Module ateams.statistics.autocorrelation

Functions

def integrated(X, isNormalized=False)

Computes (an estimate of) the integrated autocorrelation time $\tau_X$ of the observable $X$ by $$\tau_X = \frac 12 \sum_{t=-M}^M \overline \rho_X(t) = \frac 12 + \sum_{t=1}^M \overline \rho_X(t),$$ where $\overline \rho_X(t)$ is the normalized autocorrelation for $X$. Computes $\tau_X$ for all $1 \leq M \leq N-1$ for $N = |X|$ the number of observations.

Args

X : np.array

A NumPy array of numerical values.

isNormalized : boolean=False

Are the data passed to X already normalized autocorrelation times? If not, we first compute $\overline \rho_X$.

Returns

(Estimates of) the integrated autocorrelation times.

def normalized(X)

Computes the vector $\overline \rho_X$ of normalized autocorrelations $\overline \rho_X(t) = \rho_X(t)/\rho_X(0)$ for the observable $X$.

Args

X : np.array

A NumPy array of numerical values.

Returns

A NumPy array of normalized autocorrelation values.

def unnormalized(X)

Computes the vector $\rho_X$ of unnormalized autocorrelations $\rho_X(t)$ for the set of (numerical) observations $X$ by $$\rho_X(t) = \frac{1}{N-|t|} \sum_{i=1}^{N-|t|}(X_i - \overline X)(X_{i+t} - \overline X),$$ where $\overline X$ is the sample mean and $N = |X|$ is the number of samples in the observed data.

Args

X : np.array

A NumPy array of numerical values.

Returns

A NumPy array of unnormalized autocorrelation values.