Lagged correlation - Revision history

Guillaume at 10:26, 8 March 2012

2012-03-08T10:26:19Z

Guillaume at 10:25, 8 March 2012

2012-03-08T10:25:18Z

Guillaume: 1 revision

2008-12-03T10:27:11Z

1 revision

192.168.20.177: /* Matlab functions */

2008-08-28T10:25:21Z

‎Matlab functions

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[[Convolution]] or [[cross-correlation]] are similar operands that can be applied to any square-integrable real functions <mathtex>f</mathtex> and <mathtex>g</mathtex>.

==Convolution==

: <mathtex> \left( f \, * \, g \right)(\tau) \equiv \int_{-\infty}^{\infty} f(t) \, g(\tau-t) \,dt </mathtex>.

==Cross-correlation==

: <mathtex> \left( f \, \star \, g \right)(\tau) \equiv \int_{-\infty}^{\infty} f^{*}(t) \, g(\tau+t) \,dt \equiv C_{fg}(\tau)</mathtex>.

: In particular <mathtex> f \, \star \, f </mathtex> is designated the [http://mathworld.wolfram.com/Autocorrelation.html autocorrelation].

===Normalized===

: <mathtex>\widehat{C}_{fg}(\tau) \equiv \frac{\int_{-\infty}^{\infty} F^{*}(t) \, G(\tau+t) \,dt}{\int_{-\infty}^{\infty} F^{*}(t) \, F(t) \,dt \, \int_{-\infty}^{\infty} G^{*}(t) \, G(t) \,dt} = \left< \frac{F(t)}{\|F\|} ,\, \frac{G(\tau+t)}{\|G\|} \right></mathtex>, where <mathtex> F \equiv f - \overline{f} </mathtex>. The idea is to remove the average and, optionally, other linear or physical trends from <mathtex> f </mathtex>.

: In particular we have

: <mathtex>\widehat{C}_{fg}(0) = \left< \frac{F}{\|F\|} ,\, \frac{G}{\|G\|} \right></mathtex>.

: <mathtex>\widehat{C}_{ff}(0) = 1 </mathtex>.

===Discretized===

: <mathtex>\widehat{C}_{fg}(k) \equiv \frac{1}{n-1} \sum_{i} \frac{ \left( f_i - \overline{f} \right)^{*} \left( g_{k+i} - \overline{g} \right) }{\sigma_f \sigma_g} </mathtex>, where <mathtex>\sigma_f</mathtex> is the variance of <mathtex>f</mathtex>.

: In particular <mathtex> \widehat{C}_{fg}(0) </mathtex> is the famous [http://en.wikipedia.org/wiki/Correlation correlation] coefficient.

==Example==

When both <mathtex>f</mathtex> and <mathtex>g</mathtex> are [http://en.wikipedia.org/wiki/Even_and_odd_functions even] functions then the [[convolution]] and [[cross-correlation]] products are rigorously equivalent. Below, we show these products when applied to box functions (left figure) and to gaussian functions (right figure). In both cases, they are [http://en.wikipedia.org/wiki/Even_and_odd_functions even] functions. <mathtex>f</mathtex> is represented by the red line, <mathtex>g</mathtex> is represented by the blue line, the product <mathtex>f(t) \, g(\tau - t)</mathtex> is represented by the solid blue line. The solid blue area is the convolution product for lag-time <mathtex>\tau</mathtex> and the whole convolution product function (for all <mathtex>\tau</mathtex>) is represented by the green line.

http://mathworld.wolfram.com/images/gifs/convrect.gif http://mathworld.wolfram.com/images/gifs/convgaus.gif

The big conclusions we can make is that if <mathtex>f</mathtex> and <mathtex>g</mathtex> are the same functions, but with a lag of <mathtex>\epsilon</mathtex> between them, then the [[cross-correlation]] product is maximum when <mathtex>\tau = \epsilon</mathtex>.

==Matlab functions==

===xcorr===
Calculates the cross correlation from two signals.

%Consider arrays t, y1(t), y2(t)
N=length(t);
tlag = linspace(-t(N/2), t(N/2), N+1);
crosscorr = xcorr( y1, y2, N/2, 'coeff');
plot(tlag,crosscorr); xlabel('Phase in time units');ylabel('coefficient');

===cpsd===
Calculates the cross power spectral density (PSD) function of signals <mathtex> x(t)</mathtex> and <mathtex> y(t)</mathtex>.

===periodogram or pwelch===
Calculates the power spectral density (PSD) function of signal <mathtex> x(t)</mathtex>. The [[#periodogram|periodogram]] may use several window functions as option. The [[#pwelch|pwelch]] will use the welch window.

===mscoherence===
Calculates the coherence-squared spectrum function. If you apply an inverse Fourier transform (IFT), then you get the cross-correlation function in a very efficient way.

==See also==
*[http://mathworld.wolfram.com/Cross-Correlation.html Cross-correlation in MathWorld],
*[http://mathworld.wolfram.com/Convolution.html Convolution in MathWorld],
*[http://en.wikipedia.org/wiki/Phase_correlation Phase correlation on Wikipedia]

@@ Line 17: / Line 17: @@
 : In particular we have
-: <mathtex>\widehat{C}_{fg}(0) = \left< \frac{F}{\|F\|} ,\, \frac{G}{\|G\|} \right></mathtex>.
+: <math>\widehat{C}_{fg}(0) = \left< \frac{F}{\|F\|} ,\, \frac{G}{\|G\|} \right></math>.
-: <mathtex>\widehat{C}_{ff}(0) = 1 </mathtex>.
+: <math>\widehat{C}_{ff}(0) = 1 </math>.
 ===Discretized===

@@ Line 1: / Line 1: @@
-[[Convolution]] or [[cross-correlation]] are similar operands that can be applied to any square-integrable real functions <mathtex>f</mathtex> and <mathtex>g</mathtex>.
+[[Convolution]] or [[cross-correlation]] are similar operands that can be applied to any square-integrable real functions <math>f</math> and <math>g</math>.
 ==Convolution==
-: <mathtex> \left( f \, * \, g \right)(\tau) \equiv \int_{-\infty}^{\infty} f(t) \, g(\tau-t) \,dt  </mathtex>.
+: <math> \left( f \, * \, g \right)(\tau) \equiv \int_{-\infty}^{\infty} f(t) \, g(\tau-t) \,dt  </math>.
 ==Cross-correlation==
-: <mathtex> \left( f \, \star \, g \right)(\tau) \equiv \int_{-\infty}^{\infty} f^{*}(t) \, g(\tau+t) \,dt \equiv C_{fg}(\tau)</mathtex>.
+: <math> \left( f \, \star \, g \right)(\tau) \equiv \int_{-\infty}^{\infty} f^{*}(t) \, g(\tau+t) \,dt \equiv C_{fg}(\tau)</math>.
-: In particular <mathtex> f \, \star \, f </mathtex> is designated the [http://mathworld.wolfram.com/Autocorrelation.html autocorrelation].
+: In particular <math> f \, \star \, f </math> is designated the [http://mathworld.wolfram.com/Autocorrelation.html autocorrelation].
 ===Normalized===
-: <mathtex>\widehat{C}_{fg}(\tau) \equiv \frac{\int_{-\infty}^{\infty} F^{*}(t) \, G(\tau+t) \,dt}{\int_{-\infty}^{\infty} F^{*}(t) \, F(t) \,dt \, \int_{-\infty}^{\infty} G^{*}(t) \, G(t) \,dt} = \left< \frac{F(t)}{\|F\|} ,\, \frac{G(\tau+t)}{\|G\|} \right></mathtex>, where <mathtex> F \equiv f - \overline{f} </mathtex>. The idea is to remove the average and, optionally, other linear or physical trends from <mathtex> f </mathtex>.
+: <math>\widehat{C}_{fg}(\tau) \equiv \frac{\int_{-\infty}^{\infty} F^{*}(t) \, G(\tau+t) \,dt}{\int_{-\infty}^{\infty} F^{*}(t) \, F(t) \,dt \, \int_{-\infty}^{\infty} G^{*}(t) \, G(t) \,dt} = \left< \frac{F(t)}{\|F\|} ,\, \frac{G(\tau+t)}{\|G\|} \right></math>, where <math> F \equiv f - \overline{f} </math>. The idea is to remove the average and, optionally, other linear or physical trends from <math> f </math>.
 : In particular we have
@@ Line 23: / Line 23: @@
 ===Discretized===
-: <mathtex>\widehat{C}_{fg}(k) \equiv \frac{1}{n-1} \sum_{i} \frac{ \left( f_i - \overline{f} \right)^{*} \left( g_{k+i} - \overline{g} \right) }{\sigma_f \sigma_g} </mathtex>, where <mathtex>\sigma_f</mathtex> is the variance of <mathtex>f</mathtex>.
+: <math>\widehat{C}_{fg}(k) \equiv \frac{1}{n-1} \sum_{i} \frac{ \left( f_i - \overline{f} \right)^{*} \left( g_{k+i} - \overline{g} \right) }{\sigma_f \sigma_g} </math>, where <math>\sigma_f</math> is the variance of <math>f</math>.
-: In particular <mathtex> \widehat{C}_{fg}(0) </mathtex> is the famous [http://en.wikipedia.org/wiki/Correlation correlation] coefficient.
+: In particular <math> \widehat{C}_{fg}(0) </math> is the famous [http://en.wikipedia.org/wiki/Correlation correlation] coefficient.
 ==Example==
-When both <mathtex>f</mathtex> and <mathtex>g</mathtex> are [http://en.wikipedia.org/wiki/Even_and_odd_functions even] functions then the [[convolution]] and [[cross-correlation]] products are rigorously equivalent. Below, we show these products when applied to box functions (left figure) and to gaussian functions (right figure). In both cases, they are [http://en.wikipedia.org/wiki/Even_and_odd_functions even] functions. <mathtex>f</mathtex> is represented by the red line, <mathtex>g</mathtex> is represented by the blue line, the product <mathtex>f(t) \, g(\tau - t)</mathtex> is represented by the solid blue line. The solid blue area is the convolution product for lag-time <mathtex>\tau</mathtex> and the whole convolution product function (for all <mathtex>\tau</mathtex>) is represented by the green line.
+When both <math>f</math> and <math>g</math> are [http://en.wikipedia.org/wiki/Even_and_odd_functions even] functions then the [[convolution]] and [[cross-correlation]] products are rigorously equivalent. Below, we show these products when applied to box functions (left figure) and to gaussian functions (right figure). In both cases, they are [http://en.wikipedia.org/wiki/Even_and_odd_functions even] functions. <math>f</math> is represented by the red line, <math>g</math> is represented by the blue line, the product <math>f(t) \, g(\tau - t)</math> is represented by the solid blue line. The solid blue area is the convolution product for lag-time <math>\tau</math> and the whole convolution product function (for all <math>\tau</math>) is represented by the green line.
 http://mathworld.wolfram.com/images/gifs/convrect.gif http://mathworld.wolfram.com/images/gifs/convgaus.gif
-The big conclusions we can make is that if <mathtex>f</mathtex> and <mathtex>g</mathtex> are the same functions, but with a lag of <mathtex>\epsilon</mathtex> between them, then the [[cross-correlation]] product is maximum when <mathtex>\tau = \epsilon</mathtex>.
+The big conclusions we can make is that if <math>f</math> and <math>g</math> are the same functions, but with a lag of <math>\epsilon</math> between them, then the [[cross-correlation]] product is maximum when <math>\tau = \epsilon</math>.
 ==Matlab functions==
@@ Line 47: / Line 47: @@
 ===cpsd===
-Calculates the cross power spectral density (PSD) function of signals <mathtex> x(t)</mathtex> and <mathtex> y(t)</mathtex>.
+Calculates the cross power spectral density (PSD) function of signals <math> x(t)</math> and <math> y(t)</math>.
 ===periodogram or pwelch===
-Calculates the power spectral density (PSD) function of signal <mathtex> x(t)</mathtex>. The [[#periodogram|periodogram]] may use several window functions as option. The [[#pwelch|pwelch]] will use the welch window.
+Calculates the power spectral density (PSD) function of signal <math> x(t)</math>. The [[#periodogram|periodogram]] may use several window functions as option. The [[#pwelch|pwelch]] will use the welch window.
 ===mscoherence===

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