Language Reference

DISTANCE Function

DISTANCE (x, <, method> );

The DISTANCE function computes the pairwise distances between rows of x. The distances depend on the metric specified by the method argument. The arguments are as follows:

x

specifies an $n\times p$ numerical matrix that contains n points in p-dimensional space.

method

is an optional argument that specifies the method used to specify the distance between pairs of points. The method argument is either a numeric value, method$\geq 1$, or a case-insensitive character value. Only the first four character values are used. The following are valid options:

"L2"

specifies that the function compute the Euclidean ($L_2$) distance between two points. This is the default value. An equivalent alias is "Euclidean".

"L1"

specifies that the function compute the Manhattan ($L_1$) distance between two points. An equivalent alias is "CityBlock" or "Manhattan".

"LInf"

specifies that the function compute the Chebyshev ($L_\infty $) distance between two points. An equivalent alias is "Chebyshev".

p

is a numeric value, $p \geq 1$, that specifies the $L_ p$-norm.

The DISTANCE function returns an $n \times n$ symmetric matrix. The $(i,j)$th element is the distance between the ith and jth rows of x.

If u and v are two p-dimensional points, then the following formulas are used to compute the distance between u and v:

  • The Euclidean distance: $\| u - v \| _2 = (\Sigma _ k |u_ k - v_ k |^2 )^{1/2}$.

  • The $L_1$ distance: $\| u - v \| _1 = \Sigma _ k |u_ k - v_ k |$,

  • The $L_\infty $ distance: $\| u - v \| _\infty = \max (|u_1 - v_1 |,|u_2 - v_2 |,\ldots ,|u_ p - v_ p |)$.

  • The $L_ p$ distance: $\| u - v \| _ p = (\Sigma _ k |u_ k - v_ k |^ p )^{1/p}$.

The following statements illustrate the DISTANCE function: 

x = {1 0,
     0 1,
    -1 0,
     0 -1};
d2 = distance(x, "L2");
print d2[format=best5.];

Figure 24.106: Euclidean Distance Between Pairs of Points

d2
0 1.414 2 1.414
1.414 0 1.414 2
2 1.414 0 1.414
1.414 2 1.414 0 




            

            


            
The ith column of d2 contains the distances between the ith row of x and the other rows. Notice that the d2 matrix has zeros along the diagonal. 
            

            
You can also compute non-Euclidean distances, as follows: 


d1 = distance(x, "L1");
dInf = distance(x, "LInfinity");
print d1, dInf;



   

            


Figure 24.107: Distance Between Pairs of Points

               

                  

                     
                     

                        
                        
                        

                           
                           

                              
                              









d1
                                       
                                    




0
                                       
                                       2
                                       
                                       2
                                       
                                       2
                                       
                                    



2
                                       
                                       0
                                       
                                       2
                                       
                                       2
                                       
                                    



2
                                       
                                       2
                                       
                                       0
                                       
                                       2
                                       
                                    



2
                                       
                                       2
                                       
                                       2
                                       
                                       0
                                       
                                    






                           
                        

                        



                     
                     
                  

                  

                     
                     

                        
                        
                        

                           
                           

                              
                              









dInf
                                       
                                    




0
                                       
                                       1
                                       
                                       2
                                       
                                       1
                                       
                                    



1
                                       
                                       0
                                       
                                       1
                                       
                                       2
                                       
                                    



2
                                       
                                       1
                                       
                                       0
                                       
                                       1
                                       
                                    



1
                                       
                                       2
                                       
                                       1
                                       
                                       0
                                       
                                    






                           
                        

                        



                     
                     
                  





            

            


            
If a row contains a missing value, all distances that involve that row are assigned a missing value.