Lossy Intervals Distribution

A lossy intervals distribution is an aggregation of two interval distributions having only intervals existing in both.

Lossy Interval Distribution is an intersection 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 excluding the first/last intervals from the distribution.

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

Lossy Interval Distribution will contain only interval 4 as it exists in both distributions.

Mathematical Definition

Let \(ID\) is Intervals distribution

Define Lossy Interval Distribution

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

\[LID(ID_1, ID_2)(i) = min \{ID_1(i), ID_2(i) \}\]