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[[File:Comparison_convolution_correlation.svg|Comparison convolution correlation|thumb]] [[File:Cross_correlation_animation.gif|Cross correlation animation|thumb|left]] [[File:Cross_Correlation_Animation.gif|Cross Correlation Animation|thumb|left]] '''Cross-correlation''' is a statistical method used to measure the similarity between two [[time series]] by calculating the correlation of one series with another, as one is shifted in time relative to the other. This technique is widely used in various fields such as [[signal processing]], [[time series analysis]], [[finance]], and [[image processing]]. Cross-correlation helps in identifying the time lag between two time-dependent signals, which is essential for understanding the relationship between them.

==Definition==
Cross-correlation is defined for two real-valued functions, \(f(t)\) and \(g(t)\), as the integral of the product of \(f(t)\) and a shifted version of \(g(t)\) over all time. Mathematically, the cross-correlation \(R_{fg}(\tau)\) at lag \(\tau\) is given by:

\[ R_{fg}(\tau) = \int_{-\infty}^{\infty} f(t) \cdot g(t + \tau) \, dt \]

where \(t\) represents time, and \(\tau\) represents the lag.

For discrete functions or [[time series]], the cross-correlation is similarly defined as the sum over the product of \(f(t)\) and \(g(t + \tau)\) for all time points, which can be represented as:

\[ R_{fg}(\tau) = \sum_{t} f(t) \cdot g(t + \tau) \]

==Applications==
Cross-correlation has a wide range of applications across different domains:

* In [[signal processing]], it is used to find the time delay between two signals, which is crucial for [[echo]] detection and synchronization.
* In [[time series analysis]], cross-correlation helps in identifying the lead-lag relationships between two time series, which is useful in economic and financial analysis.
* In [[finance]], it is applied to compare the returns of different [[financial instruments]] or market indices to identify potential [[investment]] opportunities or risks.
* In [[image processing]], cross-correlation is used for [[template matching]], where a small image or template is located within a larger image.

==Properties==
Cross-correlation has several important properties:

* Symmetry: The cross-correlation \(R_{fg}(\tau)\) is not necessarily symmetric; \(R_{fg}(\tau) \neq R_{gf}(\tau)\) in general, which means the cross-correlation of \(f\) with \(g\) is not the same as \(g\) with \(f\) when \(\tau\) is not zero.
* Normalization: To compare cross-correlations between different pairs of signals, it is often useful to normalize the cross-correlation function, so that the values range between -1 and 1.
* Maximum at zero lag: If two signals are identical, their cross-correlation will reach its maximum value at zero lag, indicating perfect correlation.

==Limitations==
While cross-correlation is a powerful tool, it has limitations:

* It assumes linear relationships between the time series and may not capture nonlinear interactions.
* The presence of noise in the signals can significantly affect the accuracy of the cross-correlation measurement.
* It does not provide information about the causality between the two time series.

==See Also==
* [[Autocorrelation]]
* [[Convolution]]
* [[Correlation coefficient]]
* [[Signal processing]]

[[Category:Statistics]]
[[Category:Signal processing]]
[[Category:Time series analysis]]

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