In the analysis of data, acorrelogram is achart ofcorrelation statistics. For example, intime series analysis, a plot of the sampleautocorrelations${\displaystyle r_h}$ versus${\displaystyle h}$ (the time lags) is anautocorrelogram. Ifcross-correlation is plotted, the result is called across-correlogram.

The correlogram is a commonly used tool for checkingrandomness in adata set. If random, autocorrelations should be near zero for any and all time-lag separations. If non-random, then one or more of the autocorrelations will be significantly non-zero.

In addition, correlograms are used in themodel identification stage forBox–Jenkinsautoregressive moving averagetime series models. Autocorrelations should be near-zero for randomness; if the analyst does not check for randomness, then the validity of many of the statistical conclusions becomes suspect. The correlogram is an excellent way of checking for such randomness.

Inmultivariate analysis,correlation matrices shown ascolor-mapped images may also be called "correlograms" or "corrgrams".^[1][2][3]

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 serieswhite 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${\displaystyle s_{\overline{Y}}=s/\sqrt{N}}$ 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 standardstatistical 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 thestandard error of the sample mean:

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

wheres is thestandard 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 lagh is given by

${\displaystyle r_h=c_h/c_0}$

wherec_h is theautocovariance function

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

andc₀ is thevariance function

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

The resulting value ofr_h will range between −1 and +1.

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

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

Although this definition has lessbias, 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

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.

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 isGaussian.

In the above,z_1−α/2 is the quantile of thenormal distribution; SE is the standard error, which can be computed byBartlett'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 thenull hypothesis that there is no autocorrelation between time-points which are separated by lags up to 4. For most longer periods one cannot reject thenull 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 notime dependence in the data), the following formula is recommended:

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

whereN is thesample size,z is thequantile function of thestandard normal distribution and α is thesignificance 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 fittingARIMA models. In this case, amoving average model is assumed for the data and the following confidence bands should be generated:

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

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

Correlograms are available in most general purpose statistical libraries.

Correlograms:

• pythonpandas:pandas.plotting.autocorrelation_plot^[5]
• R: functionsacf andpacf

Corrgrams:

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

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

• Hanke, John E.; Reitsch, Arthur G.; Wichern, Dean W.Business forecasting (7th ed.). Upper Saddle River, NJ: Prentice Hall.
• Box, G. E. P.; Jenkins, G. (1976).Time Series Analysis: Forecasting and Control. Holden-Day.
• Chatfield, C. (1989).The Analysis of Time Series: An Introduction (Fourth ed.). New York, NY: Chapman & Hall.

• Autocorrelation Plot

 This article incorporatespublic domain material from the National Institute of Standards and Technology

• Statistical charts and diagrams
• Autocorrelation

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