Covariance and correlation

<h2 id="multiple-random-variables">Multiple random variables</h2>
With any number of random variables in excess of 1, the variables can be stacked into a <a href="/facts/Random_vector/qMfooyVf">random vector</a> whose i th element is the i th random variable. Then the variances and covariances can be placed in a <a href="/facts/Covariance_matrix/kLhgjHC6">covariance matrix</a>, in which the (i, j) element is the covariance between the i th random variable and the j th one. Likewise, the correlations can be placed in a <a href="/facts/Correlation_matrix/egkluAEm">correlation matrix</a>.

<h2 id="time-series-analysis">Time series analysis</h2>
In the case of a <a href="/facts/Time_series/fSXPR817">time series</a> which is <a href="/facts/Stationary_process/vozdgfEX">stationary</a> in the wide sense, both the means and variances are constant over time (E(Xn+m) = E(Xn) = μX and var(Xn+m) = var(Xn) and likewise for the variable Y). In this case the cross-covariance and cross-correlation are functions of the time difference:

<a href="/facts/Cross-covariance/nBl7HhGY">cross-covariance</a>

σ
          
            X
            Y
          
        
        (
        m
        )
        =
        E
        [
        (
        
          X
          
            n
          
        
        −
        
          μ
          
            X
          
        
        )
        
        (
        
          Y
          
            n
            +
            m
          
        
        −
        
          μ
          
            Y
          
        
        )
        ]
        ,
      
    
    {\displaystyle \sigma _{XY}(m)=E[(X_{n}-\mu _{X})\,(Y_{n+m}-\mu _{Y})],}

<a href="/facts/Cross-correlation/Fs1hY3br">cross-correlation</a>

ρ
          
            X
            Y
          
        
        (
        m
        )
        =
        E
        [
        (
        
          X
          
            n
          
        
        −
        
          μ
          
            X
          
        
        )
        
        (
        
          Y
          
            n
            +
            m
          
        
        −
        
          μ
          
            Y
          
        
        )
        ]
        
          /
        
        (
        
          σ
          
            X
          
        
        
          σ
          
            Y
          
        
        )
        .
      
    
    {\displaystyle \rho _{XY}(m)=E[(X_{n}-\mu _{X})\,(Y_{n+m}-\mu _{Y})]/(\sigma _{X}\sigma _{Y}).}

If Y is the same variable as X, the above expressions are called the autocovariance and autocorrelation:

<a href="/facts/Autocovariance/3uM1sVVV">autocovariance</a>

σ
          
            X
            X
          
        
        (
        m
        )
        =
        E
        [
        (
        
          X
          
            n
          
        
        −
        
          μ
          
            X
          
        
        )
        
        (
        
          X
          
            n
            +
            m
          
        
        −
        
          μ
          
            X
          
        
        )
        ]
        ,
      
    
    {\displaystyle \sigma _{XX}(m)=E[(X_{n}-\mu _{X})\,(X_{n+m}-\mu _{X})],}

<a href="/facts/Autocorrelation/dm4MqK0A">autocorrelation</a>

ρ
          
            X
            X
          
        
        (
        m
        )
        =
        E
        [
        (
        
          X
          
            n
          
        
        −
        
          μ
          
            X
          
        
        )
        
        (
        
          X
          
            n
            +
            m
          
        
        −
        
          μ
          
            X
          
        
        )
        ]
        
          /
        
        (
        
          σ
          
            X
          
          
            2
          
        
        )
        .
      
    
    {\displaystyle \rho _{XX}(m)=E[(X_{n}-\mu _{X})\,(X_{n+m}-\mu _{X})]/(\sigma _{X}^{2}).}

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Weisstein, Eric W. "Covariance". MathWorld. <a href="/wiki/Eric_W._Weisstein" target="_blank">/wiki/Eric_W._Weisstein</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Weisstein, Eric W. "Statistical Correlation". MathWorld. <a href="/wiki/Eric_W._Weisstein" target="_blank">/wiki/Eric_W._Weisstein</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
</ol>

Covariance and correlation open-in-new

Covariance and correlation