One More Geometric Interpretation of Pearson’s Correlation

  • Ildar Batyrshin Centro de Investigacion en Computacion (CIC) Instituto Politecnico Nacional (IPN), Mexico, D.F.
  • Vladik Kreinovich Department of Computer Science University of Texas at El Paso, El Paso, TX 79968, USA

Abstract

It is known that Pearson’s correlation coefficient r is equal to the cosine of the angle between the vectors ⃗x and ⃗y describing the centered data. We use this known fact to show that r describes how closer the corresponding unit vector ⃗ex = \frac{\vec{x}}{\left \|\vec{x}\right \|} is to the y-unit vector ⃗ey\frac{\vec{y}}{\left \| \vec{y} \right \|} than to its opposite −⃗ey: namely, r is equal to the (scaled) difference between the squared distances d2(⃗ex, −⃗ey) and d2(⃗ex, ⃗ey). In particular, the correlation is positive if ⃗ex is closer to ⃗ey and negative if ⃗ex is closer to −⃗ey.

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