Correlogram

The correlogram can help provide answers to the following questions:^[4]

• Are the data random?
• Is an observation related to an adjacent observation?
• Is an observation related to an observation twice-removed? (etc.)
• Is the observed time series white noise?
• Is the observed time series sinusoidal?
• Is the observed time series autoregressive?
• What is an appropriate model for the observed time series?
• Is the model

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

valid and sufficient?

• Is the formula ${\displaystyle s_{\bar {Y}}=s/{\sqrt {N}}}$ valid?

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:

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

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.

The autocorrelation coefficient at lag h is given by

${\displaystyle r_{h}=c_{h}/c_{0}\,}$

where c_h is the autocovariance function

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

and c₀ is the variance function

${\displaystyle c_{0}={\frac {1}{N}}\sum _{t=1}^{N}\left(Y_{t}-{\bar {Y}}\right)^{2}}$

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

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

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

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

${\displaystyle X={\begin{bmatrix}Y_{1}-{\bar {Y}}&\cdots &Y_{N}-{\bar {Y}}\end{bmatrix}}\in \mathbb {R} ^{n\times N}}$

We then compute the Gram matrix ${\displaystyle Q=X^{\top }X}$. 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.

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.

Correlograms are available in most general purpose statistical libraries.

Correlograms:

• python pandas: pandas.plotting.autocorrelation_plot^[5]
• R: functions acf and pacf

Corrgrams:

• python seaborn: heatmap, pairplot
• R: corrgram^[2][3]

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