# Distance metrics¶

Common distance functions.

## euclidean¶

numpy_ml.utils.distance_metrics.euclidean(x, y)[source]

Compute the Euclidean (L2) distance between two real vectors

Notes

The Euclidean distance between two vectors x and y is

$d(\mathbf{x}, \mathbf{y}) = \sqrt{ \sum_i (x_i - y_i)^2 }$
Parameters: x,y (ndarray s of shape (N,)) – The two vectors to compute the distance between d (float) – The L2 distance between x and y.

## chebyshev¶

numpy_ml.utils.distance_metrics.chebyshev(x, y)[source]

Compute the Chebyshev ($$L_\infty$$) distance between two real vectors

Notes

The Chebyshev distance between two vectors x and y is

$d(\mathbf{x}, \mathbf{y}) = \max_i |x_i - y_i|$
Parameters: x,y (ndarray s of shape (N,)) – The two vectors to compute the distance between d (float) – The Chebyshev distance between x and y.

## hamming¶

numpy_ml.utils.distance_metrics.hamming(x, y)[source]

Compute the Hamming distance between two integer-valued vectors.

Notes

The Hamming distance between two vectors x and y is

$d(\mathbf{x}, \mathbf{y}) = \frac{1}{N} \sum_i \mathbb{1}_{x_i \neq y_i}$
Parameters: x,y (ndarray s of shape (N,)) – The two vectors to compute the distance between. Both vectors should be integer-valued. d (float) – The Hamming distance between x and y.

## manhattan¶

numpy_ml.utils.distance_metrics.manhattan(x, y)[source]

Compute the Manhattan (L1) distance between two real vectors

Notes

The Manhattan distance between two vectors x and y is

$d(\mathbf{x}, \mathbf{y}) = \sum_i |x_i - y_i|$
Parameters: x,y (ndarray s of shape (N,)) – The two vectors to compute the distance between d (float) – The L1 distance between x and y.

## minkowski¶

numpy_ml.utils.distance_metrics.minkowski(x, y, p)[source]

Compute the Minkowski-p distance between two real vectors.

Notes

The Minkowski-p distance between two vectors x and y is

$d(\mathbf{x}, \mathbf{y}) = \left( \sum_i |x_i - y_i|^p \right)^{1/p}$
Parameters: x,y (ndarray s of shape (N,)) – The two vectors to compute the distance between p (float > 1) – The parameter of the distance function. When p = 1, this is the L1 distance, and when p=2, this is the L2 distance. For p < 1, Minkowski-p does not satisfy the triangle inequality and hence is not a valid distance metric. d (float) – The Minkowski-p distance between x and y.