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).

Example:

Output

[[5.65685425 8.48528137]
[2.82842712 5.65685425]]

Explanation: This computes the Euclidean distance (default, p =2) between each pair of points in A and B. For instance, the first element 5.65685425 is the distance between [1, 2] and [5, 6].

Syntax of scipy.spatial.distance_matrix()

scipy.spatial.distance_matrix(XA, XB, p=2)

Parameters:

• XA (array_like): An m x n array of m points in n-dimensional space.
• XB (array_like): A k x n array of k points in the same n-dimensional space.
• p (float, optional): The Minkowski distance metric to use. Default is 2 (Euclidean distance).

Returns: distances (ndarray) – An m × k matrix where each element [i, j] represents the distance between XA[i] and XB[j].

Raises:

• ValueError: If input arrays have incompatible shapes or dimensions.
• TypeError: If input is not array-like or if the p parameter is not valid.

Note: The number of columns (dimensions) in both XA and XB must be the same. You can use different values of p to compute various distance metrics:

• p = 1 → Manhattan Distance
• p = 2 → Euclidean Distance (default)
• p = ∞ → Chebyshev Distance

Examples

Example 1: Manhattan Distance (p =1)

Output

[[4. 8.]]

Explanation: Here, the Manhattan distance is used. The first value 4 is computed as |1-3| + |2-4| = 2 + 2.

Example 2: Chebyshev Distance (p =∞)

Output

[[4. 6.]]

Explanation: The Chebyshev distance takes the maximum absolute difference in any dimension. For [1, 2] and [4, 6], it's max(|1-4|, |2-6|) = max(3, 4) = 4.

Example 3: Custom Distance (p =3)

Output

[[4.49794145]]

Explanation: This uses Minkowski distance with p=3, which gives a value between the Manhattan and Euclidean distances.

Using scipy.spatial.distance.cdist()

While distance_matrix() is useful for computing pairwise distances using the Minkowski metric, cdist() from scipy.spatial.distance provides greater flexibility by supporting a wide range of distance metrics (e.g., cosine, correlation, cityblock, etc.)

Syntax

scipy.spatial.distance.cdist(XA, XB, metric='euclidean')

Parameters:

• XA (array_like): An m × n array of m points.
• XB (array_like): A k × n array of k points.
• metric (str or function): Distance metric to use (e.g., 'euclidean', 'cosine', 'cityblock', etc.).

Returns: distances (ndarray): An m × k matrix of distances.

Examples

Example 1: Cosine distance

Output

[[0.29289322]
[0.29289322]]

Explanation: Cosine distance measures the angular difference between vectors. Both vectors in A form a 45° angle with the vector in B, hence the equal distances.

Example 2: Cityblock (Manhattan) Distance

Output

[[7. 5.]]

Explanation: This uses the cityblock (Manhattan) metric. For [1, 2] and [4, 6], distance = |1−4| + |2−6| = 3 + 4 = 7.