One More Geometric Interpretation of Pearson’s Correlation
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 = is to the y-unit vector ⃗ey = 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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