/usr/share/pyshared/cogent/cluster/procrustes.py is in python-cogent 1.5.3-2.
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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 | #!/usr/bin/env python
"""Procrustes analysis. Main fn: procrustes
See for example:
Principles of Multivariate analysis, by Krzanowski
"""
from numpy.linalg import svd
from numpy import array, sqrt, sum, zeros, trace, dot, transpose,\
divide, square, subtract, shape, any, abs, mean
from numpy import append as numpy_append
__author__ = "Justin Kuczynski"
__copyright__ = "Copyright 2007-2012, The Cogent Project"
__credits__ = ["Justin Kuczynski"]
__license__ = "GPL"
__version__ = "1.5.3"
__maintainer__ = "Justin Kuczynski"
__email__ = "justinak@gmail.com"
__status__ = "Production"
def 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 identially sized matrices, procrustes standardizes both
such that:
- trace(AA') = 1 (A' is the transpose, and the product is
a standard matrix product).
- Both sets of points are centered around the origin
Procrustes then applies the optimal transform to the second matrix
(including scaling/dilation, rotations, and reflections) to minimize
M^2 = sum(square(mtx1 - mtx2)), or the sum of the squares of the pointwise
differences between the two input datasets
If two data sets have different dimensionality (different number of
columns), simply add columns of zeros the the smaller of the two.
This function was not designed to handle datasets with different numbers of
datapoints (rows)
Arguments:
- data1: 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
- data2: n rows of data in k space to be fit to data1. Must be the
same shape (numrows, numcols) as data1
- both must have >1 unique points
Returns:
- mtx1: a standardized version of data1
- mtx2: the orientation of data2 that best fits data1.
centered, but not necessarily trace(mtx2*mtx2') = 1
- disparity: a metric for the dissimilarity of the two datasets,
disparity = M^2 defined above
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 trace(AA') = 1.
- duplicate datapoints are generally ok, duplicating a data point will
increase it's effect on the procrustes fit.
- the disparity scales as the number of points per input matrix
"""
SMALL_NUM = 1e-6 # used to check for zero values in added dimension
# make local copies
# mtx1 = array(data1.copy(),'d')
# mtx2 = array(data2.copy(),'d')
num_rows, num_cols = shape(data1)
if (num_rows, num_cols) != shape(data2):
raise ValueError("input matrices must be of same shape")
if (num_rows == 0 or num_cols == 0):
raise ValueError("input matrices must be >0 rows, >0 cols")
# add a dimension to allow reflections (rotations in n + 1 dimensions)
mtx1 = numpy_append(data1, zeros((num_rows, 1)), 1)
mtx2 = numpy_append(data2, zeros((num_rows, 1)), 1)
# standardize each matrix
mtx1 = center(mtx1)
mtx2 = center(mtx2)
if ((not any(mtx1)) or (not any(mtx2))):
raise ValueError("input matrices must contain >1 unique points")
mtx1 = normalize(mtx1)
mtx2 = normalize(mtx2)
# transform mtx2 to minimize disparity (sum( (mtx1[i,j] - mtx2[i,j])^2) )
mtx2 = match_points(mtx1, mtx2)
# WARNING: I haven't proven that after matching the matrices, no point has
# a nonzero component in the added dimension. I believe it is true,
# though, since the unchanged matrix has no points extending into
# that dimension
if any(abs(mtx2[:,-1]) > SMALL_NUM):
raise StandardError("we have accidentially added a dimension to \
the matrix, and the vectors have nonzero components in that dimension")
# strip extra dimension which was added to allow reflections
mtx1 = mtx1[:,:-1]
mtx2 = mtx2[:,:-1]
disparity = get_disparity(mtx1, mtx2)
return mtx1, mtx2, disparity
def center(mtx):
"""translate all data (rows of the matrix) to center on the origin
returns a shifted version of the input data. The new matrix is such that
the center of mass of the row vectors is centered at the origin.
Returns a numpy float ('d') array
"""
result = array(mtx, 'd')
result -= mean(result, 0)
# subtract each column's mean from each element in that column
return result
def normalize(mtx):
"""change scaling of data (in rows) such that trace(mtx*mtx') = 1
mtx' denotes the transpose of mtx """
result = array(mtx, 'd')
num_pts, num_dims = shape(result)
mag = trace(dot(result, transpose(result)))
norm = sqrt(mag)
result /= norm
return result
def match_points(mtx1, mtx2):
"""returns a transformed mtx2 that matches mtx1.
returns a new matrix which is a transform of mtx2. Scales and rotates
a copy of mtx 2. See procrustes docs for details.
"""
u,s,vh = svd(dot(transpose(mtx1), mtx2))
q = dot(transpose(vh), transpose(u))
new_mtx2 = dot(mtx2, q)
new_mtx2 *= sum(s)
return new_mtx2
def get_disparity(mtx1, mtx2):
""" returns a measure of the dissimilarity between two data sets
returns M^2 = sum(square(mtx1 - mtx2)), the pointwise sum of squared
differences"""
return(sum(square(mtx1 - mtx2)))
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