Editing Cross-correlation (section)

===Explanation===
As an example, consider two real valued functions <math>f</math> and <math>g</math> differing only by an unknown shift along the x-axis. One can use the cross-correlation to find how much <math>g</math>  must be shifted along the x-axis to make it identical to <math>f</math>. The formula essentially slides the <math>g</math> function along the x-axis, calculating the integral of their product at each position. When the functions match, the value of <math>(f\star g)</math> is maximized. This is because when peaks (positive areas) are aligned, they make a large contribution to the integral. Similarly, when troughs (negative areas) align, they also make a positive contribution to the integral because the product of two negative numbers is positive.

[[File:Cross correlation animation.gif|center|thumb|500x500px|Animation of how cross-correlation is calculated. The left graph shows a green function G that is phase-shifted relative to function F by a time displacement of 𝜏. The middle graph shows the function F and the phase-shifted G represented together as a [[Lissajous curve]]. Integrating F multiplied by the phase-shifted G produces the right graph, the cross-correlation across all values of 𝜏.]]

With [[complex-valued function]]s <math>f</math> and <math>g</math>, taking the [[Complex conjugate|conjugate]] of <math>f</math> ensures that aligned peaks (or aligned troughs) with imaginary components will contribute positively to the integral.

In [[econometrics]], lagged cross-correlation is sometimes referred to as cross-autocorrelation.<ref>{{cite book |last1=Campbell |last2=Lo |last3=MacKinlay |year=1996 |title=The Econometrics of Financial Markets |location=NJ |publisher=Princeton University Press |isbn=0691043019 }}</ref>{{rp|p. 74}}