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Çok gruplu ayırma analizinin bazı yeni kriterler yardımıyla gerçekleştirilmesi ve bir uygulama

Discriminant analysis between two or more groups with respect to the some new criteria and an application

  1. Tez No: 14132
  2. Yazar: HALİL BAYRAKÇI
  3. Danışmanlar: DOÇ.DR. AZİZ BENER
  4. Tez Türü: Doktora
  5. Konular: Matematik, Mathematics
  6. Anahtar Kelimeler: Belirtilmemiş.
  7. Yıl: 1990
  8. Dil: Türkçe
  9. Üniversite: İstanbul Teknik Üniversitesi
  10. Enstitü: Fen Bilimleri Enstitüsü
  11. Ana Bilim Dalı: Belirtilmemiş.
  12. Bilim Dalı: Belirtilmemiş.
  13. Sayfa Sayısı: 112

Özet

ÖZET Bu çalışmada veri tablosundan elde edilen bireyler kümesinin iki ve üç gruplu ayırma analizleri incelenmiş ve ayırmayı sağlayan bazı kriterler önerilerek, bu kriterler yardımıyla elde edilen ayırmanın, verilerin tabiatında mevcut bulunan gruplanma ile uygunluğunu ölçen testler tanım¬ lanmıştır. Ayırma analizini gerçekleştiren yöntemler genel olarak; Sebastiyen yaklaşımı ve Fisher yaklaşımı olmak üze¬ re, farklı iki tarzda ele alınır, ilk yaklaşımda yakın veya benzer bireyleri, aynı bir gruba atayan bir fonksiyon belirlenmeye çalışılır ki; bu da bir metrik tanımına götürür. İkinci yaklaşımda ise, ayırıcı faktorel eksenlerin belirlenmesi sözkonusudur. Bu eksenler ise, W,B ve V sırasiyle sınıf içi, sınıf arası ve toplam atalet kuadratik formlarına karşılık gelen matrisleri göstermek üzere; V~lfi (veya W~lB) nin en büyük Özdeğerlerine karşılık gelen öz vektörler olarak belirlenir. Burada, üç gruplu hal için, q. adımdaki gruplanmayı sağlayan en etkili değişken olarak Iz(V-lBq) yu maksi¬ mum yapan değişken seçimi, kriter olarak alınmıştır. İki gruplu halde, ayırıcı fonksiyon, bir katsayı farkıyla Maha- lanobis uzaklığına dönüşmektedir. İki gruplu halde tanımla¬ nan kararsızlık aralığının, bir ayırma kriteri olarak alına¬ bileceği gösterilmiştir. Ayrıca iki gruplu halde, grupların ayrıklığı ile ilgili olmak üzere kovaryans matrislerinin eşitliğine dayanan bir test tanımlanmıştır. Üç gruplu halde ise ayırıcı eksenlerin gerçek sayısı ile ilgili olmak üzere, Rao tarafından önerilen testten faydalanılmıştır. Üç gruplu ayırmayı adım adım (stepwise) gerçekleşti¬ ren algoritma ile iki gruplu ayırmayı karar fonksiyonu yardımıyla gerçekleştiren algoritmaya uygun olarak, FORTRAN IV dilinde kodlanmış birer program hazırlanmıştır. Bu programlar yardımıyla, Haydarpaşa Göğüs Cerrahi Kalp Damar Cerrahi Merkezi Hastahanesi'inde tedavi gören kalp hastaları üzerin¬ de yapılan ölçümlerle oluşturulan veri tablolarına, ayırma analizleri uygulanmış ve bu analizlerden elde edilen sonuçlar, karşılaştırmalı olarak yorumlanmıştır. Bu analiz sonucuda, kalp-hastalığı şikayetiyle başvuran bir birey üzerin¬ de yapılan ölçümlere dayanarak %90'lara varan bir güvenilirlikle karar vermek mümkün olabilmektedir. Ayrıca, analiz sonuçlarının güvenilirliği, ölçümlerin sağlıklı yapılmış olmasına ve gruptaki birey sayılarına da yakından bağlıdır.

Özet (Çeviri)

SUMMARY DISCRIMINANT ANALYSIS BETWEEN TWO OR MORE GROUPS WITH RESPECT TO THE SOME NEW CRITERIA AND AN APPLICATION The methods used in the discrimination may generally be considered in two different groups: 1) Sebastien approach discrimination, 2) Factorial discriminant analysis or Fisherian approach discriminant analysis. Let the individu als be given as groupped in advance. The main goal of the first approach which depends on the decision process is the research of determining of the criters according to which an individual will be included to a group among many others. To realise this goal for each group, a function is defined such that the similarity or the proximity of any individual to any group might measured by means of this function. Once a function with this property is defined, the rule which is used for assigning a new individual doesn'nt belong to the data table at present is determined. Since the individual is included to a group for which its similarity or proximity is maximum. An individual is determined by means of p different variables each of which take a certain value on the indivi dual in question. So each individual may be considered as a point or a vector in the R^ space. The similarity or the proximity function in RP space may be expressed as a metric. In fact the measure for similarity which is considered here is a quantity changing in reverse proportionality with similarity. In other words it is a measure of distance. The distances between an individual to be groupped and all other individuals are measured and it is included in to the group for which the minimum distance (the maximum similarity) is obtained. So the problem reduced to determining a distance, i.e. metric. One may think that Euclid metric can be used. But in Euclid metric the weight attaigned to each variables is measured on the set of individuals so they play a role to a certain degree in the determination of the group to which the individual will be included. This means that to give a weight to each variable, in other words to apply a transformation on the variables. Such kind of transformation corresponds to a metric represented by a diagonal matrix. The discriminant analysis realised with this approach constituted the subject of the first chapter. Since the met ric defined is related to the data table at the first step, the algebric structure of the data table is examined. In this context, the transformation defined with respect to the data table and their properties are examined in details. - IXThe data table is represented by X(nxp), the indivi- duals and variables are, respectively, by x.£RP(i=l,2n), FeR ( j = l, 2,...,p). The matrix X' represents a mapping of ^n p ^ p n R into RF and X represents a mapping of R ^ into R, where.^ p ^n d n R v and R are the dual spaces of respectively R^ and R. The mapping related to these two spaces are resumed on the fallowing duallity scheme, where D“ is the weights matrix, V is the covariance matrix and M xs the metric defined on the R? space: R1 R X' W N=D RJ According to this scheme V=X'D X and W=XMX'. B p The mean of the squares of the distances among all of the elements öf a certain Sa (a=l, 2,...,k) group with res pect to a M metric is called as groupping index of that 2 group and is represented by D. na is the member of elements in SQ and d2 = I a na(na-l) x.eSa XjeSa I d2(x\,5r.,) it is clear that d (x., x.) = x iMx. »From now on, we will try to find a M metric for each S such that the value of the 2 sum Da evaluated with respect to this metric is minimum. In order to determine M, we impose a restriction such that the determinant of M is 1. So under the normality condition | M | =1, it has been proved that the M metric which minimise Dl for each a is M= JV | a 1/Pv-l The choise of the metric is considered both for the general case and the case which corresponds to the diagonal matrix for the sake of the simplicity in practice and so the measure of similirity by this metric became easier. It is proved that the diagonal elements t, (k=l, 2,..., p) of M is given by - x -t, = p n k=l.k] l/p Here, s. is the standart deviation of k the variable calcu lated with respect to the group in question. The similarity between a newly coining element â* who may be called as anonym individual and a certain Sa(a=l, 2,...,k) group is represen ted by B(a,S ) and is defined as the mean of the square of the distances between a and all of the individuals in Sa, with respect to the M metric, which we have defined: B (*'V - T- k d2(^^i) a x eS i a Here, the distance d is the measure of the similarity with respect to M metric that has been determined for each group seperatly. At the end of the first chapter under the tittle of geometric interpretation, the relation between discrimi nant analysis and principal component is investigated. In the second chapter, factorial discriminant analysis or Fisherian approach to discriminant analysis is considered. In this chapter, V quadratic form of intertia and W(within), B(between) quadratic forms of inertia related to V are defi ned in accordance with the algebric structure of data table that has been examened in Chapter I. In this context two theorems were expressed and proved. The algebric structures of quadratic forms V,W and B are examened, the relation between these quadratic forms and their duality scheme is presented. If the matrix composed of the barycentres g eRp(a =1,2,...,k) of the groups is G(pxk), then its trans- pose matrix G'(kxp) will be composed of g^eR. If D is a diagonal matrix of (kxk) order with elements - **, the duality n J sheme of the mappings determined by these matrices is as: Here, B is the matrix corresponding to the quadratic form of inertia between groups and B=G D G'. On the other hand, xi -if the total quadratic form of inertia with respect to A -»?» and the quadratic forms of inertia between and within groups are represented by respectively V,W and B, then it was pro ved that the following relations are equivalent: M(u,VMu) = M(u,WMu) + M(u\BMu~) V(v) = W(v) + B(v) v*'Vv = v'Wv + v'Bv After that, the different definition and interpreta tion of the quadratic form of inertia for a set of individu als with respect to any direction are given. In addition to these, the relation between the number of variables and groups with the positive definite property of the quadratic form of inertia is investigated and it was proved that the dimension of linear variety H determined by the barycentres of the groups in question, is in general k-1, i.e. less 1 than member of groups. In the last part of this chapter the prob lem of determination of the discriminant factoral axes, takes place. Principal component method is used for the determination of these axes. But for the first discriminant axe, the fallowing criter is prefered: The process of discrimination will be more successful when the groups are more separated from each other and the individuals of the same gr^up are more closer to each other. To achive this a vector u must be found such that the ratio of the variance between groups to the variance within group is maximum. It was shown that u vector, under the restriction j|u|| =1, maximises IA, = u'MBMu u (1) Due to the relations Bv^u”- AÎ and which are obtained from (1), it was proved that the 3 vector can be obtained as an eigenvector corresponding to the greatest eigenvalue of BV. After the comparaison of the -1 -1 eigenvector of V B and W B, which determine the axes, the test proposed by Rao, about the number of discriminating axes, in the real sens, is mentionned. Finally, a criter which makes possible clustering of an anonym individual coming from out of the data table with respect to V metric is defined. According to this criter an anonym individual a is included to Sa group if - xix -d (ga0»a) - min { d (ga, a){

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