Volume

The Volume is a product of all intervals, which is sensitive to interval ratios and their count. The volume value has exponential grow by increasing the interval count (length of the sequence). This fact makes it useless in computational models due to overflow error. Use Depth as an equivalent measure on a logarithmic scale.

Mathematical Definition

The volume can be calculated

from Intervals Chain

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

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

\[V=\prod_{i=1}^{n} \Delta_{i}\]

from Intervals Distribution

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

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

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

Properties

Volume can be calculated base of geometric mean

\[V = (\Delta_{g})^{n}\]