Autocorrelation

Correlation and covariance of random vectorsCorrelation and covariance of stochastic processesCorrelation and covariance of deterministic signalsAutocorrelation, 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. 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. R X X ⁡ ( t 1 , t 2 ) = E ⁡ [ X t 1 X t 2 ¯ ] {displaystyle operatorname {R} _{XX}(t_{1},t_{2})=operatorname {E} }     (Eq.1) K X X ⁡ ( t 1 , t 2 ) = E ⁡ [ ( X t 1 − μ t 1 ) ( X t 2 − μ t 2 ) ¯ ] = E ⁡ [ X t 1 X t 2 ¯ ] − μ t 1 μ t 2 ¯ {displaystyle operatorname {K} _{XX}(t_{1},t_{2})=operatorname {E} =operatorname {E} -mu _{t_{1}}{overline {mu _{t_{2}}}}}     (Eq.2) R X X ⁡ ( τ ) = E ⁡ [ X t X t + τ ¯ ] {displaystyle operatorname {R} _{XX}( au )=operatorname {E} }     (Eq.3) K X X ⁡ ( τ ) = E ⁡ [ ( X t − μ ) ( X t + τ − μ ) ¯ ] = E ⁡ [ X t X t + τ ¯ ] − μ μ ¯ {displaystyle operatorname {K} _{XX}( au )=operatorname {E} =operatorname {E} -mu {overline {mu }}}     (Eq.4) R X X ≜   E ⁡ [ X X T ] {displaystyle operatorname {R} _{mathbf {X} mathbf {X} } riangleq operatorname {E} }     (Eq.5) R f f ( τ ) = ∫ − ∞ ∞ f ( t + τ ) f ( t ) ¯ d t = ∫ − ∞ ∞ f ( t ) f ( t − τ ) ¯ d t {displaystyle R_{ff}( au )=int _{-infty }^{infty }f(t+ au ){overline {f(t)}},{ m {d}}t=int _{-infty }^{infty }f(t){overline {f(t- au )}},{ m {d}}t}     (Eq.6) R y y ( ℓ ) = ∑ n ∈ Z y ( n ) y ( n − ℓ ) ¯ {displaystyle R_{yy}(ell )=sum _{nin Z}y(n),{overline {y(n-ell )}}}     (Eq.7) 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. 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. 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. Unit root processes, trend stationary processes, autoregressive processes, and moving average processes are specific forms of processes with autocorrelation. In statistics, the autocorrelation of a real or complex random process is the Pearson correlation between values of the process at different times, as a function of the two times or of the time lag . Let { X t } {displaystyle left{X_{t} ight}} be a random process, and t {displaystyle t} be any point in time ( t {displaystyle t} may be an integer for a discrete-time process or a real number for a continuous-time process). Then X t {displaystyle X_{t}} is the value (or realization) produced by a given run of the process at time t {displaystyle t} . Suppose that the process has mean μ t {displaystyle mu _{t}} and variance σ t 2 {displaystyle sigma _{t}^{2}} at time t {displaystyle t} , for each t {displaystyle t} . Then the definition of the auto-correlation function between times t 1 {displaystyle t_{1}} and t 2 {displaystyle t_{2}} is:p.388:p.165 where E {displaystyle operatorname {E} } is the expected value operator and the bar represents complex conjugation. Note that the expectation may be not well defined. Subtracting the mean before multiplication yields the auto-covariance function between times t 1 {displaystyle t_{1}} and t 2 {displaystyle t_{2}} ::p.392:p.168