Up: Three stage period analysis
4 Bootstrap
Theoretical (e.g. )
and numerical (e.g. Monte Carlo)
error estimates for nonlinear models are discussed,
e.g. by Press et al. (1988).
We estimate the
errors with
the bootstrap for the regression coefficients
(Efron & Tibshirani 1986).
This bootstrap also allows estimates of "other'' model parameters
(and their errors)
that would be difficult to solve analytically for
,
e.g. the total amplitude (A) and the epochs of the minima
(
t_{min,1},
t_{min,2},... ,
t_{min,K}).
The six stages of our bootstrap are:
 1.
Minimizing
(Eq.
8)
for
with
gives

(11) 
i.e. the "empirical distribution of residuals''.
 2.
A random sample
is selected
from
.
The number of the
same
entering into
may vary between 0 and
nin this random sample with replacement.
The
determine a unique random sample
,
where
the connection of
w_{i} being the weight
of
is preserved.
 3.
A random sample
is obtained.
 4.
Minimizing
(Eq.
8)
for
with
gives one estimate for
,
as well as for the "other'' parameters
of higher order models (
).
For example, measuring
the difference between the minimum and maximum of
gives a
numerical
A estimate.
 5.
The bootstrap returns to the 2nd stage, until
S estimates of
and "other'' parameters have been obtained.
 6.
The expectation value and variance for any
component are the mean and variance
of its
S estimates in
.
The same applies to the
S estimates of "other'' model parameters.
Note that the ,
,
and remain unchanged,
while ,
and
are changing,
and
determines both
and .
We introduce a few notations useful in testing
the "Gaussian hypothesis''


H_{G}: An arbitrary
with S components represents a random sample drawn from a
Gaussian distribution.
These
(e.g.
,
,
,
or
)
are first arranged into an ascending (i.e. rank) order
and then transformed to
,
where m_{x} and s_{x} are
the mean and standard deviation of .
The cumulative distribution function is

(12) 
and the cumulative Gaussian distribution function is

(13) 
The preassigned significance level
for rejecting
H_{G} determines an upper limit
for the KolmogorovSmirnov test statistic
.
H_{G} is rejected if, and only if,

(14) 
One might (correctly) argue that preserving the connection between
and w_{i} during the 2nd stage of the
above bootstrap distorts the statistics in case of inhomogeneous
data quality.
In that case some low quality data with large
and small w_{i} will be randomly distributed among high quality
data in each bootstrap sample.
But this problem is eliminated by checking if the
distribution is Gaussian.
If
H_{G} is rejected
for
with Eq. (14),
i.e. the data quality is inhomogeneous, then the modelling is rejected
(see
R_{I} in Sect. 6.3).
If, on the other hand,
H_{G}
is not rejected for
,
preserving the connection between
and w_{i}in our bootstrap is justified.
Up: Three stage period analysis
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