On 11 Jul 2002, Jussi Piitulainen wrote in part: > What I worked out is, various "means" can be taken to be functions > from something like vectors (of varying lengths) to real numbers as > follows. This is indeed elegant. > > A (x1,...,xn) = sum_k (1/n) xk (arithmetic mean) > G = exp o A o log (geometric mean) > H = reciprocal o A o reciprocal (harmonic mean) > R = squareroot o A o square (root-mean-square) > > And in general, I would now take `conjugate of X under transform f' to > mean f^-1 o X o f where f is applied to elements of vector, X takes a > vector to a scalar, and f^-1 "undoes" f. > > Is this all right? Is there a better way to understand this concept? > What else might I find in place of X and f?
You appear to have understood the basic concept, although I think your way of expressing it may be somewhat misleading. For what it may be worth, I would describe the situations thus: Given a set of numbers X_k (k=1,...n), one calculates a mean by (1) applying a transformation T to each of the numbers; (2) summing all the transformed numbers; (3) dividing that sum by n; (4) applying the inverse transformation T' to the quotient of (3). In the case of the arithmetic mean, T = T' = the identity transform; for the geometric mean, T = the logarithm (to any convenient base b) and T' = the antilogarithm (to the same base), so that if T = ln, T' = exp; for the harmonic mean, T = T' = the reciprocal; for the RMS (root mean square), T = square and T' = square root. This schema has the advantage that the arithmetic mean is more clearly seen to behave just like the others, with a suitable choice of transformation (namely, the identity). In language similar to the notation you used above, we might write mean = T'(sum_k (1/n) Txk) and the particular mean one gets depends on one's choice of T. Cheers! -- Don. ----------------------------------------------------------------------- Donald F. Burrill [EMAIL PROTECTED] 184 Nashua Road, Bedford, NH 03110 (603) 471-7128 . . ================================================================= Instructions for joining and leaving this list, remarks about the problem of INAPPROPRIATE MESSAGES, and archives are available at: . http://jse.stat.ncsu.edu/ . =================================================================
