scipy.spatial.procrustes

scipy.spatial.procrustes(data1, data2)

Procrustes analysis, a similarity test for two data sets.

Each input matrix is a set of points or vectors (the rows of the matrix). The dimension of the space is the number of columns of each matrix. Given two identically sized matrices, procrustes standardizes both such that:

• \(tr(AA^{T}) = 1\).
• Both sets of points are centered around the origin.

Procrustes ([R309], [R310]) then applies the optimal transform to the second matrix (including scaling/dilation, rotations, and reflections) to minimize \(M^{2}=\sum(data1-data2)^{2}\), or the sum of the squares of the pointwise differences between the two input datasets.

This function was not designed to handle datasets with different numbers of datapoints (rows). If two data sets have different dimensionality (different number of columns), simply add columns of zeros to the smaller of the two.

Parameters:

data1 : array_like

Matrix, n rows represent points in k (columns) space data1 is the reference data, after it is standardised, the data from data2 will be transformed to fit the pattern in data1 (must have >1 unique points).

data2 : array_like

n rows of data in k space to be fit to data1. Must be the same shape (numrows, numcols) as data1 (must have >1 unique points).

Returns:

mtx1 : array_like

A standardized version of data1.

mtx2 : array_like

The orientation of data2 that best fits data1. Centered, but not necessarily \(tr(AA^{T}) = 1\).

disparity : float

\(M^{2}\) as defined above.

Raises:

ValueError

If the input arrays are not two-dimensional. If the shape of the input arrays is different. If the input arrays have zero columns or zero rows.

See also

scipy.linalg.orthogonal_procrustes

Notes

• The disparity should not depend on the order of the input matrices, but the output matrices will, as only the first output matrix is guaranteed to be scaled such that \(tr(AA^{T}) = 1\).
• Duplicate data points are generally ok, duplicating a data point will increase its effect on the procrustes fit.
• The disparity scales as the number of points per input matrix.

References

[R309]

(1, 2) Krzanowski, W. J. (2000). “Principles of Multivariate analysis”.

[R310]

(1, 2) Gower, J. C. (1975). “Generalized procrustes analysis”.

Examples

>>> from scipy.spatial import procrustes

The matrix b is a rotated, shifted, scaled and mirrored version of a here:

>>> a = np.array([[1, 3], [1, 2], [1, 1], [2, 1]], 'd')
>>> b = np.array([[4, -2], [4, -4], [4, -6], [2, -6]], 'd')
>>> mtx1, mtx2, disparity = procrustes(a, b)
>>> round(disparity)
0.0