Biology Reference
In-Depth Information
specifically designed to statistically address such limitations have been
developed and applied to microarray time-series analysis. Note that
it is imperative that such algorithms possess high levels of automation
and efficiency, given the huge number of genes on the microarray
that are being examined for circadian behavior.
C. Statistical Assessment of Daily Rhythms in Microarray Data
The COSOPT algorithm described in this section has been used
successfully in several studies for the analysis of microarray data in
Arabidopsis, Drosophila, and mammalian systems (see Edwards et al.
[2006]; Ceriani et al. [2002]; and Panda et al. [2002]). More methods for
statistical assessment of circadian rhythms in gene expression patterning
can be found in Straume (2004).
COSOPT accommodates variable weighting of individual time points,
such as standard errors of the mean (SEMs) from replicate
measurements or errors derived from preprocessing. COSOPT utilizes
user-provided estimates of the circadian period entered as the value
and range of the assumed period. Test periods are then calculated,
uniformly spaced in the assumed range. The computational process
begins by importing the time series on which an arithmetic linear
regression detrending is performed. The mean and SD of the detrended
time series are then calculated.
For each test period
t
, 101 test cosine basis functions
of unit amplitude are considered, varying over a
2
pð
t
þ jÞ
t
y
b
ð
t
Þ¼
cos
h
i
. The number of cosine
2
;
2
range of phase values
j
between
functions is chosen to allow that phase be considered in increments of
1% of each test period. Next, for each test cosine basis function y
b
(t),
COSOPT calculates the least-squares optimized linear correspondence
between the linear-regression-detrended data, y
lr
(t), according to the
model
y
lr
ð
t
Þ¼a þ b
y
b
ð
t
Þ:
The optimization is across all values of t, in terms of the parameters,
a
b
. The quality of optimization possible by each test cosine basis
function is quantitatively characterized by the sum of squared residuals
w
and
2
between y
lr
(t) and the model given by
a þ b
y
b
(t).
2
are used to identify the optimal phase with the smallest
The values of
w
2
providing the optimal correspondence between y
lr
(t) and
y
b
(t) (see Figure 12-14). The values for
value for
w
, least-squares fit for the
optimal phase value, now represent the optimized measures of the
average expression and magnitude of the oscillatory amplitude
expressed by y
lr
(t) (as modeled by a cosine wave of the corresponding
period).
a
and
b
