Average Remoteness

The Average Remoteness is equivalent of the geometric mean interval on a logarithmic scale. It is sensitive to interval ratios and is preferable from computational point of view.

Mathematical Definition

The Average Remoteness interval can be calculated

from Intervals Chain

Let \(IC\) is Intervals Chain described as n-tuple

\[IC = <\Delta_1, \Delta_2, ..., \Delta_n> | \forall j \in \{1,...,n\} \exists \Delta_j \in \{1,...,n\}\]

\[g = \frac{1}{n} * \sum_{i=1}^{n} \log_2 \Delta_{i}\]

from Intervals Distribution

Let \(ID\) is Intervals Distribution described as function

\[ID : \{1,...,n\} \longrightarrow \{1,...,n\}\]

\[g(ID) = \frac{\sum_{i=1}^{n} \big(i \times \log_2 ID(i)\big)}{\sum_{i=1}^{n} ID(i)}\]

Properties

Geometric mean equals 2 power average remotness

\[\Delta_g = 2^{g}\]