def besttransformation_weighted(set1, set2, weights=[1.0]):
"""This finds the besttransformation rotation matrix with predetermined
weights. The weights are used to give some coordinates more influence than
others.
"""
assert len(set1) == len(set2)
length = len(set1)
assert length > 0
if len(weights) == len(set1):
diagonal = numpy.diag(weights)
else:
diagonal = numpy.diag(numpy.ones(len(set1)))
mean1 = numpy.sum(set1, axis=0) / float(length)
mean2 = numpy.sum(set2, axis=0) / float(length)
dev1 = set1 - mean1
dev2 = set2 - mean2
A = numpy.dot(numpy.transpose(dev2), numpy.dot(diagonal, dev1))
V, diagS, Wt = numpy.linalg.svd(A)
I = numpy.matrix(numpy.identity(3))
d = numpy.linalg.det(numpy.dot(numpy.transpose(Wt), numpy.transpose(V)))
if numpy.isclose(d, -1.0):
I[2, 2] = d
U = numpy.dot(numpy.dot(numpy.transpose(Wt), I), numpy.transpose(V))
new1 = numpy.dot(dev1, U)
new2 = dev2
rmsd = RMSD(new1, new2)
sse = sumsquarederror(new1, new2)
rotation_matrix = U
return rotation_matrix, new1, mean1, rmsd, sse
def besttransformation_weighted(set1, set2, weights=[1.0]):
"""This finds the besttransformation rotation matrix with predetermined
weights. The weights are used to give some coordinates more influence than
others.
"""
assert len(set1) == len(set2)
length = len(set1)
assert length > 0
if len(weights) == len(set1):
diagonal = numpy.diag(weights)
else:
diagonal = numpy.diag(numpy.ones(len(set1)))
mean1 = numpy.sum(set1, axis=0) / float(length)
mean2 = numpy.sum(set2, axis=0) / float(length)
dev1 = set1 - mean1
dev2 = set2 - mean2
A = numpy.dot(numpy.transpose(dev2), numpy.dot(diagonal, dev1))
V, diagS, Wt = numpy.linalg.svd(A)
I = numpy.matrix(numpy.identity(3))
d = numpy.linalg.det(numpy.dot(numpy.transpose(Wt), numpy.transpose(V)))
if numpy.isclose(d, -1.0):
I[2, 2] = d
U = numpy.dot(numpy.dot(numpy.transpose(Wt), I), numpy.transpose(V))
new1 = numpy.dot(dev1, U)
new2 = dev2
rmsd = RMSD(new1, new2)
sse = sumsquarederror(new1, new2)
rotation_matrix = U
return rotation_matrix, new1, mean1, rmsd, sse
def besttransformation(set1, set2):
"""This finds the 3x3 rotation matrix which optimally superimposes
the nx3 matrix of points in set1 onto the nx3 matrix of points set2.
One reference is this: http://en.wikipedia.org/wiki/Kabsch_algorithm
Another is a python implementation that goes with pymol, see
http://www.pymolwiki.org/index.php/Kabsch
The arguments should be lists or numpy arrays of the form exampled below:
set1=[[1.0, 2.0, 3.0], [4,5,2], [9,2,4], ....,[3,1,6]]
sel2=[[4.0, 4.1, 5.0], [2,3,1], [3,1,4], ....,[1, 7, 7]]
:set1: A list or a numpy array of (n, 3) coordinates.
:set2: A list or a numpy array of (n, 3) coordinates.
:returns: The transformation matrix, the new coordinates for the two
set of coordinates, respectively.
"""
# Check to make sure same number of (x,y,z) coordinates both sets.
# If condition is not true this program stops.
assert len(set1) == len(set2), 'Lengths must match'
length = len(set1)
assert length > 0, 'Must not give empty matrices'
# Translation Step is beginning.
# These add all x_{ij}'s, y_{ij}'s, z_{ij}'s for each element in
# setj j=1,2
# i=1,2,3,..length
# /float(length) divides by length
# This creates a [mean x_j, mean y_j, mean z_j] for both sets of
# coordinates setj j=1, or 2, or the centroid of both sets.
mean1 = numpy.sum(set1, axis=0) / float(length)
mean2 = numpy.sum(set2, axis=0) / float(length)
# Next,
# Subtract x_{ij} by the mean of x_j's. Here i=1,2,..length.
# Same for y, and z.
dev1 = set1 - mean1
dev2 = set2 - mean2
# Thus, both sets are translated, so that their centroid coincides with
# the origin of the coordinate system.
# Translation Step is now completed.
# Begin Step to Compute the 3X3 Covariance Matrix, A.
A = numpy.dot(numpy.transpose(dev2), dev1)
# Covariance Matrix, A, is now calculated
# Begin of the Computation of the optimal rotation matrix using
# Singular Value Decomposition (SVD)
V, diagS, Wt = numpy.linalg.svd(A)
# V and Wt are 3x3 orthonormal bases, diagS is the diagonal elements of
# a 3x3 diagonal matrix, S, in regular SVD. In SVD, recall that the
# Covariance matrix, A, is A=V*S*transpose(W) (matrix multiplication).
#S=numpy.diag(diagS)
#A=numpy.dot(V,numpy.dot(S,Wt)
# The next step is to decide whether we need to correct our rotation
# matrix to ensure a right-handed coordinate system
# we just need to check for reflections and then produce
# the rotation. V and Wt are orthonormal, so their det's
# are +/-1.
I = numpy.matrix(numpy.identity(3))
d = numpy.linalg.det(numpy.dot(numpy.transpose(Wt), numpy.transpose(V)))
if numpy.isclose(d, -1.0):
I[2, 2] = d
# End of the Computation of the optimal rotation matrix
#The transformation matrix, U, is now V*Wt
U = numpy.dot(numpy.dot(numpy.transpose(Wt), I), numpy.transpose(V))
# rotate and translate the molecule
#sel2 = numpy.dot((set2 - Mean2), U)
new1 = numpy.dot(dev1, U)
new2 = dev2
rmsd = RMSD(new1, new2)
sse = sumsquarederror(new1, new2)
#Return the transformation matrix, the new coordinates for the two
#set of coordinates, respectively.
return U, new1, mean1, rmsd, sse, mean2
def besttransformation(set1, set2):
"""This finds the 3x3 rotation matrix which optimally superimposes
the nx3 matrix of points in set1 onto the nx3 matrix of points set2.
One reference is this: http://en.wikipedia.org/wiki/Kabsch_algorithm
Another is a python implementation that goes with pymol, see
http://www.pymolwiki.org/index.php/Kabsch
The arguments should be lists or numpy arrays of the form exampled below:
set1=[[1.0, 2.0, 3.0], [4,5,2], [9,2,4], ....,[3,1,6]]
sel2=[[4.0, 4.1, 5.0], [2,3,1], [3,1,4], ....,[1, 7, 7]]
:set1: A list or a numpy array of (n, 3) coordinates.
:set2: A list or a numpy array of (n, 3) coordinates.
:returns: The transformation matrix, the new coordinates for the two
set of coordinates, respectively.
"""
# Check to make sure same number of (x,y,z) coordinates both sets.
#If condition is not true this program stops.
assert len(set1) == len(set2)
length = len(set1)
assert length > 0
# Translation Step is beginning.
# These add all x_{ij}'s, y_{ij}'s, z_{ij}'s for each element in
# setj j=1,2
# i=1,2,3,..length
# /float(length) divides by length
# This creates a [mean x_j, mean y_j, mean z_j] for both sets of
# coordinates setj j=1, or 2, or the centroid of both sets.
mean1 = numpy.sum(set1, axis=0) / float(length)
mean2 = numpy.sum(set2, axis=0) / float(length)
# Next,
# Subtract x_{ij} by the mean of x_j's. Here i=1,2,..length.
# Same for y, and z.
dev1 = set1 - mean1
dev2 = set2 - mean2
# Thus, both sets are translated, so that their centroid coincides with
# the origin of the coordinate system.
# Translation Step is now completed.
# Begin Step to Compute the 3X3 Covariance Matrix, A.
A = numpy.dot(numpy.transpose(dev2), dev1)
# Covariance Matrix, A, is now calculated
# Begin of the Computation of the optimal rotation matrix using
# Singular Value Decomposition (SVD)
V, diagS, Wt = numpy.linalg.svd(A)
# V and Wt are 3x3 orthonormal bases, diagS is the diagonal elements of
# a 3x3 diagonal matrix, S, in regular SVD. In SVD, recall that the
# Covariance matrix, A, is A=V*S*transpose(W) (matrix multiplication).
#S=numpy.diag(diagS)
#A=numpy.dot(V,numpy.dot(S,Wt)
# The next step is to decide whether we need to correct our rotation
# matrix to ensure a right-handed coordinate system
# we just need to check for reflections and then produce
# the rotation. V and Wt are orthonormal, so their det's
# are +/-1.
I = numpy.matrix(numpy.identity(3))
d = numpy.linalg.det(numpy.dot(numpy.transpose(Wt), numpy.transpose(V)))
if numpy.isclose(d, -1.0):
I[2, 2] = d
# End of the Computation of the optimal rotation matrix
#The transformation matrix, U, is now V*Wt
U = numpy.dot(numpy.dot(numpy.transpose(Wt), I), numpy.transpose(V))
# rotate and translate the molecule
#sel2 = numpy.dot((set2 - Mean2), U)
new1 = numpy.dot(dev1, U)
new2 = dev2
rmsd = RMSD(new1, new2)
sse = sumsquarederror(new1, new2)
#Return the transformation matrix, the new coordinates for the two
#set of coordinates, respectively.
return U, new1, mean1, rmsd, sse
