The article presents a new normalisation method of diagnostic variables - normalisation with respect to the pattern. The normalisation preserves some important descriptive characteristics of variables: skewness, kurtosis and the Pearson correlation coeffcients. It is particularly useful in dynamical analysis, when we work with the whole population of objects not a sample, for example in regional studies. After proposed transformation variables are comparable not only between themselves but also across time. Then we can use them, for example, to construct

composite variables

Müller-Frączek, Iwona

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Normalisation with respect to patternIwona Müller-Fr¡czek∗Nicolaus Copernicus University in Toru«, PolandAbstractThe article presents a new normalisation method of diagnostic vari-ables - normalisation with respect to the pattern. The normalisationpreserves some important descriptive characteristics of variables: skew-ness, kurtosis and the Pearson correlation coecients. It is particularlyuseful in dynamical analysis, when we work with the whole populationof objects not a sample, for example in regional studies. After proposedtransformation variables are comparable not only between themselvesbut also across time. Then we can use them, for example, to constructcomposite variables.keywords: normalisation, standardisation, composite variable, syn-thetic measure1 IntroductionIn regional studies we often need to compare regions (objects) with respectto analyzed complex (or composite) phenomenon. Complex phenomenon is aqualitative phenomenon, that is characterized by some quantitative features,called diagnostic variables. Each object is then identied with a point ofthe multidimensional real space. One of the tools of regional research arecomposite variable (or synthetic measure). Composite variable is created toreect multidimensional points (objets) in the one-dimensional space.Many advanced methods of constructing synthetic variables have beendeveloped, however the simplest methods are often used in practice. There∗Author's Address: I. Müller-Fr¡czek, Faculty of Economic Sciences and Manage-ment, Nicolaus Copernicus University, ul. Gagarina 13 a, 87-100 Toru«, Poland, e-mail:muller@umk.pl1are a lot of such examples (see [2]), one of them is very popular Human Devel-opment Index (HDI), which ranks countries into four tiers of socio-economicdevelopment. Until 2010 HDI was a uniformly weighted sum of three indica-tors describing: life expectancy, education, and income per capita.One of the step of the construction of synthetic measure is bringing diag-nostic variables to comparability, called normalisation or standardisation.Normalisation deprives variables their units and unies their ranges. Thereare a lot of normalisation formulas (see [4], [5], [8]). Choosing a propermethod is important because normalisation inuences on results of objectordering.The usual stochastic approach can be used to determine parametersneeded to normalisation. Then we treat values of variable (observations)as a randomly selected sample of the population. This approach should notbe used in regional research, where we work with the whole population ofobjects. In this case we should use a descriptive (deterministic) approach.Normalisation formulas are most often given for static analysis, this isfor a xed point in time. A normalisation problem appears when we wantto compare situations of regions at several time points. Then the variablesshould also be comparable across time. To achieve this eect in the stochas-tic approach one can use all values of variable (both for objects and for time)to determine parameters needed for normalisation. However, this solution iscontroversial in descriptive approach (see [9]), in addition, it requires incesantconversion of results when later observations occur. In this case we shouldrather use current observations, so after usual normalisation variables arenot comparable across time. Then we can not compare the values of syn-thetic measures, we can only compare rankings. To solve this problem in thementioned Human Development Index, the parameters of feature scaling arexed on levels, that are not related to variable distribution. The levels arejustied by substantive reasons. For example, the age of 85 was establishedas the maximum life expectancy at birth.The article proposes a new method of feature normalisation - normali-sation with respect to the pattern (or pattern normalisation for short). Thisname was inspired by the Hellwig's paper (see [3], [1]). The method is con-sistent with the static approach, but it can be used to compare objects atdierent time points. The method meets the requirements of normalisationthat are suggested in literature (see [4], [6]). It preserves skewness and kur-tosis. Moreover, the absolute values of the Pearson correlation coecientsare not changed after normalisation.In the rs step of the pattern normalisation the nature of variable is de-termined in the context of analyzed complex phenomenon. We distinguishstimulants and destimulants. Stimulant is a diagnostic variable that has a2positive impact on the analyzed complex phenomenon, while destimulantnegative. In regional research determining the nature of variables is natu-ral. Most often, before normalisation, we turn destimulants into stimulantsusing their inverse values. Unfortunately, the variables after conversion losetheir interpretation and their distributions are changed. In the presentedmethod, we do not converse destimulant before normalisation. Destimulantsand stimulants are normalized in dierent ways.Determining the nature of variable allows us to choose the most benecialobservation among all values of the variable, maximum for stimulant andminimum for destimulant. We call this value a pattern. Next we convert allvalues with respect to this pattern. After transformation we get comparablevariables. All of them are destimulants with clear interpretation. Patternnormalisation can be used in common constructiion of composite variablesinstead of other methods of normalisation. A possible application is shownin [7].2 Denition of pattern normalisationSuppose that a complex phenomenon observed for n ∈ N regions is ana-lyzed. Assume that we cannot measure this phenomenon, whereas we knowa collection of measurable diagnostic variables that characterize it.Assume that diagnostic variables meet both substantive and statisticalrequirements, for more details see for example [9]. Let us consider one suchvariable x = (x1, x2, . . . , xn) ∈ Rn, which is a stimulant (then we write x ∈ S,S denotes the set of stimulants) or a destimulant (x ∈ D analogously).In the rst step we choose a pattern - the most benecial of all values ofthe variable x. The pattern is unique for all objects and is described by theformula:x+ =maxi xi if x ∈ S,minixi if x ∈ D.(1)After specifying the pattern x+ we can consider a new variable u+ insteadof the variable x given by:u+i =|xi − x+|∑nj=1 |xj − x+|=x+ − xi∑nj=1(x+ − xj)if x ∈ S,xi − x+∑nj=1(xj − x+)if x ∈ D.(2)3The formula (2) determines a certain transformation of initial variablex = (x1, x2, . . . , xn) into a new variable u+ = (u+1 , u+2 , . . . , u+n ). We call it anormalisation with respect to the pattern. After this transformation the newvariable describes the same aspect of complex phenomenon as described byx. So u+ is a diagnostic variable of this phenomenon.The pattern normalisation (2) is not just a technical procedure. Newvariable has a clear interpretation, u+i species the share of distance betweenthe i-th object and the pattern in the total distance of all objects from thepattern. The situation of the i-th object is better when the value of u+i islower.The values of variable u+ characterize the positions of objects in the wholesystem. This is the same as for other forms of normalisation, but the systemis specied in a dierent way. In the case of the pattern normalisation thesystem is represented by the sum of distances between objects and pattern,while in common normalisations descriptive characteristics of the distributionof x are used for this purpose.3 Properties of variable after normalisationThe quantitative description of an immeasurable (qualitative) phenome-non is obtained using synthetic measures. Bringing diagnostic variables tocomparability is the rst step in the construction of such measure. Thepattern normalisation can be used for this purpose.Assume that diagnostic variables are transformed with respect to theirpatterns. Then the new set of variables has advantages, which are expectedfor creating synthetic variables. These properties and some proofs are pre-sented below.A. Basic propertiesA1. All variables after pattern normalisation are unitless, non-negative andlimited to interval [0, 1]. Because of that, the new set of diagnosticvariables contains comparable elements.A2. Irrespective of the initial nature, variable after the pattern normali-sation becomes destimulant. It means that the situation of the i-thobject is better when the value u+i is lower. In this sense the patternnormalisation unies the nature of diagnostic variables.A3. Transforming of variables does not aect the ordering of objects.B. Extreme values after pattern normalisation4B1. The variable u+ can take zero value only for the pattern object. Sincethe pattern is chosen among values of the variable x, zero value is taken.u+i = 0⇔ xi = x+.Proof.u+i = 0⇔|xi − x+|∑nj=1 |xj − x+|= 0⇔ |xi − x+| = 0⇔ xi = x+B2. The value u+i equals 1 when all objects are patterns except the i-thobject. This situation is rather unrealistic.u+i = 1⇔∧j 6=ixj = x+.Proof.u+i = 1⇔|xi − x+|∑nj=1 |xj − x+|= 1⇔ |xi − x+| =n∑j=1|xj − x+| ⇔∧j 6=ixj = x+B3. The maximum value of u+ depends on the nature of variable x and itis expressed by:maxiu+i =maxi xi −mini xi∑nj=1(maxi xi − xj)if x ∈ S,maxi xi −mini xi∑nj=1(xj −mini xi)if x ∈ D.Proof.If x ∈ S, then:maxiu+i =maxi(x+ − xi)∑nj=1(x+ − xj)=x+ −mini xi∑nj=1(x+ − xj)=maxi xi −mini xi∑nj=1(maxi xi − xj).If x ∈ D, then:maxiu+i =maxi(xi − x+)∑nj=1(xj − x+)=maxi xi − x+∑nj=1(xj − x+)=maxi xi −mini xi∑nj=1(xj −mini xi).5C. Descriptive characteristics of normalised variablesC1. The mean value of u+ depends only on the number of objects and isinversely proportional to this number. It is expressed by:u+def=1nn∑i=1u+i =1n.Proof.u+ =1nn∑i=1|xi − x+|∑nj=1 |xj − x+|=1n∑ni=1 |xi − x+|∑nj=1 |xj − x+|=1nC2. The variance of u+ is described by:S2(u+)def=1nn∑i=1(u+i − u+)2 =S2(x)n2(x+ − x)2.Proof.S2(u+) =1nn∑i=1(|x+ − xi|∑nj=1 |x+ − xj|− 1n)2If x ∈ S, then:S2(u+) =1nn∑i=1(x+ − xi∑nj=1(x+ − xj)− 1n)2=1n3n∑i=1(x+ − xix+ − 1n∑nj=1 xj− 1)2=1n3n∑i=1(x+ − xix+ − x− 1)2=1n3n∑i=1(x− xix+ − x)2=1n∑ni=1(x− xi)2n2(x+ − x)2=S2(x)n2(x+ − x)2The proof is similar when x ∈ D.C3. The standard deviation of u+ depends on the nature of variable x andit is expressed by:S(u+)def=√S2(u+) =S(x)n(x+ − x)if x ∈ S,S(x)n(x− x+)if x ∈ D.6C4. The coecient of variation of u+ is given by:CV (u+)def=S(u+)u+=S(x)x+ − xif x ∈ S,S(x)x− x+if x ∈ D.C5. The 3-rd central moment of u+ is given by:µ3(u+)def=1nn∑i=1(u+i − u+)3 =µ3(x)n3(x+ − x)3.Proof.µ3(u+) =1nn∑i=1(|x+ − xi|∑nj=1 |x+ − xj|− 1n)3If x ∈ S, then:µ3(u+) =1nn∑i=1(x+ − xi∑nj=1(x+ − xj)− 1n)3=1n4n∑i=1(x+ − xix+ − 1n∑nj=1 xj− 1)3=1n4n∑i=1(x+ − xix+ − x− 1)3=1n4n∑i=1(xi − xx− x+)3=µ3(x)n3(x− x+)3The proof is similar when x ∈ D.C6. The absolute value of the coecient of skewness does not change afterthe pattern normalisation:A(u+)def=µ3(u+)S3(u+)={−A(x) if x ∈ S,A(x) if x ∈ D.C7. The 4-th central moment of u+ is given by:µ4(u+)def=1nn∑i=1(u+i − u+)4=µ4(x)n4(x+ − x)4.7Proof.µ4(u+) =1nn∑i=1(x+ − xi∑nj=1(x+ − xj)− 1n)4If x ∈ S, then:µ4(u+) =1nn∑i=1(x+ − xi∑nj=1(x+ − xj)− 1n)4=1n5n∑i=1(x+ − xix+ − 1n∑nj=1 xj− 1)4=1n5n∑i=1(xi − xx− x+)4=1n5n∑i=1(x− xix+ − x− 1)4=µ4(x)n3(x− x+)4The proof is similar when x ∈ D.C8. The kurtosis of u+ does not change after the pattern normalisation:K(u+)def=µ4(u+)S4(u+)= K(x).D. Linear relation between variables after normalisationAssume that two diagnostics variables x1, x2 are transformed with respectto their patterns. Denote by u+1 and u+2 variables after normalisation.D1. The covariance between u+1 and u+2 equals:cov(u21, u+2 )def=1nn∑i=1(u+i1 − u+1)(u+i2 − u+2)=cov(x1, x2)n2(x+1 − x1)(x+2 − x2)if x1, x2 ∈ S or x1, x2 ∈ D,−cov(x1, x2)n2(x+1 − x1)(x+2 − x2)otherwise.Proof.cov(u21, u+2 ) =1nn∑i=1(|xi1 − x+1 |∑nj=1 |xj1 − x+1 |− 1n)(|xi2 − x+2 |∑nj=1 |xj2 − x+2 |− 1n)8Assume that x1 and x2 are stimulants. The proof in other cases is similar.cov(u21, u+2 ) =1nn∑i=1(x+1 − xi1∑nj=1(xj1 − x+1 )− 1n)(x+2 − xi2∑nj=1(xj2 − x+2 )− 1n)=1n3n∑i=1(x+1 − xi1x+1 − 1n∑nj=1 xj1− 1)(x+2 − xi2x+2 − 1n∑nj=1 xj2− 1)=1n3n∑i=1(x+1 − xi1x+1 − x1− 1)(x+2 − xi2x+2 − x2− 1)=1n3n∑i=1(x1 − xi1x+1 − x1· x2 − xi2x+2 − x2)=1n∑ni=1(xi1 − x1)(xi2 − x2)n2(x+1 − x1)(x+2 − x2)=cov(x1, x2)n2(x+1 − x1)(x+2 − x2)D2. The absolute value of the Pearson correlation coecient of diagnosticvariables is preserved after the normalisation:corr(u+1 , u+2 )def=cov(u21, u+2 )S(u+1 )S(u+2 )={corr(x1, x2) if x1, x2 ∈ S or x1, x2 ∈ D,−corr(x1, x2) otherwise.E. Dynamic approachAssume that the diagnostic variable x is observed in two periods of time(then we write x1 and x2 respectively). For each period we choose a patternand transform x1 and x2 into u1+ and u2+ according to the formula (2).E1. The values of variables u1+ and u2+ are comparable.Substantiation.The system is characterized by the sum of distances between objects andthe pattern. It changes over time. For given object, if the value of the trans-formed variable increases over time, this means that the share of distancefrom this object to the pattern in the sum of all distances increases, so thesituation of this object becomes worse (in comparison with the situations ofother objects).94 SummaryThe normalisation of diagnostic variables described by formula (2) playsa double role in the construction of synthetic measure. First, it unies thenature of variables (A2). Secondly, it brings variables to comparability (A1).So, after pattern normalisation diagnostic variables become comparable des-timulants. The normalisation with respect to the pattern preserves two im-portant characteristics of the distribution of diagnostic variables - skewness(C6) and kurtosis (C8). Moreover, this conversion does not disrupt linearrelation between variables - the absolute value of the Pearson correlationcoecient is not changed (D2). This advantages are expected for normalisa-tions used for bringing variables to comparability.Unlike other methods the pattern normalisation is not just a technicalprocedure, it has clear interpretation. However, the major advantage of thepattern normalisation over other normalisation methods appears in dynamicapproach. Although the current data are the sole data used to convert vari-ables, the transformed variables are comparable in time (E1).The normalisation with respect to the pattern seems to be a useful tool inmultidimensional comparative analysis. It can be applied whenever variablesneed to be comparable, for example in the synthetic analysis of complexphenomenon.The proposed construction can have various modications, for examplewe can change the measure of distance or the method of choosing pattern.References[1] FANCHETTE, S. (1972) "Synchronic and diachronic approaches in theUnesco project on human resources indicators - Wroclaw taxonomyand bivariate diachronic analysis", UNESCO document, SHS/WS/209,Paris.[2] FREUDENBERG, M. (2003), "Composite Indicators of Country Per-formance: A Critical Assessment", OECD Science, Technology and In-dustry Working Papers, No. 2003/16, OECD Publishing, Paris.[3] HELLWIG, Z. (1968), "Procedure of Evaluating High-Level ManpowerData And Typology of Countries by Means of the Taxonomic Method",unpublished UNESCO working paper, COM/WS/91, Paris.[4] JAJUGA, K., WALESIAK, M. (2000), "Stamdardisation of Data SetUnder Dierent Measurement Scales", in Classication and Informa-tion Processing at the Turn of the Millennium. Studies in Classication,10Data Analysis, and Knowledge Organization, eds. Decker R., Gaul W.,Springer-Verlag, Berlin, Heidelberg, 105-112.[5] MILLIGAN, G.W., COOPER, M.C. (1988), "A Study of Standardiza-tion of Variables in Cluster Analysis", Journal of Classication 5, 181-204.[6] MODAK, A. (2006), "Multirateral Normalisations of Diagnostic Fea-tures", Statistics In Transition 7(5), 1125-1139.[7] MÜLLER-FRCZEK, I. (2017), "Propozycja miary syntetycznej"[Proposition of Synthetic Measure], Przegl¡d Statystyczny, 64(4), 413-428.[8] STEINLEY, D. (2004), "Standardizing Variables in K -means Cluster-ing" in Classication, Clustering, and Data Mining Applications. Stud-ies in Classication, Data Analysis, and Knowledge Organisation, eds.Banks D., McMorris F.R., Arabie P., Gaul W., Springer, Berlin, Heidel-berg.[9] ZELIA A. (2002), "Some Notes of the Selection of Normalisation ofDiagnostic Variables", Statistics In Transition 5(5), 787-802.11

Normalisation with respect to pattern

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Normalisation with respect to pattern

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