Interval as a Basic Information Unit

Intervals serve as a fundamental unit of information by measuring the number of different items, events, or symbols that occur between reseated in a sequence.

The intervals for symbol A in the following sequence would be [3, 3, 1, 1, 2, 1, 1]

block-beta
  columns 12
  s1["A"] s2["C"] s3["T"] s4["A"] s5["C"] s6["G"] s7["A"] s8["A"] s9["A"] s10["T"] s11["A"] s12["A"]
  i1["3"]:3 i2["3"]:3 i3["1"]:1 i4["1"]:1 i5["2"]:2 i6["1"]:1 i7["1"]:1

  classDef c3 fill:#2ca02c,color:#fff;
  classDef c4 fill:#98df8a,color:#000;
  class s1,s4,s7,s8,s9,s11,s12 c3
  class i1,i2,i3,i4,i5,i6,i7 c4

In general, a sequence does not necessarily end with the same symbol it begins with. To cover all cases, we consider the sequence as a looped sequence representing an infinite pattern with the same characteristics as the original data This cyclic approach corresponds to the idea of ​​representativeness heuristic.

The intervals for symbol C in the following cycled sequence would be [3, 9]

block-beta
  columns 15
  s1["A"] s2["C"] s3["T"] s4["A"] s5["C"] s6["G"] s7["A"] s8["A"] s9["A"] s10["T"] s11["A"] s12["A"] space s13["T"] s14["C"]
  space i1["3"]:3 i2["9"]:10
  s12 --> s13

  classDef c3 fill:#2ca02c,color:#fff;
  classDef c4 fill:#98df8a,color:#000;
  class s2,s5,s14 c3
  class i1,i2 c4

The circular pattern preserves both the statistical properties and the order of elements. Moreover, the average interval length is the inverse of the probability of an event, which directly relates intervals to probability.

\[\begin{array}{|c|c|c|} \hline & \Delta_a & P \\ \hline A & \frac{3 + 3 + 1 + 1 + 2 + 1 + 1}{7} = \frac{12}{7} \approx 1.7142; & \frac{7}{12} = (\frac{12}{7})^{-1} = \Delta_a^{-1} \\ \hline C & \frac{3 + 9}{2} = \frac{12}{2} = 6 & \frac{2}{12} = \frac{1}{6} = 6^{-1} = \Delta_a^{-1} \\ \hline \end{array}\]

This makes intervals a crucial informational unit that offers deeper insights into the sequence than individual occurrences alone.