Oceanography 540--Marine Geological Processes--Winter Quarter 2001
MATLAB Time Series Example
The MATLAB script timeseries.m can be used to repeat this example or used as a starting point for
further study.
In this example we will use a synthetic data set. To create our synthetic data we will make our unit of
time 1000 years = 1 ky and sample a 500,000 year record in 2 ky increments. We first create a vector
containing the times:
t=0:2:500;
We want this record to have multiple periodic components. To mimic the orbital forcing of the
Earth's climate, we create a vector d, of the same length as vector t and with components that have
periods of 100 ky, 41 ky and 21 ky, weighted 2.5:1.5:1, then plot it
echo off
axis ([0 500 -5 +8])
echo on
d=2.5*sin(2*pi*t/100)+1.5*sin(2*pi*t/41)+1*sin(2*pi*t/21);
plot(t,d)
()
We add some normally distributed random noise to the record to create a new record dn, and plot it
on the same graph:
dn=d+0.5*randn(size(d));
hold
plot(t,dn,'g')
()
We also add a long term trend to create a third record dnt, and again plot it on the same graph
dnt=dn+3*t/500;
plot(t,dnt,'r')
()
Our time series has 251 elements. To improve computation efficiency we will pad it with zeros and
extended it to 256 elements (a power of 2). This is done implicitly with the fft command. Then taking
the Fourier transform and plotting its real part:
DNT=fft(dnt,256);
hold off
plot(real(DNT))
()
By convention, the transform has elements equally spaced in frequency, with the first element
corresponding to f=0, then up to the Nyquist frequency at the middle of the record, then the negative
frequencies from large to small in the remaining elements. The negative frequencies contain
redundant information so we want to look only at the positive frequencies. These frequencies are
evenly spaced within the Nyquist interval (for our 2 ky sampling interval the Nyquist frequency is 1/4
ky):
f=0.25*(0:128)/128;
The power spectrum of the record is estimated as:
PNT=DNT .* conj(DNT)/(256*256);
(The norm of a complex number is the number multiplied by its complex conjugate.) We are
interested in the parts of this spectrum corresponding to positive frequencies
hold off
plot (f,PNT(1:129))
()
We easily see the peaks at frequencies of about .01, .025 and .05 per ky with the power in
approximate proportion to the amplitude squared (25:9:4 or approximately 6:2:1). Notice the power
present as the frequency goes to 0 corresponding to the long term (d.c.) trend. This can be removed
by detrending the data which for us is equivalent to working with the dn record. Applying the same
steps:
DN=fft(dn,256);
PN=DN .* conj(DN)/(256*256);
hold
plot (f,PN(1:129),'g')
()
We see that the low frequency contamination is removed. We will work with detrended dn record
from now on. Next we will apply a window-- a Hamming window in particular
z=hamming(251);
plot(z)
()
Multiplying the dn record by this window:
dnz=dn .* z';
plot(dnz)
()
We now take the power spectra of this windowed time series (PNZ) and compare to the situation with
no window (PN)
DNZ=fft(dnz,256);
PNZ=DNZ .* conj(DNZ)/(256*sum(z));
plot (f,PNZ(1:129))
hold
plot (f,PN(1:129),'g')
()
We are also interested in details of the time series itself. To extract parts of record, say the 21 ky
band, we need bandpass filter to let through frequencies between say .04 to .06/ky. Here we use a
Butterworth filter, one of the many possibilities:
[b,a]=butter(5,[.04/.25 .06/.25]);
plot (filter(b,a,dn))
()
We find ~500/21=24 cycles of approximately the right amplitude (1) once we are 5 cycles into series.
(There are ways of minimizing these end effects).
We also wanted to look at spectral characteristics of autocorrelated data:
dna=xcorr(dn);
plot (dna)
()
Zero lag is plotted in the middle of diagram, with negative lag to the left and positive to the right. The
plot is symmetric because the record is compared to itself. We find the power spectra of the
autocorrelated data:
DNA=fft(dna,512);
PNA=DNA .* conj(DNA)/(512*512);
f2=.25*(0:256)/256;
plot(f2,PNA(1:257,:))
()
We find the same spectral character working with the autocorrelation, the difference being that the
power is a measure of the variance due to that periodic component, i.e., most of the variance is
explained by the 100 ky period.
Lecture Index | Search the Ocean 540 Pages | Ocean 540 Home
Oceanography 540 Pages
Pages Maintained by Russ McDuff (mcduff@ocean.washington.edu)
Copyright (©) 1994-2001 Russell E. McDuff and G. Ross Heath; Copyright Notice
Content Last Modified 2/12/2001 | Page Last Built 2/12/2001
Fourier Transform Autocorrelation and Power Spectrum
Cuthbert Nyack
For a signal extending from -inf to + inf, the Autocorrelation is defined by the following expression.
The integral is evaluated for increasing values of T and the result is averaged. If the average is not
taken then the Autocorrelation would tend to infinity.
If the signal is of finite duration, then averaging it would yield a result of zero. Instead the following
definition of the Autocorrelation function is used.
Since the Autocorrelation function is even, then the following definition can also be used.
The Fourier Transform of the Autocorrelation Function is the Power Spectrum, So the
Autocorrelation function and Power Spectrum form a Fourier pair below. The power spectrum
removes the phase information from the Fourier Transform. For RANDOM SIGNALS the
autocorrelation - Power Spectrum pair is the most useful representation. Most spectrum analysers will
display either the power spectrum or the magnitude of the transform. In either case, the phase is not
displayed.
With the angular frequency replaced by the cyclic frequency, the pair becomes.
Since the Autocorrelation function is even, then the following definition for the pair can be used.
Return to main page
Return to page index
COPYRIGHT © 1996 Cuthbert Nyack.