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Autocorrelation
  CATEGORIES ABOUT AUTOCORRELATION
   
covariance and correlation
   
fourier analysis
   
regression analysis
   
signal processing
   
time series analysis
   
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DEFINITIONS


Different definitions of autocorrelation are in use depending on the field of study which is being considered and not all of them are equivalent. In some fields, the term is used interchangeably with Autocovariance .


Statistics


In Statistics , the autocorrelation of a discrete Time Series or a process ''X''''t'' is simply the Correlation of the process against a time-shifted version of itself. If ''X''''t'' is Second-order Stationary with mean μ then this definition is

:R(k) = rac{E - \mu)(X_{i+k} - \mu) }{\sigma^2}

where E is the Expected Value and ''k'' is the time shift being considered (usually referred to as the lag). This function has the attractive property of being in the range {Link without Title} with 1 indicating perfect correlation (the signals exactly overlap when time shifted by ''k'') and −1 indicating perfect anti-correlation. It is common practice in many disciplines to drop the normalisation by σ2 and use the term ''autocorrelation'' interchangeably with ''autocovariance''.


Signal processing


In signal processing, given a signal f(t), the continuous autocorrelation ''R''''f''(τ) is the continuous cross-correlation of ''f''(''t'') with itself, at lag ''τ'', and is defined as:

  • (- au) \circ f( au) = \int_{-\infty}^{\infty} f(t+ au)f^---(t)\, dt = \int_{-\infty}^{\infty} f(t)f^---(t- au)\, dt



Formally, the discrete autocorrelation ''R'' at lag ''j'' for signal ''xn'' is


:R(j) = \sum_n (x_n-m)(x_{n-j}-m )\,

where ''m'' is the Average value ( Expected Value ) of ''xn''. Quite frequently, autocorrelations are calculated for zero-centered signals, that is, for signals with zero mean. The autocorrelation definition then becomes


:R(j) = \sum_n x_n x_{n-j}.\,

Multi- Dimension al autocorrelation is defined similarly. For example, in three dimensions the autocorrelation would be defined as


:R(j,k,\ell) = \sum_{n,q,r} (x_{n,q,r}-m)(x_{n-j,q-k,r-\ell}-m).


PROPERTIES


In the following, we will describe properties of one-dimensional autocorrelations only, since most properties are easily transferred from the one-dimensional case to the multi-dimensional cases.

  • A fundamental property of the autocorrelation is symmetry, ''R''(''i'') = ''R''(−''i''), which is easy to prove from the definition. In the continuous case, the autocorrelation is an even function


::R_f(- au) = R_f( au)\,

:when ''f'' is a real function and the autocorrelation is a Hermitian function

  • ( au)\,


:when ''f'' is a complex function.