/usr/lib/python2.7/dist-packages/csb/bio/utils/__init__.py is in python-csb 1.2.3+dfsg-1.
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Computational utility functions.
This module defines a number of low-level, numerical, high-performance utility
functions like L{rmsd} for example.
"""
import numpy
import numpy.random
import csb.numeric
def fit(X, Y):
"""
Return the translation vector and the rotation matrix
minimizing the RMSD between two sets of d-dimensional
vectors, i.e. if
>>> R,t = fit(X,Y)
then
>>> Y = dot(Y, transpose(R)) + t
will be the fitted configuration.
@param X: (n, d) input vector
@type X: numpy array
@param Y: (n, d) input vector
@type Y: numpy array
@return: (d, d) rotation matrix and (d,) translation vector
@rtype: tuple
"""
from numpy.linalg import svd, det
from numpy import dot
## center configurations
x = X.mean(0)
y = Y.mean(0)
## SVD of correlation matrix
V, _L, U = svd(dot((X - x).T, Y - y))
## calculate rotation and translation
R = dot(V, U)
if det(R) < 0.:
U[-1] *= -1
R = dot(V, U)
t = x - dot(R, y)
return R, t
def wfit(X, Y, w):
"""
Return the translation vector and the rotation matrix
minimizing the weighted RMSD between two sets of d-dimensional
vectors, i.e. if
>>> R,t = fit(X,Y)
then
>>> Y = dot(Y, transpose(R)) + t
will be the fitted configuration.
@param X: (n, d) input vector
@type X: numpy array
@param Y: (n, d) input vector
@type Y: numpy array
@param w: input weights
@type w: numpy array
@return: (d, d) rotation matrix and (d,) translation vector
@rtype: tuple
"""
from numpy.linalg import svd, det
from numpy import dot, sum, average
## center configurations
norm = sum(w)
x = dot(w, X) / norm
y = dot(w, Y) / norm
## SVD of correlation matrix
V, _L, U = svd(dot((X - x).T * w, Y - y))
## calculate rotation and translation
R = dot(V, U)
if det(R) < 0.:
U[2] *= -1
R = dot(V, U)
t = x - dot(R, y)
return R, t
def scale_and_fit(X, Y, check_mirror_image=False):
"""
Return the translation vector, the rotation matrix and a
global scaling factor minimizing the RMSD between two sets
of d-dimensional vectors, i.e. if
>>> R, t, s = scale_and_fit(X, Y)
then
>>> Y = s * (dot(Y, R.T) + t)
will be the fitted configuration.
@param X: (n, d) input vector
@type X: numpy array
@param Y: (n, d) input vector
@type Y: numpy array
@return: (d, d) rotation matrix and (d,) translation vector
@rtype: tuple
"""
from numpy.linalg import svd, det
from numpy import dot, trace
## centers
x, y = X.mean(0), Y.mean(0)
## SVD of correlation matrix
V, L, U = svd(dot((X - x).T, Y - y))
## calculate rotation, scale and translation
R = dot(V, U)
if check_mirror_image and det(R) < 0:
U[-1] *= -1
L[-1] *= -1
R = dot(V, U)
s = (L.sum() / ((Y-y)**2).sum())
t = x / s - dot(R, y)
return R, t, s
def probabilistic_fit(X, Y, w=None, niter=10):
"""
Generate a superposition of X, Y where::
R ~ exp(trace(dot(transpose(dot(transpose(X-t), Y)), R)))
t ~ N(t_opt, 1 / sqrt(N))
@rtype: tuple
"""
from csb.statistics.rand import random_rotation
from numpy import dot, transpose, average
if w is None:
R, t = fit(X, Y)
else:
R, t = wfit(X, Y, w)
N = len(X)
for i in range(niter):
## sample rotation
if w is None:
A = dot(transpose(X - t), Y)
else:
A = dot(transpose(X - t) * w, Y)
R = random_rotation(A)
## sample translation (without prior so far)
if w is None:
mu = average(X - dot(Y, transpose(R)), 0)
t = numpy.random.standard_normal(len(mu)) / numpy.sqrt(N) + mu
else:
mu = dot(w, X - dot(Y, transpose(R))) / numpy.sum(w)
t = numpy.random.standard_normal(len(mu)) / numpy.sqrt(numpy.sum(w)) + mu
return R, t
def fit_wellordered(X, Y, n_iter=None, n_stdv=2, tol_rmsd=.5,
tol_stdv=0.05, full_output=False):
"""
Match two arrays onto each other by iteratively throwing out
highly deviating entries.
(Reference: Nilges et al.: Delineating well-ordered regions in
protein structure ensembles).
@param X: (n, d) input vector
@type X: numpy array
@param Y: (n, d) input vector
@type Y: numpy array
@param n_stdv: number of standard deviations above which points are considered to be outliers
@param tol_rmsd: tolerance in rmsd
@param tol_stdv: tolerance in standard deviations
@param full_output: also return full history of values calculated by the algorithm
@rtype: tuple
"""
from numpy import ones, compress, dot, sqrt, sum, nonzero, std, average
rmsd_list = []
rmsd_old = 0.
stdv_old = 0.
n = 0
converged = False
mask = ones(X.shape[0])
while not converged:
## find transformation for best match
R, t = fit(compress(mask, X, 0), compress(mask, Y, 0))
## calculate RMSD profile
d = sqrt(sum((X - dot(Y, R.T) - t) ** 2, 1))
## calculate rmsd and stdv
rmsd = sqrt(average(compress(mask, d) ** 2, 0))
stdv = std(compress(mask, d))
## check conditions for convergence
if stdv < 1e-10: break
d_rmsd = abs(rmsd - rmsd_old)
d_stdv = abs(1 - stdv_old / stdv)
if d_rmsd < tol_rmsd:
if d_stdv < tol_stdv:
converged = 1
else:
stdv_old = stdv
else:
rmsd_old = rmsd
stdv_old = stdv
## store result
perc = average(1. * mask)
## throw out non-matching rows
new_mask = mask * (d < rmsd + n_stdv * stdv)
outliers = nonzero(mask - new_mask)
rmsd_list.append([perc, rmsd, outliers])
mask = new_mask
if n_iter and n >= n_iter: break
n += 1
if full_output:
return (R, t), rmsd_list
else:
return (R, t)
def bfit(X, Y, n_iter=10, distribution='student', em=False, full_output=False):
"""
Robust superposition of two coordinate arrays. Models non-rigid
displacements with outlier-tolerant probability distributions.
@param X: (n, 3) input vector
@type X: numpy.array
@param Y: (n, 3) input vector
@type Y: numpy.array
@param n_iter: number of iterations
@type n_iter: int
@param distribution: student or k
@type distribution: str
@param em: use maximum a posteriori probability (MAP) estimator
@type em: bool
@param full_output: if true, return ((R, t), scales)
@type full_output: bool
@rtype: tuple
"""
from csb.statistics import scalemixture as sm
if distribution == 'student':
prior = sm.GammaPrior()
if em:
prior.estimator = sm.GammaPosteriorMAP()
elif distribution == 'k':
prior = sm.InvGammaPrior()
if em:
prior.estimator = sm.InvGammaPosteriorMAP()
else:
raise AttributeError('distribution')
mixture = sm.ScaleMixture(scales=X.shape[0], prior=prior, d=3)
R, t = fit(X, Y)
for _ in range(n_iter):
data = distance(X, transform(Y, R, t))
mixture.estimate(data)
R, t = probabilistic_fit(X, Y, mixture.scales)
if full_output:
return (R, t), mixture.scales
else:
return (R, t)
def xfit(X, Y, n_iter=10, seed=False, full_output=False):
"""
Maximum likelihood superposition of two coordinate arrays. Works similar
to U{Theseus<http://theseus3d.org>} and to L{bfit}.
@param X: (n, 3) input vector
@type X: numpy.array
@param Y: (n, 3) input vector
@type Y: numpy.array
@param n_iter: number of EM iterations
@type n_iter: int
@type seed: bool
@param full_output: if true, return ((R, t), scales)
@type full_output: bool
@rtype: tuple
"""
if seed:
R, t = numpy.identity(3), numpy.zeros(3)
else:
R, t = fit(X, Y)
for _ in range(n_iter):
data = distance_sq(X, transform(Y, R, t))
scales = 1.0 / data.clip(1e-9)
R, t = wfit(X, Y, scales)
if full_output:
return (R, t), scales
else:
return (R, t)
def transform(Y, R, t, s=None, invert=False):
"""
Transform C{Y} by rotation C{R} and translation C{t}. Optionally scale by C{s}.
>>> R, t = fit(X, Y)
>>> Y_fitted = transform(Y, R, t)
@param Y: (n, d) input vector
@type Y: numpy.array
@param R: (d, d) rotation matrix
@type R: numpy.array
@param t: (d,) translation vector
@type t: numpy.array
@param s: scaling factor
@type s: float
@param invert: if True, apply the inverse transformation
@type invert: bool
@return: transformed input vector
@rtype: numpy.array
"""
if invert:
x = numpy.dot(Y - t, R)
if s is not None:
s = 1. / s
else:
x = numpy.dot(Y, R.T) + t
if s is not None:
x *= s
return x
def fit_transform(X, Y, fit=fit, *args):
"""
Return Y superposed on X.
@type X: (n,3) numpy.array
@type Y: (n,3) numpy.array
@type fit: function
@rtype: (n,3) numpy.array
"""
return transform(Y, *fit(X, Y, *args))
def rmsd(X, Y):
"""
Calculate the root mean squared deviation (RMSD) using Kabsch' formula.
@param X: (n, d) input vector
@type X: numpy array
@param Y: (n, d) input vector
@type Y: numpy array
@return: rmsd value between the input vectors
@rtype: float
"""
from numpy import sum, dot, sqrt, clip, average
from numpy.linalg import svd, det
X = X - X.mean(0)
Y = Y - Y.mean(0)
R_x = sum(X ** 2)
R_y = sum(Y ** 2)
V, L, U = svd(dot(Y.T, X))
if det(dot(V, U)) < 0.:
L[-1] *= -1
return sqrt(clip(R_x + R_y - 2 * sum(L), 0., 1e300) / len(X))
def rmsd_cur(X, Y):
"""
Calculate the RMSD of two conformations as they are (no fitting is done).
For details, see L{rmsd}.
@return: rmsd value between the input vectors
@rtype: float
"""
return distance_sq(X, Y).mean() ** 0.5
def wrmsd(X, Y, w):
"""
Calculate the weighted root mean squared deviation (wRMSD) using Kabsch'
formula.
@param X: (n, d) input vector
@type X: numpy array
@param Y: (n, d) input vector
@type Y: numpy array
@param w: input weights
@type w: numpy array
@return: rmsd value between the input vectors
@rtype: float
"""
from numpy import sum, dot, sqrt, clip, average
from numpy.linalg import svd
## normalize weights
w = w / w.sum()
X = X - dot(w, X)
Y = Y - dot(w, Y)
R_x = sum(X.T ** 2 * w)
R_y = sum(Y.T ** 2 * w)
L = svd(dot(Y.T * w, X))[1]
return sqrt(clip(R_x + R_y - 2 * sum(L), 0., 1e300))
def torsion_rmsd(x, y):
"""
Compute the circular RMSD of two phi/psi angle sets.
@param x: query phi/psi angles (Nx2 array, in radians)
@type x: array
@param y: subject phi/psi angles (Nx2 array, in radians)
@type y: array
@rtype: float
"""
from numpy import array, sin, cos, sqrt
phi, psi = (x - y).T
assert len(phi) == len(psi)
r = sin(phi).sum() ** 2 + cos(phi).sum() ** 2 + sin(psi).sum() ** 2 + cos(psi).sum() ** 2
return 1 - (1.0 / len(phi)) * sqrt(r / 2.0)
def _tm_d0(Lmin):
from numpy import power
if Lmin > 15:
d0 = 1.24 * power(Lmin - 15.0, 1.0 / 3.0) - 1.8
else:
d0 = 0.5
return max(0.5, d0)
def tm_score(x, y, L=None, d0=None):
"""
Evaluate the TM-score of two conformations as they are (no fitting is done).
@param x: 3 x n input array
@type x: numpy array
@param y: 3 x n input array
@type y: numpy array
@param L: length for normalization (default: C{len(x)})
@type L: int
@param d0: d0 in Angstroms (default: calculate from C{L})
@type d0: float
@return: computed TM-score
@rtype: float
"""
from numpy import sum
if not L:
L = len(x)
if not d0:
d0 = _tm_d0(L)
d = distance(x, y)
return sum(1 / (1 + (d / d0) ** 2)) / L
def tm_superimpose(x, y, fit_method=fit, L=None, d0=None, L_ini_min=4, iL_step=1):
"""
Compute the TM-score of two protein coordinate vector sets.
Reference: http://zhanglab.ccmb.med.umich.edu/TM-score
@param x: 3 x n input vector
@type x: numpy.array
@param y: 3 x n input vector
@type y: numpy.array
@param fit_method: a reference to a proper fitting function, e.g. fit
or fit_wellordered
@type fit_method: function
@param L: length for normalization (default: C{len(x)})
@type L: int
@param d0: d0 in Angstroms (default: calculate from C{L})
@type d0: float
@param L_ini_min: minimum length of initial alignment window (increase
to speed up but loose precision, a value of 0 disables local alignment
initialization)
@type L_ini_min: int
@param iL_step: initial alignment window shift (increase to speed up
but loose precision)
@type iL_step: int
@return: rotation matrix, translation vector, TM-score
@rtype: tuple
"""
from numpy import asarray, sum, dot, zeros, clip
x, y = asarray(x), asarray(y)
if not L:
L = len(x)
if not d0:
d0 = _tm_d0(L)
d0_search = clip(d0, 4.5, 8.0)
best = None, None, 0.0
L_ini_min = min(L, L_ini_min) if L_ini_min else L
L_ini = [L_ini_min] + list(filter(lambda x: x > L_ini_min,
[L // (2 ** n_init) for n_init in range(6)]))
# the outer two loops define a sliding window of different sizes for the
# initial local alignment (disabled with L_ini_min=0)
for L_init in L_ini:
for iL in range(0, L - L_init + 1, min(L_init, iL_step)):
mask = zeros(L, bool)
mask[iL:iL + L_init] = True
# refine mask until convergence, similar to fit_wellordered
for i in range(20):
R, t = fit_method(x[mask], y[mask])
d = distance(x, dot(y, R.T) + t)
score = sum(1 / (1 + (d / d0) ** 2)) / L
if score > best[-1]:
best = R, t, score
mask_prev = mask
cutoff = d0_search + (-1 if i == 0 else 1)
while True:
mask = d < cutoff
if sum(mask) >= 3 or 3 >= len(mask):
break
cutoff += 0.5
if (mask == mask_prev).all():
break
return best
def center_of_mass(x, m=None):
"""
Compute the mean of a set of (optionally weighted) points.
@param x: array of rank (n,d) where n is the number of points
and d the dimension
@type x: numpy.array
@param m: rank (n,) array of masses / weights
@type m: numpy.array
@return: center of mass
@rtype: (d,) numpy.array
"""
if m is None:
return x.mean(0)
else:
from numpy import dot
return dot(m, x) / m.sum()
def radius_of_gyration(x, m=None):
"""
Compute the radius of gyration of a set of (optionally weighted) points.
@param x: array of rank (n,d) where n is the number of points
and d the dimension
@type x: numpy.array
@param m: rank (n,) array of masses / weights
@type m: numpy.array
@return: center of mass
@rtype: (d,) numpy.array
"""
from numpy import sqrt, dot
x = x - center_of_mass(x, m)
if m is None:
return sqrt((x ** 2).sum() / len(x))
else:
return sqrt(dot(m, (x ** 2).sum(1)) / m.sum())
def second_moments(x, m=None):
"""
Compute the tensor second moments of a set of (optionally weighted) points.
@param x: array of rank (n,d) where n is the number of points
and d the dimension
@type x: numpy.array
@param m: rank (n,) array of masses / weights
@type m: numpy.array
@return: second moments
@rtype: (d,d) numpy.array
"""
from numpy import dot
x = (x - center_of_mass(x, m)).T
if m is not None:
return dot(x * m, x.T)
else:
return dot(x, x.T)
def inertia_tensor(x, m=None):
"""
Compute the inertia tensor of a set of (optionally weighted) points.
@param x: array of rank (n,d) where n is the number of points
and d the dimension
@type x: numpy.array
@param m: rank (n,) array of masses / weights
@type m: numpy.array
@return: inertia tensor
@rtype: (d,d) numpy.array
"""
from numpy import dot, eye
x = (x - center_of_mass(x, m)).T
r2 = (x ** 2).sum(0)
if m is not None:
return eye(x.shape[0]) * dot(m, r2) - dot(x * m, x.T)
else:
return eye(x.shape[0]) * r2.sum() - dot(x, x.T)
def find_pairs(cutoff, X, Y=None):
"""
Find pairs with euclidean distance below C{cutoff}. Either between
C{X} and C{Y}, or within C{X} if C{Y} is C{None}.
Uses a KDTree and thus is memory efficient and reasonable fast.
@type cutoff: float
@type X: (m,n) numpy.array
@type Y: (k,n) numpy.array
@return: set of index tuples
@rtype: iterable
"""
try:
from scipy.spatial import cKDTree as KDTree
KDTree.query_pairs
KDTree.query_ball_tree
except (ImportError, AttributeError):
from scipy.spatial import KDTree
tree = KDTree(X, len(X))
if Y is None:
return tree.query_pairs(cutoff)
other = KDTree(Y, len(Y))
contacts = tree.query_ball_tree(other, cutoff)
return ((i, j) for (i, js) in enumerate(contacts) for j in js)
def distance_matrix(X, Y=None):
"""
Calculates a matrix of pairwise distances
@param X: m x n input vector
@type X: numpy array
@param Y: k x n input vector or None, which defaults to Y=X
@type Y: numpy array
@return: m x k distance matrix
@rtype: numpy array
"""
from numpy import add, clip, sqrt, dot, transpose, sum
if Y is None: Y = X
if X.ndim < 2: X = X.reshape((1, -1))
if Y.ndim < 2: Y = Y.reshape((1, -1))
C = dot(X, transpose(Y))
S = add.outer(sum(X ** 2, 1), sum(Y ** 2, 1))
return sqrt(clip(S - 2 * C, 0., 1e300))
def distance_sq(X, Y):
"""
Squared distance between C{X} and C{Y} along the last axis. For details, see L{distance}.
@return: scalar or array of length m
@rtype: (m,) numpy.array
"""
return ((numpy.asarray(X) - Y) ** 2).sum(-1)
def distance(X, Y):
"""
Distance between C{X} and C{Y} along the last axis.
@param X: m x n input vector
@type X: numpy array
@param Y: m x n input vector
@type Y: numpy array
@return: scalar or array of length m
@rtype: (m,) numpy.array
"""
return distance_sq(X, Y) ** 0.5
def average_structure(X):
"""
Calculate an average structure from an ensemble of structures
(i.e. X is a rank-3 tensor: X[i] is a (N,3) configuration matrix).
@param X: m x n x 3 input vector
@type X: numpy array
@return: average structure
@rtype: (n,3) numpy.array
"""
from numpy.linalg import eigh
B = csb.numeric.gower_matrix(X)
v, U = eigh(B)
if numpy.iscomplex(v).any():
v = v.real
if numpy.iscomplex(U).any():
U = U.real
indices = numpy.argsort(v)[-3:]
v = numpy.take(v, indices, 0)
U = numpy.take(U, indices, 1)
x = U * numpy.sqrt(v)
i = 0
while is_mirror_image(x, X[0]) and i < 2:
x[:, i] *= -1
i += 1
return x
def is_mirror_image(X, Y):
"""
Check if two configurations X and Y are mirror images
(i.e. their optimal superposition involves a reflection).
@param X: n x 3 input vector
@type X: numpy array
@param Y: n x 3 input vector
@type Y: numpy array
@rtype: bool
"""
from numpy.linalg import det, svd
## center configurations
X = X - numpy.mean(X, 0)
Y = Y - numpy.mean(Y, 0)
## SVD of correlation matrix
V, L, U = svd(numpy.dot(numpy.transpose(X), Y)) #@UnusedVariable
R = numpy.dot(V, U)
return det(R) < 0
def deg(x):
"""
Convert an array of torsion angles in radians to torsion degrees
ranging from -180 to 180.
@param x: array of angles
@type x: numpy array
@rtype: numpy array
"""
from csb.bio.structure import TorsionAngles
func = numpy.vectorize(TorsionAngles.deg)
return func(x)
def rad(x):
"""
Convert an array of torsion angles in torsion degrees to radians.
@param x: array of angles
@type x: numpy array
@rtype: numpy array
"""
from csb.bio.structure import TorsionAngles
func = numpy.vectorize(TorsionAngles.rad)
return func(x)
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