Transformation to Achieve Perfect Correlation

Abstract

Correlation and linear regression are common means to evaluate association and empirical relationships between two or more variables. Such relationships often show significant departure of |r_XY | from unity. Existing transformations to increase correlation fail to achieve perfect correlation. For a bivariate data, the paper proposes transforming Y to y=G.‖x‖‖y‖, which gives r_(X y)=1 where G is the G-inverse of the matrix A=x.x^Tand x, y denote vectors of deviation scores. The concept is extended to perfect linearity between a dependent variable (Y) and a set of independent variables (Multiple linear regressions) or between set of dependent variables and set of independent variables (Canonical regression), avoiding problems of insignificant beta coefficients in univariate and multivariate regression models and outliers. Empirical illustration of G-inverse and extensions for multiple linear regressions and Canonical regressions are also given. The proposed transformation is a novel method of introducing perfect correlation between two variables. Extension of the concept in multiple linear regressions and canonical regression will go a long way in empirical researches in various branches of science. Future studies may include finding distribution of the proposed perfect correlations and comparison of efficacy of our suggested approach against other traditional ones by providing quantitative evidences.

Keywords

• Correlation coefficient
• Generalized inverse
• Multiple linear regressions
• Canonical regression
• Normal distribution

Mükemmel Korelasyona Ulaşmak için Dönüşüm

Abstract

Korelasyon ve doğrusal regresyon, iki veya daha fazla değişken arasındaki ilişkiyi ve ampirik ilişkileri değerlendirmek için yaygın araçlardır. Bu tür ilişkiler genellikle |r_XY |'nin birlikten önemli ölçüde saptığını gösterir. Korelasyonu artırmak için yapılan mevcut dönüşümler mükemmel korelasyona ulaşmada başarısız olur. İki değişkenli veriler için, makale Y'yi y=G.‖x‖‖y‖'ye dönüştürmeyi önerir; bu da r_(X y)=1'i verir; burada G, A=x.x^T matrisinin G-tersidir ve x, y sapma puanlarının vektörlerini belirtir. Kavram, bağımlı değişken (Y) ile bağımsız değişkenler kümesi (Çoklu doğrusal regresyonlar) veya bağımlı değişkenler kümesi ile bağımsız değişkenler kümesi (Kanonik regresyon) arasındaki mükemmel doğrusallığa genişletilir ve tek değişkenli ve çok değişkenli regresyon modellerinde ve aykırı değerlerde önemsiz beta katsayıları sorunlarından kaçınılır. G-tersinin ampirik gösterimi ve çoklu doğrusal regresyonlar ve Kanonik regresyonlar için uzantılar da verilmiştir. Önerilen dönüşüm, iki değişken arasında mükemmel korelasyon tanıtmanın yeni bir yöntemidir. Kavramın çoklu doğrusal regresyonlarda ve kanonik regresyonda genişletilmesi, çeşitli bilim dallarındaki ampirik araştırmalarda uzun bir yol kat edecektir. Gelecekteki çalışmalar, önerilen mükemmel korelasyonların dağılımını bulmayı ve nicel kanıtlar sağlayarak önerilen yaklaşımımızın etkinliğinin diğer geleneksel yaklaşımlarla karşılaştırılmasını içerebilir.

Keywords

• Korelasyon katsayısı
• Genelleştirilmiş ters
• Çoklu doğrusal regresyonlar
• Kanonik regresyon
• Normal dağılım

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