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It is assumed that distances on an attractor in phase space
obey the cumulative distribution function (see Chapter 5)

(B.1) 
Following Ellner [Ellner, 1988], we also assume that the correlation
integral
, and that the distribution does only hold
for distances inside the scaling region. The scaling region
is a straight part in a loglog plot of vs. ;
that is why the cumulative distribution should have the form

(B.2) 
However, it seems more logical to assume that the probability density
function does only hold for distances inside the scaling region
, since the cumulative distribution depends on the density
for distances in .
So suppose that distances in have some unknown
distribution and that the probability of finding a
distance in that region is .
Furthermore, assume that distances in are
distributed according to the probability density function

(B.3) 
Hence, in the scaling region, the cumulative distribution function
should have the form

(B.4) 
If we discard distances in , we should have .
Therefore, after solving for and dividing all distances
by , we obtain

(B.5) 
The likelihood function for this set of data is:

(B.6) 
The maximum likelihood estimator, obtained from this likelihood
function is exactly the same
as from the ``doubly truncated'' case (see section 5.2.3).
Note that the distances do not give any information about .
However, the reason that Ellner's estimator has smaller variance
than the ``doubly truncated'' one is related to
the fact that we are looking for a (straight) scaling region.
We then have to assume that ; however, is unknown!
A loglog plot will not show a straight line
if
.
This means that if we indeed see a sufficiently straight scaling region,
the chosen may be larger than the one for which eq. (B.3)
holds. Any errors caused by a deviation of from
have then died out. In that case, the distances
do indeed contain information about , so that it will be
better to use the Ellner estimator instead of the ``doubly truncated''
case. The alternative is to find another way of identifying the
region for which eq. (B.3) holds. However, the purpose
of this appendix is only to clarify the contribution of the
distances to the estimation of the dimension.
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Up: The identification of strange
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