Auto-Correlation Measure

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An Auto-Correlation Measure is a cross-correlation measure on the same sequential data (typically a timeseries).

References

2016

• (Wikipedia, 2016) ⇒ http://wikipedia.org/wiki/autocorrelation Retrieved:2016-4-1.
• Autocorrelation, also known as serial correlation or cross-autocorrelation, is the cross-correlation of a signal with itself at different points in time (that is what the cross stands for). Informally, it is the similarity between observations as a function of the time lag between them. It 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.

2008

• (Upton & Cook, 2008) ⇒ Graham Upton, and Ian Cook. (2008). “A Dictionary of Statistics, 2nd edition revised." Oxford University Press. ISBN:0199541450
• QUOTE: Autocorrelation: A measure of the linear relationship between two separate instances of the same random variable, as distinct from correlation, which refers to the linear relationship between two random variables. As with correlation the possible values lie between - 1 and 1 inclusive, with unrelated instances having a theoretical autocorrelation of 0. In the case of a time series, autocorrelation measures the extent of the linear relation between values at time points that are a listed interval (the lag) apart. Similarly, spatial autocorrelation quantifies the linear relationship between values at points in space that are a fixed distance apart (in any direction in the case of an *isotropic process). It is usually found that spatial autocorrelation is near 1 for points close togetherand decays to 0 as the distance increases - thus the daily rainfalls at the Lords and Oval cricket grounds in London will resemble each other closely, but will bear little or no resemblance to the rainfalls at the Kensington Oval in the West Indies. For a random variable X at time (or location) t, the population autocorrelation function (ACF) for lag $l, p_l$, is given by