Correlation sum

In <a href="/facts/Chaos_theory/LTC17vsF">chaos theory</a>, the correlation sum is the estimator of the <a href="/facts/Correlation_integral/pXfSVTXr">correlation integral</a>, which reflects the mean probability that the states at two different times are close:

C
        (
        ε
        )
        =
        
          
            1
            
              N
              
                2
              
            
          
        
        
          ∑
          
            
              
                
                  i
                  ≠
                  j
                
                
                  i
                  ,
                  j
                  =
                  1
                
              
            
          
          
            N
          
        
        Θ
        (
        ε
        −
        ‖
        
          
            
              x
              →
            
          
        
        (
        i
        )
        −
        
          
            
              x
              →
            
          
        
        (
        j
        )
        ‖
        )
        ,
        
        
          
            
              x
              →
            
          
        
        (
        i
        )
        ∈
        
          
            R
          
          
            m
          
        
        ,
      
    
    {\displaystyle C(\varepsilon )={\frac {1}{N^{2}}}\sum _{\stackrel {i,j=1}{i\neq j}}^{N}\Theta (\varepsilon -\|{\vec {x}}(i)-{\vec {x}}(j)\|),\quad {\vec {x}}(i)\in \mathbb {R} ^{m},}

where 
 
 
 
 N
 
 
 {\displaystyle N}
 
 is the number of considered states 
 
 
 
 
 
 
 x
 →
 
 
 
 (
 i
 )
 
 
 {\displaystyle {\vec {x}}(i)}
 
, 
 
 
 
 ε
 
 
 {\displaystyle \varepsilon }
 
 is a threshold distance, 
 
 
 
 ‖
 ⋅
 ‖
 
 
 {\displaystyle \|\cdot \|}
 
 a norm (e.g. <a href="/facts/Euclidean_norm/R2UbzmzM">Euclidean norm</a>) and 
 
 
 
 Θ
 (
 ⋅
 )
 
 
 {\displaystyle \Theta (\cdot )}
 
 the <a href="/facts/Heaviside_step_function/bgsUtxgP">Heaviside step function</a>. If only a <a href="/facts/Time_series/fSXPR817">time series</a> is available, the phase space can be reconstructed by using a time delay embedding (see <a href="/facts/Takens%2527_theorem/9U2aDkG1">Takens' theorem</a>):

x
              →
            
          
        
        (
        i
        )
        =
        (
        u
        (
        i
        )
        ,
        u
        (
        i
        +
        τ
        )
        ,
        …
        ,
        u
        (
        i
        +
        τ
        (
        m
        −
        1
        )
        )
        ,
      
    
    {\displaystyle {\vec {x}}(i)=(u(i),u(i+\tau ),\ldots ,u(i+\tau (m-1)),}

where 
 
 
 
 u
 (
 i
 )
 
 
 {\displaystyle u(i)}
 
 is the time series, 
 
 
 
 m
 
 
 {\displaystyle m}
 
 the embedding dimension and 
 
 
 
 τ
 
 
 {\displaystyle \tau }
 
 the time delay.
The correlation sum is used to estimate the <a href="/facts/Correlation_dimension/4Syu0Y30">correlation dimension</a>.

Correlation sum open-in-new

Correlation sum