Autocorrelation of a sequence of vectors

Question

Let's say I have a sequence of 2-d vectors and I want to calculate autocorrelation of this sequence of vectors. If $V_i$ where i = 1:n is the list of vectors then acf as a function of time lag 't' is given by c(t) = $\sum_{i=1}^{n-t} ( V_i . V_{i+t} )$ My question is should these vectors be subtracted by the average vector ? For scalars one usually divides the list of values by the standard deviation after subtracting them by the mean of values. What is the equivalent of that in case of vectors ? Thank you

Answer 1

For vectors $\mathbf{x}$ and $\mathbf{y}$, $$ \text{Corr}(\mathbf{x},\mathbf{y}) = \frac{(\mathbf{x}-\mu_x\mathbf{1})\cdot(\mathbf{y}-\mu_y\mathbf{1})}{\|\mathbf{x}-\bar{x}\mathbf{1}\|_2\|\mathbf{y}-\bar{y}\mathbf{1}\|_2} $$ where $\mu_x,\mu_y$ are the population means, and $\bar{x}, \bar{y}$ are the sample means. Think of scalars as 1-D vectors.