Cross-correlation

Comparison convolution correlation

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Cross correlation animation

Cross Correlation Animation

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