## Autocorrelation

Above: A plot of a series of 100 random numbers concealing a sine function. Below: The sine function revealed in a correlogram produced by autocorrelation.

Visual comparison of convolution, cross-correlation and autocorrelation.

Autocorrelation, also known as serial correlation, is the correlation of a signal with itself at different points in time. Informally, it is the similarity between observations as a function of the time lag between them. The analysis of autocorrelation 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.

Unit root processes, trend stationary processes, autoregressive processes, and moving average processes are specific forms of processes with autocorrelation.

Contents

1 Definitions

1.1 Statistics

1.2 Signal processing

2 Properties

3 Efficient computation

4 Estimation

5 Regression analysis

6 Applications

7 Serial dependence

8 See also

9 References

10 Further reading

11 External links

Definitions[edit]

Different fields of study define autocorrelation differently, and not all of these definitions are equivalent. In some fields, the term is used interchangeably with autocovariance.

Statistics[edit]

In statistics, the autocorrelation of a random process is the correlation between values of the process at different times, as a function of the two times or of the time lag. Let X be a stochastic process, and t be any point in time. (t may be an integer for a discrete-time process or a real number for a continuous-time process.) Then Xt is the value (or realization) produced by a given run of the process at time t. Suppose that the process has mean μt and variance σt2 at time t, for each t. Then the definition of the autocorrelation between times s and t is

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