Interval Index

R←X⍸Y

Classic Edition: the symbol ⍸ (Iota Underbar) is not available in Classic Edition, and Interval Index is instead represented by ⎕U2378.

X is an ordered non-scalar array that represents a set of intervals or ranges.

Note that the i^th interval starts at X[i], then includes all subsequent values up to but not including X[i+1].

For example, if X is (1 3 5) it defines 4 intervals numbered 0 to 3 as follows.

0	less than 1	<1
1	between 1 and 3	(≥1)∧(<3)
2	between 3 and 5	(≥3)∧(<5)
3	greater than or equal to 5	≥5

If X is 'AEIOU' it defines 6 intervals numbered 0 to 5 as follows:

0	before A	⎕UCS[0,⍳64]
1	between A and E	ABCD
2	between E and I	EFGH
3	between I and O	IJKLMN
4	between O and U	OPQREST
5	U and after	UVWXYZ...

Y is an array of the same type (numeric or character) as X.

The result R is an integer array that identifies into which interval the corresponding value in Y falls.

Like dyadic ⍳ (see Index Of), Interval Index works with major cells. For a vector these are its elements; for a matrix its rows, and so forth.

X and Y are compared using the same logic as monadic ⍋ (see Grade Up (Monadic)) which is independent of ⎕CT and ⎕DCT.

⎕IO is an implicit arguments of Interval Index. In all the following examples, ⎕IO is 1.

Examples

      10 20 30⍸11 1 31 21
1 0 3 2

In the above example:

11 is between X[1] and X[2] so the answer is 1.
1 is less than X[1] so the answer is 0
31 is greater than X[⍴X] so the answer is 3
21 is between X[2] and X[3] so the answer is 2.

      'AEIOU' ⍸ 'DYALOG'
1 5 1 3 4 2

And in the alphabetic example above:

"D" is between X[1] and X[2], so the answer is 1
"Y" is after X[⍴X] so the answer is 5
"A" is between X[1] and X[2], so the answer is 1
as so on ...

Example (Classification)

Commercially, olive oil is graded as follows:

if its acidity is less than 0.8%, as "Extra Virgin"
if its acidity is less than 2%, as "Virgin"
if its acidity is less than 3.3%, as "Ordinary"
otherwise, as "Lampante"

     grades←'Extra Virgin' 'Virgin' 'Ordinary' 'Lampante'
     acidity←0.8 2 3.3

     samples←1.3 1.9 0.7 4 .6 3.2
     acidity⍸samples
1 1 0 3 0 2
     samples,⍪grades[1+acidity⍸samples]
┌───┬────────────┐
│1.3│Virgin      │
├───┼────────────┤
│1.9│Virgin      │
├───┼────────────┤
│0.7│Extra Virgin│
├───┼────────────┤
│4  │Lampante    │
├───┼────────────┤
│0.6│Extra Virgin│
├───┼────────────┤
│3.2│Ordinary    │
└───┴────────────┘

Example (Data Consolidation by Interval)

x represents some data sampled in chronological order at timestamps t.

      ⍴x
200000
      x
3984300 2020650 819000 1677100 3959200 2177250 3431800 ...

      ⍴t
200000 3

      (10↑t) (¯10↑t)
┌─────┬────────┐
│0 0 0│23 59 54│
│0 0 0│23 59 55│
│0 0 0│23 59 56│
│0 0 0│23 59 56│
│0 0 0│23 59 58│
│0 0 2│23 59 58│
│0 0 3│23 59 59│
│0 0 3│23 59 59│
│0 0 4│23 59 59│
│0 0 5│23 59 59│
└─────┴────────┘

u represents timestamps for 5-minute intervals:

      ⍴u
288 3
      (10↑u) (¯10↑u)
┌──────┬───────┐
│0  0 0│23 10 0│
│0  5 0│23 15 0│
│0 10 0│23 20 0│
│0 15 0│23 25 0│
│0 20 0│23 30 0│
│0 25 0│23 35 0│
│0 30 0│23 40 0│
│0 35 0│23 45 0│
│0 40 0│23 50 0│
│0 45 0│23 55 0│
└──────┴───────┘

Therefore, the expression (u⍸t){+/⍵}⌸x summarises x in 5-minute intervals.

      u ⍸ t
1 1 1 1 1 1 1 1 1 1 ... 288 288 288 288 288 288

      (u⍸t) {+/⍵}⌸ x
1339083050 1365108650 1541944750 1393476000 1454347100 ...

      (u⍸t) {(⍺⌷u),+/⍵}⌸ x
 0  0 0 1339083050
 0  5 0 1365108650
 0 10 0 1541944750
 0 15 0 1393476000
   ...
23 45 0 1388823150
23 50 0 1453472350
23 55 0 1492078850

Higher-Rank Left Argument

If X is a higher rank array, the function compares sub-arrays in Y with the major cells of X, where a major cell is a sub-array on the leading dimension of X with shape 1↓⍴X. In this case, the shape of the result R is (1-⍴⍴X)↓⍴Y.

Example

      x ← ↑ 'Fi' 'Jay' 'John' 'Morten' 'Roger'
      x
Fi
Jay   
John  
Morten
Roger 
      ⍴x
5 6
      y ← x ⍪ ↑ 'JD' 'Jd' 'Geoff' 'Alpha' 'Omega' 'Zeus  '
      y
Fi
Jay   
John  
Morten
Roger 
JD    
Jd    
Geoff 
Alpha 
Omega 
Zeus  

      x ⍸ y
1 2 3 4 5 1 2 1 0 4 5
      y ,⍪ x⍸y
Fi     1
Jay    2
John   3
Morten 4
Roger  5
JD     1
Jd     2
Geoff  1
Alpha  0
Omega  4
Zeus   5

Further Example

      ⍴x
5 6
      ⍴y
3 3 6
      x
Fi    
Jay   
John  
Morten
Roger 
      y
Fi    
Jay   
John  
      
Morten
Roger 
JD    
      
Jd    
Geoff 
Alpha 
      x⍸y
1 2 3
4 5 1
2 1 0

Nested Array Example

A card-player likes to sort a hand into suits spades, hearts, diamond, clubs (fortunately alphabetic) and high-to-low within each suit.

      suits←'Clubs' 'Diamonds' 'Hearts' 'Spades'
      pack←,(⊂¨suits)∘.,1↓14 ⍝ 11=Jack ... 14=Ace
      hand←↑(,pack)[7?52]
      hand←hand[⍒hand;]
      hand
┌────────┬──┐
│Spades  │12│
├────────┼──┤
│Hearts  │12│
├────────┼──┤
│Hearts  │7 │
├────────┼──┤
│Hearts  │2 │
├────────┼──┤
│Diamonds│11│
├────────┼──┤
│Diamonds│9 │
├────────┼──┤
│Clubs   │8 │
└────────┴──┘

Another card, the 10 of diamonds is dealt. Where must it go in the hand ?

      (⊖hand)⍸'Diamonds' 10 ⍝ left arg must be sorted up
2
      (¯2↓hand)⍪'Diamonds' 10⍪¯2↑hand
┌────────┬──┐
│Spades  │12│
├────────┼──┤
│Hearts  │12│
├────────┼──┤
│Hearts  │7 │
├────────┼──┤
│Hearts  │2 │
├────────┼──┤
│Diamonds│11│
├────────┼──┤
│Diamonds│10│
├────────┼──┤
│Diamonds│9 │
├────────┼──┤
│Clubs   │8 │
└────────┴──┘

Note that if (∧/Y∊X) and X is sorted and ⎕CT=0 ,then x⍸y is the same as x⍳y.