def __findTransformation(self, x, y):
"""
Match two arrays by rotation and translation. Returns the
rotation matrix and the translation vector.
Back transformation:
for atom i new coordinates will be::
y_new[i] = N0.dot(r, y[i]) + t
for all atoms in one step::
y_new = N0.dot(y, N0.transpose(r)) + t
@param x: coordinates
@type x: array
@param y: coordinates
@type y: array
@return: rotation matrix, translation vector
@rtype: array, array
@author: Michael Habeck
"""
from numpy.linalg import svd
## center configurations
x_av = N0.sum(x) / len(x)
y_av = N0.sum(y) / len(y)
x = x - x_av
y = y - y_av
## svd of correlation matrix
v, l, u = svd(N0.dot(N0.transpose(x), y))
## build rotation matrix and translation vector
r = N0.dot(v, u)
t = x_av - N0.dot(r, y_av)
return r, t
def findTransformation(x, y):
"""
Match two arrays by rotation and translation. Returns the
rotation matrix and the translation vector.
:param x: first set of coordinates
:type x: array('f')
:param y: second set of coordinates
:type y: array('f')
:return: rotation matrix (3x3) and translation vector (1x3)
:rtype: array, array
"""
## center configurations
x_av = N0.average(x)
y_av = N0.average(y)
x = x - x_av
y = y - y_av
## svd of correlation matrix
v, l, u = svd(N0.dot(N0.transpose(x), y))
## build rotation matrix and translation vector
r = N0.dot(v, u)
t = x_av - N0.dot(r, y_av)
return r, t
def __pairwiseDistances(self, u, v):
"""
pairwise distance between 2 3-D numpy arrays of atom coordinates.
@param u: coordinates
@type u: array
@param v: coordinates
@type v: array
@return: Numpy array len(u) x len(v)
@rtype:array
@author: Wolfgang Rieping.
"""
## check input
if not type( u ) == arraytype or\
not type( v ) == arraytype:
raise ComplexError('unsupported argument type ' + \
str( type(u) ) + ' or ' + str( type(v) ) )
diag1 = N0.diagonal(N0.dot(u, N0.transpose(u)))
diag2 = N0.diagonal(N0.dot(v, N0.transpose(v)))
dist = -N0.dot(v, N0.transpose(u)) - N0.transpose(
N0.dot(u, N0.transpose(v)))
dist= N0.transpose(N0.asarray(list(map(lambda column,a:column+a, \
N0.transpose(dist), diag1))))
return N0.transpose(
N0.sqrt(N0.asarray(list(map(lambda row, a: row + a, dist,
diag2)))))
def __pairwiseDistances(self, u, v):
"""
pairwise distance between 2 3-D numpy arrays of atom coordinates.
@param u: coordinates
@type u: array
@param v: coordinates
@type v: array
@return: Numpy array len(u) x len(v)
@rtype:array
@author: Wolfgang Rieping.
"""
## check input
if not type( u ) == arraytype or\
not type( v ) == arraytype:
raise ComplexError('unsupported argument type ' + \
str( type(u) ) + ' or ' + str( type(v) ) )
diag1= N0.diagonal(N0.dot(u,N0.transpose(u)))
diag2= N0.diagonal(N0.dot(v,N0.transpose(v)))
dist= -N0.dot(v,N0.transpose(u))-N0.transpose(N0.dot(u,N0.transpose(v)))
dist= N0.transpose(N0.asarray(list(map(lambda column,a:column+a, \
N0.transpose(dist), diag1))))
return N0.transpose(N0.sqrt(N0.asarray(
list(map(lambda row,a: row+a, dist, diag2)))))
def findTransformation(x, y):
"""
Match two arrays by rotation and translation. Returns the
rotation matrix and the translation vector.
:param x: first set of coordinates
:type x: array('f')
:param y: second set of coordinates
:type y: array('f')
:return: rotation matrix (3x3) and translation vector (1x3)
:rtype: array, array
"""
## center configurations
x_av = N0.average(x)
y_av = N0.average(y)
x = x - x_av
y = y - y_av
## svd of correlation matrix
v, l, u = svd(N0.dot(N0.transpose(x), y))
## build rotation matrix and translation vector
r = N0.dot(v, u)
t = x_av - N0.dot(r, y_av)
return r, t
def __findTransformation(self, x, y):
"""
Match two arrays by rotation and translation. Returns the
rotation matrix and the translation vector.
Back transformation:
for atom i new coordinates will be::
y_new[i] = N0.dot(r, y[i]) + t
for all atoms in one step::
y_new = N0.dot(y, N0.transpose(r)) + t
@param x: coordinates
@type x: array
@param y: coordinates
@type y: array
@return: rotation matrix, translation vector
@rtype: array, array
@author: Michael Habeck
"""
from numpy.linalg import svd
## center configurations
x_av = N0.sum(x) / len(x)
y_av = N0.sum(y) / len(y)
x = x - x_av
y = y - y_av
## svd of correlation matrix
v, l, u = svd(N0.dot(N0.transpose(x), y))
## build rotation matrix and translation vector
r = N0.dot(v, u)
t = x_av - N0.dot(r, y_av)
return r, t
def squared_distance_matrix(x, y):
d1 = N0.diagonal(N0.dot(x, N0.transpose(x)))
d2 = N0.diagonal(N0.dot(y, N0.transpose(y)))
a1 = N0.add.outer(d1, d2)
a2 = N0.dot(x, N0.transpose(y))
return a1 - 2 * a2
def squared_distance_matrix(x, y):
d1 = N0.diagonal(N0.dot(x, N0.transpose(x)))
d2 = N0.diagonal(N0.dot(y, N0.transpose(y)))
a1 = N0.add.outer(d1,d2)
a2 = N0.dot(x, N0.transpose(y))
return a1 - 2 * a2
def pairwiseDistances(u, v):
"""
Pairwise distances between two arrays.
:param u: first array
:type u: array
:param v: second array
:type v: array
:return: array( len(u) x len(v) ) of double
:rtype: array
"""
diag1 = N0.diagonal(N0.dot(u, N0.transpose(u)))
diag2 = N0.diagonal(N0.dot(v, N0.transpose(v)))
dist = -N0.dot( v,N0.transpose(u) )\
-N0.transpose( N0.dot( u, N0.transpose(v) ) )
dist = N0.transpose( N0.asarray( list(map( lambda column,a:column+a, \
N0.transpose(dist), diag1)) ) )
return N0.transpose(
N0.sqrt(N0.asarray(list(map(lambda row, a: row + a, dist, diag2)))))
def pairwiseDistances(u, v):
"""
Pairwise distances between two arrays.
:param u: first array
:type u: array
:param v: second array
:type v: array
:return: array( len(u) x len(v) ) of double
:rtype: array
"""
diag1 = N0.diagonal( N0.dot( u, N0.transpose(u) ) )
diag2 = N0.diagonal( N0.dot( v, N0.transpose(v) ) )
dist = -N0.dot( v,N0.transpose(u) )\
-N0.transpose( N0.dot( u, N0.transpose(v) ) )
dist = N0.transpose( N0.asarray( list(map( lambda column,a:column+a, \
N0.transpose(dist), diag1)) ) )
return N0.transpose( N0.sqrt( N0.asarray(
list(map( lambda row,a: row+a, dist, diag2 ) ) )))
def error(self, msm, d2):
"""
@param msm: membership matrix
@type msm: array('f')
@param d2: distance from data to the centers
@type d2: array('f')
@return: weighted error
@rtype: float
"""
p = N0.power(msm, self.w)
product = N0.dot(p, N0.transpose(d2))
return N0.trace(product)
def error(self, msm, d2):
"""
@param msm: membership matrix
@type msm: array('f')
@param d2: distance from data to the centers
@type d2: array('f')
@return: weighted error
@rtype: float
"""
p = N0.power(msm, self.w)
product = N0.dot(p, N0.transpose(d2))
return N0.trace(product)
def transform( self, *rt ):
"""
Apply given transformation to all frames (in place).
:param rt: rotation translation matrix
:type rt: array( 4 x 4 ) OR array(3 x 3), array(3 x 1)
"""
if len(rt) == 2:
r, t = rt[0], rt[1]
else:
rt = rt[0]
r, t = (rt[0:3,0:3], rt[0:3, 3])
r = N0.transpose( r )
r = r.astype(N0.Float32)
t = t.astype(N0.Float32)
for i in range( len( self.frames ) ):
self.frames[ i ] = N0.array( N0.dot( self.frames[i], r ) ) + t
def rowDistances( x, y ):
"""
Calculate the distances between the items of two arrays (of same shape)
after least-squares superpositioning.
:param x: first set of coordinates
:type x: array('f')
:param y: second set of coordinates
:type y: array('f')
:return: array( len(x), 'f' ), distance between x[i] and y[i] for all i
:rtype: array
"""
## find transformation for best match
r, t = findTransformation(x, y)
## transform coordinates
z = N0.dot(y, N0.transpose(r)) + t
## calculate row distances
return N0.sqrt(N0.sum(N0.power(x - z, 2), 1))
def rowDistances(x, y):
"""
Calculate the distances between the items of two arrays (of same shape)
after least-squares superpositioning.
:param x: first set of coordinates
:type x: array('f')
:param y: second set of coordinates
:type y: array('f')
:return: array( len(x), 'f' ), distance between x[i] and y[i] for all i
:rtype: array
"""
## find transformation for best match
r, t = findTransformation(x, y)
## transform coordinates
z = N0.dot(y, N0.transpose(r)) + t
## calculate row distances
return N0.sqrt(N0.sum(N0.power(x - z, 2), 1))
def clusterEntropy(self):
centropy = N0.diagonal(N0.dot(self.msm,
N0.transpose(N0.log(self.msm))))
return -1 / float(self.npoints) * centropy
def match(x, y, n_iterations=1, z=2, eps_rmsd=0.5, eps_stdv=0.05):
"""
Matches two arrays onto each other, while iteratively removing outliers.
Superimposed array y would be C{ N0.dot(y, N0.transpose(r)) + t }.
:param n_iterations: number of calculations::
1 .. no iteration
0 .. until convergence
:type n_iterations: 1|0
:param z: number of standard deviations for outlier definition (default: 2)
:type z: float
:param eps_rmsd: tolerance in rmsd (default: 0.5)
:type eps_rmsd: float
:param eps_stdv: tolerance in standard deviations (default: 0.05)
:type eps_stdv: float
:return: (r,t), [ [percent_considered, rmsd_for_it, outliers] ]
:rtype: (array, array), [float, float, int]
"""
iter_trace = []
rmsd_old = 0
stdv_old = 0
n = 0
converged = 0
mask = N0.ones(len(y), N0.Int32)
while not converged:
## find transformation for best match
r, t = findTransformation(N0.compress(mask, x, 0),
N0.compress(mask, y, 0))
## transform coordinates
xt = N0.dot(y, N0.transpose(r)) + t
## calculate row distances
d = N0.sqrt(N0.sum(N0.power(x - xt, 2), 1)) * mask
## calculate rmsd and stdv
rmsd = N0.sqrt(N0.average(N0.compress(mask, d)**2))
stdv = MU.SD(N0.compress(mask, d))
## check conditions for convergence
d_rmsd = abs(rmsd - rmsd_old)
d_stdv = abs(1 - stdv_old / stdv)
if d_rmsd < eps_rmsd and d_stdv < eps_stdv:
converged = 1
else:
rmsd_old = rmsd
stdv_old = stdv
## store result
perc = round(float(N0.sum(mask)) / float(len(mask)), 2)
## throw out non-matching rows
mask = N0.logical_and(mask, N0.less(d, rmsd + z * stdv))
outliers = N0.nonzero(N0.logical_not(mask))
iter_trace.append([perc, round(rmsd, 3), outliers])
n += 1
if n_iterations and n >= n_iterations:
break
return (r, t), iter_trace
def match(x, y, n_iterations=1, z=2, eps_rmsd=0.5, eps_stdv=0.05):
"""
Matches two arrays onto each other, while iteratively removing outliers.
Superimposed array y would be C{ N0.dot(y, N0.transpose(r)) + t }.
:param n_iterations: number of calculations::
1 .. no iteration
0 .. until convergence
:type n_iterations: 1|0
:param z: number of standard deviations for outlier definition (default: 2)
:type z: float
:param eps_rmsd: tolerance in rmsd (default: 0.5)
:type eps_rmsd: float
:param eps_stdv: tolerance in standard deviations (default: 0.05)
:type eps_stdv: float
:return: (r,t), [ [percent_considered, rmsd_for_it, outliers] ]
:rtype: (array, array), [float, float, int]
"""
iter_trace = []
rmsd_old = 0
stdv_old = 0
n = 0
converged = 0
mask = N0.ones(len(y), N0.Int32 )
while not converged:
## find transformation for best match
r, t = findTransformation(N0.compress(mask, x, 0),
N0.compress(mask, y, 0))
## transform coordinates
xt = N0.dot(y, N0.transpose(r)) + t
## calculate row distances
d = N0.sqrt(N0.sum(N0.power(x - xt, 2), 1)) * mask
## calculate rmsd and stdv
rmsd = N0.sqrt(N0.average(N0.compress(mask, d)**2))
stdv = MU.SD(N0.compress(mask, d))
## check conditions for convergence
d_rmsd = abs(rmsd - rmsd_old)
d_stdv = abs(1 - stdv_old / stdv)
if d_rmsd < eps_rmsd and d_stdv < eps_stdv:
converged = 1
else:
rmsd_old = rmsd
stdv_old = stdv
## store result
perc = round(float(N0.sum(mask)) / float(len(mask)), 2)
## throw out non-matching rows
mask = N0.logical_and(mask, N0.less(d, rmsd + z * stdv))
outliers = N0.nonzero( N0.logical_not( mask ) )
iter_trace.append([perc, round(rmsd, 3), outliers])
n += 1
if n_iterations and n >= n_iterations:
break
return (r, t), iter_trace
def clusterEntropy(self):
centropy = N0.diagonal(N0.dot(self.msm,
N0.transpose(N0.log(self.msm))))
return -1/float(self.npoints)*centropy
def pcMovie( self, ev, steps, factor=1., ref=0, morph=1 ):
"""
Morph between the two extreme values of a single principal
component.
:param ev: EigenVector to visualize
:type ev: int
:param steps: number of intermediate frames
:type steps: int
:param factor: exageration factor (default: 1 = No exageration)
:type factor: float
:param ref: take other eigenvecors from this frame (default: 1)
:type ref: int
:param morph: morph between min and max (1) or take real values (0)
(default: 1)
:type morph: 1|0
:return: Trajectory with frames visualizing the morphing.
:rtype: Trajectory
"""
fit = 1
if self.pc is not None:
fit = self.pc['fit']
pc = self.getPca( fit=fit )
## eigenvectors (rows)
U = pc['u']
## raveled and centered frames
x_avg = N0.average(self.frames, 0)
X = N0.array( [N0.ravel(x) for x in self.frames - x_avg] )
## ev'th eigenvector of reference frame
alpha_0 = N0.dot( X[ref], U[ev] )
## list of deviations of ev'th eigenvector of each frame from ref
alpha_range = N0.dot(X, U[ev]) - alpha_0
## get some representative alphas...
if morph:
a_min = factor * min(alpha_range)
a_max = factor * max(alpha_range)
delta = (a_max - a_min) / steps
alpha_range = [ a_min + i*(delta) for i in range(0, steps) ]
else:
alpha_range = N0.sort( alpha_range )
delta = len(alpha_range) / (steps * 1.0)
alpha_range = [ alpha_range[ int(round( i*delta )) ]
for i in range(0,steps) ]
## scale ev'th eigenvector of ref with different alphas
Y = N0.array( [ X[ref] + alpha * U[ev] for alpha in alpha_range] )
## back convert to N x 3 coordinates
Y = N0.reshape(Y, (Y.shape[0], -1, 3))
Y = x_avg + Y
result = self.__class__()
result.ref = self.ref
result.frames = Y
return result
def fit( self, mask=None, ref=None, n_it=1,
prof='rms', verbose=1, fit=1, **profInfos ):
"""
Superimpose all coordinate frames on reference coordinates. Put rms
values in a profile. If n_it > 1, the fraction of atoms considered
for the fit is put into a profile called |prof|_considered
(i.e. by default 'rms_considered').
:param mask: atom mask, atoms to consider default: [all]
:type mask: [1|0]
:param ref: use as reference, default: None, average Structure
:type ref: PDBModel
:param n_it: number of fit iterations, kicking out outliers on the way
1 -> classic single fit, 0 -> until convergence
(default: 1)
:type n_it: int
:param prof: save rms per frame in profile of this name, ['rms']
:type prof: str
:param verbose: print progress info to STDERR (default: 1)
:type verbose: 1|0
:param fit: transform frames after match, otherwise just calc rms
(default: 1)
:type fit: 1|0
:param profInfos: additional key=value pairs for rms profile info []
:type profInfos: key=value
"""
if ref is None:
refxyz = N0.average( self.frames, 0 )
else:
refxyz = ref.getXyz()
if mask is None:
mask = N0.ones( len( refxyz ), N0.Int32 )
refxyz = N0.compress( mask, refxyz, 0 )
if verbose: T.errWrite( "rmsd fitting..." )
rms = [] ## rms value of each frame
non_outliers = [] ## fraction of atoms considered for rms and fit
iterations = [] ## number of iterations performed on each frame
for i in range(0, len( self.frames) ):
xyz = self.frames[i]
if n_it != 1:
(r, t), rmsdList = rmsFit.match( refxyz,
N0.compress( mask, xyz, 0), n_it)
iterations.append( len( rmsdList ) )
non_outliers.append( rmsdList[-1][0] )
xyz_transformed = N0.dot( xyz, N0.transpose(r)) + t
rms += [ rmsdList[-1][1] ]
else:
r, t = rmsFit.findTransformation( refxyz,
N0.compress( mask, xyz, 0))
xyz_transformed = N0.dot( xyz, N0.transpose(r)) + t
d = N0.sqrt(N0.sum(N0.power( N0.compress(mask, xyz_transformed,0)\
- refxyz, 2), 1))
rms += [ N0.sqrt( N0.average(d**2) ) ]
if fit:
self.frames[i] = xyz_transformed.astype(N0.Float32)
if verbose and i%100 == 0:
T.errWrite( '#' )
self.setProfile( prof, rms, n_iterations=n_it, **profInfos )
if non_outliers:
self.setProfile( prof+'_considered', non_outliers,
n_iterations=n_it,
comment='fraction of atoms considered for iterative fit' )
if verbose: T.errWrite( 'done\n' )
def calc_cluster_center(self, msm):
p = N0.power(msm, self.w)
ccenter = N0.transpose(N0.dot(p, self.data))
return N0.transpose(ccenter / N0.sum(p, 1))
def calc_cluster_center(self, msm):
p = N0.power(msm, self.w)
ccenter = N0.transpose(N0.dot(p, self.data))
return N0.transpose(ccenter / N0.sum(p, 1))
