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2. A Short Average
x⌹x=x
If x is a vector, x⌹x=x is a shorter phrase for the average
of vector x . Note that + / ⌿ ÷ ⍴ ≢ are not used.
The idea originated with Timo Seppälä at APL82
[14].
To see why it works, start with the definition
y⌹x ←→ (⌹(+⍉x)+.×x)+.×(+⍉x)+.×y
which defines the rectangular case in terms of the square case.
| x⌹x=x |
| x⌹w | w←x=x |
| (⌹(+⍉w)+.×w)+.×(+⍉w)+.×x | definition of ⌹ |
| (⌹(⍉w)+.×w)+.×(⍉w)+.×x | w is non-complex |
| (÷≢x)+.×w+.×x | w is (≢x)⍴1; w+.×w is ≢x |
| (÷≢x)+.×+⌿x | ((≢x)⍴1)+.×x ←→ +⌿x |
| (÷≢x)×+⌿x | LHS and RHS of +.× are scalars |
| (+⌿x)÷≢x |
| QED |
Alternatively, y⌹x computes a linear regression for y wherein the constant term
is the mean of y ; that is, the best least square
“fit” of a vector by a single number,
is its mean.
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