RMSE between two arrays of matrices#

ahrs.utils.metrics.rmse_matrices(A: ndarray, B: ndarray, element_wise: bool = False) → ndarray#

The Root Mean Square Error (RMSE) between two arrays of matrices \(A\) and \(B\) is calculated as the square root of the mean of the squared differences between the elements of the two arrays.

If the input arrays are 2-dimensional matrices, the RMSE is calculated as:

\[RMSE = \sqrt{\frac{1}{MN}\sum_{i=1}^{M}\sum_{j=1}^{N}(A_{ij} - B_{ij})^2}\]

If the input arrays are arrays of 2-dimensional matrices (3-dimensional array), the RMSE is calculated as:

\[RMSE = \sqrt{\frac{1}{k}\sum_{l=1}^{k}\frac{1}{MN}\sum_{i=1}^{M}\sum_{j=1}^{N}(A_{ij}^{(l)} - B_{ij}^{(l)})^2}\]

where \(k\) is the number of matrices in the arrays.

If the option element_wise is set to True, the RMSE is calculated element-wise, and an M-by-N array of RMSEs is returned. The following calls are equivalent:

rmse = rmse_matrices(A, B, element_wise=True)
rmse = np.sqrt(np.nanmean((A-B)**2, axis=0))

If the inputs are arrays of matrices (3-dimensional arrays), its call is also equivalent to:

rmse = np.zeros_like(A[0])
for i in range(A.shape[1]):
    for j in range(A.shape[2]):
        rmse[i, j] = np.sqrt(np.nanmean((A[:, i, j]-B[:, i, j])**2))

If the inputs are 2-dimensional matrices, the following calls would return the same result:

rmse_matrices(A, B)
rmse_matrices(A, B, element_wise=False)
rmse_matrices(A, B, element_wise=True)

Parameters:

A (numpy.ndarray) – First M-by-N matrix or array of k M-by-N matrices.
B (numpy.ndarray) – Second M-by-N matrix or array of k M-by-N matrices.
element_wise (bool, default: False) – If True, calculate RMSE element-wise, and return an M-by-N array of RMSEs.

Returns:

rmse – Root Mean Square Error between the two matrices, or array of k RMSEs between the two arrays of matrices.

Return type:

float or np.ndarray

Raises:

ValueError – If the comparing arrays do not have the same shape.

Examples

>>> C = np.random.random((4, 3, 2))     # Array of four 3-by-2 matrices
>>> C.view()
array([[[0.2816407 , 0.30850589],
        [0.44618209, 0.33081522],
        [0.7994625 , 0.07377569]],

       [[0.35549399, 0.47050713],
        [0.94168683, 0.50388058],
        [0.70023837, 0.77216167]],

       [[0.79897129, 0.28555452],
        [0.892488  , 0.71476669],
        [0.19071524, 0.4123666 ]],

       [[0.86301978, 0.14686002],
        [0.98784823, 0.26129908],
        [0.46982206, 0.88037599]]])
>>> D = np.random.random((4, 3, 2))     # Array of four 3-by-2 matrices
>>> D.view()
array([[[0.71560918, 0.34100321],
        [0.92518341, 0.50741267],
        [0.30730944, 0.19173378]],

       [[0.31846657, 0.08578454],
        [0.62643489, 0.84014104],
        [0.7111152 , 0.95428613]],

       [[0.8101591 , 0.9584096 ],
        [0.91118705, 0.71203119],
        [0.58217189, 0.45598271]],

       [[0.79837603, 0.09954558],
        [0.26532781, 0.55711476],
        [0.03909648, 0.10787888]]])
>>> rmse_matrices(C[0], D[0])       # RMSE between first matrices
0.3430603410873006
>>> rmse_matrices(C, D)             # RMSE between each of the four matrices
array([0.34306034, 0.25662067, 0.31842239, 0.48274156])
>>> rmse_matrices(C, D, element_wise=True)  # RMSE element-wise along first dimension
array([[0.22022923, 0.3886001 ],
       [0.46130561, 0.2407136 ],
       [0.38114819, 0.40178899]])
>>> rmse_matrices(C[0], D[0], element_wise=True)
0.3430603410873006
>>> rmse_matrices(C[0], D[0])
0.3430603410873006

RMSE between two arrays of matrices#

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