librosa.util.normalize

librosa.util.normalize(S, norm=inf, axis=0, threshold=None, fill=None)

Normalize an array along a chosen axis.

Given a norm (described below) and a target axis, the input array is scaled so that

norm(S, axis=axis) == 1

For example, axis=0 normalizes each column of a 2-d array by aggregating over the rows (0-axis). Similarly, axis=1 normalizes each row of a 2-d array.

This function also supports thresholding small-norm slices: any slice (i.e., row or column) with norm below a specified threshold can be left un-normalized, set to all-zeros, or filled with uniform non-zero values that normalize to 1.

Note: the semantics of this function differ from scipy.linalg.norm in two ways: multi-dimensional arrays are supported, but matrix-norms are not.

Parameters:

S : np.ndarray

The matrix to normalize

norm : {np.inf, -np.inf, 0, float > 0, None}

• np.inf : maximum absolute value
• -np.inf : mininum absolute value
• 0 : number of non-zeros (the support)

• float : corresponding l_p norm

  See scipy.linalg.norm for details.

• None : no normalization is performed

axis : int [scalar]

Axis along which to compute the norm.

threshold : number > 0 [optional]

Only the columns (or rows) with norm at least threshold are normalized.

By default, the threshold is determined from the numerical precision of S.dtype.

fill : None or bool

If None, then columns (or rows) with norm below threshold are left as is.

If False, then columns (rows) with norm below threshold are set to 0.

If True, then columns (rows) with norm below threshold are filled uniformly such that the corresponding norm is 1.

Note

fill=True is incompatible with norm=0 because no uniform vector exists with l0 “norm” equal to 1.

Returns:

S_norm : np.ndarray [shape=S.shape]

Normalized array

Raises:

ParameterError

If norm is not among the valid types defined above

If S is not finite

If fill=True and norm=0

See also

scipy.linalg.norm

Notes

This function caches at level 40.

Examples

>>> # Construct an example matrix
>>> S = np.vander(np.arange(-2.0, 2.0))
>>> S
array([[-8.,  4., -2.,  1.],
       [-1.,  1., -1.,  1.],
       [ 0.,  0.,  0.,  1.],
       [ 1.,  1.,  1.,  1.]])
>>> # Max (l-infinity)-normalize the columns
>>> librosa.util.normalize(S)
array([[-1.   ,  1.   , -1.   ,  1.   ],
       [-0.125,  0.25 , -0.5  ,  1.   ],
       [ 0.   ,  0.   ,  0.   ,  1.   ],
       [ 0.125,  0.25 ,  0.5  ,  1.   ]])
>>> # Max (l-infinity)-normalize the rows
>>> librosa.util.normalize(S, axis=1)
array([[-1.   ,  0.5  , -0.25 ,  0.125],
       [-1.   ,  1.   , -1.   ,  1.   ],
       [ 0.   ,  0.   ,  0.   ,  1.   ],
       [ 1.   ,  1.   ,  1.   ,  1.   ]])
>>> # l1-normalize the columns
>>> librosa.util.normalize(S, norm=1)
array([[-0.8  ,  0.667, -0.5  ,  0.25 ],
       [-0.1  ,  0.167, -0.25 ,  0.25 ],
       [ 0.   ,  0.   ,  0.   ,  0.25 ],
       [ 0.1  ,  0.167,  0.25 ,  0.25 ]])
>>> # l2-normalize the columns
>>> librosa.util.normalize(S, norm=2)
array([[-0.985,  0.943, -0.816,  0.5  ],
       [-0.123,  0.236, -0.408,  0.5  ],
       [ 0.   ,  0.   ,  0.   ,  0.5  ],
       [ 0.123,  0.236,  0.408,  0.5  ]])

>>> # Thresholding and filling
>>> S[:, -1] = 1e-308
>>> S
array([[ -8.000e+000,   4.000e+000,  -2.000e+000,
          1.000e-308],
       [ -1.000e+000,   1.000e+000,  -1.000e+000,
          1.000e-308],
       [  0.000e+000,   0.000e+000,   0.000e+000,
          1.000e-308],
       [  1.000e+000,   1.000e+000,   1.000e+000,
          1.000e-308]])

>>> # By default, small-norm columns are left untouched
>>> librosa.util.normalize(S)
array([[ -1.000e+000,   1.000e+000,  -1.000e+000,
          1.000e-308],
       [ -1.250e-001,   2.500e-001,  -5.000e-001,
          1.000e-308],
       [  0.000e+000,   0.000e+000,   0.000e+000,
          1.000e-308],
       [  1.250e-001,   2.500e-001,   5.000e-001,
          1.000e-308]])
>>> # Small-norm columns can be zeroed out
>>> librosa.util.normalize(S, fill=False)
array([[-1.   ,  1.   , -1.   ,  0.   ],
       [-0.125,  0.25 , -0.5  ,  0.   ],
       [ 0.   ,  0.   ,  0.   ,  0.   ],
       [ 0.125,  0.25 ,  0.5  ,  0.   ]])
>>> # Or set to constant with unit-norm
>>> librosa.util.normalize(S, fill=True)
array([[-1.   ,  1.   , -1.   ,  1.   ],
       [-0.125,  0.25 , -0.5  ,  1.   ],
       [ 0.   ,  0.   ,  0.   ,  1.   ],
       [ 0.125,  0.25 ,  0.5  ,  1.   ]])
>>> # With an l1 norm instead of max-norm
>>> librosa.util.normalize(S, norm=1, fill=True)
array([[-0.8  ,  0.667, -0.5  ,  0.25 ],
       [-0.1  ,  0.167, -0.25 ,  0.25 ],
       [ 0.   ,  0.   ,  0.   ,  0.25 ],
       [ 0.1  ,  0.167,  0.25 ,  0.25 ]])