Key: Difference between revisions

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[[Monadic]]ally, Key will group identical [[major cell]]s together and applies the [[function]] operand once for each unique major cell. The function is applied with the unique major cell as left argument, while the right argument is the indices of major cells that match it:  
[[Monadic]]ally, Key will group identical [[major cell]]s together and applies the [[function]] operand once for each unique major cell. The function is applied with the unique major cell as left argument, while the right argument is the indices of major cells that match it:  


<source lang=apl>
<syntaxhighlight lang=apl>
       {⍺⍵}⌸'Mississippi'
       {⍺⍵}⌸'Mississippi'
┌─┬────────┐
┌─┬────────┐
Line 19: Line 19:
In the [[dyadic]] case, Key applies the function to collections of major cells from the right argument corresponding to unique elements of the left argument:
In the [[dyadic]] case, Key applies the function to collections of major cells from the right argument corresponding to unique elements of the left argument:


<source lang=apl>
<syntaxhighlight lang=apl>
       'Mississippi'{⍺⍵}⌸'ABCDEFGHIJK'  
       'Mississippi'{⍺⍵}⌸'ABCDEFGHIJK'  
┌─┬────┐
┌─┬────┐
Line 32: Line 32:
</syntaxhighlight>
</syntaxhighlight>


The monadic case, <source lang=apl inline>f⌸Y</syntaxhighlight> is equivalent to <source lang=apl inline>Y f⌸ ⍳≢Y</syntaxhighlight>.
The monadic case, <syntaxhighlight lang=apl inline>f⌸Y</syntaxhighlight> is equivalent to <syntaxhighlight lang=apl inline>Y f⌸ ⍳≢Y</syntaxhighlight>.


== Problems ==
== Problems ==
=== Vocabulary ===
=== Vocabulary ===
A common problem with Key is the inability to control the order of the result (as Key will use the order of appearance) and the "vocabulary" (as Key will never include information for a major cell that doesn't occur). For example, here we want to count occurrences of the letters A, C, G, T:
A common problem with Key is the inability to control the order of the result (as Key will use the order of appearance) and the "vocabulary" (as Key will never include information for a major cell that doesn't occur). For example, here we want to count occurrences of the letters A, C, G, T:
<source lang=apl>
<syntaxhighlight lang=apl>
       {⍺,≢⍵}⌸'TCCGCGGTGGCG'
       {⍺,≢⍵}⌸'TCCGCGGTGGCG'
T 2
T 2
Line 44: Line 44:
</syntaxhighlight>
</syntaxhighlight>
Since A is entirely missing in the argument, it isn't mentioned in the result either. Likewise, the result is mis-ordered due to G and T appearing before the first C. A common solution is to inject the vocabulary before the actual data, and then decrement from the counts:
Since A is entirely missing in the argument, it isn't mentioned in the result either. Likewise, the result is mis-ordered due to G and T appearing before the first C. A common solution is to inject the vocabulary before the actual data, and then decrement from the counts:
<source lang=apl>      {⍺,¯1+≢⍵}⌸'ACGT','TCCGCGGTGGCG'
<syntaxhighlight lang=apl>      {⍺,¯1+≢⍵}⌸'ACGT','TCCGCGGTGGCG'
A 0
A 0
C 4
C 4
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</syntaxhighlight>
</syntaxhighlight>
Now that the meaning of each count is known, the operand's left argument can be ignored, and the decrementing can be factored out from the operand:
Now that the meaning of each count is known, the operand's left argument can be ignored, and the decrementing can be factored out from the operand:
<source lang=apl>
<syntaxhighlight lang=apl>
       ¯1+{≢⍵}⌸'ACGT','TCCGCGGTGGCG'
       ¯1+{≢⍵}⌸'ACGT','TCCGCGGTGGCG'
0 4 6 2
0 4 6 2
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=== Computing the unique ===
=== Computing the unique ===
Key computes the set of [[unique]] major cells. Often, this collection is needed separately from the occurrence information, but can be hard to extract. For example, to get the most frequently occurring letter:
Key computes the set of [[unique]] major cells. Often, this collection is needed separately from the occurrence information, but can be hard to extract. For example, to get the most frequently occurring letter:
<source lang=apl>
<syntaxhighlight lang=apl>
       ⊃⍒{≢⍵}⌸'TCCGCGGTGGCG'
       ⊃⍒{≢⍵}⌸'TCCGCGGTGGCG'
3
3
</syntaxhighlight>
</syntaxhighlight>
Notice that 3 is the index in the unique set of letters, and so it is tempting to write:
Notice that 3 is the index in the unique set of letters, and so it is tempting to write:
<source lang=apl>
<syntaxhighlight lang=apl>
       {(⊃⍒{≢⍵}⌸⍵)⌷∪⍵}'TCCGCGGTGGCG'
       {(⊃⍒{≢⍵}⌸⍵)⌷∪⍵}'TCCGCGGTGGCG'
G
G
</syntaxhighlight>
</syntaxhighlight>
However, while this code works, it is inefficient in that the unique is computed twice. This can be avoided by letting Key return the unique and using that:
However, while this code works, it is inefficient in that the unique is computed twice. This can be avoided by letting Key return the unique and using that:
<source lang=apl>
<syntaxhighlight lang=apl>
       (keys counts)←,⌿{⍺,≢⍵}⌸'TCCGCGGTGGCG'
       (keys counts)←,⌿{⍺,≢⍵}⌸'TCCGCGGTGGCG'
       keys⌷⍨⊃⍒counts
       keys⌷⍨⊃⍒counts
Line 73: Line 73:
</syntaxhighlight>
</syntaxhighlight>
Unfortunately, this can introduce a different inefficiency, in that the result of Key's operand can end up being a [[heterogeneous array]] (containing multiple [[datatype]]s), and these are stored as pointer arrays, consuming memory for one pointer per element, and forcing "pointer chasing" when addressing the data. A possible work-around is to collect the unique keys separately from the result of counts:
Unfortunately, this can introduce a different inefficiency, in that the result of Key's operand can end up being a [[heterogeneous array]] (containing multiple [[datatype]]s), and these are stored as pointer arrays, consuming memory for one pointer per element, and forcing "pointer chasing" when addressing the data. A possible work-around is to collect the unique keys separately from the result of counts:
<source lang=apl>
<syntaxhighlight lang=apl>
       data←'TCCGCGGTGGCG'
       data←'TCCGCGGTGGCG'
       keys←0⌿data
       keys←0⌿data
Line 80: Line 80:
G
G
</syntaxhighlight>
</syntaxhighlight>
If there are a large number of unique values, the repeated updating of the accumulating <source lang=apl inline>keys</syntaxhighlight> variable can be an issue in itself.
If there are a large number of unique values, the repeated updating of the accumulating <syntaxhighlight lang=apl inline>keys</syntaxhighlight> variable can be an issue in itself.


== External links ==
== External links ==

Revision as of 22:18, 10 September 2022

⌸

Key (⌸) is a primitive monadic operator which takes a dyadic function operand where specified keys group the indices or major cells of an argument. It was introduced in Dyalog APL version 14.0 and is commonly compared to SQL's GROUP BY statement.

Description

Monadically, Key will group identical major cells together and applies the function operand once for each unique major cell. The function is applied with the unique major cell as left argument, while the right argument is the indices of major cells that match it: 

      {⍺⍵}⌸'Mississippi'
┌─┬────────┐
│M│1       │
├─┼────────┤
│i│2 5 8 11│
├─┼────────┤
│s│3 4 6 7 │
├─┼────────┤
│p│9 10    │
└─┴────────┘

In the dyadic case, Key applies the function to collections of major cells from the right argument corresponding to unique elements of the left argument: 

      'Mississippi'{⍺⍵}⌸'ABCDEFGHIJK' 
┌─┬────┐
│M│A   │
├─┼────┤
│i│BEHK│
├─┼────┤
│s│CDFG│
├─┼────┤
│p│IJ  │
└─┴────┘

The monadic case, f⌸Y is equivalent to Y f⌸ ⍳≢Y.

Problems

Vocabulary

A common problem with Key is the inability to control the order of the result (as Key will use the order of appearance) and the "vocabulary" (as Key will never include information for a major cell that doesn't occur). For example, here we want to count occurrences of the letters A, C, G, T: 

      {⍺,≢⍵}⌸'TCCGCGGTGGCG'
T 2
C 4
G 6

Since A is entirely missing in the argument, it isn't mentioned in the result either. Likewise, the result is mis-ordered due to G and T appearing before the first C. A common solution is to inject the vocabulary before the actual data, and then decrement from the counts: 

      {⍺,¯1+≢⍵}⌸'ACGT','TCCGCGGTGGCG'
A 0
C 4
G 6
T 2

Now that the meaning of each count is known, the operand's left argument can be ignored, and the decrementing can be factored out from the operand: 

      ¯1+{≢⍵}⌸'ACGT','TCCGCGGTGGCG'
0 4 6 2

Computing the unique

Key computes the set of unique major cells. Often, this collection is needed separately from the occurrence information, but can be hard to extract. For example, to get the most frequently occurring letter: 

      ⊃⍒{≢⍵}⌸'TCCGCGGTGGCG'
3

Notice that 3 is the index in the unique set of letters, and so it is tempting to write: 

      {(⊃⍒{≢⍵}⌸⍵)⌷∪⍵}'TCCGCGGTGGCG'
G

However, while this code works, it is inefficient in that the unique is computed twice. This can be avoided by letting Key return the unique and using that: 

      (keys counts)←,⌿{⍺,≢⍵}⌸'TCCGCGGTGGCG'
      keys⌷⍨⊃⍒counts
G

Unfortunately, this can introduce a different inefficiency, in that the result of Key's operand can end up being a heterogeneous array (containing multiple datatypes), and these are stored as pointer arrays, consuming memory for one pointer per element, and forcing "pointer chasing" when addressing the data. A possible work-around is to collect the unique keys separately from the result of counts: 

      data←'TCCGCGGTGGCG'
      keys←0⌿data
      counts←{keys⍪←⍺ ⋄ ≢⍵}⌸data
      keys⌷⍨⊃⍒counts
G

If there are a large number of unique values, the repeated updating of the accumulating keys variable can be an issue in itself.

External links

Lessons

Documentation


APL built-ins [edit]
Primitives (Timeline) Functions
Scalar
Monadic Conjugate ∙ Negate ∙ Signum ∙ Reciprocal ∙ Magnitude ∙ Exponential ∙ Natural Logarithm ∙ Floor ∙ Ceiling ∙ Factorial ∙ Not ∙ Pi Times ∙ Roll ∙ Type ∙ Imaginary ∙ Square Root ∙ Round
Dyadic Add ∙ Subtract ∙ Times ∙ Divide ∙ Residue ∙ Power ∙ Logarithm ∙ Minimum ∙ Maximum ∙ Binomial ∙ Comparison functions ∙ Boolean functions (And, Or, Nand, Nor) ∙ GCD ∙ LCM ∙ Circular ∙ Complex ∙ Root
Non-Scalar
Structural Shape ∙ Reshape ∙ Tally ∙ Depth ∙ Ravel ∙ Enlist ∙ Table ∙ Catenate ∙ Reverse ∙ Rotate ∙ Transpose ∙ Raze ∙ Mix ∙ Split ∙ Enclose ∙ Nest ∙ Cut (K) ∙ Pair ∙ Link ∙ Partitioned Enclose ∙ Partition
Selection First ∙ Pick ∙ Take ∙ Drop ∙ Unique ∙ Identity ∙ Stop ∙ Select ∙ Replicate ∙ Expand ∙ Set functions (Intersection ∙ Union ∙ Without) ∙ Bracket indexing ∙ Index ∙ Cartesian Product ∙ Sort
Selector Index generator ∙ Grade ∙ Index Of ∙ Interval Index ∙ Indices ∙ Deal ∙ Prefix and suffix vectors
Computational Match ∙ Not Match ∙ Membership ∙ Find ∙ Nub Sieve ∙ Encode ∙ Decode ∙ Matrix Inverse ∙ Matrix Divide ∙ Format ∙ Execute ∙ Materialise ∙ Range
Operators Monadic Each ∙ Commute ∙ Constant ∙ Replicate ∙ Expand ∙ Reduce ∙ Windowed Reduce ∙ Scan ∙ Outer Product ∙ Key ∙ I-Beam ∙ Spawn ∙ Function axis ∙ Identity (Null, Ident)
Dyadic Bind ∙ Compositions (Compose, Reverse Compose, Beside, Withe, Atop, Over) ∙ Inner Product ∙ Determinant ∙ Power ∙ At ∙ Under ∙ Rank ∙ Depth ∙ Variant ∙ Stencil ∙ Cut ∙ Direct definition (operator) ∙ Identity (Lev, Dex)
Quad names Index origin ∙ Comparison tolerance ∙ Migration level ∙ Atomic vector
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