# Toward a Theory of Normalizing Function of Interestingness Measure of Binary Association Rules

Authors: Armand Armand; André Totohasina; Daniel Rajaonasy Feno
Journal: International Journal of Mathematics and Mathematical Sciences
 (2018)

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## Abstract

Regarding the existence of more than sixty interestingness measures proposed in the literature since 1993 till today in the topics of association rules mining and facing the importance these last one, the research on normalization probabilistic quality measures of association rules has already led to many tangible results to consolidate the various existing measures in the literature. This article recommends a simple way to perform this normalization. In the interest of a unified presentation, the article offers also a new concept of normalization function as an effective tool for resolution of the problem of normalization measures that have already their own normalization functions.


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## Body

## 1. Introduction

### 1.1. Definitions and Notations

We always put ourselves in the framework of a context of binary data mining (see, for example, [1–3], which illustrate the importance of association rules mining based on choosing some quality measure) D=(T,I,R), where I is a nonempty finite set of attributes or variables, T a finite set of entities or objects, R a binary relation from T to I, and P discrete uniform probability in the probabilistic space (T,P(T)) [4, 5].In the next sections, we use the following notation for two itemsetsX, Y: X′=e∈T/∀x∈X;eRx, i.e., the set of all transactions containing the X pattern that is the dual of a pattern X of I(X⊆I) [4, 6, 7]. n=T represents the size of the total sample size; nX=X′ represents the number of transactions satisfying pattern X; nXY=X′∩Y′ represents the number of transactions satisfying both X and Y; nX¯=|T|-T-TX′, where X¯ is the logical negation of X; nX/n=TX′/|T| represents the support of the pattern X.Hereafter, our work is divided into three sections. Section2 gives the definition of normalization function. Section 3 recommends the raw results of normalizing function of some probabilistic quality measures. Finally Section 4, [5, 8–10] sets out the conclusion and perspectives.


## 1.1. Definitions and Notations

We always put ourselves in the framework of a context of binary data mining (see, for example, [1–3], which illustrate the importance of association rules mining based on choosing some quality measure) D=(T,I,R), where I is a nonempty finite set of attributes or variables, T a finite set of entities or objects, R a binary relation from T to I, and P discrete uniform probability in the probabilistic space (T,P(T)) [4, 5].In the next sections, we use the following notation for two itemsetsX, Y: X′=e∈T/∀x∈X;eRx, i.e., the set of all transactions containing the X pattern that is the dual of a pattern X of I(X⊆I) [4, 6, 7]. n=T represents the size of the total sample size; nX=X′ represents the number of transactions satisfying pattern X; nXY=X′∩Y′ represents the number of transactions satisfying both X and Y; nX¯=|T|-T-TX′, where X¯ is the logical negation of X; nX/n=TX′/|T| represents the support of the pattern X.Hereafter, our work is divided into three sections. Section2 gives the definition of normalization function. Section 3 recommends the raw results of normalizing function of some probabilistic quality measures. Finally Section 4, [5, 8–10] sets out the conclusion and perspectives.


## 2. Normalizing FunctionFn

### 2.1. Motivations

The theory and practice of normalization probabilistic quality measures (see, for example, [4, 5, 11], (Totohasina et al. [12]), [6, 7]) have been resolving included in the list of tools for problems concerning the data mining. This is done in the view of regrouping [3, 4, 6, 12–17] different existing measures available from the literature. Let us notice that [4] proves existence of infinity quality measures through the concept of the so-called normalized quality measure under five conditions, but recently [16] still proposes a novel interestingness measure. By opening the door to the possibility of creating definitions of new concepts in the context of data mining, perhaps, this will bring to following a new reflection among researchers in this field. This is the normalizing function. What is meant by a normalizing function? The following section will attempt to answer such question. Remember that this paper is the logical continuation of the paper [18].


### 2.2. Proposal Approach

Definition 1.
We consider a probabilistic measure of interestμ. We call normalizing function of μ the piecewise continuous function that can normalize directly the measure μ and is defined as(1)Fnx=Fnfx,ifXfavorsYandx∈xind;ximp;Fndx,ifXdisfavorsYandx∈xinc;xind.It takes the following particular values:(2)Fanx=xfx+yf,ifXfavorsY;xdx+yd,ifXdisfavorsY.(3)Fshndx=xfx+m+yf,ifXfavorsY;xdx+yd,ifXdisfavorsY.(4)Fshngx=xfx+yf,ifXfavorsY;xdx+m+yd,ifXdisfavorsY.(5)Fhnx=xfx+m+yf,ifXfavorsY;xdx+m+yd,ifXdisfavorsY,where x is value of μ; ximp is value of μ at logical implication; xind is value of μ at independence; xinc is value of μ at incompatibility; Fnf is normalizing function, if X favors Y, that is to say, in case where PY′/X′>P(Y′); Fnd is normalizing function, if X disfavors Y, that is to say, in case where P(Y′/X′)<P(Y′); Fan is normalizing function of measure μ, that is, affine normalizable; Fhn is normalizing function of measure μ, that is, homographic normalizable; Fshnd is normalizing function of measure μ, that is, semihomographic-normalizable to right; Fshng is μ normalizing function of measure μ, that is, semihomographic-normalizable to left; xf, yf, xd, and yd are μ of normalization coefficient x; m is real and R¯=-∞;+∞.The definition of normalizing function is thus determined according to the quality measures. Note that this is a function as any other, with respect to the variablex; it is a numerical function of a real variable; only, this variable x is as follows:x = f 1 n , P X ′ , P Y ′ , P Y ′ ∩ X ′. The four coefficients of normalization of probabilistic measures quality have to meet the following conditions:x f = f 2 n , P X ′ , P Y ′, yf=f3n,PX′,PY′, xd=f4n,PX′,PY′, and yd=f5n,PX′,PY′, where f1,f2,f3,f4, and f5 are the five real functions. We note in passing that there is also a group of measures with the same normalization coefficients. The normalizing function is also one of the means that allow to give an interpretation of a normalized measurement after providing the respective values of normalization coefficients xf, yf, xd, and yd. It also expresses an opportunity of value for all normalized measures. According to what we have just written, any normalization of function reflecting the objective in normalization must necessarily have the following properties.Property 2 (necessary conditions).
P 1: Fnf is a continuous function, positive and strictly increasing on the interval xind;ximp and realizes a bijection on the interval xind;ximp→0;1; that is to say, Fnf must have the following: a limit 1 to the point ximp; that is to say, limx→ximpFnf(x)=1; a limit 0 to the point xind; that is to say, limx→xindFnf(x)=0.
P 2: Fnd is a continuous function, negative, and strictly increasing on the interval xinc;xind and realizes a bijection on the interval xinc;xind→-1;0; that is to say, Fnd must have the following: a limit 0 to the point xind; that is to say, limx→xindFnd(x)=0; a limit -1 to the point xinc; that is to say, limx→xincFnd(x)=-1. As a recap, we have the following.
P 3: Fn is a continuous function at the point xind and increasing on the interval xinc;ximp and realizes a bijection on the interval xinc;ximp→-1;1; that is to say, Fn must have the following: a limit 1 to the point ximp; that is to say, limx→ximpFn(x)=1; a limit 0 to the point xind; that is to say, limx→xindFn(x)=0 and Fn(xind)=0; limit -1 to the point xind; that is to say, limx→xincFn(x)=-1.


## 2.1. Motivations

The theory and practice of normalization probabilistic quality measures (see, for example, [4, 5, 11], (Totohasina et al. [12]), [6, 7]) have been resolving included in the list of tools for problems concerning the data mining. This is done in the view of regrouping [3, 4, 6, 12–17] different existing measures available from the literature. Let us notice that [4] proves existence of infinity quality measures through the concept of the so-called normalized quality measure under five conditions, but recently [16] still proposes a novel interestingness measure. By opening the door to the possibility of creating definitions of new concepts in the context of data mining, perhaps, this will bring to following a new reflection among researchers in this field. This is the normalizing function. What is meant by a normalizing function? The following section will attempt to answer such question. Remember that this paper is the logical continuation of the paper [18].


## 2.2. Proposal Approach

Definition 1.
We consider a probabilistic measure of interestμ. We call normalizing function of μ the piecewise continuous function that can normalize directly the measure μ and is defined as(1)Fnx=Fnfx,ifXfavorsYandx∈xind;ximp;Fndx,ifXdisfavorsYandx∈xinc;xind.It takes the following particular values:(2)Fanx=xfx+yf,ifXfavorsY;xdx+yd,ifXdisfavorsY.(3)Fshndx=xfx+m+yf,ifXfavorsY;xdx+yd,ifXdisfavorsY.(4)Fshngx=xfx+yf,ifXfavorsY;xdx+m+yd,ifXdisfavorsY.(5)Fhnx=xfx+m+yf,ifXfavorsY;xdx+m+yd,ifXdisfavorsY,where x is value of μ; ximp is value of μ at logical implication; xind is value of μ at independence; xinc is value of μ at incompatibility; Fnf is normalizing function, if X favors Y, that is to say, in case where PY′/X′>P(Y′); Fnd is normalizing function, if X disfavors Y, that is to say, in case where P(Y′/X′)<P(Y′); Fan is normalizing function of measure μ, that is, affine normalizable; Fhn is normalizing function of measure μ, that is, homographic normalizable; Fshnd is normalizing function of measure μ, that is, semihomographic-normalizable to right; Fshng is μ normalizing function of measure μ, that is, semihomographic-normalizable to left; xf, yf, xd, and yd are μ of normalization coefficient x; m is real and R¯=-∞;+∞.The definition of normalizing function is thus determined according to the quality measures. Note that this is a function as any other, with respect to the variablex; it is a numerical function of a real variable; only, this variable x is as follows:x = f 1 n , P X ′ , P Y ′ , P Y ′ ∩ X ′. The four coefficients of normalization of probabilistic measures quality have to meet the following conditions:x f = f 2 n , P X ′ , P Y ′, yf=f3n,PX′,PY′, xd=f4n,PX′,PY′, and yd=f5n,PX′,PY′, where f1,f2,f3,f4, and f5 are the five real functions. We note in passing that there is also a group of measures with the same normalization coefficients. The normalizing function is also one of the means that allow to give an interpretation of a normalized measurement after providing the respective values of normalization coefficients xf, yf, xd, and yd. It also expresses an opportunity of value for all normalized measures. According to what we have just written, any normalization of function reflecting the objective in normalization must necessarily have the following properties.Property 2 (necessary conditions).
P 1: Fnf is a continuous function, positive and strictly increasing on the interval xind;ximp and realizes a bijection on the interval xind;ximp→0;1; that is to say, Fnf must have the following: a limit 1 to the point ximp; that is to say, limx→ximpFnf(x)=1; a limit 0 to the point xind; that is to say, limx→xindFnf(x)=0.
P 2: Fnd is a continuous function, negative, and strictly increasing on the interval xinc;xind and realizes a bijection on the interval xinc;xind→-1;0; that is to say, Fnd must have the following: a limit 0 to the point xind; that is to say, limx→xindFnd(x)=0; a limit -1 to the point xinc; that is to say, limx→xincFnd(x)=-1. As a recap, we have the following.
P 3: Fn is a continuous function at the point xind and increasing on the interval xinc;ximp and realizes a bijection on the interval xinc;ximp→-1;1; that is to say, Fn must have the following: a limit 1 to the point ximp; that is to say, limx→ximpFn(x)=1; a limit 0 to the point xind; that is to say, limx→xindFn(x)=0 and Fn(xind)=0; limit -1 to the point xind; that is to say, limx→xincFn(x)=-1.


## 3. Applications

We recall in Table1 the respective definitions of the various measures that lead to the results below:(1)
Cost multiplying: x=PX′∩Y′PY′¯/PX′∩Y′¯PY′=1-PY′PY′/X′/1-PY′/X′PY′ which is such that ximp=+∞, xind=1, xinc=0. Using research theories of normalization coefficients xf, yf, xd, and yd in [18] we have xf=-1, yf=1, xd=1, and yd=-1. As a result, by replacing x, xf, yf, xd, and yd by their values in the expression (3) its normalizing function is such that (6)Fshndx=-1x+1,ifXfavorsYandx∈1;+∞x-1,ifXdisfavorsYandx∈0;1.It is easy to see that this function is continuous piecewise, in particular pointxind=1.Finally, it represents a function that exposed the necessary and sufficient conditions for the normalized and continuous measures.Its tables of variation are respectively presented in Figures1 and 2.The graphic illustration and the variation table ofFn of the measure Cost Multiplying reveal that Fnf is strictly increasing, is positive, and realizes a bijection on [1;+∞[→[0;1] and even fd is strictly increasing and negative and realizes a bijection on [0;1]→[-1;0]. These results show that Fn is strictly increasing and it realizes a bijection on [0;+∞[→[-1;1].Note that, in the following, for the search of the four normalization coefficients, the same principle is used with the measure “dimension multiplier” for the other following measures because they are already expressed in Table1.(2)
Example counter-example: its normalizing function is such that using the expression (4) for m=0 with xf=PY′/(1-PY′), yf=(1-2PY′)/(1-PY′), xd=(2PY′-1)/PY′ and yd=-1 so(7)Fshngx=PY′1-PY′x+1-2PY′1-PY′,ifXfavorsY2PY′-1PY′1x-1,ifXdisfavorsY.(3)
Informal gain: its normalizing function is such that using expression (4) for m=1 with xf=-1/log⁡P(Y′), yf=0, xd=1, and yd=-1 so (8)Fshngx=-1log⁡PY′x,ifXfavorsY1x+1-1,ifXdisfavorsY.(4)
Odd-ratio: its normalizing function is such that using expression (3) for m=0 with xf=-1, yf=1, xd=1, and yd=-1 so(9)Fshndx=1x+1,ifXfavorsYandx∈1;+∞x-1,ifXdisfavorsYandx∈0;1.(5)
Conviction: its normalizing function is such that using expression (5) for m=0 with xf=-1, yf=1, xd=-(1-PY′)/PY′, and yd=(1-PY′)/PY′ so(10)Fhnx=-1x+1,ifXfavorsY-1-PY′PY′1x+1-PY′PY′,ifXdisfavorsY.(6)
Sebag: its normalizing function is such that using expression (3) for m=0 with xf=-PY′/(1-PY′), yf=1, xd=(1-PY′)/PY′, and yd=-1 so(11)Fshndx=-PY′1-PY′1x+1,ifXfavorsY1-PY′PY′x-1,ifXdisfavorsY.(7)
MGK: its normalizing function is such that using expression (2) with xf=1, yf=0, xd=1, and yd=0 so(12)Fanx=xavecx=PY′/X′-PY′1-PY′,ifXfavorsYandx∈0;1PY′/X′-PY′PY′,ifXdisfavorsYandx∈-1;0.(8)
Support; its normalizing function is such that using expression (2) with xf=1/PX′1-PY′, yf=-PY′/(1-PY′), xd=1/PX′PY′, and yd=-1 so(13)Fanx=1PX′1-PY′x-PY′1-PY′,ifXfavorsY1PX′PY′x-1,ifXdisfavorsY.(9)
Confidence: its normalizing function is such that using expression (2) with xf=1/1-PY′, yf=-PY′/1-PY′, xd=1/PY′, and yd=-1 so(14)Fanx=11-PY′x-PY′1-PY′,ifXfavorsY1PY′x-1,ifXdisfavorsY.(10)
Recall: its normalizing function is such that using expression (2) with xf=PY′/PX′1-PY′, yf=-PY′/(1-PY′), xd=1/PX′, and yd=-1 so(15)Fanx=PY′PX′1-PY′x-PY′1-PY′,ifXfavorsY.1PX′x-1,ifXdisfavorsY(11)
Lift: its normalizing function is such that using expression (2) with xf=PY′/1-PY′, yf=-PY′/1-PY′, xd=1, and yd=-1 so(16)Fanx=PY′1-PY′x-PY′1-PY′,ifXfavorsYx-1,ifXdisfavorsY.(12)
Laverage: its normalizing function is such that using expression (2) with xf=PY′/1-PY′, yf=-PY′PX′-1/(1-PY′), xd=1/PY′, and yd=-PY′PX′-1/PY′ so (17)Fanx=PY′1-PY′x-PY′PX′-11-PY′,ifXfavorsY1PY′x-PY′PX′-1PY′,ifXdisfavorsY.(13)
Centered confidence: its normalizing function is such that using expression (2) with xf=1/(1-PY′), yf=0, xd=1/PY′, and yd=0 so (18)Fanx=11-PY′x,ifXfavorsY1PY′x,ifXdisfavorsY.(14)
Featured confirmed confidence: its normalizing function is such that using expression (2) with xf=1/21-PY′, yf=(1-2PY′)/2PY′, xd=1/2PY′, and yd=(1-2PY′)/2PY′ so(19)Fanx=121-PY′x+1-2PY′21-PY′,ifXfavorsY12PY′x+1-2PY′2PY′,ifXdisfavorsY.(15)
Certainty factor: its normalizing function is such that using expression (2) with xf=1, yf=0, xd=(1-PX′)/PY′, and yd=0 so(20)Fanx=x,ifXfavorsY1-PX′PY′x,ifXdisfavorsY.(16)
Gras implication index: its normalizing function is such that using expression (2) with xf=PX′PY′/nPX′1-PY′, yf=0, xd=(PX′PY′/nPX′PY′)x, and yd=0 so(21)Fanx=PX′PY′nPX′1-PY′x,ifXfavorsYPX′PY′nPX′PY′x,ifXdisfavorsY.(17)
Piatetsky-Shapiro: its normalizing function is such that using expression (2) with xf=1/nPX′1-PY′, yf=0, xd=1/nPX′PY′, and yd=0 so(22)Fanx=1nPX′1-PY′x,ifXfavorsY1nPX′PY′x,ifXdisfavorsY.(18)
Cosinus: its normalizing function is such that using expression (2) with xf=PX′PY′/PX′1-PY′, yf=-PX′/(1-PY′), xd=PX′PY′/PX′PY′, and yd=-1 so(23)Fanx=PX′PY′PX′1-PY′x-PX′1-PY′,ifXfavorsYPX′PY′PX′PY′x-1,ifXdisfavorsY.(19)
Loevinger: its normalizing function is such that using expression (2) with xf=1, yf=0, xd=(1-PY′)/PY′, and yd=0 so(24)Fanx=x,iffavorsY1-PY′PY′x,ifXdisfavorsY.(20)
Cohen ou Kappa: its normalizing function is such that using expression (2) with xf=(PX′+PY′-2PX′PY′)/2PX′1-PY′, yf=0, xd=PX′+PY′-2PX′PY′/2PX′PY′, and yd=0 so(25)Fanx=PX′+PY′-2PX′PY′2PX′1-PY′x,ifXfavorsYPX′+PY′-2PX′PY′2PX′PY′x,ifXdisfavorsY.(21)
Addiction: its normalizing function is such that using expression (2) with xf=1/(1-PY′), yf=0, xd=1/PY′, and yd=-1 so(26)Fanx=11-PY′x,1PY′x-1.(22)
Novelty: its normalizing function is such that using expression (2) with xf=1/PX′1-PY′, yf=0, xd=1/PX′PY′, and yd=0 so(27)Fanx=1PX′1-PY′x1PX′PY′x.(23)
Czekanowski-Dice ou F-measure: its normalizing function is such that using expression (2) with xf=(PX′+PY′)/2PX′1-PY′, yf=-2PX′PY′/2PX′(1-PY′, xd=(PX′+PY′)/2PX′PY′, and yd=-1 so(28)Fanx=PX′+PY′2PX′1-PY′x-2PX′PY′2PX′1-PY′PX′+PY′2PX′PY′x-1.(24)
Relative risk: its normalizing function is such that using expression (2) with xf=(PY′-PX′)/(1-PY′), yf=-(PY′-PX′)/(1-PY′), xd=1, and yd=-1 so(29)Fanx=PY′-PX′1-PY′x-PY′-PX′1-PY′x-1,.(25)
Negative reliability: its normalizing function is such that using expression (2) with xf=1/PX′, yf=-(1-PX′-PY′+PY′PX′)/PX′1-PY′, xd=(1-PY′)/PX′Y′, and yd=-(1-PX′-PY′+PY′PX′)/PX′PY′ so(30)Fanx=1PX′x-1-PX′-PY′+PY′PX′PX′1-PY′1-PY′PX′Y′x-1-PX′-PY′+PY′PX′PX′PY′.Researches out on the expressions of measurement normalizing functions prove that certain measures have identical normalizing functions, leading to what state the theorem below.Table 1
Probabilistic quality measures of expression.
N o Measuresμ Expressions ofx=μX→Y 1 Cost multiplying x = P X ′ ∩ Y ′ P Y ¯ ′ P X ′ ∩ Y ¯ ′ P Y ′  2 Example counter-example x = P X ′ ∩ Y ′ - P X ′ ∩ Y ¯ ′ P X ′ ∩ Y ′  3 Informal gain x = log ⁡ P X ′ ∩ Y ′ P X ′ P Y ′  4 Odd-Ratio x = P X ′ ∩ Y ′ P X ¯ ′ ∩ Y ¯ ′ P X ¯ ′ ∩ Y ′ P X ′ ∩ Y ¯ ′  5 Conviction x = P X ′ P Y ¯ ′ P X ′ ∩ Y ¯ ′  6 Sebag x = P X ′ ∩ Y ′ P X ′ P Y ¯ ′  7 M G K x = P Y ′ / X ′ - P Y ′ 1 - P Y ′ P Y ′ / X ′ - P Y ′ P Y ′  8 Rucel and Rao Index x = P X ′ ∩ Y ′  9 Confidence or precision x = P Y ′ X ′  10 Recall x = P X ′ Y ′  11 Interest or Lift x = P Y ′ / X ′ P Y ′  12 Laverage x = P Y ′ X ′  13 Centered confidence x = P Y ′ X ′ - P Y ′  14 Confirmed confidence x = 1 - 2 P Y ¯ ′ X ′  15 Certainty factor x = P Y ′ / X ′ - P Y ′ 1 - P Y ′  16 Gras implication x = n P X ′ ∩ Y ′ - P X ′ P Y ′ P X ′ P Y ′  17 Piatesky-Shapiro x = n ( P X ′ ∩ Y ′  18 Cosinus x = P X ′ ∩ Y ′ P X ′ P Y ′  19 Loevinger x = 1 - P X ′ ∩ Y ¯ ′ P X ′ P Y ¯ ′  20 Cohen ou Kappa x = 2 P X ′ ∩ Y ′ - P X ′ P Y ′ P X ′ + P Y ′ - 2 P X ′ P Y ′  21 Addiction x = P Y ¯ ′ - P Y ¯ ′ X ′  22 Novelty x = P X ′ Y ′ - P X ′ P Y ′  23 Czekanowski-Dice x = 2 P X ′ ∩ Y ′ P X ′ ∩ Y ′ + 1 - P X ¯ ′ ∩ Y ¯ ′  24 Relative risk x = P X ′ / Y ′ P X ′ / Y ¯ ′  25 Negative reliability x = P X ¯ ′ Y ¯ ′Figure 1
Change in measure normalizing function “Cost Multiplying”.Figure 2
Geometric interpretation of the normalization function of the “Cost Multiplying” measure.Theorem 3.
(i) All measures having the same baseline(ximp,xind,xinc) have the same normalizing function.
(ii) All measures affine normalizable are homographic normalizable, but the converse is false.Proof.
(i) Suppose the four possible forms ofFn.(a)
μis affine normalizable:(31)Fanx=xfx+yf,ifXfavorsYxdx+yd,ifXdisfavorsYwith(32)xf=1ximp-xind,yf=-1ximp-xindxind,xd=1xind-xincand(33)yd=-1xind-xincxindso we have(34)Fanx=1ximp-xindx-1ximp-xindxind,ifXfavorsY1xind-xincx-1xind-xincxind,ifXdisfavorsY.(b)
μis to right semihomographic normalizable:in the c
ase whereximp=+∞, that is to say, limx→+∞(xf/(x+m))=0, the expression is used: (35)Fshndx=xfx+m+yf,ifXfavorisYxdx+yd,ifXdisfavorsYwith(36)xf=-x+m,yf=1,xd=1xind-xincand(37)yd=-1xind-xincxindso(38)Fshndx=-xind+mx+m+1,ifXfavorsY1xind-xincx+1xind-xincxinc,siXdisfavorsY.(c)
μis to left semihomographic normalizable: in the case where xinc=-∞; that is to say, limx→-∞(xf/(xinc+m))=0. This time we use the following expression:(39)Fshngx=xfx+yf,ifXfavorsYxdx+m+yd,ifXdisfavorsYsuch that(40)xf=1ximp-xind,yf=-1ximp-xindxind,xd=xind+mand(41)yd=-1so(42)Fshndx=1ximp-xindx-1ximp-xindxind,ifXfavorsYxind+mx+m-1,ifXdisfavorsY.(d)
μis homographic normalizable:in the case wherexinc=-∞, ximp=+∞ and xind∈R∗; that is to say,(43)limx→-∞⁡xfxinc+m=0and(44)limx→-∞⁡xfxinc+m=0.Here we use the following expression:(45)Fhnx=xfx+m+yf,ifXfavorsYxdx+m+yd,siXdisfavorsYwhere(46)xf=-xind+m,yf=1,xd=xind+mand(47)yd=-1,so(48)Fhnx=-xind+mx+m+1,ifXfavorsYxind+mx+m-1,ifXdisfavorsY.
(ii) (a) It is seen that ifx∈R and (x+m)∈R∗, then the four terms of the function of normalizations are well defined; therefore, there is no problem for calculating the normalization coefficients.
(b) We see that ifx=∞, then we can always get a projective application R¯ in R for the functions Fshn, Fshnd, and Fshng and therefore the four coefficients of normalization are always calculable; by constraint, if x=∞, then you can never get a projective application over the interval R¯ in R for the function Fan; therefore, we can not calculate these four coefficients. The theorem is stated.Table1 recalls the respective definitions of the various measures.


## 4. Conclusion and Perspectives

This study showed that normalization of probabilistic quality measures with a homographic homeomorphism is more powerful than the normalization homeomorphism refines initiated by André Totohasina. Indeed, we showed that any measure affine-normalizable is homographic normalizable, while the converse is false. Besides, this work has explained the process of normalization by homographic function and combination with an affine function by trying to sweep the present main measures in the literature with the aim of a presentation easier to understand. The database has several branches; the purpose of this research is the normalization of quality measures. We always say, in the context of the database, the study on association rules knows an important development, added to the measures called interest; yet probabilistic quality measures have an important place in the context of data mining. Thereafter, the probabilistic measure of quality and its normalization must be complement. As shown by research on normalization probabilistic quality measures realizing the normalization operation requires passing through a relatively complex theory. We can consider several possible ways to carry out its standardization process. In our opinion, the use of normalizing function seems the simplest way.In future work, we understand the positive impact of consideration of these normalizing functions in the development of the bases of the rules in search of binary data.



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*Source: 4814716-2018-11-14.xml*