Cross-correlation

In signal processing, cross-correlation is a measure of similarity of two series as a function of the displacement of one relative to the other. This is also known as a sliding dot product or sliding inner-product. It is commonly used for searching a long signal for a shorter, known feature. It has applications in pattern recognition, single particle analysis, electron tomography, averaging, cryptanalysis, and neurophysiology. The cross-correlation is similar in nature to the convolution of two functions. In an autocorrelation, which is the cross-correlation of a signal with itself, there will always be a peak at a lag of zero, and its size will be the signal energy. In probability and statistics, the term cross-correlations refers to the correlations between the entries of two random vectors and , while the correlations of a random vector are the correlations between the entries of itself, those forming the correlation matrix of . If each of and is a scalar random variable which is realized repeatedly in a time series, then the correlations of the various temporal instances of are known as autocorrelations of , and the cross-correlations of with across time are temporal cross-correlations. In probability and statistics, the definition of correlation always includes a standardising factor in such a way that correlations have values between −1 and +1. If and are two independent random variables with probability density functions and , respectively, then the probability density of the difference is formally given by the cross-correlation (in the signal-processing sense) ; however, this terminology is not used in probability and statistics. In contrast, the convolution (equivalent to the cross-correlation of and ) gives the probability density function of the sum . For continuous functions and , the cross-correlation is defined as:which is equivalent towhere denotes the complex conjugate of , and is called displacement or lag. For highly-correlated and which have a maximum cross-correlation at a particular , a feature in at also occurs later in at , hence could be described to lag by .