def InnerProduct(self,vector,other): """Innerproduct for vectors""" if vector.GetSpace()==other.GetSpace(): from Numeric import matrixmultiply,transpose # (v,o)=(v_coor^T*basis^T*basis*o_coor) metric=matrixmultiply(self.GetBasis(),transpose(self.GetBasis())) return dot(vector.GetCoordinates(),matrixmultiply(metric,other.GetCoordinates())) else: raise ValueError, 'Innerproduct of two vectors in different vector space not implemented'
def InnerProduct(self, vector, other):
"""Innerproduct for vectors"""
if vector.GetSpace() == other.GetSpace():
from Numeric import matrixmultiply, transpose
# (v,o)=(v_coor^T*basis^T*basis*o_coor)
metric = matrixmultiply(self.GetBasis(),
transpose(self.GetBasis()))
return dot(vector.GetCoordinates(),
matrixmultiply(metric, other.GetCoordinates()))
else:
raise ValueError, 'Innerproduct of two vectors in different vector space not implemented'
def corrcoef(*args):
"""
corrcoef(X) where X is a matrix returns a matrix of correlation
coefficients for each row of X.
corrcoef(x,y) where x and y are vectors returns the matrix or
correlation coefficients for x and y.
Numeric arrays can be real or complex
The correlation matrix is defined from the covariance matrix C as
r(i,j) = C[i,j] / (C[i,i]*C[j,j])
"""
if len(args) == 2:
X = transpose(array([args[0]] + [args[1]]))
elif len(args == 1):
X = args[0]
else:
raise RuntimeError, 'Only expecting 1 or 2 arguments'
C = cov(X)
d = resize(diagonal(C), (2, 1))
r = divide(C, sqrt(matrixmultiply(d, transpose(d))))[0, 1]
try:
return r.real
except AttributeError:
return r
def render(self):
s = self.svgDrawing
if not s:
return
x, y, z = self.body.getPosition()
modelview = glGetDoublev(GL_MODELVIEW_MATRIX)
projection = glGetDoublev(GL_PROJECTION_MATRIX)
viewport = glGetIntegerv(GL_VIEWPORT)
# calculate the z coordinate
m = transpose(reshape(modelview, (4, 4)))
wz = -matrixmultiply(m, reshape((x, y, z, 1), (4, 1)))[2][0]
# don't draw anything if we're behind the viewer
if wz < 0.1:
return
# calculate the screen-space x and y coordinates
x, y, z = gluProject(x, y, z, modelview, projection, viewport)
scale = self.scale / wz
s.transform.reset()
s.transform.translate(x, y)
s.transform.scale(scale, -scale)
s.transform.rotate(self.rotation)
s.draw()
def render(self):
s = self.svgDrawing
if not s:
return
x, y, z = self.body.getPosition()
modelview = glGetDoublev(GL_MODELVIEW_MATRIX)
projection = glGetDoublev(GL_PROJECTION_MATRIX)
viewport = glGetIntegerv(GL_VIEWPORT)
# calculate the z coordinate
m = transpose(reshape(modelview, (4, 4)))
wz = -matrixmultiply(m, reshape((x, y, z, 1), (4, 1)))[2][0]
# don't draw anything if we're behind the viewer
if wz < 0.1:
return
# calculate the screen-space x and y coordinates
x, y, z = gluProject(x, y, z, modelview, projection, viewport)
scale = self.scale / wz
s.transform.reset()
s.transform.translate(x, y)
s.transform.scale(scale, -scale)
s.transform.rotate(self.rotation)
s.draw()
def eigenvector_for_largest_eigenvalue(matrix):
"""Returns eigenvector corresponding to largest eigenvalue of matrix.
Implements a numerical method for finding an eigenvector by repeated
application of the matrix to a starting vector. For a matrix A the
process w(k) <-- A*w(k-1) converges to eigenvector w with the largest
eigenvalue. Because distance matrix D has all entries >= 0, a theorem
due to Perron and Frobenius on nonnegative matrices guarantees good
behavior of this method, excepting degenerate cases where the
iteration oscillates. For distance matrices a remedy is to add the
identity matrix to A, permitting the iteration to converge to the
eigenvector. (From Sander and Schneider (1991), and Vingron and
Sibbald (1993))
Note: Only works on square matrices.
"""
#always add the identity matrix to avoid oscillating behavior
matrix = matrix + identity(len(matrix))
#v is a random vector (chosen as the normalized vector of ones)
v = ones(len(matrix))/len(matrix)
#iterate until convergence
for i in range(1000):
new_v = matrixmultiply(matrix,v)
new_v = new_v/sum(new_v) #normalize
if sum(map(abs,new_v-v)) > 1e-9:
v = new_v #not converged yet
continue
else: #converged
break
return new_v
def _alpha(self,obsIndices):
""" computes forward values"""
B = self.B
A = self.A
alpha = [B[obsIndices[0]] * self.pi] # (19)
for o in obsIndices[1:]:
alpha.append(matrixmultiply(alpha[-1],A)*B[o]) # (20)
return alpha
def rotate(self, angle): m = identity(3, typecode = Float) s = sin(angle) c = cos(angle) m[0, 0] = c m[0, 1] = -s m[1, 0] = s m[1, 1] = c self.matrix = matrixmultiply(self.matrix, m)
def rotate(self, angle):
m = identity(3, typecode=Float)
s = sin(angle)
c = cos(angle)
m[0, 0] = c
m[0, 1] = -s
m[1, 0] = s
m[1, 1] = c
self.matrix = matrixmultiply(self.matrix, m)
def get_transformed(self):
"Get the transformed coordinate set."
if self.coords is None or self.reference_coords is None:
raise Exception, "No coordinates set."
if self.rot is None:
raise Exception, "Nothing superimposed yet."
if self.transformed_coords is None:
self.transformed_coords=matrixmultiply(self.coords, self.rot)+self.tran
return self.transformed_coords
def _beta(self,obsIndices,scale_factors=None):
""" computes backward values"""
B = self.B
A = self.A
scale_factors = scale_factors or list(ones(len(obsIndices),Float))
beta = [ones(self.N,Float)*scale_factors[-1]] # (24)
for t in range(len(obsIndices)-2,-1,-1):
beta.append(matrixmultiply(A,(1./scale_factors[t])*B[obsIndices[t+1]]*beta[-1])) # (25)
beta.reverse()
return beta
def __rmul__(self,other):
import copy
from Numeric import asarray,matrixmultiply
a=copy.copy(self)
if asarray(other).shape == ():
a.SetBasis(other*self.GetBasis())
elif asarray(other).shape == self.GetBasis().shape:
a.SetBasis(matrixmultiply(other,self.GetBasis()))
else:
raise ValueError, 'matrices are not aligned'
return a
def run(self):
"Superimpose the coordinate sets."
if self.coords is None or self.reference_coords is None:
raise Exception, "No coordinates set."
coords=self.coords
reference_coords=self.reference_coords
# center on centroid
av1=sum(coords)/self.n
av2=sum(reference_coords)/self.n
coords=coords-av1
reference_coords=reference_coords-av2
# correlation matrix
a=matrixmultiply(transpose(coords), reference_coords)
u, d, vt=singular_value_decomposition(a)
self.rot=transpose(matrixmultiply(transpose(vt), transpose(u)))
# check if we have found a reflection
if determinant(self.rot)<0:
vt[2]=-vt[2]
self.rot=transpose(matrixmultiply(transpose(vt), transpose(u)))
self.tran=av2-matrixmultiply(av1, self.rot)
def __rmul__(self, other):
import copy
from Numeric import asarray, matrixmultiply
a = copy.copy(self)
if asarray(other).shape == ():
a.SetBasis(other * self.GetBasis())
elif asarray(other).shape == self.GetBasis().shape:
a.SetBasis(matrixmultiply(other, self.GetBasis()))
else:
raise ValueError, 'matrices are not aligned'
return a
def _alpha_scaled(self,obsIndices):
""" computes forward values"""
B = self.B
A = self.A
alpha_t = B[obsIndices[0]] * self.pi # (19)
scaling_factors = [sum(alpha_t)]
alpha_scaled = [alpha_t/scaling_factors[-1]]
for o in obsIndices[1:]:
alpha_t = matrixmultiply(alpha_scaled[-1],A)*B[o] # (92a)
scaling_factors.append(sum(alpha_t))
alpha_scaled.append(alpha_t/scaling_factors[-1]) # (92b)
return alpha_scaled,scaling_factors
def applyAttributes(self, attrs, key="transform"):
transform = attrs.get(key)
if transform:
m = re.match(r"translate\(\s*(.+?)\s*,(.+?)\s*\)", transform)
if m:
dx, dy = [float(c) for c in m.groups()]
self.matrix[0, 2] += dx
self.matrix[1, 2] += dy
m = re.match(r"matrix\(\s*" + "\s*,\s*".join(["(.+?)"] * 6) + r"\s*\)", transform)
if m:
e = [float(c) for c in m.groups()]
e = [e[0], e[2], e[4], e[1], e[3], e[5], 0, 0, 1]
m = reshape(e, (3, 3))
self.matrix = matrixmultiply(self.matrix, m)
def savitzky_golay(window_size=None,order=2):
if window_size is None:
window_size = order + 2
if window_size % 2 != 1 or window_size < 1:
raise TypeError("window size must be a positive odd number")
if window_size < order + 2:
raise TypeError("window size is too small for the polynomial")
# A second order polynomial has 3 coefficients
order_range = range(order+1)
half_window = (window_size-1)//2
B = array(
[ [k**i for i in order_range] for k in range(-half_window, half_window+1)] )
# -1
# [ T ] T
# [ B * B ] * B
M = matrixmultiply(
inverse( matrixmultiply(transpose(B), B)),
transpose(B)
)
return M
def applyAttributes(self, attrs, key = "transform"):
transform = attrs.get(key)
if transform:
m = re.match(r"translate\(\s*(.+?)\s*,(.+?)\s*\)", transform)
if m:
dx, dy = [float(c) for c in m.groups()]
self.matrix[0, 2] += dx
self.matrix[1, 2] += dy
m = re.match(r"matrix\(\s*" + "\s*,\s*".join(["(.+?)"] * 6) + r"\s*\)", transform)
if m:
e = [float(c) for c in m.groups()]
e = [e[0], e[2], e[4], e[1], e[3], e[5], 0, 0, 1]
m = reshape(e, (3, 3))
self.matrix = matrixmultiply(self.matrix, m)
def arbitaryRotation(coord, angle, axis):
"""arbitaryRotation(coord, angle, vector) --> rotates point by angle around vector
!!NB!! the axis is always through the origin!! must do offset first"""
u, v, w = normalise(axis)
oldCoord = array([coord[0], coord[1], coord[2]])
cth = cos(angle/180.*pi)
sth = sin(angle/180.*pi)
R = array([[cth + u**2*(1-cth), -w*sth+u*v*(1-cth), v*sth+u*w*(1-cth)],
[ w*sth+u*v*(1-cth), cth+v**2*(1-cth), -u*sth+v*w*(1-cth)],
[-v*sth+u*w*(1-cth), u*sth+v*w*(1-cth), cth+w**2*(1-cth)]])
newCoord = matrixmultiply(R,oldCoord)
return [newCoord[0], newCoord[1], newCoord[2]]
def transform(self, rot, tran):
"""
Apply rotation and translation to the atomic coordinates.
Example:
>>> rotation=rotmat(pi, Vector(1,0,0))
>>> translation=array((0,0,1), 'f')
>>> atom.transform(rotation, translation)
@param rot: A right multiplying rotation matrix
@type rot: 3x3 Numeric array
@param tran: the translation vector
@type tran: size 3 Numeric array
"""
self.coord=matrixmultiply(self.coord, rot)+tran
def rotmat(p,q):
"""
Return a (left multiplying) matrix that rotates p onto q.
Example:
>>> r=rotmat(p,q)
>>> print q, p.left_multiply(r)
@param p: moving vector
@type p: L{Vector}
@param q: fixed vector
@type q: L{Vector}
@return: rotation matrix that rotates p onto q
@rtype: 3x3 Numeric array
"""
rot=matrixmultiply(refmat(q, -p), refmat(p, -p))
return rot
def refmat(p,q):
"""
Return a (left multiplying) matrix that mirrors p onto q.
Example:
>>> mirror=refmat(p,q)
>>> qq=p.left_multiply(mirror)
>>> print q, qq # q and qq should be the same
@type p,q: L{Vector}
@return: The mirror operation, a 3x3 Numeric array.
"""
p.normalize()
q.normalize()
if (p-q).norm()<1e-5:
return eye(3)
pq=p-q
pq.normalize()
b=pq.get_array()
b.shape=(3, 1)
i=eye(3)
ref=i-2*matrixmultiply(b, transpose(b))
return ref
def transform(self, transform):
self.matrix = matrixmultiply(self.matrix, transform.matrix)
# start!
sup=SVDSuperimposer()
# set the coords
# y will be rotated and translated on x
sup.set(x, y)
# do the lsq fit
sup.run()
# get the rmsd
rms=sup.get_rms()
# get rotation (right multiplying!) and the translation
rot, tran=sup.get_rotran()
# rotate y on x
y_on_x1=matrixmultiply(y, rot)+tran
# same thing
y_on_x2=sup.get_transformed()
print y_on_x1
print
print y_on_x2
print
print "%.2f" % rms
def scale(self, sx, sy):
m = identity(3, typecode=Float)
m[0, 0] = sx
m[1, 1] = sy
self.matrix = matrixmultiply(self.matrix, m)
def applyTransform(self, transform):
m = matrixmultiply(transform.matrix, self.transform.matrix)
self.gradientDesc.SetMatrix(transform.getGMatrix(m))
def transform(self, transform): self.matrix = matrixmultiply(self.matrix, transform.matrix)
def scale(self, sx, sy): m = identity(3, typecode = Float) m[0, 0] = sx m[1, 1] = sy self.matrix = matrixmultiply(self.matrix, m)
def right_multiply(self, matrix):
"Return Vector=Vector x Matrix"
a=matrixmultiply(self._ar, matrix)
return Vector(a)
def left_multiply(self, matrix):
"Return Vector=Matrix x Vector"
a=matrixmultiply(matrix, self._ar)
return Vector(a)
def ACL(tree):
"""Returns a normalized dictionary of sequence weights {seq_id: weight}
tree: a PhyloNode object
The ACL method is named after Altschul, Carroll and Lipman, who published a
paper on sequence weighting in 1989.
The ACL method is based on an idea of Felsenstein (1973). Imagine
electrical current flows from the root of the tree down the edges and
out the leaves. If the edge lengths are proportional to their electrical
resistances, current flowing out each leaf equals the leaf weight.
The first step in the calculation of the weight vector is calculating a
variance-covariance matrix. The variance of a leaf is proportional to the
distance from the root to that leaf. The covariance of two leaves is
proportional to the distance from the root to the last common ancestor
of the two leaves.
The second step in the calculation results in a vector of weights. Suppose
there are n leaves on the tree. Let i be the vector of size n, all of whose
elements are 1.0. The weight vector is calculated as:
w = (inverse(M)*i)/(transpose(i)*inverse(M)*i)
See Altschul 1989
"""
#clip branch lengths to avoid error due to negative or zero branch lengths
_clip_branch_lengths(tree)
#get a list of sequence IDs (in the order that the tree will be traversed)
seqs = []
for n in tree.TerminalDescendants:
seqs.append(n.Data)
#initialize the variance-covariance matrix
m = zeros([len(seqs),len(seqs)],Float64)
#calculate (co)variances
#variance of a node is defined as the distance from the root to the leaf
#covariance of two nodes is defined as the distance from the root to the
#last common ancestor of the two leaves.
for x in tree.TerminalDescendants:
for y in tree.TerminalDescendants:
idx_x = seqs.index(x.Data)
idx_y = seqs.index(y.Data)
if idx_x == idx_y:
m[idx_x,idx_y] = x.distance(tree)
else:
lca = x.lastCommonAncestor(y)
dist_lca_root = lca.distance(tree)
m[idx_x,idx_y] = dist_lca_root
m[idx_y,idx_x] = dist_lca_root
#get the inverse of the variance-covariance matrix
inv = inverse(m)
#build vector i (vector or ones, length = # of leaves in the tree)
i = ones(len(seqs),Float64)
numerator = matrixmultiply(inv, i)
denominator = matrixmultiply(matrixmultiply(transpose(i),inv),i)
weight_vector = numerator/denominator
#return a Weights object (is dict {seq_id: weight})
return Weights(dict(zip(seqs,weight_vector)))
def iagaussian(s,mu,sigma):
""" o Purpose
Generate a 2D Gaussian image.
o Synopsis
g = iagaussian(s,mu,sigma)
o Input
s: [rows columns]
mu: Mean vector. 2D point (x;y). Point of maximum value.
sigma: covariance matrix (square). [ Sx^2 Sxy; Syx Sy^2]
o Output
g:
o Description
A 2D Gaussian image is an image with a Gaussian distribution. It can be used to generate test patterns or Gaussian filters both for spatial and frequency domain. The integral of the gaussian function is 1.0.
o Examples
import Numeric
f = iagaussian([8,4], [3,1], [[1,0],[0,1]])
print Numeric.array2string(f, precision=4, suppress_small=1)
g = ianormalize(f, [0,255]).astype(Numeric.UnsignedInt8)
print g
f = iagaussian(100, 50, 10*10)
g = ianormalize(f, [0,1])
g,d = iaplot(g)
showfig(g)
f = iagaussian([50,50], [25,10], [[10*10,0],[0,20*20]])
g = ianormalize(f, [0,255]).astype(Numeric.UnsignedInt8)
iashow(g)
"""
from Numeric import asarray,product,arange,NewAxis,transpose,matrixmultiply,reshape,concatenate,resize,sum,zeros,Float,ravel,pi,sqrt,exp
from LinearAlgebra import inverse,determinant
if type(sigma).__name__ in ['int', 'float', 'complex']: sigma = [sigma]
s, mu, sigma = asarray(s), asarray(mu), asarray(sigma)
if (product(s) == max(s)):
x = arange(product(s))
d = x - mu
if len(d.shape) == 1:
tmp1 = d[:,NewAxis]
tmp3 = d
else:
tmp1 = transpose(d)
tmp3 = tmp1
if len(sigma) == 1:
tmp2 = 1./sigma
else:
tmp2 = inverse(sigma)
k = matrixmultiply(tmp1, tmp2) * tmp3
else:
aux = arange(product(s))
x, y = iaind2sub(s, aux)
xx = reshape(concatenate((x,y)), (2, product(x.shape)))
d = transpose(xx) - resize(reshape(mu,(len(mu),1)), (s[0]*s[1],len(mu)))
if len(sigma) == 1:
tmp = 1./sigma
else:
tmp = inverse(sigma)
k = matrixmultiply(d, tmp) * d
k = sum(transpose(k))
g = zeros(s, Float)
aux = ravel(g)
if len(sigma) == 1:
tmp = sigma
else:
tmp = determinant(sigma)
aux[:] = 1./(2*pi*sqrt(tmp)) * exp(-1./2 * k)
return g
def applyTransform(self, transform): m = matrixmultiply(transform.matrix, self.transform.matrix) self.gradientDesc.SetMatrix(transform.getGMatrix(m))