133 lines
4.3 KiB
Python
133 lines
4.3 KiB
Python
|
"""
|
||
|
|
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
|
||
|
|
--------
|
||
|
|
>>> import numpy as np
|
||
|
|
>>> 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.float64, copy=True)
|
||
|
|
mtx2 = np.array(data2, dtype=np.float64, 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
|
||
|
|
|