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Wikipedia / 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, while the correlations itself, those forming the correlation matrix is a scalar random variable which is realized repeatedly in a time series are known as autocorrelations 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.

are two independent random variables with probability density functions; however, this terminology is not used in probability and statistics. In contrast, the convolution.

Cross-correlation of deterministic signals

, the cross-correlation is defined as: denotes the complex conjugate is called displacement or lag.

could be described to lag.

Similarly, for discrete functions, the cross-correlation is defined as:, the (circular) cross-correlation is defined as:, the kernel cross-correlation is defined as: is an affine transform.

can be circular translation transform, rotation transform, or scale transform, etc. The kernel cross-correlation extends cross-correlation from linear space to kernel space. Cross-correlation is equivariant to translation; kernel cross-correlation is equivariant to any affine transforms, including translation, rotation, and scale, etc.

Explanation

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.

With complex-valued functions, taking the conjugate 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.

Properties

Cross-correlation of random vectors

Definition

For random vectors, each containing random elements whose expected value and variance exist, the cross-correlation matrix is defined by is the expectation value.

Example

.

Definition for complex random vectors

are complex random vectors denotes Hermitian transposition.

Cross-correlation of stochastic processes

In time series analysis and statistics, the cross-correlation of a pair of random process may be an integer for a discrete-time process or a real number for a continuous-time is the value (or realization.

Cross-correlation function

is is the expected value operator. Note that this expression may be not defined.

Cross-covariance function

:Note that this expression is not well-defined for all time series or processes, because the mean or variance may not exist.

Definition for wide-sense stationary stochastic process

represent a pair of stochastic processes that are jointly wide-sense stationary. Then the cross-covariance function (not absolute time), and are given as follows.

Cross-correlation function

This definition is used by Newland. Note that some references including Gubner, which is equivalent to reversing the sign of the time lag and (for complex-valued processes) taking the complex conjugate. The resulting spectrum will be the complex conjugate of one obtained based on the other convention.

Cross-covariance function

indicates the expected value are jointly wide-sense stationary.

The cross-correlation of a pair of jointly wide sense stationary stochastic processes can be estimated by averaging the product of samples measured from one process and samples measured from the other (and its time shifts). The samples included in the average can be an arbitrary subset of all the samples in the signal (e.g., samples within a finite time window or a sub-sampling of one of the signals). For a large number of samples, the average converges to the true cross-correlation.

Normalization

It is common practice in some disciplines (e.g. statistics and time series analysis) to normalize the cross-correlation function to get a time-dependent Pearson correlation coefficient. However, in other disciplines (e.g. engineering) the normalization is usually dropped and the terms "cross-correlation" and "cross-covariance" are used interchangeably.

, with 1 indicating perfect correlation and −1 indicating perfect anti-correlation.

The normalization is important both because the interpretation of the autocorrelation as a correlation provides a scale-free measure of the strength of statistical dependence, and because the normalization has an effect on the statistical properties of the estimated autocorrelations.

Properties

Symmetry property

For jointly wide-sense stationary stochastic processes, the cross-correlation function has the following symmetry property:

Time delay analysis

Cross-correlations are useful for determining the time delay between two signals, e.g., for determining time delays for the propagation of acoustic signals across a microphone array. After calculating the cross-correlation between the two signals, the maximum (or minimum if the signals are negatively correlated) of the cross-correlation function indicates the point in time where the signals are best aligned; i.e., the time delay between the two signals is determined by the argument of the maximum, or arg max of the cross-correlation, as in

Terminology in image processing

Zero-normalized cross-correlation (ZNCC)

For image-processing applications in which the brightness of the image and template can vary due to lighting and exposure conditions, the images can be first normalized. This is typically done at every step by subtracting the mean and dividing by the standard deviation is

is standard deviation.

In functional analysis terms, this can be thought of as the dot product of two normalized vectors is the inner product is the L² norm. Cauchy–Schwarz.

multiplied by a positive scalar.

Normalized correlation is one of the methods used for template matching, a process used for finding instances of a pattern or object within an image. It is also the 2-dimensional version of Pearson product-moment correlation coefficient.

Normalized cross-correlation (NCC)

Nonlinear systems

Caution must be applied when using cross correlation function which assumes Gaussian variance for nonlinear systems. In certain circumstances, which depend on the properties of the input, cross correlation between the input and output of a system with nonlinear dynamics can be completely blind to certain nonlinear effects. This problem arises because some quadratic moments can equal zero and this can incorrectly suggest that there is little "correlation" (in the sense of statistical dependence) between two signals, when in fact the two signals are strongly related by nonlinear dynamics.

See also

References

Further reading