Cosines and correlation

correlationcosinelaw-of-cosinesstatisticsgeometry

Abstraction: Correlation as geometric cosine angle between random variable vectors

Key points:

For independent X and Y: Var(X+Y) = Var(X) + Var(Y), analogous to Pythagorean theorem c^2 = a^2 + b^2
For correlated X and Y: Var(X+Y) = Var(X) + Var(Y) + 2*Cov(X,Y), analogous to law of cosines with cos(theta) = -rho
Restated with supplementary angle phi: cos(phi) = rho = Corr(X,Y); phi near 0 means high positive correlation, near pi means high negative
In an inner product space where <X,Y> = E(XY), correlation is literally the cosine of the angle between X and Y as vectors
Standard deviations of independent variables add like sides of a right triangle; correlated variables add like law of cosines

Connections: Correlation · Cosine Similarity · Inner Product · Probability Theory