|
|
| Line 1: |
Line 1: |
| In [[mathematics]] and [[statistics]], the '''quasi-arithmetic mean''' or '''generalised ''f''-mean''' is one generalisation of the more familiar [[mean]]s such as the [[arithmetic mean]] and the [[geometric mean]], using a function <math>f</math>. It is also called '''Kolmogorov mean''' after Russian scientist [[Andrey Kolmogorov]].
| |
|
| |
|
| ==Definition==
| |
|
| |
|
| If ''f'' is a function which maps an interval <math>I</math> of the real line to the [[real number]]s, and is both [[continuous function|continuous]] and [[injective function|injective]] then we can define the '''''f''-mean of two numbers'''
| | One particular author is known through name of Gabrielle Lattimer though she doesn't really like being called in that way. For years she's been working because a library assistant. To bake is something that my wife been doing for growth cycles. For years she's been tough to adapt in Massachusetts. She is running and examining a blog here: http://[http://www.Wired.com/search?query=circuspartypanama circuspartypanama].com<br><br>Also visit my web blog ... [http://circuspartypanama.com triche clash of clans] |
| :<math>x_1, x_2 \in I</math>
| |
| as
| |
| :<math>M_f(x_1,x_2) = f^{-1}\left( \frac{f(x_1)+f(x_2)}2 \right).</math>
| |
| | |
| For <math>n</math> numbers | |
| :<math>x_1, \dots, x_n \in I</math>,
| |
| the '''f-mean''' is
| |
| :<math>M_f(x_1, \dots, x_n) = f^{-1}\left( \frac{f(x_1)+ \cdots + f(x_n)}n \right).</math>
| |
| | |
| We require ''f'' to be injective in order for the [[inverse function]] <math>f^{-1}</math> to exist. Since <math>f</math> is defined over an interval, <math>\frac{f\left(x_1\right) + f\left(x_2\right)}2</math>
| |
| lies within the domain of <math>f^{-1}</math>.
| |
| | |
| 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 <math>x</math> nor smaller than the smallest number in <math>x</math>.
| |
| | |
| == Examples ==
| |
| | |
| * If we take <math>I</math> to be the real line and <math>f = \mathrm{id}</math>, (or indeed any linear function <math>x\mapsto a\cdot x + b</math>, <math>a</math> not equal to 0) then the ''f''-mean corresponds to the [[arithmetic mean]].
| |
| | |
| * If we take <math>I</math> to be the set of positive real numbers and <math>f(x) = \log(x)</math>, 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 we take <math>I</math> to be the set of positive real numbers and <math>f(x) = \frac{1}{x}</math>, then the ''f''-mean corresponds to the [[harmonic mean]].
| |
| | |
| * If we take <math>I</math> to be the set of positive real numbers and <math>f(x) = x^p</math>, then the ''f''-mean corresponds to the [[power mean]] with exponent <math>p</math>.
| |
| | |
| == Properties ==
| |
| | |
| * [[Partition of a set|Partitioning]]: The computation of the mean can be split into computations of equal sized sub-blocks.
| |
| :: <math> | |
| M_f(x_1,\dots,x_{n\cdot k}) =
| |
| M_f(M_f(x_1,\dots,x_{k}),
| |
| M_f(x_{k+1},\dots,x_{2\cdot k}),
| |
| \dots,
| |
| M_f(x_{(n-1)\cdot k + 1},\dots,x_{n\cdot k}))
| |
| </math>
| |
| * Subsets of elements can be averaged a priori, without altering the mean, given that the multiplicity of elements is maintained.
| |
| :With <math>m=M_f(x_1,\dots,x_k)</math> it holds
| |
| ::<math>M_f(x_1,\dots,x_k,x_{k+1},\dots,x_n) = M_f(\underbrace{m,\dots,m}_{k \text{ times}},x_{k+1},\dots,x_n)</math>
| |
| * The quasi-arithmetic mean is invariant with respect to offsets and scaling of <math>f</math>:
| |
| ::<math>\forall a\ \forall b\ne0 ((\forall t\ g(t)=a+b\cdot f(t)) \Rightarrow \forall x\ M_f (x) = M_g (x)</math>.
| |
| * If <math>f</math> is [[Monotonic function|monotonic]], then <math>M_f</math> is monotonic.
| |
| * Any quasi-arithmetic mean <math>M</math> of two variables has the ''mediality property'' <math>M(M(x,y),M(z,w))=M(M(x,z),M(y,w))</math> and the ''self-distributivity'' property <math>M(x,M(y,z))=M(M(x,y),M(x,z))</math>. Moreover, any of those properties is essentially sufficient to characterize quasi-arithmetic means; see Aczél–Dhombres, Chapter 17.
| |
| * Any quasi-arithmetic mean <math>M</math> of two variables has the ''balancing property'' <math>M\big(M(x, M(x, y)), M(y, M(x, y))\big)=M(x, y)</math>. An interesting problem is whether this condition (together with fixed-point, symmetry, monotonicity and continuity properties) implies that the mean is quasi-arthmetic. [[Georg Aumann]] showed in the 1930s that the answer is no in general,<ref>{{cite journal|last=Aumann|first=Georg|title=Vollkommene Funktionalmittel und gewisse Kegelschnitteigenschaften|journal=[[Journal für die reine und angewandte Mathematik]]|year=1937|volume=176|pages=49–55}}</ref> but that if one additionally assumes <math>M</math> to be an [[analytic function]] then the answer is positive.<ref>{{cite journal|last=Aumann|first=Georg|title=Grundlegung der Theorie der analytischen Analytische Mittelwerte|journal=Sitzungsberichte der Bayerischen Akademie der Wissenschaften|year=1934|pages=45–81}}</ref>
| |
| | |
| == Homogeneity ==
| |
| | |
| [[Mean]]s are usually [[Homogeneous function|homogeneous]], | |
| but for most functions <math>f</math>, the ''f''-mean is not.
| |
| Indeed, the only homogeneous quasi-arithmetic means are the [[power mean]]s 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 <math>C</math>.
| |
| :<math>M_{f,C} x = C x \cdot f^{-1}\left( \frac{f\left(\frac{x_1}{C x}\right) + \cdots + f\left(\frac{x_n}{C x}\right)}{n} \right)</math>
| |
| However this modification may violate [[Monotonic function|monotonicity]] and the partitioning property of the mean.
| |
| | |
| == References ==
| |
| {{reflist}}
| |
| * Aczél, J.; Dhombres, J. G. (1989) Functional equations in several variables. With applications to mathematics, information theory and to the natural and social sciences. Encyclopedia of Mathematics and its Applications, 31. Cambridge Univ. Press, Cambridge, 1989.
| |
| * Andrey Kolmogorov (1930) “On the Notion of Mean”, in “Mathematics and Mechanics” (Kluwer 1991) — pp. 144–146.
| |
| * Andrey Kolmogorov (1930) Sur la notion de la moyenne. Atti Accad. Naz. Lincei 12, pp. 388–391.
| |
| * John Bibby (1974) “Axiomatisations of the average and a further generalisation of monotonic sequences,” Glasgow Mathematical Journal, vol. 15, pp. 63–65.
| |
| * Hardy, G. H.; Littlewood, J. E.; Pólya, G. (1952) Inequalities. 2nd ed. Cambridge Univ. Press, Cambridge, 1952.
| |
| | |
| == See also ==
| |
| * [[Generalized mean]]
| |
| * [[Jensen's inequality]]
| |
| | |
| {{DEFAULTSORT:Quasi-Arithmetic Mean}}
| |
| [[Category:Means]]
| |
One particular author is known through name of Gabrielle Lattimer though she doesn't really like being called in that way. For years she's been working because a library assistant. To bake is something that my wife been doing for growth cycles. For years she's been tough to adapt in Massachusetts. She is running and examining a blog here: http://circuspartypanama.com
Also visit my web blog ... triche clash of clans