This document describes using MATLAB to analyze a synthetic time series dataset representing climate data over 500,000 years. The time series contains periodic signals at 100ky, 41ky and 21ky. Random noise and a long term trend are added. Fourier analysis is used to identify the dominant periodic components in the frequency domain. A Hamming window and bandpass filter are applied to further analyze specific frequency bands like the 21ky signal. Autocorrelation is also examined to identify cyclic patterns in the time series.

1. 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')

2. ()
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))

3. ()
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))

4. ()
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

5. 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

6. 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);

7. 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.

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