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Arithmetic mean interval

Arithmetic mean interval is an additive measure indifferent to interval ratios. The arithmetic mean could be used as a reference for comparing with the geometric mean. The geometric mean and the arithmetic mean are equal when all intervals are equal, which is true only for the periodic appearance of the elements in the sequence.

Mathematical Definition

The arithmetic mean interval can be calculated

from Intervals Chain

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

\[IC = <\Delta_1, \Delta_2, ..., \Delta_n> | \forall j \in \{1,...,n\} \exists \Delta_j \in \{1,...,n\}\]
\[\Delta_a = \frac{1}{n} \times \sum_{i=1}^{n} \Delta_{i}\]

from Intervals Distribution

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

\[ID : \{1,...,n\} \longrightarrow \{1,...,n\}\]
\[\Delta_a(ID) = \frac{\sum_{i=1}^{n} \big(i \times ID(i)\big)}{\sum_{i=1}^{n} ID(i)}\]

Properties

With Cycle Bindings, the arithmetic mean interval equals the cardinality of an alphabet

\(IC\) produced by Cycled Binding

\(\Delta_a = |alphabet(Intervals^{-1}(IC))|\)

Let Sequence \(S\)

\(A=alphabet(S)\) is Alpabet

\(IC = Intervals(S)\) produced by Cycled Binding

then \(\Delta_a = |A|\)