# Autocorrelation Time

### Summary

In time-series analysis, auto-correlation time refers to the degree of correlation between the values of the same variables across different observations in the data or by time lags.

For example, one might expect the air temperature on the 1st day of the month to be more similar to the temperature on the second day compared to the 31st day. If the temperature values that occurred closer together in time are, in-fact, more similar than the temperature values that occurred farther apart in time, the data would be auto-correlated.

• It is a mathematical representation of the degree of similarity between a given time series and a lagged version of itself over succesive time intervals.

• It is given by Box and Jenkins in 1976. The purposes of this are:

1. To detect non-randomness in data
2. To identify an appropriate time series model if the data are not random.
• Given measurements $X_{1}, Y_{2}, \ldots, Y_{N}$ at time $T_{1}, T_{2}. \ldots, T_{N}$, and let $k$ be the lag then the autocorrelation function is defined as \begin{align*} r_{k} = \frac{\sum_{i=1}^{N-k} (X_{i} - \bar{X}) (X_{i+k} - \bar{X})}{\sum_{i=1}^{N} (X_{i} - \bar{X})^{2}} \end{align*} Note that X_{1} means X_{1}(T_{1}). Auto-correlation is a correlation coefficient. When the Auto-correlation is used to detect non-randomness, it is Usually only the first (lag 1) Auto-correlation that is of interest. Randomness is one of the key assumptions in determing if a univariate statistical process is in control.

• It can be used to answer the following questions:

1. Was the sample data set generated from a random process?
2. Would a non-linear or time series model be a more appropriate model for these data than a simple constant plus error model?
• In Monte-Carlo simulation based on statistical ensemble, the system (or configuration) should be in the thermally equilibrium. If you measure some observable in the thermally equilibrium system, the observable are purely independent i.e. If you look at the autocorrelation plot then, the future will depend upon the past data.

• All Monte-carlo has the same general structure: given some probability measure $\pi$ on some configuration space $S$. we wish to generate many random samples from $\pi$.

• Auto-correlation ($\tau$) is the number of steps that are needed before the chain “forgets” where it started.

Published on Aug 8, 2021

Last revised on Jan 18, 2023

References