Cfrgtkky

Suppose $v$ is an $n$-ary vector with entries from the set ${0,1}$ (i.e. a vector of ones and zeros).

A paper I am reading defines the "auto-correlation sequences" $$v*v$$ where $*$ denotes the correlation operator.

1) What is an auto-correlation sequence of a vector?

2) What is the correlation operator? (I'm assuming it can be applied to two distinct vectors too)

My first guess was that to auto-correlate a vector you try all the possible rotational permutations of the vector and measure the cosine of the angle between each permuted vector with the original. However, Mathematica's CorrelationFunction on ${1,0}$ with $lag=0$ returns 1 and with $lag=1$ returns $-frac{1}{2}$, which shoots down my theory since I would expect orthogonal vectors to have $0$ correlation. So what is Mathematica doing here?

The sample correlation of vectors $(X_1, dots, X_n)$ and
$(Y_1, dots, Y_n)$ is

$$rho_{(X,Y)} = frac{frac{1}{n-1}sum_{i=1}^n (X_i - bar X)(Y_i - bar Y) }{S_XS_Y},$$
where $bar X, bar Y$ are the respective sample means and $S_XS_Y$ are the respective sample standard deviations.

Roughly speaking, the sample autocorrelation of lag $ell$ of a vector $(X_1, dots X_n)$ is the sample correlation of the
vector $(X_1, dots, X_{n-ell})$ and and the lagged vector
$(X_ell, X_{ell + 1}, dots, X_n).$

Various refinements are used in specific applications.
Perhaps the one you are looking for is of the following
form:

$$rho_ell = frac{sum_{i=1}^{n-ell} (X_1 - bar X)(X_ell - bar X) }{(n-1)S_X^2},$$
Notice that $bar X$ and $S_X^2$ are based on the
entire sequence. Also, when $ell=0,$ we have $rho_ell = 1.$
See Wikipedia
at the last bullet under Estimation.

As I recall, this is used in the R function acf:

set.seed(1115)

x = round(rnorm(10,200,15))-20*(1:10); x

 [1] 176 169 127  99  96  92  45  70  10  12

acf(x)

acf(x, plot=F)



Autocorrelations of series ‘x’, by lag



     0      1      2      3      4      5      6      7      8      9 

 1.000  0.625  0.299  0.138 -0.060 -0.177 -0.304 -0.363 -0.434 -0.223