Biology Reference
In-Depth Information
V. METHODS FOR RHYTHM ANALYSIS
AND ANALYTICAL STRATEGIES
Our objective in this chapter is to present methods for detecting rhythms
in confounded data, whether stationary original data series or
preprocessed nonstationary data that have been made (more closely)
stationary. Presented here are a few analytical strategies that have found
common usage for these purposes.
A. Model-Dependent Algorithms
Model-dependent algorithms assume a specific analytic form for the
rhythmic wave and then employ nonlinear least-squares (NLLS)
techniques, as described in Chapter 8, for estimating the values of the
model parameters from the data. We consider two such algorithms, out
of several in the literature.
1. Cosin2nl
This is an algorithm to assess the period, phase, and amplitude of
a one-component cosine function with a linear trend of the form:
2
pð
t
þ fÞ
t
y
ð
t
Þ¼
c
0
þ
c
1
t
þ a
cos
;
where y(t) is the time series on which analysis is being performed, c
0
is
a constant offset term, c
1
is a slope of the linear trend, t is time, and
a
are the amplitude, phase, and period, respectively, of the
cosine function. The parameters of this function are then estimated by
nonlinear least-squares minimization of the Gauss-Newton type, as
described in Chapter 8. The procedure allows for nonlinear asymmetric
joint confidence limits for all parameters to be calculated, if desired,
at any user-specified confidence probability level. The details of the
procedure can be found in Straume et al. (1991).
,
f
, and
t
An amplitude term significantly different from zero indicates
a statistically significant rhythm at the specified level of confidence
probability. If the confidence limits of the amplitude term encompass
zero, however, the rhythm is not statistically significant at the specified
level of confidence probability. As expected from a NLLS algorithm,
COSIN2NL requires user-specified initialization in the form of initial
guesses for the values of the parameters of the cosine model. Thus, it is
not a fully objective analytical strategy because it may be susceptible
to the influence of user-introduced bias.
Figures 11-20 and 11-21 illustrate the results of applying this algorithm
to the data sets presented in Figures 11-14 and 11-15. The estimates
for the model parameters, together with their 95% confidence intervals,
are also presented. The specific details appear in the figure legends.
