Rational quadratic covariance function

In <a href="/facts/Statistics/DNPsKGYU">statistics</a>, the rational quadratic <a href="/facts/Covariance_function/pqUTwANG">covariance function</a> is used in <a href="/facts/Spatial_statistics/R6rlDzRl">spatial statistics</a>, <a href="/facts/Geostatistics/UeChfgNG">geostatistics</a>, <a href="/facts/Machine_learning/e0w0XJTu">machine learning</a>, image analysis, and other fields where multivariate statistical analysis is conducted on <a href="/facts/Metric_space/gnBBRXtc">metric spaces</a>. It is commonly used to define the statistical <a href="/facts/Covariance/hILocGNH">covariance</a> between measurements made at two points that are d units distant from each other. Since the covariance only depends on distances between points, it is <a href="/facts/Stationary_process/vozdgfEX">stationary</a>. If the distance is <a href="/facts/Euclidean_distance/9qDoQKQe">Euclidean distance</a>, the rational quadratic covariance function is also <a href="/facts/Isotropic/mc3QEY2x">isotropic</a>.
The rational quadratic covariance between two points separated by d distance units is given by

C
        (
        d
        )
        =
        
          
            (
          
        
        1
        +
        
          
            
              d
              
                2
              
            
            
              2
              α
              
                k
                
                  2
                
              
            
          
        
        
          
            
              )
            
          
          
            −
            α
          
        
      
    
    {\displaystyle C(d)={\Bigg (}1+{\frac {d^{2}}{2\alpha k^{2}}}{\Bigg )}^{-\alpha }}

where α and k are non-negative <a href="/facts/Parameter/zFvVFMv9">parameters</a> of the covariance.

Rational quadratic covariance function open-in-new

Rational quadratic covariance function