# Array Operations

Manipulating sample data as arrays

## Overview

The `array-operations` system contains a collection of functions and macros for manipulating Common Lisp arrays and performing numerical calculations with them.

Array-operations is a ‘generic’ way of operating on array like data structures. Several `aops` functions have been implemented for `data-frame`. For those that haven’t, you can transform arrays to data frames using the `df:matrix-df` function, and a data-frame to an array using `df:as-array`. This make it convenient to work with the data sets using either system.

## Quick look

Arrays can be created with numbers from a statistical distribution:

``````(rand '(2 2)) ; => #2A((0.62944734 0.2709539) (0.81158376 0.6700171))
``````

in linear ranges:

``````(linspace 1 10 7) ; => #(1 5/2 4 11/2 7 17/2 10)
``````

or generated using a function, optionally given index position

``````(generate #'identity '(2 3) :position) ; => #2A((0 1 2) (3 4 5))
``````

They can also be transformed and manipulated:

``````(defparameter A #2A((1 2)
(3 4)))
(defparameter B #2A((2 3)
(4 5)))

;; split along any dimension
(split A 1)  ; => #(#(1 2) #(3 4))

;; stack along any dimension
(stack 1 A B) ; => #2A((1 2 2 3)
;        (3 4 4 5))

;; element-wise function map
(each #'+ #(0 1 2) #(2 3 5)) ; => #(2 4 7)

;; element-wise expressions
(vectorize (A B) (* A (sqrt B))) ; => #2A((1.4142135 3.4641016)
;        (6.0       8.944272))

;; index operations e.g. matrix-matrix multiply:
(each-index (i j)
(sum-index k
(* (aref A i k) (aref B k j)))) ; => #2A((10 13)
;        (22 29))
``````

## Array shorthand

The library defines the following short function names that are synonyms for Common Lisp operations:

array-operations Common Lisp
size array-total-size
rank array-rank
dim array-dimension
dims array-dimensions
nrow number of rows in matrix
ncol number of columns in matrix

The `array-operations` package has the nickname `aops`, so you can use, for example, `(aops:size my-array)` without `use`‘ing the package.

## Displaced arrays

According to the Common Lisp specification, a displaced array is:

An array which has no storage of its own, but which is instead indirected to the storage of another array, called its target, at a specified offset, in such a way that any attempt to access the displaced array implicitly references the target array.

Displaced arrays are one of the niftiest features of Common Lisp. When an array is displaced to another array, it shares structure with (part of) that array. The two arrays do not need to have the same dimensions, in fact, the dimensions do not be related at all as long as the displaced array fits inside the original one. The row-major index of the former in the latter is called the offset of the displacement.

### displace

Displaced arrays are usually constructed using `make-array`, but this library also provides `displace` for that purpose:

``````(defparameter *a* #2A((1 2 3)
(4 5 6)))
(aops:displace *a* 2 1) ; => #(2 3)
``````

Here’s an example of using displace to implement a sliding window over some set of values, say perhaps a time-series of stock prices:

``````(defparameter stocks (aops:linspace 1 100 100))
(loop for i from 0 to (- (length stocks) 20)
do (format t "~A~%" (aops:displace stocks 20 i)))
;#(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20)
;#(2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21)
;#(3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22)
``````

### flatten

`flatten` displaces to a row-major array:

``````(aops:flatten *a*) ; => #(1 2 3 4 5 6)
``````

### split

The real fun starts with `split`, which splits off sub-arrays nested within a given axis:

``````(aops:split *a* 1) ; => #(#(1 2 3) #(4 5 6))
(defparameter *b* #3A(((0 1) (2 3))
((4 5) (6 7))))
(aops:split *b* 0) ; => #3A(((0 1) (2 3)) ((4 5) (6 7)))
(aops:split *b* 1) ; => #(#2A((0 1) (2 3)) #2A((4 5) (6 7)))
(aops:split *b* 2) ; => #2A((#(0 1) #(2 3)) (#(4 5) #(6 7)))
(aops:split *b* 3) ; => #3A(((0 1) (2 3)) ((4 5) (6 7)))
``````

Note how splitting at `0` and the rank of the array returns the array itself.

### sub

Now consider `sub`, which returns a specific array, composed of the elements that would start with given subscripts:

``````(aops:sub *b* 0) ; => #2A((0 1)
;        (2 3))
(aops:sub *b* 0 1) ; => #(2 3)
(aops:sub *b* 0 1 0) ; => 2
``````

In the case of vectors, `sub` works like `aref`:

``````(aops:sub #(1 2 3 4 5) 1) ; => 2
``````

There is also a `(setf sub)` function.

### partition

`partition` returns a consecutive chunk of an array separated along its first subscript:

``````(aops:partition #2A((0 1)
(2 3)
(4 5)
(6 7)
(8 9))
1 3) ; => #2A((2 3)
;        (4 5))
``````

and also has a `(setf partition)` pair.

### combine

`combine` is the opposite of `split`:

``````(aops:combine #(#(0 1) #(2 3))) ; => #2A((0 1)
;        (2 3))
``````

### subvec

`subvec` returns a displaced subvector:

``````(aops:subvec #(0 1 2 3 4) 2 4) ; => #(2 3)
``````

There is also a `(setf subvec)` function, which is like `(setf subseq)` except for demanding matching lengths.

### reshape

Finally, `reshape` can be used to displace arrays into a different shape:

``````(aops:reshape #2A((1 2 3)
(4 5 6)) '(3 2))
; => #2A((1 2)
;        (3 4)
;        (5 6))
``````

You can use `t` for one of the dimensions, to be filled in automatically:

``````(aops:reshape *b* '(1 t)) ; => #2A((0 1 2 3 4 5 6 7))
``````

`reshape-col` and `reshape-row` reshape your array into a column or row matrix, respectively:

``````(defparameter *a* #2A((0 1)
(2 3)
(4 5)))
(aops:reshape-row *a*) ;=> #2A((0 1 2 3 4 5))
(aops:reshape-col *a*) ;=> #2A((0) (1) (2) (3) (4) (5))
``````

## Specifying dimensions

Functions in the library accept the following in place of dimensions:

• a list of dimensions (as for `make-array`),
• a positive integer, which is used as a single-element list,
• another array, the dimensions of which are used.

The last one allows you to specify dimensions with other arrays. For example, to reshape an array `a1` to look like `a2`, you can use

``````(aops:reshape a1 a2)
``````

``````(aops:reshape a1 (aops:dims a2))
``````

## Creation & transformation

Use the functions in this section to create commonly used arrays types. When the resulting element type cannot be inferred from an existing array or vector, you can pass the element type as an optional argument. The default is elements of type `T`.

Element traversal order of these functions is unspecified. The reason for this is that the library may use parallel code in the future, so it is unsafe to rely on a particular element traversal order.

The following functions all make a new array, taking the dimensions as input. There are also versions ending in `!` which do not make a new array, but take an array as first argument, which is modified and returned.

Function Description
zeros Filled with zeros
ones Filled with ones
rand Filled with uniformly distributed random numbers between 0 and 1
randn Normally distributed with mean 0 and standard deviation 1
linspace Evenly spaced numbers in given range

For example:

``````(aops:zeros 3)
; => #(0 0 0)

(aops:zeros 3 'double-float)
; => #(0.0d0 0.0d0 0.0d0)

(aops:rand '(2 2))
; => #2A((0.6686077 0.59425664)
;        (0.7987722 0.6930506))

(aops:rand '(2 2) 'single-float)
; => #2A((0.39332366 0.5557821)
;        (0.48831415 0.10924244))

(let ((a (make-array '(2 2) :element-type 'double-float)))
;; Modify array A, filling with random numbers
;; element type is taken from existing array
(aops:rand! a))
; => #2A((0.6324615478515625d0 0.4636608362197876d0)
;        (0.4145939350128174d0 0.5124958753585815d0))
``````
``````(linspace 0 4 5)   ;=> #(0 1 2 3 4)
(linspace 1 3 5)   ;=> #(0 1/2 1 3/2 2)
(linspace 1 3 5 'double-float) ;=> #(1.0d0 1.5d0 2.0d0 2.5d0 3.0d0)
(linspace 0 4d0 3) ;=> #(0.0d0 2.0d0 4.0d0)
``````

### generate

`generate` (and `generate*`) allow you to generate arrays using functions. The function signatures are:

``````generate* (element-type function dimensions &optional arguments)
generate (function dimensions &optional arguments)
``````

Where `arguments` are passed to `function`. Possible arguments are:

• no arguments, when ARGUMENTS is nil
• the position (= row major index), when ARGUMENTS is :POSITION
• a list of subscripts, when ARGUMENTS is :SUBSCRIPTS
• both when ARGUMENTS is :POSITION-AND-SUBSCRIPTS
``````(aops:generate (lambda () (random 10)) 3) ; => #(6 9 5)

(aops:generate #'identity '(2 3) :position) ; => #2A((0 1 2)
;        (3 4 5))

(aops:generate #'identity '(2 2) :subscripts)
; => #2A(((0 0) (0 1))
;        ((1 0) (1 1)))

(aops:generate #'cons '(2 2) :position-and-subscripts)
; => #2A(((0 0 0) (1 0 1))
;        ((2 1 0) (3 1 1)))
``````

### permute

`permute` can permute subscripts (you can also invert, complement, and complete permutations, look at the docstring and the unit tests). Transposing is a special case of permute:

``````(defparameter *a* #2A((1 2 3)
(4 5 6)))
(aops:permute '(0 1) *a*) ; => #2A((1 2 3)
;        (4 5 6))
(aops:permute '(1 0) *a*) ; => #2A((1 4)
;        (2 5)
;        (3 6))
``````

### each

`each` applies a function to its one dimensional array arguments elementwise. It essentially is an element-wise function map on each of the vectors:

``````(aops:each #'+ #(0 1 2)
#(2 3 5)
#(1 1 1)
; => #(3 5 8)
``````

### vectorize

`vectorize` is a macro which performs elementwise operations

``````(defparameter a #(1 2 3 4))
(aops:vectorize (a) (* 2 a)) ; => #(2 4 6 8)

(defparameter b #(2 3 4 5))
(aops:vectorize (a b) (* a (sin b)))
; => #(0.9092974 0.28224 -2.2704074 -3.8356972)
``````

There is also a version `vectorize*` which takes a type argument for the resulting array, and a version `vectorize!` which sets elements in a given array.

### margin

The semantics of `margin` are more difficult to explain, so perhaps an example will be more useful. Suppose that you want to calculate column sums in a matrix. You could `permute` (transpose) the matrix, `split` its sub-arrays at rank one (so you get a vector for each row), and apply the function that calculates the sum. `margin` automates that for you:

``````(aops:margin (lambda (column)
(reduce #'+ column))
#2A((0 1)
(2 3)
(5 7)) 0) ; => #(7 11)
``````

But the function is more general than this: the arguments `inner` and `outer` allow arbitrary permutations before splitting.

### recycle

Finally, `recycle` allows you to reuse the elements of the first argument, `object`, to create new arrays by extending the dimensions. The `:outer` keyword repeats the original `object` and `:inner` keyword argument repeats the elements of `object`. When both `:inner` and `:outer` are `nil`, `object` is returned as is. Non-array `objects` are intepreted as rank 0 arrays, following the usual semantics.

``````(aops:recycle #(2 3) :inner 2 :outer 4)
; => #3A(((2 2) (3 3))
((2 2) (3 3))
((2 2) (3 3))
((2 2) (3 3)))
``````

Three dimensional arrays can be tough to get your head around. In the example above, `:outer` asks for 4 2-element vectors, composed of repeating the elements of `object` twice, i.e. repeat ‘2’ twice and repeat ‘3’ twice. Compare this with `:inner` as 3:

``````(aops:recycle #(2 3) :inner 3 :outer 4)
; #3A(((2 2 2) (3 3 3))
((2 2 2) (3 3 3))
((2 2 2) (3 3 3))
((2 2 2) (3 3 3)))
``````

The most common use case for `recycle` is to ‘stretch’ a vector so that it can be an operand for an array of compatible dimensions. In Python, this would be known as ‘broadcasting’. See the Numpy broadcasting basics for other use cases.

For example, suppose we wish to multiply array `a`, a size 4x3 with vector `b` of size 3, as in the figure below:

We can do that by `recycling` array `b` like this:

``````(recycle #(1 2 3) :outer 4)
;#2A((1 2 3)
;	 (1 2 3)
;	 (1 2 3)
;	 (1 2 3))
``````

In a similar manner, the figure below (also from the Numpy page) shows how we might stretch a vector horizontally to create an array compatible with the one created above.

To create that array from a vector, use the `:inner` keyword:

``````(recycle #(0 10 20 30) :inner 3)
;#2A((0 0 0)
;	 (10 10 10)
;	 (20 20 20)
;	 (30 30 30))
``````

### turn

`turn` rotates an array by a specified number of clockwise 90° rotations. The axis of rotation is specified by RANK-1 (defaulting to 0) and RANK-2 (defaulting to 1). In the first example, we’ll rotate by 90°:

``````(defparameter array-1 #2A((1 0 0)
(2 0 0)
(3 0 0)))
(aops:turn array-1 1)
;; #2A((3 2 1)
;;     (0 0 0)
;;	   (0 0 0))
``````

and if we rotate it twice (180°):

``````(aops:turn array-1 2)
;; #2A((0 0 3)
;;     (0 0 2)
;;     (0 0 1))
``````

finally, rotate it three times (270°):

``````(aops:turn array-1 3)
;; #2A((0 0 0)
;;     (0 0 0)
;;     (1 2 3))
``````

### map-array

`map-array` maps a function over the elements of an array.

``````(aops:map-array #2A((1.7 2.1 4.3 5.4)
(0.3 0.4 0.5 0.6))
#'log)
; #2A(( 0.53062826 0.7419373  1.4586151  1.686399)
;     (-1.2039728 -0.9162907 -0.6931472 -0.5108256))
``````

### outer

`outer` is generalized outer product of arrays using a provided function.

Lambda list: `(function &rest arrays)`

The resulting array has the concatenated dimensions of arrays

The examples below return the outer product of vectors or arrays. This is the outer product you get in most linear algebra packages.

``````(defparameter a #(2 3 5))
(defparameter b #(7 11))
(defparameter c #2A((7 11)
(13 17)))

(outer #'* a b)
;#2A((14 22)
;    (21 33)
;    (35 55))

(outer #'* c a)
;#3A(((14 21 35) (22 33 55))
;    ((26 39 65) (34 51 85)))
``````

## Indexing operations

### nested-loop

`nested-loop` is a simple macro which iterates over a set of indices with a given range

``````(defparameter A #2A((1 2) (3 4)))

(aops:nested-loop (i j) (array-dimensions A)
(setf (aref A i j) (* 2 (aref A i j))))
A ; => #2A((2 4) (6 8))

(aops:nested-loop (i j) '(2 3)
(format t "(~a ~a) " i j)) ; => (0 0) (0 1) (0 2) (1 0) (1 1) (1 2)
``````

### sum-index

`sum-index` is a macro which uses a code walker to determine the dimension sizes, summing over the given index or indices

``````(defparameter A #2A((1 2) (3 4)))

;; Trace
(aops:sum-index i (aref A i i)) ; => 5

;; Sum array
(aops:sum-index (i j) (aref A i j)) ; => 10

;; Sum array
(aops:sum-index i (row-major-aref A i)) ; => 10
``````

The main use for `sum-index` is in combination with `each-index`.

### each-index

`each-index` is a macro which creates an array and iterates over the elements. Like `sum-index` it is given one or more index symbols, and uses a code walker to find array dimensions.

``````(defparameter A #2A((1 2)
(3 4)))
(defparameter B #2A((5 6)
(7 8)))

;; Transpose
(aops:each-index (i j) (aref A j i)) ; => #2A((1 3)
;        (2 4))

;; Sum columns
(aops:each-index i
(aops:sum-index j
(aref A j i))) ; => #(4 6)

;; Matrix-matrix multiply
(aops:each-index (i j)
(aops:sum-index k
(* (aref A i k) (aref B k j)))) ; => #2A((19 22)
;        (43 50))
``````

### reduce-index

`reduce-index` is a more general version of `sum-index`; it applies a reduction operation over one or more indices.

``````(defparameter A #2A((1 2)
(3 4)))

;; Sum all values in an array
(aops:reduce-index #'+ i (row-major-aref A i)) ; => 10

;; Maximum value in each row
(aops:each-index i
(aops:reduce-index #'max j
(aref A i j)))  ; => #(2 4)
``````

## Reducing

Some reductions over array elements can be done using the Common Lisp `reduce` function, together with `aops:flatten`, which returns a displaced vector:

``````(defparameter a #2A((1 2)
(3 4)))
(reduce #'max (aops:flatten a)) ; => 4
``````

### argmax & argmin

`argmax` and `argmin` find the `row-major-aref` index where an array value is maximum or minimum. They both return two values: the first value is the index; the second is the array value at that index.

``````(defparameter a #(1 2 5 4 2))
(aops:argmax a) ; => 2 5
(aops:argmin a) ; => 0 1
``````

### vectorize-reduce

More complicated reductions can be done with `vectorize-reduce`, for example the maximum absolute difference between arrays:

``````(defparameter a #2A((1 2)
(3 4)))
(defparameter b #2A((2 2)
(1 3)))

(aops:vectorize-reduce #'max (a b) (abs (- a b))) ; => 2
``````

### best

`best` compares two arrays according to a function and returns the ‘best’ value found. The function, `FN` must accept two inputs and return true/false. This function is applied to elements of ARRAY. The row-major-aref index is returned.

Example: The index of the maximum is

``````   * (best #'> #(1 2 3 4))
3   ; row-major index
4   ; value
``````

### most

`most` finds the element of ARRAY that returns the value closest to positive infinity when FN is applied to the array value. Returns the row-major-aref index, and the winning value.

Example: The maximum of an array is:

``````     (most #'identity #(1 2 3))
-> 2    (row-major index)
3    (value)
``````

and the minimum of an array is:

``````      (most #'- #(1 2 3))
0
-1
``````

See also `reduce-index` above.

## Scalar values

Library functions treat non-array objects as if they were equivalent to 0-dimensional arrays: for example, `(aops:split array (rank array))` returns an array that effectively equivalent (`eq`) to array. Another example is `recycle`:

``````(aops:recycle 4 :inner '(2 2)) ; => #2A((4 4)
;        (4 4))
``````

## Stacking

You can stack compatible arrays by column or row. Metaphorically you can think of these operations as stacking blocks. For example stacking two row vectors yields a 2x2 array:

``````(stack-rows #(1 2) #(3 4))
;; #2A((1 2)
;;     (3 4))
``````

Like other functions, there are two versions: generalised stacking, with rows and columns of type `T` and specialised versions where the element-type is specified. The versions allowing you to specialise the element type end in `*`.

The stack functions use object dimensions (as returned by `dims` to determine how to use the object.

• when the object has 0 dimensions, fill a column with the element
• when the object has 1 dimension, use it as a column
• when the object has 2 dimensions, use it as a matrix

`copy-row-major-block` is a utility function in the `stacking` package that does what it suggests; it copies elements from one array to another. This function should be used to implement copying of contiguous row-major blocks of elements.

### rows

`stack-rows-copy` is the method used to implement the copying of objects in `stack-row*`, by copying the elements of `source` to `destination`, starting with the row index `start-row` in the latter. Elements are coerced to `element-type`.

`stack-rows` and `stack-rows*` stack `objects` row-wise into an array of the given `element-type`, coercing if necessary. Always return a simple array of rank 2. `stack-rows` always returns an array with elements of type `T`, `stack-rows*` coerces elements to the specified type.

### columns

`stack-cols-copy` is a method used to implement the copying of objects in `stack-col*`, by copying the elements of `source` to `destination`, starting with the column index `start-col` in the latter. Elements are coerced to `element-type`.

`stack-cols` and `stack-cols*` stack `objects` column-wise into an array of the given `element-type`, coercing if necessary. Always return a simple array of rank 2. `stack-cols` always returns an array with elements of type `T`, `stack-cols*` coerces elements to the specified type.

### arbitrary

`stack` and `stack*` stack array arguments along `axis`. `element-type` determines the element-type of the result.

``````(defparameter *a1* #(0 1 2))
(defparameter *a2* #(3 5 7))
(aops:stack 0 *a1* *a2*) ; => #(0 1 2 3 5 7)
(aops:stack 1
(aops:reshape-col *a1*)
(aops:reshape-col *a2*)) ; => #2A((0 3)
;        (1 5)
;        (2 7))
``````