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| In [[signal processing]], '''cross-correlation''' is a measure of similarity of two [[waveforms]] as a function of a time-lag applied to one of them. 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 tomographic averaging, [[cryptanalysis]], and [[neurophysiology]].
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| For continuous functions ''f'' and ''g'', the cross-correlation is defined as''':'''
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| : <math>(f \star g)(t)\ \stackrel{\mathrm{def}}{=} \int_{-\infty}^{\infty} f^*(\tau)\ g(\tau+t)\,d\tau,</math>
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| where ''f''* denotes the [[complex conjugate]] of ''f'' and ''t'' is the time lag.
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| Similarly, for discrete functions, the cross-correlation is defined as''':'''
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| : <math>(f \star g)[n]\ \stackrel{\mathrm{def}}{=} \sum_{m=-\infty}^{\infty} f^*[m]\ g[m+n].</math>
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| [[File:Comparison convolution correlation.svg|thumb|300px|Visual comparison of [[convolution]], cross-correlation and [[autocorrelation]].]]
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| The cross-correlation is similar in nature to the [[convolution]] of two functions.
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| In an [[autocorrelation]], which is the cross-correlation of a signal with itself, there will always be a peak at a lag of zero unless the signal is a trivial zero signal.
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| In [[probability theory]] and [[statistics]], ''correlation'' is always used to include a standardising factor in such a way that correlations have values between −1 and +1, and the term '''cross-correlation''' is used for referring to the [[covariance and correlation|correlation]] corr(''X'', ''Y'') between two [[random variables]] ''X'' and ''Y'', while the "correlation" of a random vector ''X'' is considered to be the [[correlation matrix]] (matrix of correlations) between the scalar elements of ''X''.
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| If <math>X</math> and <math>Y</math> are two [[independent (probability)|independent]] [[random variable]]s with [[probability density function]]s ''f'' and ''g'', respectively, then the probability density of the difference <math>Y - X</math> is formally given by the cross-correlation (in the signal-processing sense) <math>f \star g</math>; however this terminology is not used in probability and statistics. In contrast, the [[convolution]] <math>f * g</math> (equivalent to the cross-correlation of ''f''(''t'') and ''g''(−''t'') ) gives the probability density function of the sum <math>X + Y</math>.
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| ==Explanation==
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| As an example, consider two real valued functions <math>f</math> and <math>g</math> differing only by an unknown shift along the x-axis. One can use the cross-correlation to find how much <math>g</math> must be shifted along the x-axis to make it identical to <math>f</math>. The formula essentially slides the <math>g</math> function along the x-axis, calculating the integral of their product at each position. When the functions match, the value of <math>(f\star g)</math> 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.
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| With [[complex-valued function]]s <math>f</math> and <math>g</math>, taking the [[Complex conjugate|conjugate]] of <math>f</math> ensures that aligned peaks (or aligned troughs) with imaginary components will contribute positively to the integral.
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| In [[econometrics]], lagged cross-correlation is sometimes referred to as cross-autocorrelation.<ref>{{cite book |last=Campbell |last2=Lo |last3=MacKinlay |year=1996 |title=The Econometrics of Financial Markets |location=NJ |publisher=Princeton University Press |isbn=0691043019 }}</ref>
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| ==Properties==
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| * The cross-correlation of functions ''f''(''t'') and ''g''(''t'') is equivalent to the [[convolution]] of ''f''*(−''t'') and ''g''(''t''). I.e.''':'''
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| :: <math>f\star g = f^*(-t)*g.</math>
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| * If ''f'' is [[Hermitian function|Hermitian]], then <math>f\star g = f*g.</math>
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| * <math>(f\star g)\star(f\star g)=(f\star f)\star (g\star g)</math>
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| * Analogous to the [[convolution theorem]], the cross-correlation satisfies''':'''
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| :: <math>\mathcal{F}\{f\star g\}=(\mathcal{F}\{f\})^* \cdot \mathcal{F}\{g\},</math>
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| where <math>\mathcal{F}</math> denotes the [[Fourier transform]], and an asterisk again indicates the complex conjugate. Coupled with [[fast Fourier transform]] algorithms, this property is often exploited for the efficient numerical computation of cross-correlations. (see [[Discrete_Fourier_transform#Circular_convolution_theorem_and_cross-correlation_theorem|circular cross-correlation]])
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| * The cross-correlation is related to the [[spectral density]]. (see [[Wiener–Khinchin theorem]])
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| * The cross correlation of a convolution of ''f'' and ''h'' with a function ''g'' is the convolution of the cross-correlation of ''f'' and ''g'' with the kernel ''h'':
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| :: <math>(f * h) \star g = h(-)*(f \star g)</math>
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| ==Time series analysis==
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| In [[time series analysis]], as applied in [[statistics]] and [[signal processing]], the cross correlation between two time series describes the normalized cross covariance function.
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| Let <math>(X_t,Y_t)</math> represent a pair of [[stochastic process]]es that are jointly [[wide sense stationary]]. Then the cross correlation is given by
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| : <math>\gamma_{xy}(\tau) = \operatorname{E}[(X_t - \mu_x)(Y_{t+\tau} - \mu_y)],</math>
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| where <math>\mu_x</math> and <math>\mu_y</math> are the means of <math>X_t</math> and <math>Y_t</math> respectively.
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| The cross correlation of a pair of jointly [[wide sense stationary]] [[stochastic process]] 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.
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| ==Time delay analysis==
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| '''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.<ref>{{cite journal|last=Rhudy|first=Matthew|coauthors=Brian Bucci, Jeffrey Vipperman, Jeffrey Allanach, and Bruce Abraham|title=Microphone Array Analysis Methods Using Cross-Correlations|journal=Proceedings of 2009 ASME International Mechanical Engineering Congress, Lake Buena Vista, FL|year=2009|month=November}}</ref><ref>{{cite journal|last=Rhudy|first=Matthew|title=Real Time Implementation of a Military Impulse Classifier|journal=University of Pittsburgh, Master's Thesis|year=2009|month=November}}</ref> 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
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| : <math>\tau_\mathrm{delay}=\underset{t}{\operatorname{arg\,max}}((f \star g)(t))</math>
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| ==Normalized cross-correlation==
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| 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, <math>t(x,y)</math> with a subimage <math>f(x,y)</math> is
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| : <math>\frac{1}{n} \sum_{x,y}\frac{(f(x,y) - \overline{f})(t(x,y) - \overline{t})}{\sigma_f \sigma_t}</math>.
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| where <math>n</math> is the number of pixels in <math>t(x,y)</math> and <math>f(x,y)</math>,
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| <math>\overline{f}</math> is the average of ''f'' and
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| <math>\sigma_f</math> is [[standard deviation]] of ''f''.
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| In [[functional analysis]] terms, this can be thought of as the dot product of two [[Unit vector|normalized vectors]]. That is, if
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| : <math>F(x,y) = f(x,y) - \overline{f}</math>
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| and
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| : <math>T(x,y) = t(x,y) - \overline{t}</math>
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| then the above sum is equal to
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| : <math>\left\langle\frac{F}{\|F\|},\frac{T}{\|T\|}\right\rangle</math>
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| where <math>\langle\cdot,\cdot\rangle</math> is the [[inner product]] and <math>\|\cdot\|</math> is the [[Lp space|''L''² norm]].
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| 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.
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| Normalized correlation is one of the methods used for [[template matching]], a process used for finding incidences of a pattern or object within an image. It is also the 2-dimensional version of [[Pearson product-moment correlation coefficient]].
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| ==Nonlinear systems==
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| Caution must be applied when using cross correlation 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.<ref name="SAB1">Billings S.A. "Nonlinear System Identification: NARMAX Methods in the Time, Frequency, and Spatio-Temporal Domains". Wiley, 2013</ref> This problem arises because some moments can go to zero and this can incorrectly suggest that there is little correlation between two signals when in fact the two signals are strongly related by nonlinear dynamics.
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| ==See also==
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| * [[Autocorrelation]]
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| * [[Autocovariance]]
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| * [[Coherence (signal processing)]]
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| * [[Convolution]]
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| * [[Correlation]]
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| * [[Cross-covariance]]
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| * [[Cross-spectrum]]
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| * [[Digital image correlation]]
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| * [[Phase correlation]]
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| * [[Scaled correlation]]
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| * [[Spectral density]]
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| * [[Wiener–Khinchin theorem]]
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| ==References==
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| {{Reflist}}
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| ==Further reading==
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| *{{cite journal |last=Tahmasebi |first=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 }}
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| ==External links==
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| * [http://mathworld.wolfram.com/Cross-Correlation.html Cross Correlation from Mathworld]
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| * http://scribblethink.org/Work/nvisionInterface/nip.html
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| * http://www.phys.ufl.edu/LIGO/stochastic/sign05.pdf
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| * http://www.staff.ncl.ac.uk/oliver.hinton/eee305/Chapter6.pdf
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| {{Statistics|analysis}}
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| [[Category:Bilinear operators]]
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| [[Category:Covariance and correlation]]
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| [[Category:Signal processing]]
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| [[Category:Time domain analysis]]
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Live Cattle, like corn is a terrific market for beginning commodity traders. The meats complex also includes Pork Bellies, Lean Hogs and Feeder Cattle. Here's some hints and kinks to provide you off to be able to good start trading the meats!
We have covered a few basic reasons for having selling a home, consequently they are important for consider inside your research. Surely we strongly suggest you learn more about the parties. We know built terrific which can aid you in pursuit for help. Getting a high altitude snapshot in order to be of immense benefit a person. But we now kept mindful yourself . for last, and search for know whatever we mean as soon as you've got read along.
If attempt to landscape while budgeting, remember a person simply can finish a project in stages. In fact, it can often a good idea to break your project up into different steps and even seasons. This is easier attain this in financial terms. Write each component of the process down and select ones possess important to complete first.
But today, it's systems. Really bright fellows with slide rules (well, maybe little calculators now) and degrees decide what will be the "buy area" and sell area for a basket of stocks, depending all styles of parameters, quite a few of which you'd never learn of. For instance are generally three basic programs made to kick in when the futures get too raised. Some kick in by analyzing the number of foreign currency that expertise bank contains. (why? as the currencies fluctuate, they buy and sell stocks as hedges against the currency) Some programs are tied to interest swaps, some to interest rate derivatives, and so forth.
You'd a bit surprised how splitting a bone . get always their old, musty shower curtains, but this could be the first thing a buyer will notice upon entering the shower. Shower curtains are relatively cheap and also they can complete the room appear brighter.
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