IntervalUnions

The type IntervalUnion implements unions of disjoint Intervals of real numbers like [0,1] ∪ [2,3] or ]0.2,2.1[ ∪ ]2.1,3.14]. The following section gives some basic examples of operations on IntervalUnions.

Example Usage

Definition

An IntervalUnion is simply an array of Intervals. The Intervals given to the constructor can be in any order and may overlap, they will be merged and ordered by the constructor. For IntervalUnions with only a single Interval, simplified constructors can be used to pass the Interval arguments directly:

julia> i = IntervalUnion(0,1)
[0,1]
julia> j = IntervalUnion([Interval(0,1,true),Interval(1,true,3,true)])
[0,1[ ∪ ]1,3[
julia> k = IntervalUnion([Interval(4.3,true,5),Interval(0,1,true),Interval(-2,0.3,true),Interval(1,true,3,true)])
[-2,1[ ∪ ]1,3[ ∪ ]4.3,5]
julia> l = IntervalUnion(-2,true,2,true)
]-2,2[

As for Intervals, an empty union of Intervals can be created with:

julia> i = IntervalUnion()
∅

Basic queries

The same basic Interval queries can be used for IntervalUnions:

julia> i = IntervalUnion([Interval(-2,0,true),Interval(0,true,2.3),Interval(3.1,true,4)])
[-2,0[ ∪ ]0,2.3] ∪ ]3.1,4]
julia> left(i)
3-element Array{Real,1}:
 -2  
  0  
  3.1
julia> right(i)
3-element Array{Real,1}:
 0  
 2.3
 4  
julia> limits(i)
5-element Array{Real,1}:
 -2  
  0  
  2.3
  3.1
  4  
julia> empty(i)
false
julia> cardinal(i)
5.199999999999999
julia> number_of_components(i)
3
julia> empty(IntervalUnion())
true

Inclusivity

It is possible to check if a given number belongs to an IntervalUnion:

julia> i = IntervalUnion([Interval(0,true,1,true),Interval(2.5,true,3),Interval(3.5,6)])
]0,1[ ∪ ]2.5,3] ∪ [3.5,6]
julia> 0.5 ∈ i
true
julia> 1 ∈ i
false
julia> 0.9999999 ∈ i
true

It is also possible to check if an IntervalUnion is included in another one:

julia> i = IntervalUnion([Interval(0,true,1,true),Interval(2.5,true,3),Interval(3.5,6)])
]0,1[ ∪ ]2.5,3] ∪ [3.5,6]
julia> j = IntervalUnion([Interval(2.6,3),Interval(3.5,4,true)])
[2.6,3] ∪ [3.5,4[
julia> j ⊆ i
true

Intersection

We can compute the intersection of two IntervalUnions:

julia> i = IntervalUnion([Interval(0,1,true),Interval(1,true,2.3)])
[0,1[ ∪ ]1,2.3]
julia> j = IntervalUnion([Interval(1,1),Interval(2.3,true,4,true)])
[1,1] ∪ ]2.3,4[
julia> i ∩ j
∅
julia> k = IntervalUnion([Interval(-1,0,true),Interval(3,4)])
[-1,0[ ∪ [3,4]
julia> j ∩ k
[3,4[
julia> i ∩ IntervalUnion()
∅

Union

We can compute the union of two IntervalUnions:

julia> i = IntervalUnion([Interval(0,1,true),Interval(1,true,2.3)])
[0,1[ ∪ ]1,2.3]
julia> j = IntervalUnion([Interval(1,1),Interval(2.3,true,4,true)])
[1,1] ∪ ]2.3,4[
julia> i ∪ j
[0,4[
julia> k = IntervalUnion([Interval(-1,0,true),Interval(3,4)])
[-1,0[ ∪ [3,4]
julia> i ∪ k
[-1,1[ ∪ ]1,2.3] ∪ [3,4]
julia> i ∪ IntervalUnion()
[0,1[ ∪ ]1,2.3]

Complement

It is possible to compute the complement of IntervalUnions:

julia> i = IntervalUnion(0,1)
[0,1]
julia> complement(i)
]-Inf,0[ ∪ ]1,Inf[
julia> j = IntervalUnion(0,0)
[0,0]
julia> complement(j)
]-Inf,0[ ∪ ]0,Inf[
julia> k = IntervalUnion(0,true,0,true)
]0,0[
julia> complement(k)
]-Inf,Inf[
julia> complement(IntervalUnion())
]-Inf,Inf[
julia> l = IntervalUnion([Interval(0,1,true),Interval(2,true,3)])
[0,1[ ∪ ]2,3]
julia> complement(l)
]-Inf,0[ ∪ [1,2] ∪ ]3,Inf[
julia> l ∩ complement(l)
∅
julia> l ∪ complement(l)
]-Inf,Inf[

Set Difference

We can compute the set difference between two IntervalUnions, that is all elements present in the first one but not in the second one:

julia> i = IntervalUnion([Interval(0,1,true),Interval(1,true,2.3)])
[0,1[ ∪ ]1,2.3]
julia> j = IntervalUnion([Interval(1,1),Interval(2.3,true,4,true)])
[1,1] ∪ ]2.3,4[
julia> k = IntervalUnion([Interval(-1,0,true),Interval(3,4)])
[-1,0[ ∪ [3,4]
julia> julia> setdiff(j,k)
[1,1] ∪ ]2.3,3[

Symmetric Difference

We can compute the symmetric difference between two IntervalUnions, that is all elements present in the first or the second one but not in both:

julia> i = IntervalUnion([Interval(0,1,true),Interval(1,true,2.3)])
[0,1[ ∪ ]1,2.3]
julia> j = IntervalUnion([Interval(1,1),Interval(2.3,true,4,true)])
[1,1] ∪ ]2.3,4[
julia> k = IntervalUnion([Interval(-1,0,true),Interval(3,4)])
[-1,0[ ∪ [3,4]
julia> symdiff(i,k)
[-1,1[ ∪ ]1,2.3] ∪ [3,4]
julia> symdiff(j,k)
[-1,0[ ∪ [1,1] ∪ ]2.3,3[ ∪ [4,4]

Compact and Superset

We can compute the compact Interval of a given IntervalUnion:

julia> i = IntervalUnion([Interval(0,true,1),Interval(2,3)])
]0,1] ∪ [2,3]
julia> compact(i)
[0,3]
julia> i = IntervalUnion([Interval(0,true,1),Interval(3,5)])
]0,1] ∪ [3,5]
julia> compactness(i)
0.6

Since we might want to keep the open/close information on the extremities, we can also compute the superset of a given IntervalUnion:

julia> i = IntervalUnion([Interval(0,true,1),Interval(2,3)])
]0,1] ∪ [2,3]
julia> superset(i)
]0,3]

The complement of the superset can be computed using the super_complement function:

julia> i = IntervalUnion([Interval(0,true,1),Interval(2,3)])
]0,1] ∪ [2,3]
julia> super_complement(i)
]-Inf,0] ∪ ]3,Inf[

We can compute the IntervalUnion complement in between the boundaries of a given IntervalUnion:

julia> i = IntervalUnion([Interval(0,1),Interval(4,5)])
[0,1] ∪ [4,5]
julia> restricted_complement(i)
]1,4[
julia> j = IntervalUnion([Interval(0,true,1,true),Interval(2.5,true,3),Interval(3.5,6)])
]0,1[ ∪ ]2.5,3] ∪ [3.5,6]
julia> restricted_complement(j)
[1,2.5] ∪ ]3,3.5[
julia> IntervalUnion([Interval(0,1),Interval(4,5)])
[0,1] ∪ [4,5]
julia> complement_cardinal(i)
3
julia> j = IntervalUnion([Interval(0,true,1,true),Interval(2.5,true,3),Interval(3.5,6)])
]0,1[ ∪ ]2.5,3] ∪ [3.5,6]
julia> complement_cardinal(j)
2

Sampling

We can sample uniform random numbers from IntervalUnions:

julia> i = IntervalUnion([Interval(0,1,true),Interval(1,true,2.3)])
[0,1[ ∪ ]1,2.3]
julia> sample(i)
0.22557298488974253
julia> i = IntervalUnion([Interval(0,1,true),Interval(1,true,2.3)])
[0,1[ ∪ ]1,2.3]
julia> sample(i,4)
4-element Array{Float64,1}:
 0.7974550091944592 
 0.36823887828930935
 1.1796141777238074 
 2.2731170537623138

Similarity metrics

Finally, we can compare two IntervalUnions with different metrics:

Jaccard coefficient: $ J(X,Y)=\frac{\left| X \cap Y \right|}{\left| X \cup Y \right|} $
Overlap coefficient: $ O\left(X,Y\right)=\frac{\left| X \cap Y \right|}{min\left( \left|X\right|,\left|Y\right| \right)} $
Dice coefficient: $ D\left(X,Y\right)=\frac{2\left| X \cap Y \right|}{\left|X\right|+\left|Y\right|} $

julia> i = IntervalUnion([Interval(0,1,true),Interval(1,true,2.3)])
[0,1[ ∪ ]1,2.3]
julia> j = IntervalUnion([Interval(1,1),Interval(2.3,true,4,true)])
[1,1] ∪ ]2.3,4[
julia> k = IntervalUnion([Interval(-1,0,true),Interval(3,4)])
[-1,0[ ∪ [3,4]
julia> jaccard(i,j)
0.0
julia> jaccard(j,k)
0.37037037037037035
julia> overlap_coefficient(i,j)
0.0
julia> overlap_coefficient(j,k)
0.588235294117647
julia> dice_coefficient(i,j)
0.0
julia> dice_coefficient(j,k)
0.5405405405405405