_procrustes.py
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"""
This module provides functions to perform full Procrustes analysis.
This code was originally written by Justin Kucynski and ported over from
scikit-bio by Yoshiki Vazquez-Baeza.
"""
import numpy as np
from scipy.linalg import orthogonal_procrustes
__all__ = ['procrustes']
def procrustes(data1, data2):
r"""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:
- :math:`tr(AA^{T}) = 1`.
- Both sets of points are centered around the origin.
Procrustes ([1]_, [2]_) then applies the optimal transform to the second
matrix (including scaling/dilation, rotations, and reflections) to minimize
:math:`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 :math:`tr(AA^{T}) = 1`.
disparity : float
:math:`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
scipy.spatial.distance.directed_hausdorff : Another similarity test
for two data sets
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 :math:`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
----------
.. [1] Krzanowski, W. J. (2000). "Principles of Multivariate analysis".
.. [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
"""
mtx1 = np.array(data1, dtype=np.double, copy=True)
mtx2 = np.array(data2, dtype=np.double, copy=True)
if mtx1.ndim != 2 or mtx2.ndim != 2:
raise ValueError("Input matrices must be two-dimensional")
if mtx1.shape != mtx2.shape:
raise ValueError("Input matrices must be of same shape")
if mtx1.size == 0:
raise ValueError("Input matrices must be >0 rows and >0 cols")
# translate all the data to the origin
mtx1 -= np.mean(mtx1, 0)
mtx2 -= np.mean(mtx2, 0)
norm1 = np.linalg.norm(mtx1)
norm2 = np.linalg.norm(mtx2)
if norm1 == 0 or norm2 == 0:
raise ValueError("Input matrices must contain >1 unique points")
# change scaling of data (in rows) such that trace(mtx*mtx') = 1
mtx1 /= norm1
mtx2 /= norm2
# transform mtx2 to minimize disparity
R, s = orthogonal_procrustes(mtx1, mtx2)
mtx2 = np.dot(mtx2, R.T) * s
# measure the dissimilarity between the two datasets
disparity = np.sum(np.square(mtx1 - mtx2))
return mtx1, mtx2, disparity