Biology Reference
In-Depth Information
time series may sometimes effectively determine the periodic
components.
6.00
5.00
4.00
Figures 9-12 and 9-13 present the power spectra of the data shown in
Figures 9-5 and 9-6, respectively, presented as a function of the period of
the sine or cosine components. The power spectra shown in Figures 9-12
and 9-13 are simply a plot of a
n
þ
3.00
2.00
1.00
b
n
as a function of L/n, for n
,N.
There are several algorithms that can be used to calculate the unknown
coefficients a
0
, a
n
, and b
n
(n
¼
1,2,
...
0.00
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
A
Frequency
) and represent the data as sine and/or
cosine series as shown in Eq. (9-4). As noted, the FFT algorithm is the most
widely used. These figures were done with the FFT function of MATLAB,
but many other software packages, including BERKELEY MADONNA,
provide similar implementations.
¼
1, 2,
...
6.00
5.00
4.00
3.00
2.00
1.00
The power spectrum of the LH example, Figure 9-12, indicates the data
contain a periodic component of approximately 230 minutes
(corresponding to L
0.00
0.00
0.50
1.00
1.50
2.00
2.50
3.00
3.50
4.00
4.50
5.00
B
Frequency
7) corresponding to the visual
impression from Figure 9-5. In contrast, the power spectrum of the GH
example, Figure 9-13, gives no clear indication of a dominant frequency
(or period). In addition, even in the more conclusive LH example, the
calculated power spectra have not actually provided much information
about the physiological mechanisms involved in hormone secretion into
the blood or the kinetics of the elimination from the blood.
¼
1440 and n
¼
6.00
5.00
4.00
3.00
2.00
1.00
0.00
0.00
0.50
1.00
1.50
2.00
2.50
3.00
3.50
4.00
4.50
5.00
C
Frequency
Summarizing, the FFT and power spectrum methods are limited to data
sets of points equally spaced in time and having equal levels of
experimental uncertainties. Also, it should be noted the power spectrum
approach only evaluates the contributions of sine and cosine waves with
integer harmonics. That is, only periods of L/n are considered, where L is
the base period (1440 minutes for the present examples) and n
FIGURE 9-11.
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 mðtÞ¼
p
sin ð2ptÞþ
p
cos ð3ptÞþ
1
p
sin ð7ptÞ: As
expected from the analytical form of m(t), three
dominant frequencies are identified at
v ¼ 1, v ¼ 3/2, and v ¼ 7/2, corresponding to the
three periodic components with periods L ¼ 1,
L ¼ 2/3, and L ¼ 2/7 of m(t). The amplitudes are
proportional to the squares of the coefficients
before the periodic components.
¼
1, 2, 3,
4,
, and, thus, only variances carried by these frequencies will be
explained by the FFT method.
...
B. Fractional Variance Methods
A different way to present these results is in terms of fractional variance,
defined as the fraction of the variance explained by the model over the
total variance of the data. Alternatively, one can consider the fraction of
remaining variance, defined as the fraction of the variance that cannot be
explained by the model over the total variance of the data. Both
quantities are smaller than 1, and are often expressed as percentages.
The two are easily related as:
fractional remaining variance
¼
1
fractional variance
:
0
500
1000
1500
Recall that we used a similar approach in Chapter 4 to quantify
heritability, considering the fractional variance of a linear model with
equal weights of the data points. In the current context, we want to
Period
FIGURE 9-12.
The power spectrum of the LH data shown in
Figure 9-5.
