132 lines
4.3 KiB
Python
132 lines
4.3 KiB
Python
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"""
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This module provides functions to perform full Procrustes analysis.
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This code was originally written by Justin Kucynski and ported over from
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scikit-bio by Yoshiki Vazquez-Baeza.
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"""
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import numpy as np
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from scipy.linalg import orthogonal_procrustes
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__all__ = ['procrustes']
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def procrustes(data1, data2):
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r"""Procrustes analysis, a similarity test for two data sets.
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Each input matrix is a set of points or vectors (the rows of the matrix).
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The dimension of the space is the number of columns of each matrix. Given
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two identically sized matrices, procrustes standardizes both such that:
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- :math:`tr(AA^{T}) = 1`.
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- Both sets of points are centered around the origin.
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Procrustes ([1]_, [2]_) then applies the optimal transform to the second
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matrix (including scaling/dilation, rotations, and reflections) to minimize
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:math:`M^{2}=\sum(data1-data2)^{2}`, or the sum of the squares of the
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pointwise differences between the two input datasets.
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This function was not designed to handle datasets with different numbers of
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datapoints (rows). If two data sets have different dimensionality
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(different number of columns), simply add columns of zeros to the smaller
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of the two.
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Parameters
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----------
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data1 : array_like
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Matrix, n rows represent points in k (columns) space `data1` is the
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reference data, after it is standardised, the data from `data2` will be
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transformed to fit the pattern in `data1` (must have >1 unique points).
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data2 : array_like
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n rows of data in k space to be fit to `data1`. Must be the same
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shape ``(numrows, numcols)`` as data1 (must have >1 unique points).
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Returns
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-------
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mtx1 : array_like
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A standardized version of `data1`.
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mtx2 : array_like
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The orientation of `data2` that best fits `data1`. Centered, but not
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necessarily :math:`tr(AA^{T}) = 1`.
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disparity : float
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:math:`M^{2}` as defined above.
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Raises
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------
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ValueError
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If the input arrays are not two-dimensional.
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If the shape of the input arrays is different.
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If the input arrays have zero columns or zero rows.
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See Also
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--------
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scipy.linalg.orthogonal_procrustes
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scipy.spatial.distance.directed_hausdorff : Another similarity test
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for two data sets
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Notes
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-----
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- The disparity should not depend on the order of the input matrices, but
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the output matrices will, as only the first output matrix is guaranteed
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to be scaled such that :math:`tr(AA^{T}) = 1`.
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- Duplicate data points are generally ok, duplicating a data point will
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increase its effect on the procrustes fit.
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- The disparity scales as the number of points per input matrix.
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References
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----------
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.. [1] Krzanowski, W. J. (2000). "Principles of Multivariate analysis".
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.. [2] Gower, J. C. (1975). "Generalized procrustes analysis".
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Examples
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--------
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>>> from scipy.spatial import procrustes
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The matrix ``b`` is a rotated, shifted, scaled and mirrored version of
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``a`` here:
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>>> a = np.array([[1, 3], [1, 2], [1, 1], [2, 1]], 'd')
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>>> b = np.array([[4, -2], [4, -4], [4, -6], [2, -6]], 'd')
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>>> mtx1, mtx2, disparity = procrustes(a, b)
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>>> round(disparity)
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0.0
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"""
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mtx1 = np.array(data1, dtype=np.double, copy=True)
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mtx2 = np.array(data2, dtype=np.double, copy=True)
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if mtx1.ndim != 2 or mtx2.ndim != 2:
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raise ValueError("Input matrices must be two-dimensional")
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if mtx1.shape != mtx2.shape:
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raise ValueError("Input matrices must be of same shape")
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if mtx1.size == 0:
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raise ValueError("Input matrices must be >0 rows and >0 cols")
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# translate all the data to the origin
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mtx1 -= np.mean(mtx1, 0)
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mtx2 -= np.mean(mtx2, 0)
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norm1 = np.linalg.norm(mtx1)
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norm2 = np.linalg.norm(mtx2)
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if norm1 == 0 or norm2 == 0:
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raise ValueError("Input matrices must contain >1 unique points")
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# change scaling of data (in rows) such that trace(mtx*mtx') = 1
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mtx1 /= norm1
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mtx2 /= norm2
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# transform mtx2 to minimize disparity
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R, s = orthogonal_procrustes(mtx1, mtx2)
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mtx2 = np.dot(mtx2, R.T) * s
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# measure the dissimilarity between the two datasets
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disparity = np.sum(np.square(mtx1 - mtx2))
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return mtx1, mtx2, disparity
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