American Journal of Computation, Communication and Control 2018; 5(3): 101-108 http://www.aascit.org/journal/ajccc ISSN: 2375-3943

Generalized fG -Mean: Derivation of Various Formulations of Average

Dhritikesh Chakrabarty

Department of Statistics, Handique Girls’ College, Gauhati University, Guwahati, India Email address

Citation

Dhritikesh Chakrabarty. Generalized fG -Mean: Derivation of Various Formulations of Average. American Journal of Computation, Communication and Control. Vol. 5, No. 3, 2018, pp. 101-108.

Received : September 25, 2018; Accepted : September 26, 2018; Published : November 22, 2018

Abstract: Average is a basic concept and averaging is a basic technique/tool behind the most of the measures associated to characteristics of data. A number of definitions/formulations of average have been developed so far. These definitions/formulations have been found suitable to be used in different situations. There are still many situations where there are scarcities of definitions/formulations of average to deal with the respective situations. It has been found possible to compose general definition of average so that the definitions/formulations of average, already developed, can be obtained/derived from it as well as more new definitions/formulations of average can be obtained/derived from it. One general definition of average, termed in this paper as Generalized fG -Mean, has been introduced with a view to obtain different definitions/formulations of average. Applying the technique, some definitions/formulations have been derived for various types of averages. This paper describes this general definition and the derivation of a number of definitions/formulations of average.

Keywords: Average, Generalized fG -Mean, Existing Means, Derivations

be n numbers of a list and/or the values assumed by a 1. Introduction variable x. Average [1, 2] is basic concept and averaging is a basic Some of the definitions /formulations of averages, already technique/tool behind the most of the measures associated to constructed [6, 7, 8, 9, 10, 11], are as follows: characteristics of data. A number of definitions/formulations Arithmetic Mean = + … … … … …. + ) (1) of averages had primarily been developed by Pythagoras [3, + 4, 5] who constructed the definitions / formulations of the Geometric Mean = ...... / (2) three most common averages namely Arithmetic Mean, Geometric Mean & Harmonic Mean which are known as provided the n numbers are non-negative. Pythagorean means [6, 7]. Harmonic Mean = { … … . ) (3) Let

, ,...... , provided the n numbers are all different from 0.

Quadratic Mean = { ……………. )/ (4) Square Root Mean = { / / ……………. /) (5) Cubic Mean = { ……………. )/ (6) Cube Root Mean = { / / ……………. /) (7) Generalized p-Mean = { ……………. / (8) American Journal of Computation, Communication and Control 2018; 5(3): 101-108 102

pth Root Mean = { ( / + / + …………….+ /)} (9) e-Mean = log e { ( + + …………….+ } (10) Scale s-Mean = { ( + + … … … … …. + )} (11) Shift a-Mean = {( ̶a) + ( ̶ ) +...... + ( ̶ a)} + a (12) ̶ ̶ ̶ shift a– Inverse Scale s -Mean = { ( + + … … … + )} + a (13) These definitions/formulations have been found suitable to the arithmetic mean of the functional values be used in different situations. There are still many situations , , …………, where there are scarcities of definitions/formulations of () () () average to deal with the respective situations. It has been is taken first and then the inverse functional value of the found possible to compose general definition of average so arithmetic obtained is taken to obtain the generalized f-mean that the definitions/formulations of average, already defined by developed, can be obtained/derived from it as well as more new definitions/formulations of average can be [ {() + () + … … … … …. + ()}] obtained/derived from it. Kolmogorov constructed one definition/formulation of average known as the Generalized f Here, f is an invertible function. -Mean [12-17]. Let us now take the geometric mean of the functional The Generalized f - Mean of values , ,...... , , is defined by (), (), …………, ()

and then the inverse functional value of the geometric mean [ {( ) + ( ) + … … … … …. + ( )}] (14) obtained. where f is an invertible function. In doing this we will obtain the formulation It has been shown that the Generalized f - Mean can be [{( )( )...... ( )}/ used to derive the definitions/formulations of the existing means and also of new means [9, 10, 11, 18]. This can be regarded one generalized definition of However, the Generalized f -Mean due to Kolmogorov has average. also been found to be not suitable for some situations. Let this average be termed as the Generalized fG - Mean in Recently, one more generalized definition of average has this article. been constructed which has been termed as Generalized f H – Thus, the Generalized fG - Mean of Mean [18]. In this paper, attempt has been made on the construction of one more general definition of average in a , ,...... , similar manner as the construction of the Generalized f - denoted by fG (, ,...... ,) can be defined as Mean by Kolmogorov. The definition constructed here has / been termed as Generalized fG -Mean. It has been shown that fG (, ,...... ,) = [{()()...... ()} (15) the existing definitions/formulations of average can be where f is any invertible function. obtained from the Generalized fG -Mean. This paper describes this general definition of average and the derivation of the existing definitions/formulations of average. 3. Properties of Generalized fG - Mean

The Generalized fG - Mean satisfies the following 2. Generalized fG - Mean: One properties: Technique of Definition of Average 1. Partitioning: The computation of the Generalized fG - Mean can be split into computations of equal k - sized sub- In constructing the generalized f-mean (also known as blocks i.e. Kolmogorov mean) of

, ,...... ,

fG ( , , ………., ) = fG { fG ( , , ……, ), fG ( , , ……, ), ………, fG ( (), (), ……, )} Proof: We have 103 Dhritikesh Chakrabarty: Generalized fG -Mean: Derivation of Various Formulations of Average

/ fG ( , , ……, ) = [{()()...... ()} , fG ( , , ……, ) = / [{()()...... ()} ,

/ fG ( (), (), ……, )= ((). (())...... ( )} Accordingly,

fG { fG ( , , ……, ), fG ( , , ……, ),...... , fG ( (), (), ……, )}

/ = [{()()...... ( )}

= fG ( , , ………., ) 2. Subsets of elements can be averaged a priori, without altering the mean, given that the multiplicity of elements is maintained. Thus, with m = fG ( , , ……, ) it holds that

fG ( , , ……, , , , ……, )= fG (, , ….,, , , ……, ) Proof: We have

/ m = fG ( , , ……, ) = {()()...... ()} Therefore,

/ () = f [{()()...... ()} ]

/ = {()()...... ()} Hence,

fG (, , ….,, , , ……, )

/ = [{ () ()...... ()()()...... ()} ]

/ = [{()()...... ()()()...... ()} ]

= fG ( , , ……, , , , ……, )

3. The Generalized fG - Mean is invariant with respect to scaling of f i.e. for all real b,

ϕ(t) = b f(t ) implies ϕG () = fG () Proof: We have

/ ϕG ( , , ……, ) = [{(). ()...... ()} ]

/ = [{(). ()...... ()} ] Now,

ϕ(t) = b f(t ) ⇒ t = { b f(t ) } ⇒ t = { b f(t )}

⇒ (y) = ( ), putting y = b f(t ) ⇒ t = ( )

⇒ : x → ( ) i.e. p maps x to ( ) Thus

/ ϕG ( , , ……, ) = [{(). ()...... ()} ]

/ = [{(). ()...... ()} ] American Journal of Computation, Communication and Control 2018; 5(3): 101-108 104

/ = [{.()...... …… ()} ]

/ = [{()()...... ()} ]

= fG ( , , ……, )

4. If f is monotonic, then fG is monotonic. Proof: We have

/ fG ( , , ……, ) = [{()()...... ()} ] If f is monotonic then

either () < ()< ……………< ()

or () > ()> ……………> () This implies,

/ /() either [{()()...... ()} < [{()()...... ()}

/ /() or [{()()...... ()} > [{()()...... ()} for all k, which implies,

either fG ( , , ……, ) < fG ( , , ……, )

or fG ( , , ……, ) < fG ( , , ……, )

Hence, fG is monotonic.. 5. Generalized fG - Mean of two variables has the mediality property namely

fG [fG {( x, fG (, )}, fG {( y, fG (, )}] = fG (, ) Proof: We have

/ fG (, ) = {(). ()} Thus,

fG {( x, fG (, )} = [(). {(). ()}]/

= [(). {(). ()}]/ Similarly,

fG {( y, fG (, )} = [(). {(). ()}]/ Now,

f f G {( x, fG (, )} = f [{(). {(). ()}]

= [(). {(). ()}]/ Similarly,

f f G {( y, fG (, )} = [{(). {(). ()}]/

= [(). {(). ()}]/ Therefore,

fG [fG {( x, fG (, )}, fG {( y, fG (, )}] 105 Dhritikesh Chakrabarty: Generalized fG -Mean: Derivation of Various Formulations of Average

= [{(). ()}]

= fG (, )

6. Generalized fG - Mean of two variables has the self-distributive property namely

fG {x, fG (, )} = fG { fG (, ), fG (, z)} Proof: We have

/ fG (, ) = {(). ()} Accordingly,

/ / fG (, ) = {(). ()} & fG (, ) = {(). ()} Now,

1/2 fG { fG (, ), fG (, z)} = [{()()}. {()()}]

1/2 = [{()()}. {()()}]

= [(). {()}1/2.{()}1/2]1/2 Thus,

1/2 fG {x, fG (, )} = { (). f fG (, ) }

1/2 = [(). {()()}]

= [(). {()}1/2.{()}1/2]1/2

= fG { fG (, ), fG (, z)}

7. fG - Mean of two variables x & y has the balancing property namely

fG [fG {x, fG (, )}, fG {( y, fG (, )}] = fG (, ) Proof: We have

/ / fG {x, fG (, )} = [(). {(). ()} ]

= [(). {(). ()}/]1/2

/ / & fG {y, fG (, )} = [(). {(). ()} ]

= [(). {(). ()}/]1/2 Therefore,

fG [fG {x, fG (, )}, fG {( y, fG (, )}]

1/2 = [fG {x, fG (, )}. f fG {y, fG (, )}]

= [ {(). ()}/}/]

= fG (, ) Remark: Then apply the function selected in the definition of This definition / formulation can be applied in searching Generalized fG - Mean defined by (15). for / constructing of a number of definitions / formulations for average. The technique can be summarized in the following steps: 4. Derivation of Various Average from Generalized Select a suitable function which is invertible. fG -Mean Then find out the inverse function of the function selected. Arithmetic Mean: American Journal of Computation, Communication and Control 2018; 5(3): 101-108 106

Let the invertible function f(.) be such that Let f(x) = exp (x 1/2 ) ̶ 1 1/2 x Then f {exp (x )} = x f: x → e ̶ 1 ) 2,, 1/2 i.e. f (y) = (log e y putting y = exp (x ) ̶ 1 2 x x This means, f maps x to (log e x) i.e. f maps x to e i.e. f(x) = e 1/2 ̶ 1 x ̶ 1 x Therefore, if f(x) = exp (x ) Here, f (e ) = x i.e. f (y) = log e y, putting y = e This implies that f ̶ 1 (.) is a function with then the Generalized fG - Mean becomes f ̶ 1: x → log x { ( / + / + …………….+ /)} e ̶ 1 ̶ 1 i.e. f maps x to log e x i.e. f (x) = log e x which is nothing but the square root mean defined by (5). x Therefore, if f(x) = e Cubic Mean: 3 then the Generalized fG - Mean becomes Let f(x) = exp (x ) ̶ 1 3 Then, f {exp (x )} = x + + … … … … …. + ) ̶ 1 ) 1/3 , 3 i.e. f (y) = (log e y putting y = exp (x ) ̶ 1 1/3 This implies, f maps x to (log e x) which is nothing but Pythagorean arithmetic mean defined by 3 Therefore, if f(x) = exp (x ) (1). then the Generalized f - Mean becomes Geometric Mean: G Let the invertible function f(.) be such that { ( + + …………….+ ) } f: x → x which is nothing but the cubic mean defined by (6). i.e. f maps x to x i.e. f(x) = x Cube Root Mean: Then f ̶ 1 (x) = x Let the invertible function f(.) be such that i.e. f ̶ 1 also maps x to x f: x → exp (x 1/3 ) Therefore, if f(x) = x 1/3 then the Generalized fG - Mean becomes i.e. f(x) = exp (x ) ̶ 1 1/3 / Then f {exp (x )} = x (...... ) ̶ 1 ) 3, 1/3 i.e. f (y) = (log e y , putting y = exp (x ) ̶ 1 which is nothing but Pythagorean geometric mean defined by Thus f (.) is a function with (2). f ̶ 1: x → (log x) 3 Harmonic Mean: e

Let the invertible function f(.) be such that Thus if in the Generalized fG -Mean, the function f(.) is selected as f: x → e 1/ x f(x) = exp (x 1/3 ) i.e. f maps x to e x i.e. f(x) = e 1/ x ̶ 1 1/ x Here, f (e ) = x then the Generalized fG -Mean becomes ̶ 1 ̶ 1, 1/ x i.e. f (y) = ( log e y) putting y = e Thus, f ̶ 1 maps x to (log x) ̶ 1 ( / + / + …………….+ /)} e Therefore, if f(x) = e 1/ x then the Generalized fG - Mean becomes This is nothing but the cube root mean defined by (7). Generalized p-Mean: { ( + + … … . + )} Let the invertible function f(.) be such that 1/p which is nothing but Pythagorean harmonic mean defined f: x → exp (x ) by (3). i.e. f(x) = exp (x 1/ p ) Quadratic Mean: ̶ 1 1/p 2 Then f {exp (x )} = x Let f(x) = exp (x ) ̶ 1 ) p, 1/ p ̶ 1 2 i.e. f (y) = (log e y , putting y = exp (x ) Then, f {exp (x )} = x ̶ 1 ̶ 1 ) 1/2 , 2 Thus f (.) is a function with i.e. f (y) = (log e y , putting y = exp (x ) ̶ 1 1/2 ̶ 1: p This means, f maps x to (log e x) f x → (log e x) Therefore, if f(x) = exp (x 2) Therefore, if f(x) = exp (x 1/ p) then the Generalized fG - Mean becomes then the Generalized fG - Mean becomes / { ( + + …………….+ )} { ( + + …………….+ } which is nothing but the quadratic mean defined by (4). Square Root Mean: This is nothing but the Generalized p-Mean defined by (8). 107 Dhritikesh Chakrabarty: Generalized fG -Mean: Derivation of Various Formulations of Average

Generalized pth Root Mean: 1 Let the invertible function f(.) be such that (| | + | |+ …………………….+ | |) f: x → exp (x 1/ p) Arithmetic Mean of () = x3 is i.e. f(x) = exp (x 1/ p) ̶ 1 1/ p 1 Then f {exp (x )} = x ( + + …………….+ ) ̶ 1 ) p, 1/ p i.e. f (y) = (log e y , putting y = exp (x ) ̶ 1 Thus f (.) is a function with Arithmetic Mean of () = x p is ̶ 1: p f x → (log e x) 1 ( + + …………….+ ) Thus if in the Generalized fG -Mean, f(.) is selected as Arithmetic Mean of () = x 1/ p is f(x) = exp (x 1/ p) 1 then the Generalized f -Mean becomes ( / + / + …………….+ /) G { ( / + / + …………….+ /)} Arithmetic Mean of () = e x is th 1 This is nothing but the Generalized p Root Mean defined ( + + …………….+) by (9). Arithmetic Mean of () = log x is 5. Generalized f -Mean of a Function G 1 ( + + …………………….+ ) From the Generalized fG -Mean, one can define the Generalized f -Mean of a function G Geometric Mean: = (. ) = () Geometric Mean of () can be obtained as

/ of x by { (). (). () …………… ()}

fG { (), (),...... ,()} Harmonic Mean: Harmonic Mean of () can be obtained as / = [{()()...... ()} ] (4.1) 1 where 1 1 1 1 { + +...... + } () () () = (), = (), … … … . , = () Quadratic Mean: 6. Some Definitions/Formulations of Quadratic Mean of () can be obtained as Average of Function { ( + + …………….+ )}/ From this definition, one can obtain the definitions / where = (), = (), … … … . , = () formulations of various means as mentioned above, for a Cubic Mean: function of variable, as follows: Cubic Mean of () can be obtained as Arithmetic Mean: / { ( + + …………….+ )} Substituting f(x) = exp {()} Generalized p Mean: in (4.1), Arithmetic Mean of () can be obtained as Generalized p Mean of () can be obtained as {( ) + ( ) + … … … … …. + ( )} { ( + + …………….+ )}/ In particular th 2 Generalized p Root Mean: Arithmetic Mean of () = x is Generalized pth Root Mean of () can be obtained as 1 ( + + …………….+ ) { ( / + / + …………….+ /)} Arithmetic Mean of () = |x| is e Mean: e Mean of () can be obtained as American Journal of Computation, Communication and Control 2018; 5(3): 101-108 108

and influence (translated by Steven Rendall in collaboration log e { ( + + …………….+ } with Christoph Riedweg and Andreas Schatzmann, Ithaca)”, ISBN 0-8014-4240-0, Cornell University Press. Scale s -Mean: Scale s Mean or simply s Mean of () can be obtained as [5] Cornelli G., McKirahan R., Macris C. (2013): “ On Pythagoreanism ”, Berlin, Walter de Gruyter. { (. + . + … … … … …. + s. )} [6] Cantrell David W. (...... ): “Pythagorean Means ”, Math World . Shift a -Mean: Shift a - Mean of () can be obtained as [7] de Carvalho Miguel (2016): “Mean, what do you Mean?”, The American Statistician , 70, 764‒776. {( ̶ a) + ( ̶ ) +...... + ( ̶ a)} + a [8] Plackett R. L. (1958): “Studies in the History of Probability and Statistics: VII. The Principle of the Arithmetic Mean”, Shift a – Inverse Scale s – Mean: Biometrika , 45 (1/2), 130-135. Shift ( a) – Inverse Scale s - Mean of () can be obtained as [9] Nastase Adrian S. (2015): “How to Derive the RMS Value of Pulse and Square Waveforms”, ̶ ̶ ̶ MasteringElectronicsDesign.com . { ( + + … … … + )} + a [10] Dhritikesh Chakrabarty (2018): “Derivation of Some Formulations of Average from One Technique of Construction 7. Conclusion of Mean”, American Journal of Mathematical and Computational Sciences , 3 (3), 62 – 68. Available in http://www.aascit.org/journal/ajmcs. One can conclude that the Generalized fG - Mean constructed here can be regarded as a source from where lots [11] Dhritikesh Chakrabarty (2018): “One Generalized Definition of definitions/formulations can be derived for various types of Average: Derivation of Formulations of Various Means”, of averages. Journal of Environmental Science, Computer Science and Different types of formulations of average are necessary Engineering & Technology , Section C, (E-ISSN: 2278 – 179 X), 7 (3), 212 – 225, Also available in www.jecet.org. for handling different types of data. That is why we need more and more formulations of average. [12] Kendall D. G. (1991): “Andrei Nikolaevich Kolmogorov. 25 The types of average, formulated here, have been derived April 1903-20 October 1987”, Biographical Memoirs of Fellows of the Royal Society , 37, 300–326. from the Generalized fG - Mean constructed here. However, this Generalized fG - Mean, for generating means, may not be [13] Kolmogorov Andrey (1930): “On the Notion of Mean”, in sufficient to yield many types of averages to deal with many “Mathematics and Mechanics ” (Kluwer 1991), 144 – 146. [7] types of data. Thus, there is necessity of further study on [14] Kolmogorov Andrey (1933): “Grundbegriffe der searching for more and more techniques of defining / Wahrscheinlichkeitsrechnung (in German) , Berlin: Julius formulating of more types of averages. Springer. [15] Parthasarathy K. R. (1988): “Obituary: Andrei Nikolaevich Kolmogorov”, Journal of Applied Probability , 25 (2), 445 – References 450.

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[3] O'Meara Dominic J. (1989): “ Pythagoras Revived: [18] Dhritikesh Chakrabarty (2018): “fH -Mean: One Generalized Mathematics and Philosophy in Late Antiquity ”, ISBN 0-19- Definition of Average”, Journal of Environmental Science, 823913-0, Clarendon Press, Oxford. Computer Science and Engineering & Technology , Section C, (E-ISSN: 2278 – 179 X), 7 (4), 301 – 314, Also available in [4] Riedweg Christoph (2005): “ Pythagoras: his life, teaching, www.jecet.org.