Biology Reference
In-Depth Information
of grasping the more complex rhythmic pattern of this simulated data
set. Regardless of its poor performance, we present the output from
COSIN2NL in Figure 11-21 in order to emphasize, as we did in
Chapter 8, that NLLS methods will calculate estimates for the
parameters of the model even when the model provides a very poor fit.
It is the researcher's role to select the algorithm that will provide the best
performance on any specific data set.
2. FFT-NLLS
FFT-NLLS is a more flexible algorithm allowing for the presence of
multiple periodic components in the form of a sum of multiple cosine
waves. In essence, it combines two fundamental techniques we have
already used multiple times—FFT methods and NLLS methods. The
FFT-NLLS procedure can be outlined as follows: time series are first
linear regression detrended to produce zero-mean, zero-slope data. An
FFT power spectrum is then calculated for the detrended data. A model
of the form:
X
n
2
pð
t
þ f
i
Þ
t
i
y
LR
ð
t
Þ¼
1
a
i
cos
i
¼
is fit to the data, where y
LR
(t) is the linear regression detrended time
series on which analysis is being performed, n is the order of fit, t is time,
and
t
i
are the amplitude, phase, and period, respectively, of
the i-th cosine component.
a
i
,
f
i
, and
The period, phase, and amplitude of the most powerful spectral peak are
used to initialize a one-component cosine function (i.e., n
1). The
parameters of this function are then estimated by nonlinear least-squares
minimization as in COSIN2NL. Upon convergence, approximate
nonlinear asymmetric joint confidence limits are estimated for all
parameters (period, phase, amplitude, and constant offset) at 95%
confidence probability. If the amplitude is significantly different from
zero, then the procedure is repeated at the next higher order. The two
most powerful FFT spectral peaks are then used to initialize a two-
component cosine function (i.e., n
¼
2) that is subsequently NLLS
minimized to the linear regression detrended data, and confidence limits
are again evaluated. This process is repeated iteratively until at least
one cosine component is identified with an amplitude that is not
statistically significant.
1
¼
The statistical significance of each derived rhythmic component is
assessed by way of the relative amplitude error (RAE), defined as the
1. Other possible scenarios for terminating the procedure along with more
detailed description of the FFT-NLLS procedure can be found in Plautz et al.
(1997).
