SciPy – Spatial Distance Matrix

A distance matrix contains the distances computed pairwise between the vectors of matrix/ matrices. scipy.spatial package provides us distance_matrix() method to compute the distance matrix. Generally matrices are in the form of 2-D array and the vectors of the matrix are matrix rows ( 1-D array).

Syntax: scipy.spatial.distance_matrix(x, y, p=2)
 
Parameters:
    x : (M, K) Matrix of M vectors, each of dimension K. 
    y : (N, K) Matrix of N vectors, each of dimension K. 
    p : float, 1 <= p <= infinity, defines which Minkowski p-norm to use.

Returns: (M, N) ndarray
/ matrix containing the distance from every vector in x to every vector in y.

Note: the column dimensions of both x, and y matrices must be same.

We can use different values for p to apply different types of the distances to compute the distance matrix.

p = 1, Manhattan Distance
p = 2, Euclidean Distance
p = ∞, Chebychev Distance

Example 1.

We compute the distance matrix for two matrices x, and y. Both matrices have same dimension (3, 2).  So the distance matrix has dimension (3,3). Using p=2, the distances are calculated as Minkowski 2-norm (or Euclidean distance).

Python3

# Python program to compute distance matrix

# import important libraries
import numpy as np
from scipy.spatial import distance_matrix

# Create the matrices
x = np.array([[1,2],[2,1],[2,2]])
y = np.array([[5,0],[1,2],[2,0]])

# Display the matrices
print("matrix x:\n", x)
print("matrix y:\n", y)

# compute the distance matrix
dist_mat = distance_matrix(x, y, p=2)

# display distance matrix
print("Distance Matrix:\n", dist_mat)

Output:

distance matrix example 1

Example 2.

We compute the distance matrix for two matrices x, and y. Both matrices have different dimensions. Matrix x has dimension (3,2) and matrix y has dimension (5,2). So the distance matrix has dimension (3,5).

Python3

# Python program to compute distance matrix

# import important libraries
import numpy as np
from scipy.spatial import distance_matrix

# Create the matrices
x = np.array([[1,2],[2,1],[2,2]])
y = np.array([[0,0],[0,0],[1,1],[1,1],[1,2]])

# Display the matrices
print("matrix x:\n", x)
print("matrix y:\n", y)

# compute the distance matrix
dist_mat = distance_matrix(x, y, p=2)

# display distance matrix
print("Distance Matrix:\n", dist_mat)

Output:

distance matrix example 2

Example 3.

We compute the distance matrix using single matrix ( i.e. x). Matrix x has dimension (3,2). Same matrix x is given as parameter y. The distance matrix has dimension (3,3).

Python3

# Python program to compute distance matrix

# import important libraries
import numpy as np
from scipy.spatial import distance_matrix

# Create the matrix
x = np.array([[1,2],[2,1],[2,2]])

# Display the matrix
print("matrix x:\n", x)

# compute the distance matrix
dist_mat = distance_matrix(x, x, p=2)

# display distance matrix
print("Distance Matrix:\n", dist_mat)

output:

distance matrix example 3

Note: Notice that the above distance matrix is a symmetric matrix. When both x, and y matrices are same, the distance matrix is a symmetric matrix.

Example 4.

We compute the distance matrix for two matrices x, and y. Both matrices have different dimensions. Matrix x has dimension (3,2) and matrix y has dimension (5,2). So the distance matrix has dimension (3,5). Using p=1, the distances are calculated as Minkowski 1-norm (or Manhattan Distance).

Python3

# Python program to compute distance matrix

# import important libraries
import numpy as np
from scipy.spatial import distance_matrix

# Create the matrices
x = np.array([[1,2],[2,1],[2,2]])
y = np.array([[5,0],[1,2],[2,0]])

# Display the matrices
print("matrix x:\n", x)
print("matrix y:\n", y)

# compute the distance matrix
dist_mat = distance_matrix(x, y, p=1)

# display distance matrix
print("Distance Matrix:\n", dist_mat)

Output:

distance matrix example 4

Example 5.

We compute the distance matrix for two matrices x, and y. Both matrices have dimension (2, 5). So the distance matrix has dimension (3,5). Using p=2, the distances are calculated as Minkowski 2-norm (or Euclidean Distance).

Python3

# Python program to compute distance matrix

# import important libraries
import numpy as np
from scipy.spatial import distance_matrix

# Create the matrices
x = np.array([[1,2,3,4,5],[2,1,0,3,4]])
y = np.array([[0,0,0,0,1],[1,1,1,1,2]])

# Display the matrices
print("matrix x:\n", x)
print("matrix y:\n", y)

# compute the distance matrix
dist_mat = distance_matrix(x, y, p=2)

# display distance matrix
print("Distance Matrix:\n", dist_mat)

Output:

distance matrix example 5