where is the embedding dimension reconstruction of phase space and (see section 2.5). This expression, however, does not include the correction function

(3.12) |

By first estimating the moments of the probability density function of nearest neighbour distances, and then using eqs (3.9) and (3.11), it is possible to estimate the dimension and entropy spectra. In practice, the weighted distances are calculated for some values of and .

There are three possibilities:

**N-method**- : Badii and Politi [Badii and Politi, 1985] use a fixed value of (in which case the correction function is a constant) and study the behaviour of eq. (3.9) as a function of .
**k-method**- : Termonia and Alexandrowicz [Termonia and Alexandrowicz, 1983] fix at its maximum value and study the behaviour of eq. (3.9) as a function of .
**k(N)-method**- :
There is a third possibility:
consider the well-known estimator of the local density

(3.13)

So we can observe the scaling behaviour of eq. (3.9) where and vary together according to eq. (3.14).

If we calculate the distances for different values of and , we are able to estimate both the dimension function and the entropy function . However, eq. (3.11) is only valid for fixed .