Correlogram

Importance

Randomness (along with fixed model, fixed variation, and fixed distribution) is one of the four assumptions that typically underlie all measurement processes. The randomness assumption is critically important for the following three reasons:

• Most standard statistical tests depend on randomness. The validity of the test conclusions is directly linked to the validity of the randomness assumption.
• Many commonly used statistical formulae depend on the randomness assumption, the most common formula being the formula for determining the standard error of the sample mean:

${\displaystyle s_{\overline{Y}}=s/\sqrt{N}}$

where s is the standard deviation of the data. Although heavily used, the results from using this formula are of no value unless the randomness assumption holds.

• For univariate data, the default model is

${\displaystyle Y=\text{constant}+\text{error}}$

If the data are not random, this model is incorrect and invalid, and the estimates for the parameters (such as the constant) become nonsensical and invalid.

Estimation of autocorrelations

The autocorrelation coefficient at lag h is given by

${\displaystyle r_h=c_h/c_0}$

where c_h is the autocovariance function

${\displaystyle c_h=\frac{1}{N}\sum _{t=1}^{N-h}\left(Y_t-\overline{Y}\right)\left(Y_{t+h}-\overline{Y}\right)}$

and c₀ is the variance function

${\displaystyle c_0=\frac{1}{N}\sum _{t=1}^N{\left(Y_t-\overline{Y}\right)}^2}$

The resulting value of r_h will range between −1 and +1.

Alternate estimate

Some sources may use the following formula for the autocovariance function:

${\displaystyle c_h=\frac{1}{N-h}\sum _{t=1}^{N-h}\left(Y_t-\overline{Y}\right)\left(Y_{t+h}-\overline{Y}\right)}$

Although this definition has less bias, the (1/N) formulation has some desirable statistical properties and is the form most commonly used in the statistics literature. See pages 20 and 49–50 in Chatfield for details.

In contrast to the definition above, this definition allows us to compute ${\displaystyle c_h}$ in a slightly more intuitive way. Consider the sample ${\displaystyle Y_1,\dots ,Y_N}$, where ${\displaystyle Y_i\in {\mathbb{R}}^n}$ for ${\displaystyle i=1,\dots ,N}$. Then, let

${\displaystyle X=\left[\begin{array}{ccc}{Y}_1-\overline{Y}& \cdots & Y_N-\overline{Y}\end{array}\right]\in {\mathbb{R}}^{n\times N}}$

We then compute the Gram matrix ${\displaystyle Q=X^{\mathrm{\top }}X}$. Finally, ${\displaystyle c_h}$ is computed as the sample mean of the ${\displaystyle h}$th diagonal of ${\displaystyle Q}$. For example, the ${\displaystyle 0}$th diagonal (the main diagonal) of ${\displaystyle Q}$ has ${\displaystyle N}$ elements, and its sample mean corresponds to ${\displaystyle c_0}$. The ${\displaystyle 1}$st diagonal (to the right of the main diagonal) of ${\displaystyle Q}$ has ${\displaystyle N-1}$ elements, and its sample mean corresponds to ${\displaystyle c_1}$, and so on.

Statistical inference with correlograms

In the same graph one can draw upper and lower bounds for autocorrelation with significance level ${\displaystyle \alpha }$:

${\displaystyle B=\pm z_{1-\alpha /2}SE(r_h)}$ with ${\displaystyle r_h}$ as the estimated autocorrelation at lag ${\displaystyle h}$.

If the autocorrelation is higher (lower) than this upper (lower) bound, the null hypothesis that there is no autocorrelation at and beyond a given lag is rejected at a significance level of ${\displaystyle \alpha }$. This test is an approximate one and assumes that the time-series is Gaussian.

In the above, z_1−α/2 is the quantile of the normal distribution; SE is the standard error, which can be computed by Bartlett's formula for MA(ℓ) processes:

${\displaystyle SE(r_1)=\frac{1}{\sqrt{N}}}$

${\displaystyle SE(r_h)=\sqrt{\frac{1+2\sum _{i=1}^{h-1}{r}_i^2}{N}}}$ for ${\displaystyle h>1.}$

In the example plotted, we can reject the null hypothesis that there is no autocorrelation between time-points which are separated by lags up to 4. For most longer periods one cannot reject the null hypothesis of no autocorrelation.

Note that there are two distinct formulas for generating the confidence bands:

1. If the correlogram is being used to test for randomness (i.e., there is no time dependence in the data), the following formula is recommended:

${\displaystyle \pm \frac{z_{1-\alpha /2}}{\sqrt{N}}}$

where N is the sample size, z is the quantile function of the standard normal distribution and α is the significance level. In this case, the confidence bands have fixed width that depends on the sample size.

2. Correlograms are also used in the model identification stage for fitting ARIMA models. In this case, a moving average model is assumed for the data and the following confidence bands should be generated:

${\displaystyle \pm z_{1-\alpha /2}\sqrt{\frac{1}{N}\left(1+2\sum _{i=1}^k{r}_i^2\right)}}$

where k is the lag. In this case, the confidence bands increase as the lag increases.

Software

Correlograms are available in most general purpose statistical libraries.

Correlograms:

• python pandas: pandas.plotting.autocorrelation_plot
• R: functions acf and pacf

Corrgrams:

• python seaborn: heatmap, pairplot
• R: corrgram

Related techniques

• Partial autocorrelation function
• Spectral plot
• Seasonal subseries plot
• Scaled Correlation
• Variogram