Жалпыланған орта шамалар: f-орта және оның қасиеттері

Generalization of means
In mathematics and statistics, the quasi arithmetic mean or generalised f mean or Kolmogorov Nagumo de Finetti mean is one generalisation of the more familiar means such as the arithmetic mean and the geometric mean, using a function It is also called Kolmogorov mean after Soviet mathematician Andrey Kolmogorov. It is a broader generalization than the regular generalized mean.

If = ℝ, the real line, and , (or indeed any linear function , not equal to 0) then the f mean corresponds to the arithmetic mean. If = ℝ+, the positive real numbers and , then the f mean corresponds to the geometric mean. According to the f mean properties, the result does not depend on the base of the logarithm as long as it is positive and not 1. If = ℝ+ and , then the f mean corresponds to the harmonic mean. If = ℝ+ and , then the f mean corresponds to the power mean with exponent If = ℝ and , then the f mean is the mean in the log semiring, which is a constant shifted version of the LogSumExp (LSE) function (which is the logarithmic sum), The corresponds to dividing by n, since logarithmic division is linear subtraction. The LogSumExp function is a smooth maximum: a smooth approximation to the maximum function.

There are several different sets of properties that characterize the quasi arithmetic mean (i. e., each function that satisfies these properties is an f mean for some function f). Mediality is essentially sufficient to characterize quasi arithmetic means. Self distributivity is essentially sufficient to characterize quasi arithmetic means. Balancing: An interesting problem is whether this condition (together with symmetry, fixed point, monotonicity and continuity properties) implies that the mean is quasi arithmetic. Georg Aumann showed in the 1930s that the answer is no in general, but that if one additionally assumes to be an analytic function then the answer is positive.

Means are usually homogeneous, but for most functions , the f mean is not. Indeed, the only homogeneous quasi arithmetic means are the power means (including the geometric mean); see Hardy–Littlewood–Pólya, page 68. The homogeneity property can be achieved by normalizing the input values by some (homogeneous) mean
However this modification may violate monotonicity and the partitioning property of the mean.