Cross-correlation#Normalized cross-correlation

For continuous functions $f$ and $g$, the cross-correlation is defined as:Bracewell, R. "Pentagram Notation for Cross Correlation." The Fourier Transform and Its Applications. New York: McGraw-Hill, pp. 46 and 243, 1965.Papoulis, A. The Fourier Integral and Its Applications. New York: McGraw-Hill, pp. 244–245 and 252-253, 1962.Weisstein, Eric W. "Cross-Correlation." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Cross-Correlation.html$$(f\; \backslash star\; g)(\backslash tau)\backslash \; \backslash triangleq\; \backslash int\_\{-\backslash infty\}^\{\backslash infty\}\; \backslash overline\{f(t)\}\; g(t+\backslash tau)\backslash ,dt$$which is equivalent to$$(f\; \backslash star\; g)(\backslash tau)\backslash \; \backslash triangleq\; \backslash int\_\{-\backslash infty\}^\{\backslash infty\}\; \backslash overline\{f(t-\backslash tau)\}\; g(t)\backslash ,dt$$where $\backslash overline\{f(t)\}$ denotes the complex conjugate of $f(t)$, and $\backslash tau$ is called displacement or lag.

For highly-correlated $f$ and $g$ which have a maximum cross-correlation at a particular $\backslash tau$, a feature in $f$ at $t$ also occurs later in $g$ at $t+\backslash tau$, hence $g$ could be described to lag $f$ by $\backslash tau$.

If $f$ and $g$ are both continuous periodic functions of period $T$, the integration from $-\backslash infty$ to $\backslash infty$ is replaced by integration over any interval $[t\_0,t\_0+T]$ of length $T$:$$(f\; \backslash star\; g)(\backslash tau)\backslash \; \backslash triangleq\; \backslash int\_\{t\_0\}^\{t\_0+T\}\; \backslash overline\{f(t)\}\; g(t\; +\; \backslash tau)\backslash ,dt$$which is equivalent to$$(f\; \backslash star\; g)(\backslash tau)\backslash \; \backslash triangleq\; \backslash int\_\{t\_0\}^\{t\_0+T\}\; \backslash overline\{f(t-\backslash tau)\}\; g(t)\backslash ,dt$$Similarly, for discrete functions, the cross-correlation is defined as:{{cite book | last1 =Rabiner | first1 =L.R. | last2 =Schafer | first2 =R.W. | title =Digital Processing of Speech Signals | publisher =Prentice Hall | series =Signal Processing Series | date =1978 | location =Upper Saddle River, NJ | pages =[https://archive.org/details/digitalprocessin00rabi_0/page/147 147–148] | isbn =0132136031 | url-access =registration | url =https://archive.org/details/digitalprocessin00rabi_0/page/147 }}{{cite book | last1 =Rabiner | first1 =Lawrence R. | last2 =Gold | first2 =Bernard | title =Theory and Application of Digital Signal Processing | publisher =Prentice-Hall | date =1975 | location =Englewood Cliffs, NJ | pages =[https://archive.org/details/theoryapplicatio00rabi/page/401 401] | isbn =0139141014 | url-access =registration | url =https://archive.org/details/theoryapplicatio00rabi/page/401 }}$$(f\; \backslash star\; g)[n]\backslash \; \backslash triangleq\; \backslash sum\_\{m=-\backslash infty\}^\{\backslash infty\}\; \backslash overline\{f[m]\}\; g[m+n]$$which is equivalent to:$$(f\; \backslash star\; g)[n]\backslash \; \backslash triangleq\; \backslash sum\_\{m=-\backslash infty\}^\{\backslash infty\}\; \backslash overline\{f[m\; -\; n]\}\; g[m]$$For finite discrete functions $f,g\backslash in\backslash mathbb\{C\}^N$, the (circular) cross-correlation is defined as:{{cite thesis | last1=Wang | first1=Chen | title=Kernel learning for visual perception, Chapter 2.2.1 | publisher =Nanyang Technological University, Singapore | type =Doctoral thesis | date =2019 | pages =[https://hdl.handle.net/10356/105527 17–18] | doi=10.32657/10220/47835 | hdl=10356/105527 | doi-access=free | hdl-access=free }}$$(f\; \backslash star\; g)[n]\backslash \; \backslash triangleq\; \backslash sum\_\{m=0\}^\{N-1\}\; \backslash overline\{f[m]\}\; g[(m+n)\_\{\backslash text\{mod\}~N\}]$$which is equivalent to:$$(f\; \backslash star\; g)[n]\backslash \; \backslash triangleq\; \backslash sum\_\{m=0\}^\{N-1\}\; \backslash overline\{f[(m-n)\_\{\backslash text\{mod\}~N\}]\}\; g[m]$$For finite discrete functions $f\backslash in\backslash mathbb\{C\}^N$, $g\backslash in\backslash mathbb\{C\}^M$, the kernel cross-correlation is defined as:{{cite journal | last1=Wang | first1=Chen | last2=Zhang | first2=Le | last3=Yuan | first3=Junsong | last4=Xie | first4=Lihua | title=Kernel Cross-Correlator | journal=Proceedings of the AAAI Conference on Artificial Intelligence | publisher=Association for the Advancement of Artificial Intelligence | series=The Thirty-second AAAI Conference On Artificial Intelligence | date =2018 | volume=32 | pages =4179–4186 | doi=10.1609/aaai.v32i1.11710 | s2cid=3544911 | url=https://ojs.aaai.org/index.php/AAAI/article/view/11710 | doi-access=free }}$$(f\; \backslash star\; g)[n]\backslash \; \backslash triangleq\; \backslash sum\_\{m=0\}^\{N-1\}\; \backslash overline\{f[m]\}\; K\_g[(m+n)\_\{\backslash text\{mod\}~N\}]$$where $K\_g\; =\; [k(g,\; T\_0(g)),\; k(g,\; T\_1(g)),\; \backslash dots,\; k(g,\; T\_\{N-1\}(g))]$ is a vector of kernel functions $k(\backslash cdot,\; \backslash cdot)\backslash colon\; \backslash mathbb\{C\}^M\; \backslash times\; \backslash mathbb\{C\}^M\; \backslash to\; \backslash mathbb\{R\}$ and $T\_i(\backslash cdot)\backslash colon\; \backslash mathbb\{C\}^M\; \backslash to\; \backslash mathbb\{C\}^M$ is an affine transform.

Specifically, $T\_i(\backslash cdot)$ 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.

As an example, consider two real valued functions $f$ and $g$ differing only by an unknown shift along the x-axis. One can use the cross-correlation to find how much $g$ must be shifted along the x-axis to make it identical to $f$. The formula essentially slides the $g$ function along the x-axis, calculating the integral of their product at each position. When the functions match, the value of $(f\backslash star\; g)$ 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. Integrating F multiplied by the phase-shifted G produces the right graph, the cross-correlation across all values of 𝜏.]]

With complex-valued functions $f$ and $g$, taking the conjugate of $f$ 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.{{cite book |last1=Campbell |last2=Lo |last3=MacKinlay |year=1996 |title=The Econometrics of Financial Markets |location=NJ |publisher=Princeton University Press |isbn=0691043019 }}{{rp|p. 74}}

{{unordered list

| The cross-correlation of functions $f(t)$ and $g(t)$ is equivalent to the convolution (denoted by $*$) of $\backslash overline\{f(-t)\}$ and $g(t)$. That is:

: $[f(t)\; \backslash star\; g(t)](t)\; =\; [\backslash overline\{f(-t)\}\; *\; g(t)](t).$

| $[f(t)\; \backslash star\; g(t)](t)\; =\; [\backslash overline\{g(t)\}\; \backslash star\; \backslash overline\{f(t)\}](-t).$

| If $f$ is a Hermitian function, then $f\; \backslash star\; g\; =\; f\; *\; g.$

| If both $f$ and $g$ are Hermitian, then $f\; \backslash star\; g\; =\; g\; \backslash star\; f$.

| $\backslash left(f\; \backslash star\; g\backslash right)\; \backslash star\; \backslash left(f\; \backslash star\; g\backslash right)\; =\; \backslash left(f\; \backslash star\; f\backslash right)\; \backslash star\; \backslash left(g\; \backslash star\; g\backslash right)$.

| Analogous to the convolution theorem, the cross-correlation satisfies

: $\backslash mathcal\{F\}\backslash left\backslash \{f\; \backslash star\; g\backslash right\backslash \}\; =\; \backslash overline\{\backslash mathcal\{F\}\; \backslash left\backslash \{f\backslash right\backslash \}\}\; \backslash cdot\; \backslash mathcal\{F\}\backslash left\backslash \{g\backslash right\backslash \},$

where $\backslash mathcal\{F\}$ denotes the Fourier transform, and an $\backslash overline\{f\}$ again indicates the complex conjugate of $f$, since $\backslash mathcal\{F\}\backslash left\backslash \{\backslash overline\{f(-t)\}\backslash right\backslash \}=\backslash overline\{\backslash mathcal\{F\}\backslash left\backslash \{f(t)\backslash right\backslash \}\}$. Coupled with fast Fourier transform algorithms, this property is often exploited for the efficient numerical computation of cross-correlations{{cite book|doi=10.1109/ICSPCS.2015.7391783|isbn=978-1-4673-8118-5|chapter = GPU implementation of cross-correlation for image generation in real time|title = 2015 9th International Conference on Signal Processing and Communication Systems (ICSPCS)|pages = 1–6|year = 2015|last1 = Kapinchev|first1 = Konstantin|last2 = Bradu|first2 = Adrian|last3 = Barnes|first3 = Frederick|last4 = Podoleanu|first4 = Adrian|s2cid=17108908 }} (see circular cross-correlation).

| The cross-correlation is related to the spectral density (see Wiener–Khinchin theorem).

| The cross-correlation of a convolution of $f$ and $h$ with a function $g$ is the convolution of the cross-correlation of $g$ and $f$ with the kernel $h$:

: $g\; \backslash star\; \backslash left(f\; *\; h\backslash right)\; =\; \backslash left(g\; \backslash star\; f\backslash right)\; *\; h$.

}}

{{main|Cross-correlation matrix}}

For random vectors $\backslash mathbf\{X\}\; =\; (X\_1,\backslash ldots,X\_m)$ and $\backslash mathbf\{Y\}\; =\; (Y\_1,\backslash ldots,Y\_n)$, each containing random elements whose expected value and variance exist, the cross-correlation matrix of $\backslash mathbf\{X\}$ and $\backslash mathbf\{Y\}$ is defined by{{cite book |first=John A. |last=Gubner |year=2006 |title=Probability and Random Processes for Electrical and Computer Engineers |publisher=Cambridge University Press |isbn=978-0-521-86470-1}}{{rp|p.337}}$$\backslash operatorname\{R\}\_\{\backslash mathbf\{X\}\backslash mathbf\{Y\}\}\; \backslash triangleq\backslash \; \backslash operatorname\{E\}\backslash left[\backslash mathbf\{X\}\; \backslash mathbf\{Y\}\backslash right]$$and has dimensions $m\; \backslash times\; n$. Written component-wise:$$\backslash operatorname\{R\}\_\{\backslash mathbf\{X\}\backslash mathbf\{Y\}\}\; =$$

\begin{bmatrix}

\operatorname{E}[X_1 Y_1] & \operatorname{E}[X_1 Y_2] & \cdots & \operatorname{E}[X_1 Y_n] \\ \\

\operatorname{E}[X_2 Y_1] & \operatorname{E}[X_2 Y_2] & \cdots & \operatorname{E}[X_2 Y_n] \\ \\

\vdots & \vdots & \ddots & \vdots \\ \\

\operatorname{E}[X_m Y_1] & \operatorname{E}[X_m Y_2] & \cdots & \operatorname{E}[X_m Y_n]

\end{bmatrix}

The random vectors $\backslash mathbf\{X\}$ and $\backslash mathbf\{Y\}$ need not have the same dimension, and either might be a scalar value.

Where $\backslash operatorname\{E\}$ is the expectation value.

For example, if $\backslash mathbf\{X\}\; =\; \backslash left(\; X\_1,X\_2,X\_3\; \backslash right)$ and $\backslash mathbf\{Y\}\; =\; \backslash left(\; Y\_1,Y\_2\; \backslash right)$ are random vectors, then $\backslash operatorname\{R\}\_\{\backslash mathbf\{X\}\backslash mathbf\{Y\}\}$ is a $3\; \backslash times\; 2$ matrix whose $(i,j)$-th entry is $\backslash operatorname\{E\}[X\_i\; Y\_j]$.

If $\backslash mathbf\{Z\}\; =\; (Z\_1,\backslash ldots,Z\_m)$ and $\backslash mathbf\{W\}\; =\; (W\_1,\backslash ldots,W\_n)$ are complex random vectors, each containing random variables whose expected value and variance exist, the cross-correlation matrix of $\backslash mathbf\{Z\}$ and $\backslash mathbf\{W\}$ is defined by$$\backslash operatorname\{R\}\_\{\backslash mathbf\{Z\}\backslash mathbf\{W\}\}\; \backslash triangleq\backslash \; \backslash operatorname\{E\}[\backslash mathbf\{Z\}\; \backslash mathbf\{W\}^\{\backslash rm\; H\}]$$where $\{\}^\{\backslash rm\; H\}$ denotes Hermitian transposition.

In time series analysis and statistics, the cross-correlation of a pair of random process is the correlation between values of the processes at different times, as a function of the two times. Let $(X\_t,\; Y\_t)$ be a pair of random processes, and $t$ be any point in time ($t$ may be an integer for a discrete-time process or a real number for a continuous-time process). Then $X\_t$ is the value (or realization) produced by a given run of the process at time $t$.

Suppose that the process has means $\backslash mu\_X(t)$ and $\backslash mu\_Y(t)$ and variances $\backslash sigma\_X^2(t)$ and $\backslash sigma\_Y^2(t)$ at time $t$, for each $t$. Then the definition of the cross-correlation between times $t\_1$ and $t\_2$ is{{rp|p.392}}$$\backslash operatorname\{R\}\_\{XY\}(t\_1,\; t\_2)\; \backslash triangleq\backslash \; \backslash operatorname\{E\}\backslash left[X\_\{t\_1\}\; \backslash overline\{Y\_\{t\_2\}\}\backslash right]$$where $\backslash operatorname\{E\}$ is the expected value operator. Note that this expression may be not defined.

Subtracting the mean before multiplication yields the cross-covariance between times $t\_1$ and $t\_2$:{{rp|p.392}}$$\backslash operatorname\{K\}\_\{XY\}(t\_1,\; t\_2)\; \backslash triangleq\backslash \; \backslash operatorname\{E\}\backslash left[\backslash left(X\_\{t\_1\}\; -\; \backslash mu\_X(t\_1)\backslash right)\backslash overline\{(Y\_\{t\_2\}\; -\; \backslash mu\_Y(t\_2))\}\backslash right]$$Note that this expression is not well-defined for all time series or processes, because the mean or variance may not exist.

Let $(X\_t,\; Y\_t)$ represent a pair of stochastic processes that are jointly wide-sense stationary. Then the cross-covariance function and the cross-correlation function are given as follows.

$$\backslash operatorname\{R\}\_\{XY\}(\backslash tau)\; \backslash triangleq\backslash \; \backslash operatorname\{E\}\backslash left[X\_t\; \backslash overline\{Y\_\{t+\backslash tau\}\}\backslash right]$$ or equivalently $$\backslash operatorname\{R\}\_\{XY\}(\backslash tau)\; =\; \backslash operatorname\{E\}\backslash left[X\_\{t-\backslash tau\}\; \backslash overline\{Y\_\{t\}\}\backslash right]$$

$$\backslash operatorname\{K\}\_\{XY\}(\backslash tau)\; \backslash triangleq\backslash \; \backslash operatorname\{E\}\backslash left[\backslash left(X\_t\; -\; \backslash mu\_X\backslash right)\backslash overline\{\backslash left(Y\_\{t+\backslash tau\}\; -\; \backslash mu\_Y\backslash right)\}\backslash right]$$ or equivalently $$\backslash operatorname\{K\}\_\{XY\}(\backslash tau)\; =\; \backslash operatorname\{E\}\backslash left[\backslash left(X\_\{t-\backslash tau\}\; -\; \backslash mu\_X\backslash right)\backslash overline\{\backslash left(Y\_\{t\}\; -\; \backslash mu\_Y\backslash right)\}\backslash right]$$where $\backslash mu\_X$ and $\backslash sigma\_X$ are the mean and standard deviation of the process $(X\_t)$, which are constant over time due to stationarity; and similarly for $(Y\_t)$, respectively. $\backslash operatorname\{E\}[\backslash \; ]$ indicates the expected value. That the cross-covariance and cross-correlation are independent of $t$ is precisely the additional information (beyond being individually wide-sense stationary) conveyed by the requirement that $(X\_t,\; Y\_t)$ 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{{which|date=May 2015}} of one of the signals). For a large number of samples, the average converges to the true cross-correlation.

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.

The definition of the normalized cross-correlation of a stochastic process is$$$$

\rho_{XX}(t_1, t_2) =

\frac{\operatorname{K}_{XX}(t_1, t_2)}{\sigma_X(t_1)\sigma_X(t_2)} =

\frac{\operatorname{E}\left[\left(X_{t_1} - \mu_{t_1}\right)\overline{\left(X_{t_2} - \mu_{t_2}\right)}\right]}{\sigma_X(t_1)\sigma_X(t_2)}

If the function $\backslash rho\_\{XX\}$ is well-defined, its value must lie in the range $[-1,1]$, with 1 indicating perfect correlation and −1 indicating perfect anti-correlation.

For jointly wide-sense stationary stochastic processes, the definition is$$$$

\rho_{XY}(\tau) =

\frac{\operatorname{K}_{XY}(\tau)}{\sigma_X \sigma_Y} =

\frac{\operatorname{E}\left[\left(X_t - \mu_X\right) \overline{\left(Y_{t+\tau} - \mu_Y\right)}\right]}{\sigma_X \sigma_Y}

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.

For jointly wide-sense stationary stochastic processes, the cross-correlation function has the following symmetry property:Kun Il Park, Fundamentals of Probability and Stochastic Processes with Applications to Communications, Springer, 2018, 978-3-319-68074-3{{rp|p.173}}$$\backslash operatorname\{R\}\_\{XY\}(t\_1,\; t\_2)\; =\; \backslash overline\{\backslash operatorname\{R\}\_\{YX\}(t\_2,\; t\_1)\}$$Respectively for jointly WSS processes:$$\backslash operatorname\{R\}\_\{XY\}(\backslash tau)\; =\; \backslash overline\{\backslash operatorname\{R\}\_\{YX\}(-\backslash tau)\}$$

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.{{cite conference|last=Rhudy|first=Matthew|author2=Brian Bucci |author3=Jeffrey Vipperman |author4=Jeffrey Allanach |author5=Bruce Abraham |title=Microphone Array Analysis Methods Using Cross-Correlations|conference=Proceedings of 2009 ASME International Mechanical Engineering Congress, Lake Buena Vista, FL|pages=281–288|date=November 2009 |doi=10.1115/IMECE2009-10798|isbn=978-0-7918-4388-8}}{{cite thesis |last=Rhudy|first=Matthew|title=Real Time Implementation of a Military Impulse Classifier |type=MS thesis |publisher=University of Pittsburgh |date=November 2009|url=http://d-scholarship.pitt.edu/9773/}}{{clarify|reason=I doubt that this definition is used for microphone arrays, since it involves an integral over all time. Perhaps integration over a time-window?|date=May 2015}} 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$$\backslash tau\_\backslash mathrm\{delay\}=\backslash underset\{t\; \backslash in\; \backslash mathbb\{R\}\}\{\backslash operatorname\{arg\backslash ,max\}\}((f\; \backslash star\; g)(t))$$Terminology in image processing

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. That is, the cross-correlation of a template $t(x,y)$ with a subimage $f(x,y)$ is

$$\backslash frac\{1\}\{n\backslash sigma\_f\; \backslash sigma\_t\}\; \backslash sum\_\{x,y\}\backslash left(f(x,y)\; -\; \backslash mu\_f\; \backslash right)\backslash left(t(x,y)\; -\; \backslash mu\_t\; \backslash right)$$

where $n$ is the number of pixels in $t(x,y)$ and $f(x,y)$,

$\backslash mu\_f$ is the average of $f$ and $\backslash sigma\_f$ is standard deviation of $f$.

In functional analysis terms, this can be thought of as the dot product of two normalized vectors. That is, if$$F(x,y)\; =\; f(x,y)\; -\; \backslash mu\_f$$and$$T(x,y)\; =\; t(x,y)\; -\; \backslash mu\_t$$then the above sum is equal to$$\backslash left\backslash langle\backslash frac\{F\}\{\backslash |F\backslash |\},\backslash frac\{T\}\{\backslash |T\backslash |\}\backslash right\backslash rangle$$where $\backslash langle\backslash cdot,\backslash cdot\backslash rangle$ is the inner product and $\backslash |\backslash cdot\backslash |$ is the L² norm. Cauchy–Schwarz then implies that ZNCC has a range of $[-1,\; 1]$.

Thus, if $f$ and $t$ are real matrices, their normalized cross-correlation equals the cosine of the angle between the unit vectors $F$ and $T$, being thus $1$ if and only if $F$ equals $T$ 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.

NCC is similar to ZNCC with the only difference of not subtracting the local mean value of intensities:$$\backslash frac\{1\}\{n\backslash sigma\_f\; \backslash sigma\_t\}\; \backslash sum\_\{x,y\}f(x,y)\; t(x,y)$$

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.{{cite book |last=Billings |first=S. A. |title=Nonlinear System Identification: NARMAX Methods in the Time, Frequency, and Spatio-Temporal Domains |publisher=Wiley |year=2013 |isbn=978-1-118-53556-1 }} 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.

{{Div col|colwidth=25em}}

• Autocorrelation
• Autocovariance
• Coherence
• Convolution
• Correlation
• Correlation function
• Cross-correlation matrix
• Cross-covariance
• Cross-spectrum
• Digital image correlation
• Phase correlation
• Scaled correlation
• Spectral density
• Wiener–Khinchin theorem

{{div col end}}

{{Reflist|30em}}

• {{cite journal |last1=Tahmasebi |first1=Pejman |last2=Hezarkhani |first2=Ardeshir |last3=Sahimi |first3=Muhammad |year=2012 |title=Multiple-point geostatistical modeling based on the cross-correlation functions |journal=Computational Geosciences |volume=16 |issue=3 |pages=779–797 |doi=10.1007/s10596-012-9287-1 |bibcode=2012CmpGe..16..779T |s2cid=62710397 }}

• [http://mathworld.wolfram.com/Cross-Correlation.html Cross Correlation from Mathworld]
• http://scribblethink.org/Work/nvisionInterface/nip.html
• http://www.staff.ncl.ac.uk/oliver.hinton/eee305/Chapter6.pdf

{{Statistics|analysis}}

Category:Bilinear maps

Category:Covariance and correlation

Category:Signal processing

Category:Time domain analysis