A Watanabe-Kraskov variable agglomeration tree. Variables are agglomerated (or "unitized") from the bottom-up, with each merge-node representing a (constructed) variable having entropy equal to the joint entropy of the agglomerating variables. Thus, the agglomeration of two m-ary variables 
  
    
      
        
          X
          
            1
          
        
      
    
    {\displaystyle X_{1}}
  
 and 
  
    
      
        
          X
          
            2
          
        
      
    
    {\displaystyle X_{2}}
  
 having individual entropies 
  
    
      
        H
        (
        
          X
          
            1
          
        
        )
      
    
    {\displaystyle H(X_{1})}
  
 and 
  
    
      
        H
        (
        
          X
          
            2
          
        
        )
      
    
    {\displaystyle H(X_{2})}
  
 yields a single 
  
    
      
        
          m
          
            2
          
        
      
    
    {\displaystyle m^{2}}
  
-ary variable 
  
    
      
        
          X
          
            1
            ,
            2
          
        
      
    
    {\displaystyle X_{1,2}}
  
 with entropy 
  
    
      
        H
        (
        
          X
          
            1
            ,
            2
          
        
        )
        =
        H
        (
        
          X
          
            1
          
        
        ,
        
          X
          
            2
          
        
        )
      
    
    {\displaystyle H(X_{1,2})=H(X_{1},X_{2})}
  
. When 
  
    
      
        
          X
          
            1
          
        
      
    
    {\displaystyle X_{1}}
  
 and 
  
    
      
        
          X
          
            2
          
        
      
    
    {\displaystyle X_{2}}
  
 are highly dependent (i.e., redundant) and have large mutual information 
  
    
      
        I
        (
        
          X
          
            1
          
        
        ;
        
          X
          
            2
          
        
        )
      
    
    {\displaystyle I(X_{1};X_{2})}
  
, then 
  
    
      
        H
        (
        
          X
          
            1
            ,
            2
          
        
        )
      
    
    {\displaystyle H(X_{1,2})}
  
 ≪ 
  
    
      
        H
        (
        
          X
          
            1
          
        
        )
        +
        H
        (
        
          X
          
            2
          
        
        )
      
    
    {\displaystyle H(X_{1})+H(X_{2})}
  
 because 
  
    
      
        H
        (
        
          X
          
            1
          
        
        ,
        
          X
          
            2
          
        
        )
        =
        H
        (
        
          X
          
            1
          
        
        )
        +
        H
        (
        
          X
          
            2
          
        
        )
        −
        I
        (
        
          X
          
            1
          
        
        ;
        
          X
          
            2
          
        
        )
      
    
    {\displaystyle H(X_{1},X_{2})=H(X_{1})+H(X_{2})-I(X_{1};X_{2})}
  
, and this would be considered a parsimonious unitization or aggregation.
English: A Watanabe-Kraskov variable aggregation tree.