Online calculator: Time Series Autocorrelation function (ACF)

Autocorrelation, also known as serial correlation, is the correlation of a signal with a delayed copy of itself as a function of delay. Informally, it is the similarity between observations as a function of the time lag between them. Autocorrelation analysis is a mathematical tool for finding repeating patterns, such as the presence of a periodic signal obscured by noise or identifying the missing fundamental frequency in a signal implied by its harmonic frequencies. It is often used in signal processing for analyzing functions or series of values, such as time-domain signals.

In statistics, the autocorrelation of a random process is the Pearson correlation between the process's values at different times, as a function of the two times or of the time lag. ¹

The sample Pearson correlation coefficient between x and y is:
$r=\frac{\sum_{i=1}^{N}(x_i - \bar x)(y_i- \bar y)}{\sqrt{\sum_{i=1}^{N}(x_i - \bar x)^2}\sqrt{\sum_{i=1}^{N}(y_i - \bar y)^2}}$

For autocorrelation, this coefficient is computed between the time series and the same time series lagged by specified number of periods. For example, for 1-period time lag, the correlation coefficient is computed between first N-1 values, i.e. $x_1, ..., x_{N-1}$ and next N-1 values (values shifted by one), i.e. $x_2, ..., x_N$ .
$r_1=\frac{\sum_{i=1}^{N-1}(x_i - \bar{x}_{(1)})(x_{i+1}- \bar{x}_{(2)})}{\sqrt{\sum_{i=1}^{N-1}(x_i - \bar{x}_{(2)})^2}\sqrt{\sum_{i=1}^{N-1}(x_{i+1} - \bar{x}_{(2)})^2}}$ ,
where $\bar{x}_{(1)}$ is the mean of the first N-1 values, and $\bar{x}_{(2)}$ is the mean of the last N-1 values.

If we ignore difference between $\bar{x}_{(1)}$ and $\bar{x}_{(2)}$ , we can simplify the formula above to
$r_1=\frac{\sum_{i=1}^{N-1}(x_i - \bar{x})(x_{i+1}- \bar{x})}{\sum_{i=1}^{N}(x_i - \bar{x})^2}$

This can be generalized for values separated by k periods as:
$r_k=\frac{\sum_{i=1}^{N-k}(x_i - \bar{x})(x_{i+k}- \bar{x})}{\sum_{i=1}^{N}(x_i - \bar{x})^2}$

The value of $r_k$ is called the autocorrelation coefficient at lag $k$ . The plot of the sample autocorrelations $r_k$ versus $k$ (the time lags) is called the correlogram or autocorrelation plot.

The correlogram is a commonly used tool for checking randomness in a data set. Computing autocorrelations ascertain this randomness for data values at varying time lags. If random, such 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.

Also, correlograms are used in the model identification stage for Box–Jenkins autoregressive moving average time 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.²

The default data for the calculator below is obtained by noising sine function using Noisy function generator, and you can clearly see a non-random pattern.

Time Series Autocorrelation function

Time Series

	Time	Value

Calculation precision

Digits after the decimal point: 2

Time Series

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Correlogram

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Time Series Autocorrelation function (ACF)

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Time Series Autocorrelation function

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Time Series Autocorrelation function

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