+1. Alternate estimate Some sources may use the following formula for the autocovariance function: 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. Statistical inference with correlograms In the same graph one can draw upper and lower bounds for autocorrelation with significance level : with as the estimated autocorrelation at lag . 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 . This test is an approximate one and assumes that the time-series is Gaussian. In the above, z1−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: for 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: 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: 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 Lag plot Spectral plot Seasonal subseries plot Scaled Correlation Variogram References Further reading External links Autocorrelation Plot Statistical charts and diagrams Autocorrelation