Next: Generalizations
Up: MAXIMUM LIKELIHOOD METHODS
Previous: The Neerven estimator
Contents
The correlation entropy can be defined
by [Eckmann and Ruelle, 1985] (see Chapter 4):

(5.39) 
In section 5.2.4 we derived an estimator for

(5.40) 
using the likelihood function for a doubly censored
set of data:

(5.41) 
To simplify the algebra (cf. appendix C), we
write the likelihood function as:
where
.
The logarithm of this likelihood function is:
To estimate the entropy, we have to estimate for at least
two embedding dimensions, since is unknown.
Therefore, we now consider the case where the sample consists of
distances calculated at embedding dimension and
distances calculated at embedding dimension .
We assume that
the and distances are independent. The likelihood
function for this case,
,
is the product of the likelihood functions
and
.
The likelihood equation for is:
The likelihood equations for and are:

(5.45) 
and

(5.46) 
For the estimators of the parameters , and
we find:

(5.47) 
and

(5.48) 
and a similar expression for
.
The estimator of the entropy is now given by:

(5.49) 
To obtain the variances of the dimension and entropy
estimators, we have to invert the information matrix

(5.50) 
where
and

(5.52) 
and a similar expression for
;
all other elements are zero due to our choice of parameters.
For the variances of and we find:

(5.53) 
Using eq. (5.27) with

(5.54) 
we obtain an expression for the variance of the ML entropy estimator:
If we substitute the estimated values
,
and
we obtain:

(5.56) 
and

(5.57) 
It is gratifying that this derivation and the one
in appendix C are consistent and
that both the estimators and their variances do not depend
on the reference distance. They should not, because the dimension and entropy
are invariant under smooth coordinate transformations [Ott et al., 1984].
In section 5.6 the validity of the expressions for the variances
is supported by Monte Carlo simulations.
It can readily be verified that the covariance between
and is
zero if
.
Finally, the estimator of the entropy (eq. (5.49)) is
equivalent to the one of the previous chapter
(eq. (4.7), with ) if we set
and if the correlation integrals are
based on independent distances.
Correlations between distances are discussed in section 5.6.2.
Next: Generalizations
Up: MAXIMUM LIKELIHOOD METHODS
Previous: The Neerven estimator
Contents
webmaster@rullf2.xs4all.nl