Intervals

The type Interval implements a basic interval of real numbers like [0,1] or ]2.1,3.14]. The following section gives some basic examples of operations on Intervals. Note that unions of disjoint Intervals cannot be computed and will throw an error. To work with disjoint intervals, use IntervalUnions instead.

Example Usage

Interval Definition

An Interval is defined as:

A left limit, which has to be a Real number.
A right limit, which has to be a Real number.
Two booleans indicating whether the interval is closed on each side.

The full constructor enables to specify each attribute directly:

julia> l = Interval(1.6,true,2.3,true)
]1.6,2.3[

Since Intervals are most of the time closed on both side, a simplified constructor can be used as a shortcut:

julia> i = Interval(0,1)
[0,1]
julia> i = Interval(0,false,1,false)
[0,1]

It is possible to omit closed limits:

julia> j = Interval(0.2,true,2)
]0.2,2]
julia> j = Interval(0.2,true,2,false)
]0.2,2]
julia> k = Interval(2,3,true)
[2,3[
julia> k = Interval(2,false,3,true)
[2,3[

Finally, the empty Interval can be constructed with no argument:

julia> m = Interval()
∅

Basic queries

Once an Interval is built, one has access to very simple queries:

julia> i = Interval(0,2)
[0,2]
julia> left(i)
0
julia> right(i)
2
julia> empty(i)
false
julia> empty(Interval())
true
julia> cardinal(i)
2
julia> cardinal(Interval(1,1))
0
julia> cardinal(Interval())
0

Order

Intervals implement a lexicographical order:

julia> Interval(0,2) < Interval(1,2,true)
true
julia> Interval(1,3) > Interval(-1,10)
true
julia> Interval(0,1) < Interval(0,1,true)
true

Inclusivity

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

julia> i = Interval(0,1)
[0,1]
julia> 0 in i
true
julia> 1.000001 in i
false
julia> j = Interval(0,1,true)
[0,1[
julia> 1 ∈ j
false
julia> 0.99999999 in j
true

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

julia> i = Interval(0,1)
[0,1]
julia> j = Interval(0,1,true)
[0,1[
julia> j ⊆ i
true
julia> i ⊆ j
false

Or if two Intervals are disjoint:

julia> i = Interval(0,1,true)
[0,1[
julia> j = Interval(1,true,2)
]1,2]
julia> disjoint(i,j)
true
julia> k = Interval(-1,0)
]-1,0]
julia> disjoint(i,k)
false

Intersection

We can compute the intersection of two Intervals:

julia> i = Interval(0,1)
[0,1]
julia> j = Interval(0.4,true,1,true)
]0.4,1[
julia> i ∩ j
]0.4,1[
julia> k = Interval(0.2,true,0.8)
]0.2,0.8]
julia> i ∩ k
]0.2,0.8]
julia> intersect(k,j)
]0.4,0.8[
julia> intersect(i,Interval())
∅

Union

We can compute the union of two Intervals:

julia> i = Interval(0,1)
[0,1]
julia> j = Interval(0.4,true,2.1,true)
]0.4,2.1[
julia> i ∪ j
[0,2.1[
julia> i ∪ Interval()
[0,1]

Sampling

We can sample uniform random numbers from Intervals:

julia> i = Interval(2,4,true)
[2,4[
julia> sample(i)
3.935796996224805
julia> i = Interval(3,true,6)
]3,6]
julia> sample(i,4)
4-element Array{Float64,1}:
 5.543827369817867 
 4.678054798740224 
 3.1740822420010355
 3.6870186624440504

Similarity metrics

Finally, we can compare two Intervals 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 = Interval(3,true,6)
]3,6]
julia> j = Interval(4,7)
[4,7]
julia> jaccard(i,j)
0.5
julia> overlap_coefficient(i,j)
0.6666666666666666
julia> dice_coefficient(i,j)
0.6666666666666666