Zero variance

I noticed today that by using complex numbers you can create a set of values whose variance is zero, even though not all the numbers are equal.

Let X be a set of real numbers. Let Y be a set of complex numbers evenly spaced along the circle whose center is at mean(X) and whose radius is stdev(X). Y doesn't have to have the same number of points as X. In fact, you can even take the limit and let Y be the set of all points on the circle. Here's a picture of what I'm talking about, where X is the set {0..10}:

The mean of Y will be equal to the mean of X, but the variance and standard deviation of Y will be 0. Furthermore, the variance of the real components of Y (ignoring the imaginary components) is equal to half the variance of X.

It would be even more interesting if you could define Y so that its real components were exactly the same as the real components of X. However in the case where X consists of two points, this require the imaginary components of Y to be +/- infinity, so I'm suspicious.

I'm not sure whether this kind of thing is actually useful or not, but it does at least provide an interesting interpretation of the meaning of variance. The fact that the variance of the real part of Y is proportional to the variance of X also makes me wonder whether there's a connection here to the idea of marginal distributions.

Here's a bit of scheme code that shows what I'm talking about.

(define pi 3.14159265358979323846)

(define (curry f x) 
  (lambda y (apply f x y)))

(define (n-downto-0 x)
  (if (= 0 x) '(0) (cons x (n-downto-0 (sub1 x)))))

;; returns a list of all the n-th roots of x
(define (roots x n)
  (map (lambda (i) (make-polar (* x (expt 1 (/ 1 n)))
                               (* 2 pi (/ i n))))
       (n-downto-0 (sub1 n))))

(define (avg vals)
  (/ (apply + vals) (length vals)))

(define (variance vals)
  (let ((mean (avg vals)))
    (avg (map (lambda (x) (sqr (- x mean))) vals))))

(define X (n-downto-0 100))
(define Y (map (curry + (avg X)) 
               (roots (sqrt (variance X)) (length X))))

(variance X)
(variance Y)
(* 2 (variance (map real-part Y)))

Posted on March 11, 2007 09:17 PM
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