasi dari Transformasi Fourier
time
terhadap fungsi domain frekuensi (ditampilkan dalam warna biru)
Visualisasi dari Transformasi Fourier
Komponen-komponen frekuensi tersebar di seluruh spektrum frekuensi
Visualisasi dari Transformasi Fourier
Komponen-komponen frekuensi tersebar di seluruh spektrum frekuensi
Visualisasi dari Transformasi Fourier
Komponen-komponen frekuensi tersebar di seluruh spektrum frekuensi
Visualisasi dari Transformasi Fourier
Yang ditampilkan sebagai puncak dalam domain frekuensi
Visualisasi dari Transformasi Fourier
frequency
Visualisasi dari Transformasi Fourier
Fungsi f dalam domain waktu
Hasil transformasi Fourier dalam domain frekuensi
Transformasi Fourier
frequency
time
14 day
27 day
27 day
14 day
Wattimena et al. (2014)
Time-series arus zonal
Time-series arus zonal