Partial Intervals Distribution

A Partial intervals distribution is an Interval distribution produced from Partial intervals chain by counting all non-empty elements (intervals) in distribution

From a partial interval chaincalculate intervals distribution

block-beta
  columns 7
  space  p1["1"] p2["2"] p3["3"] p4["4"] p5["5"] p6["6"]
  space s1["-"] s2["2"] s3["3"] s4["2"] s5["-"] s6["6"]

  classDef imaginary fill:#526cfe09,color:#000,stroke-dasharray: 10 5;
  classDef position fill:#fff,color:#000,stroke-width:0px;
  class inf,sup imaginary
  class p0,p1,p2,p3,p4,p5,p6,p7 position

  classDef c1 fill:#ff7f0e,color:#fff;
  classDef c2 fill:#ffbb78,color:#000;
  classDef c2a fill:#ffbb788a,color:#000;
  classDef c3 fill:#2ca02c,color:#fff;
  classDef c4 fill:#98df8a,color:#000;
  classDef c4a fill:#98df8a8a,color:#000;
  classDef c5 fill:#d62728,color:#fff;
  classDef c6 fill:#ff9896,color:#000;
  classDef c6a fill:#ff98968a,color:#000;
  classDef c7 fill:#9467bd,color:#fff;
  classDef c8 fill:#c5b0d5,color:#000;
  classDef c9 fill:#8c564b,color:#fff;
  classDef c10 fill:#c49c94,color:#000;
  classDef c11 fill:#e377c2,color:#fff;
  classDef c12 fill:#f7b6d2,color:#000;
  classDef c13 fill:#bcbd22,color:#fff;
  classDef c14 fill:#dbdb8d,color:#000;
  classDef c14a fill:#dbdb8d8a,color:#000;
  classDef c15 fill:#17becf,color:#fff;
  classDef c16 fill:#9edae5,color:#000;

  class pomn,p00,p01,p06,p07,p02n position
  class t1,t2,t5,e0,e1 position

Let there be a partial interval chain.

block-beta
  columns 7
  space  p1["1"] p2["2"] p3["3"] p4["4"] p5["5"] p6["6"]
  space s1["-"] s2["2"] s3["3"] s4["2"] s5["-"] s6["6"]
  space:7
  space i1["0"] i2["2"] i3["1"] i4["0"] i5["0"] i6["1"]

  classDef imaginary fill:#526cfe09,color:#000,stroke-dasharray: 10 5;
  classDef position fill:#fff,color:#000,stroke-width:0px;
  class inf,sup imaginary
  class p0,p1,p2,p3,p4,p5,p6,p7 position

  classDef c1 fill:#ff7f0e,color:#fff;
  classDef c2 fill:#ffbb78,color:#000;
  classDef c2a fill:#ffbb788a,color:#000;
  classDef c3 fill:#2ca02c,color:#fff;
  classDef c4 fill:#98df8a,color:#000;
  classDef c4a fill:#98df8a8a,color:#000;
  classDef c5 fill:#d62728,color:#fff;
  classDef c6 fill:#ff9896,color:#000;
  classDef c6a fill:#ff98968a,color:#000;
  classDef c7 fill:#9467bd,color:#fff;
  classDef c8 fill:#c5b0d5,color:#000;
  classDef c9 fill:#8c564b,color:#fff;
  classDef c10 fill:#c49c94,color:#000;
  classDef c11 fill:#e377c2,color:#fff;
  classDef c12 fill:#f7b6d2,color:#000;
  classDef c13 fill:#bcbd22,color:#fff;
  classDef c14 fill:#dbdb8d,color:#000;
  classDef c14a fill:#dbdb8d8a,color:#000;
  classDef c15 fill:#17becf,color:#fff;
  classDef c16 fill:#9edae5,color:#000;

  class pomn,p00,p01,p06,p07,p02n position
  class t1,t2,t5,e0,e1 position

  s2 --> i2
  s3 --> i3
  s4 --> i2
  s6 --> i6

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Partial Intervals distribution, as any Intervals distribution, used as an input data in calculating characteristics.

Mathematical Definition

Let \(-\) is empty element.

Let \(IC_p\) is Partial Interval Chain length of \(l\) described as function \(IC_p : \{1,...,l\} \longrightarrow \{1,...,l\} \cup \{-\}\)

Let \(ID\) is Interval Distribution length of \(l\) described as function \(ID : \big\{ IC \big\} \longrightarrow \big\{ \{1,...,l\} \longrightarrow N_0 \big\},\)

Define

\[ID_p : \big\{ IC_p \big\} \longrightarrow \big\{ \{1,...,l\} \longrightarrow N_0 \big\},\]

\[ID_p(IC_p)(i) = ID(IC_p)(i) \bigg| IC_p(i) \notin \{-\}\]