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 f . It is also called Kolmogorov mean after Russian mathematician Andrey Kolmogorov.
If f is a function which maps an interval I of the real line to the real numbers, and is both continuous and injective then we can define the f-mean of two numbers
x 1 , x 2 ∈ I as
M f ( x 1 , x 2 ) = f − 1 ( f ( x 1 ) + f ( x 2 ) 2 ) . For n numbers
x 1 , … , x n ∈ I ,
the f-mean is
M f ( x 1 , … , x n ) = f − 1 ( f ( x 1 ) + ⋯ + f ( x n ) n ) . We require f to be injective in order for the inverse function f − 1 to exist. Since f is defined over an interval, f ( x 1 ) + f ( x 2 ) 2 lies within the domain of f − 1 .
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 x nor smaller than the smallest number in x .
If I = ℝ, the real line, and f ( x ) = x , (or indeed any linear function x ↦ a ⋅ x + b , a not equal to 0) then the f-mean corresponds to the arithmetic mean.If I = ℝ+, the positive real numbers and f ( x ) = log ( x ) , 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 I = ℝ+ and f ( x ) = 1 x , then the f-mean corresponds to the harmonic mean.If I = ℝ+ and f ( x ) = x p , then the f-mean corresponds to the power mean with exponent p .If I = ℝ and f ( x ) = exp ( x ) , then the f-mean is a constant shifted version of the LogSumExp (LSE) function, M f ( x 1 , … , x n ) = L S E ( x 1 , … , x n ) − log ( n ) . The LogSumExp function is used as a smooth approximation to the maximum function.Partitioning: The computation of the mean can be split into computations of equal sized sub-blocks.Subsets of elements can be averaged a priori, without altering the mean, given that the multiplicity of elements is maintained.With
m = M f ( x 1 , … , x k ) it holds
M f ( x 1 , … , x k , x k + 1 , … , x n ) = M f ( m , … , m ⏟ k times , x k + 1 , … , x n ) The quasi-arithmetic mean is invariant with respect to offsets and scaling of f :If f is monotonic, then M f is monotonic.Any quasi-arithmetic mean M of two variables has the mediality property M ( M ( x , y ) , M ( z , w ) ) = M ( M ( x , z ) , M ( y , w ) ) and the self-distributivity property M ( x , M ( y , z ) ) = M ( M ( x , y ) , M ( x , z ) ) . Moreover, any of those properties is essentially sufficient to characterize quasi-arithmetic means; see Aczél–Dhombres, Chapter 17.Any quasi-arithmetic mean M of two variables has the balancing property M ( M ( x , M ( x , y ) ) , M ( y , M ( x , y ) ) ) = M ( x , y ) . An interesting problem is whether this condition (together with fixed-point, symmetry, 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 M to be an analytic function then the answer is positive.Under regularity conditions, a central limit theorem can be derived for the generalised f-mean, thus implying that for a large sample n { M f ( X 1 , … , X n ) − f − 1 ( E f ( X 1 , … , X n ) ) } is approximately normal.Means are usually homogeneous, but for most functions f , 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 C .
M f , C x = C x ⋅ f − 1 ( f ( x 1 C x ) + ⋯ + f ( x n C x ) n ) However this modification may violate monotonicity and the partitioning property of the mean.
