# Vocabulary/barco

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`|: y`Transpose

Rank Infinity *-- operates on x and y as a whole --*
WHY IS THIS IMPORTANT?

Reverses the axes of an array `y`

ii=: ] {. [: i. 10 #~ # NB. utility verb: make self-indexing array ]y=: ii 2 3 4 NB. sample value of shape: 2 3 4 0 1 2 3 10 11 12 13 20 21 22 23 100 101 102 103 110 111 112 113 120 121 122 123 $ y 2 3 4 $ |:y 4 3 2 |:y 0 100 10 110 20 120 1 101 11 111 21 121 2 102 12 112 22 122 3 103 13 113 23 123

### Common uses

Transpose a 2D matrix

]z=: 2 3 $ 'abcdef' NB. sample mx of shape: 2 3 abc def |:z ad be cf

`x |: y`Rearrange Axes

Rank 1 _ *-- operates on lists of x and the entirety of y --*
WHY IS THIS IMPORTANT?

Rearranges the axes of an array `y`.

- (
`x`is an atom) — the axis having index`x`becomes the new last axis - (
`x`is a list) — the axes having indices`x`become the new last axes.

ii=: ] {. [: i. 10 #~ # NB. utility verb: make self-indexing array ]y =: ii 2 3 4 0 1 2 3 10 11 12 13 20 21 22 23 100 101 102 103 110 111 112 113 120 121 122 123 0 |: y NB. Move axis 0 to end. Old axes 1 and 2 become new axes 0 and 1 0 100 1 101 2 102 3 103 10 110 11 111 12 112 13 113 20 120 21 121 22 122 23 123 $y 2 3 4 $ 0 |: y NB. New shape shows original axis 0 (size=2) is now the last axis 3 4 2

Repeat with a higher-rank y

$ y=: ii 2 3 4 5 6 2 3 4 5 6 x =: 2 1 NB. axes with sizes 4 and 3 respectively become the last axes $ x |: y 2 5 6 4 3

### More Information

1. Use a special form of `x` in ` x |: y ` to *extract the diagonal* of matrix `y`

] y =: 4 4 $ 'abcdefghijklmnop' abcd efgh ijkl mnop x =: < 0 1 x |: y afkp

### More Uses

1. To cause the cells of a noun to correspond to a desired computation.

Reordering the axes of an array has two effects:

1. The index lists used to refer to atoms and cells are correspondingly reordered; 1. The axes that make up the cells are changed.

The best way to understand reordering is to assign a meaning to each axis.
For example we could have an array `data` of information for a school district. The axes could have the following meanings:

axis 0 - the school

axis 1 - grade level

axis 2 - sex of students

axis 3 - core subject

axis 4 - a list of information for a given combination of school/grade/sex/subject

Each axis has associated with it an interpretation of the index of that axis, which might look like

SCHOOLS =: 'Ridgemont High';'Valley High';'Eastgate High' GRADES =: 9;10;11;12 SEX =: 'M';'F' SUBJECT =: 'Math';'Science';'English';'History';'Language' INFO =: 'Number of students';'25th pctile grade';'50th pctile grade';'75th pctile grade' NB. Create a data table where each atom shows its index list data =: }:@;@:((,&'_') each) each { {.@;:@": every each SCHOOLS;GRADES;SEX;SUBJECT;<INFO

With this setup, `(<1 1 0 2 3) { data` would be the 75th-pctile English grade for 10th-grade boys at Valley High.

(<1 1 0 2 3) { data +------------------------+ |Valley_10_M_English_75th| +------------------------+

Each arrangement of axes makes some computations easier than others. It is easy to compute results that correspond to **cells** of a noun. Here, the 1-cell is the information line. It would be easy to compute a statistic on each combination of school/grade/sex/subject.

It would also be easy to calculate a result for each combination of school/grade/sex, using all the subject data for that combination, because the subject information makes up the 2-cells of `data`.

(<1 1 0) { data NB. A single 2-cell +---------------------------+-------------------------+-------------------------+-------------------------+ |Valley_10_M_Math_Number |Valley_10_M_Math_25th |Valley_10_M_Math_50th |Valley_10_M_Math_75th | +---------------------------+-------------------------+-------------------------+-------------------------+ |Valley_10_M_Science_Number |Valley_10_M_Science_25th |Valley_10_M_Science_50th |Valley_10_M_Science_75th | +---------------------------+-------------------------+-------------------------+-------------------------+ |Valley_10_M_English_Number |Valley_10_M_English_25th |Valley_10_M_English_50th |Valley_10_M_English_75th | +---------------------------+-------------------------+-------------------------+-------------------------+ |Valley_10_M_History_Number |Valley_10_M_History_25th |Valley_10_M_History_50th |Valley_10_M_History_75th | +---------------------------+-------------------------+-------------------------+-------------------------+ |Valley_10_M_Language_Number|Valley_10_M_Language_25th|Valley_10_M_Language_50th|Valley_10_M_Language_75th| +---------------------------+-------------------------+-------------------------+-------------------------+

To operate on such a cell you would have

subjinfoverb =: verb define NB. y is a 5x4 table of info for each subject ... ) school_grade_sex =: subjinfoverb"2 data NB. Compute for each combination of school/grade/sex

But it would **not** be easy to compute something for every combination of school/grade/subject. That information is spread across cells.

This is where reordering the axes comes in. To do a computation for each school/grade/subject, make those axes the leading axes of the array, leaving sex/info as the last 2 axes. Then each 2-cell will have all the information for a single combination of school/grade/subject.

$data NB. Data array, school/grade/sex/subject/info 3 4 2 5 4 $ 0 1 3 2 4 |: data NB. school/grade/subject/sex/info 3 4 5 2 4 $ 2 4 |: data NB. alternate form 3 4 5 2 4 (<1 1 0) { 2 4 |: data NB. A single 2-cell of the reshaped data +-----------------------+---------------------+---------------------+---------------------+ |Valley_10_M_Math_Number|Valley_10_M_Math_25th|Valley_10_M_Math_50th|Valley_10_M_Math_75th| +-----------------------+---------------------+---------------------+---------------------+ |Valley_10_F_Math_Number|Valley_10_F_Math_25th|Valley_10_F_Math_50th|Valley_10_F_Math_75th| +-----------------------+---------------------+---------------------+---------------------+

As illustrated in the last example, you don't have to give the entire list of axes. The axes you specify are moved to be the last axes of the result.

sexinfoverb =: verb define NB. y is a 2x4 table of info for each sex ... ) school_grade_subject =: sexinfoverb"2 (0 1 3 2 4) |: data NB. Compute for each combination of school/grade/sex

Any index list into the transposed array must be transposed accordingly. If you needed to get to that 75th-pctile English grade for 10th-grade boys at Valley High, the index list into the transposed array would be `(<1 1 2 0 3)`

(<1 1 2 0 3) { 2 4 |: data +------------------------+ |Valley_10_M_English_75th| +------------------------+