Redundant Intervals Distribution

A redundant intervals distribution is an aggregation of two interval distributions having all intervals from the first one and intervals existing only the second.

Redundant Interval Distribution is a union of two distributions.

Mostly, this distribution is used to solve measure dependence on Binding direction with Bounded binding. In that case, this would be equivalent to counting both intervals - first for Start binding direction and End binding direction.

For example, there are 2 distributions for Bounded Binding - one uses Start binding direction and the other End binding direction.

block-beta
    columns 8
    p0["0"]        p1["1"] space:3         p5["5"]                p6["6"] p7["7"]
    inf["⊥"] s1["A"] s2["C"] s3["T"] s4["C"] s5["A"] s6["G"] sup["⊥"]
    space es0["Start(1) = 1"]:1 ts1["Start(5)=4"]:5 space
    space      te1["End(1) = 4"]:4 ee0["End(5) = 2"]:2 space

    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,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 s1,s5 c4
    class inf,sup,te1,ee0,ts1,es0 c4a
    class pomn,p00,p01,p06,p07,p02n position

Redundant Interval Distribution will include all intervals [1, 4, 2].

Mathematical Definition

Let \(ID\) is Intervals distribution

Define Redundunt Interval Distribution

\[RID: \{ID\} \times \{ID\} \longrightarrow \{ID\}\]

\[RID(ID_1, ID_2)(i) = max \{ID_1(i), ID_2(i) \}\]