In mathematics and statistics, the quasi-arithmetic mean or generalised f-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 Russian mathematician Andrey Kolmogorov. It is a broader generalization than the regular generalized mean.
is defined as
We require f to be injective in order for the inverse function to exist. Since is defined over an interval, lies within the domain of .
Since f is injective and continuous, it follows that f is a strictly monotonic function, and therefore that the f-mean is neither larger than the largest number of the tuple nor smaller than the smallest number in .
- 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.
The following properties hold for for any single function :
Symmetry: The value of is unchanged if its arguments are permuted.
Fixed point: for all x, .
Monotonicity: is monotonic in each of its arguments (since is monotonic).
Continuity: is continuous in each of its arguments (since is continuous).
Replacement: Subsets of elements can be averaged a priori, without altering the mean, given that the multiplicity of elements is maintained. With it holds:
Partitioning: The computation of the mean can be split into computations of equal sized sub-blocks:
Self-distributivity: For any quasi-arithmetic mean of two variables: .
Mediality: For any quasi-arithmetic mean of two variables:.
Balancing: For any quasi-arithmetic mean of two variables:.
Scale-invariance: The quasi-arithmetic mean is invariant with respect to offsets and scaling of : .
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.:chapter 17
- Self-distributivity is essentially sufficient to characterize quasi-arithmetic means.:chapter 17
- Replacement: Kolmogorov proved that the five properties: symmetry, fixed-point, monotonicity, continuity and replacement fully characterize the 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 and 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.
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