Depth

The Depth is equivalent of Volume on on a logarithmic scale. It is sensitive to interval ratios and their count (length of the sequence). The Depth is better then Volume from computational point of view

Mathematical Definition

The depth can be calculated

from Intervals Chain

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

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

\[G=\sum_{i=1}^{n} \log_2 \Delta_{i}\]

from Intervals Distribution

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

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

\[V(ID)=\prod_{i=1}^{n} i^{ID(i)}\]

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

Properties

Volume equals 2 power depth

\[V = 2^{G}\]