Vector's Correlation

Description:

• Consider 2 De-meaned Vector $\tilde{a},\tilde{b}$ from vector $a,b$
• The corresponding Correlation Coefficient is the cosine angle between the 2 de-meaned vectors:
  • $\rho (a,b):=\cos \angle (\tilde{a},\tilde{b})=\frac{{\tilde{a}}^T\tilde{b}}{||\tilde{a}|{|}_2||\tilde{b}|{|}_2}\in [-1,1]$

Uncertainty of vector $x$:

• We can model uncertainty by the way of set $U$, we say $x\in U$, otherwise unknown.
• To model the lack of knowledge about a vector, we can use the uncertainty set, $U=\{x:||x-\hat{x}||\le \rho \}$
  • where $\hat{x}$ is refered to the “nomial” model
  • $\rho$ meansures the size of uncertainty around $\hat{x}$