Autocorrelation

A correlogram, also known as an autocorrelation function, is a statistical tool that describes the correlation of a variable with itself at different points in time or space. It plays a crucial role in understanding the internal structure of time series or spatial data.

What sets correlograms apart from autocovariance functions or covariograms is their normalization. While the autocovariance function and the covariogram measure covariance, the autocorrelation function measures correlation, which is a normalized version of covariance.

The normalization ensures that the autocorrelation function is dimensionless and lies between -1 and 1, thus facilitating comparisons across different datasets or variables.

The autocorrelation function $R(t, s)$ for a lag $h$ (the difference between the two time points $t$ and $s$) can be defined as:

$$R(t, s) = \frac{C(t, s)}{\sqrt{C(t, t) \cdot C(s, s)}},$$

where:

• $C(t, s)$ is the autocovariance function, the expected value of the product of the deviations of a random variable from its expected value at two different time points, $t$ and $s$,
• $C(t, t)$ and $C(s, s)$ are the variances at times $t$ and $s$ respectively.

In essence, a correlogram is a plot of the autocorrelation function against the lag. For a time series, it shows how the autocorrelation changes with the time lag between the points in the series. Similarly, for spatial data, it shows how the autocorrelation changes with the spatial lag, or the distance and direction between the points.

The correlogram provides valuable information about the underlying patterns in a data set, such as cyclic patterns, trends, or seasonality in time series data, or spatial patterns in geostatistical data. It is a fundamental tool in many areas of research, including econometrics, climatology, neuroscience, and many more.