![\begin{displaymath}
D_q\equiv {1\over 1-q} \lim_{\epsilon\to 0} {\ln I(q,\epsilon)\over \ln\left({1\over \epsilon}\right),}
\end{displaymath}](q_46.gif) |
(1) |
where
![\begin{displaymath}
I(q,\epsilon)\equiv \sum_{i=1}^N {\mu_i}^q,
\end{displaymath}](q_47.gif) |
(2) |
is the box size, and
is the Natural Measure. If
, then
![\begin{displaymath}
D_{q_1} \leq D_{q_2}.
\end{displaymath}](q_51.gif) |
(3) |
The Capacity Dimension (a.k.a. Box Counting Dimension) is given by
,
![\begin{displaymath}
D_0 = {1\over 1-0} \lim_{\epsilon\to 0} {\ln\left({\sum_{i=1...
...= - \lim_{\epsilon\to 0} {\ln[N(\epsilon)]\over \ln \epsilon}.
\end{displaymath}](q_53.gif) |
(4) |
If all
s are equal, then the Capacity Dimension is obtained for any
. The Information Dimension
is defined by
But
![\begin{displaymath}
\lim_{q\to 1} \ln\left({\sum_{i=1}^{N(\epsilon)} {\mu_i}^q}\...
...= \ln\left({\sum_{i=1}^{N(\epsilon)} \mu_i}\right)= \ln 1 = 0,
\end{displaymath}](q_57.gif) |
(6) |
so use L'Hospital's Rule
![\begin{displaymath}
D_1 = \lim_{\epsilon\to 0}\left({{1\over\ln\epsilon} \lim_{q\to 1} {\sum q{\mu_i}^{q-1}\over\sum {\mu_i}^q}}\right).
\end{displaymath}](q_58.gif) |
(7) |
Therefore,
![\begin{displaymath}
D_1 = \lim_{\epsilon\to 0} {\sum_{i=1}^{N(\epsilon)} \mu_i\ln \mu_i\over \ln\epsilon}.
\end{displaymath}](q_59.gif) |
(8) |
is called the Correlation Dimension. The
-dimensions satisfy
![\begin{displaymath}
D_{q+1}\leq D_q.
\end{displaymath}](q_61.gif) |
(9) |
See also Fractal Dimension
© 1996-9 Eric W. Weisstein
1999-05-25