Biology Reference
In-Depth Information
According to the Fourier Theorem, a broad class of time series can be
represented as a sum of sine and/or cosine waves of the form given by
Eq. (9-3) with periods L, L/ 2, L/ 3, and so on, where L is the maximal time
range for the time series. One important prerequisite is that the time
series does not show an overall trend over time. For example, the time
series in Figures 9-5 and 9-6 do not appear to exhibit a pronounced
upward or downward trend. Conversely, a time series containing height
or weight data of a healthy child over several years will exhibit an
upward trend. Time series that preserve their statistical characteristics
such as mean and variance over time are called stationary.
2
Thus, a time
series that exhibits a trend cannot be stationary.
1.20
1.00
0.80
0.60
0.40
0.20
0.00
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
A
Frequency
1.20
1.00
0.80
0.60
The power spectrum function generated by the FFT is the sum of the
squares of the amplitudes of these sine and/or cosine terms for each
frequency (or period). The classical method for analysis of time-series
data is to analyze the power spectrum of the data and identify the peaks,
as illustrated in Figure 9-9. The peaks of the highest amplitude
represent the prevalent frequencies/periods in the composition of the
time series.
0.40
0.20
0.00
0.00
0.20
0.40
0.60
0.80
1.00
1.20
1.40
1.60
1.80
2.00
B
Frequency
1.20
1.00
0.80
0.60
Consider, as an example, the function m(t)
t) sampled 500 times
over the interval [0,10] at equally spaced time instances. A FFT analysis
should then identify the unique period L
¼
sin(2
p
0.40
0.20
¼
1, corresponding to a
0.00
0.00
0.20
0.40
0.60
0.80
1.00
1.20
1.40
1.60
1.80
2.00
frequency v
1, which is apparent in Figure 9-10(A). Sampling at the
same sampling rate over a longer time interval (e.g., [0,100] and [0,1000]
in Figures 9-10(B) and (C), respectively) sharpens the peak, more clearly
identifying the dominant frequency, as expected.
¼
C
Frequency
FIGURE 9-10.
Results of the FFT algorithm using, respectively,
500 (panel A), 5000 (panel B), and 50,000 (panel
C) data points, sampled at a rate of 50 data
points per unit time from the function m(t) ¼
sin(2pt).
The result of the FFT algorithm using, respectively, 500, 5000, and 50,000
data points, sampled at a rate of 50 data points per unit time, from the
p
sin
p
cos
1
2
function m
ð
t
Þ¼
ð
2
p
t
Þþ
ð
3
p
t
Þþ
p
ð
p
Þ
sin
7
t
is given in
Figure 9-11. In general, if Y
i
represents a discrete stationary time series of
N data points at time values X
i
(i
,N), then Y
i
can be exactly
represented as the sum of sine and cosine waves
3
with a period L equal
to the maximal time range (i.e., T
¼
1,2,
...
1440 minutes for Figures 9-5
and 9-6) and where N is an odd number as:
¼
ð
N
X
1
Þ=
2
a
0
2
þ
2
p
n
b
n
sin
2
p
n
Y
i
¼
a
n
cos
X
i
þ
X
i
:
(9-4)
L
L
n
¼
1
When N is an even number, the sum is from 1 to (N
1)/2 and either
a
¼
0orb
¼
0. Considering the power spectrum of such
ð
N
1
Þ=
2
ð
N
1
Þ=
2
2. The exact mathematical definition can be found in Box et al. (1994).
3. Equivalent representations involving only sine or only cosine functions with
phase shift parameters are also valid. We will use these alternative forms in
Chapter 11.
