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.
