def find_factors(idat, odat, k = None):
"""
A routine to compute the main predictors (linear combinations of
coordinates) in idat to predict odat.
*Parameters*
idat: d x n data matrix,
with n measurements, each of dimension d
odat: q x n data matrix
with n measurements, each of dimension q
*Returns*
**Depending on whether or not** *k* **is provided, the returned
value is different**
* if k is given, compute the first k regressors and return an orthogonal
matrix that contains the regressors in its colums,
i.e. reg[0,:] is the first regressor
* if k is not given or None, return a d-dimensional vector v(k)
explaining which fraction of the total predictable variance can be
explained using only k regressors.
**NOTE**
#. idat and odat must have zero mean
#. To interpret the regressors, it is advisable to have the
for columns of idat having the same variance
"""
# transform into z-scores
u, s, v = svd(idat, full_matrices = False)
su = dot(diag(1./s), u.T)
z = dot(su,idat)
# ! Note that the covariance of z is *NOT* 1, but 1/n; z*z.T = 1 !
# least-squares regression:
A = dot(odat, pinv(z))
uA, sigma_A, vA = svd(A, full_matrices = False)
if k is None:
vk = cumsum(sigma_A**2) / sum(sigma_A**2)
return vk
else:
# choose k predictors
sigma_A1 = sigma_A.copy()
sigma_A1[k:] = 0
A1 = reduce(dot, [uA, diag(sigma_A1), vA])
B = dot(A1, su)
uB, sigma_B, vB = svd(B, full_matrices = False)
regs = vB[:k,:].T
return regs
def pc_pm_std(data, ndim):
"""
This is a helper function.
It returns the value of +1 * std(x), where x is the ndim-th principal
component of the data
Parameters:
-----------
data: `array` (*n*-by-*d*)
the data on which the principal component analysis is performed.
ndim: `integer`
the number of the principal axis on which the analysis is performed.
**NOTE** this is zero-based, i.e. to compute the first principal
component, ndim=0
Returns:
--------
std_pc: `array` (1-by-*d*)
the vector that points in the direction of the *ndim*th principal
axis, and has the length of the standard deviation of the scores
along this axis.
"""
u,s,v = svd(data.T, full_matrices = False)
direction = u[:, ndim : ndim + 1]
scale = std(dot(direction.T, data.T))
return scale * direction.T
def homog3D (points2d, points3d):
"""
Compute a matrix relating homogeneous 3D points (4xN) to homogeneous
2D points (3xN)
Not sure why anyone would do this. Note that the returned transformation
*NOT* an isometry. But it's here... so deal with it.
"""
numPoints = points2d.shape[1]
assert (numPoints >= 4)
A = None
for i in range (0, numPoints):
xiPrime = points2d[:,i]
xi = points3d[:,i]
Ai_row0 = pl.concatenate ((pl.zeros (4,), -xiPrime[2]*xi, xiPrime[1]*xi))
Ai_row1 = pl.concatenate ((xiPrime[2]*xi, pl.zeros (4,), -xiPrime[0]*xi))
Ai = pl.row_stack ((Ai_row0, Ai_row1))
if A is None:
A = Ai
else:
A = pl.vstack ((A, Ai))
U, S, V = pl.svd (A)
V = V.T
h = V[:,-1]
P = pl.reshape (h, (3, 4))
return P
def homog2D(xPrime, x):
"""
Compute the 3x3 homography matrix mapping a set of N 2D homogeneous
points (3xN) to another set (3xN)
"""
numPoints = xPrime.shape[1]
assert numPoints >= 4
A = None
for i in range(0, numPoints):
xiPrime = xPrime[:, i]
xi = x[:, i]
Ai_row0 = pl.concatenate((pl.zeros(3), -xiPrime[2] * xi, xiPrime[1] * xi))
Ai_row1 = pl.concatenate((xiPrime[2] * xi, pl.zeros(3), -xiPrime[0] * xi))
Ai = pl.row_stack((Ai_row0, Ai_row1))
if A is None:
A = Ai
else:
A = pl.vstack((A, Ai))
U, S, V = pl.svd(A)
V = V.T
h = V[:, -1]
H = pl.reshape(h, (3, 3))
return H
def reduceDim(fullmat,n=1):
"""
reduces the dimension of a d x d - matrix to a (d-n)x(d-n) matrix,
keeping the largest eigenvalues unchanged.
"""
u,s,v = svd(fullmat)
return dot(u[:-n,:-n],dot(diag(s[:-n]),v[:-n,:-n]))
def reduceDim(fullmat, n=1):
"""
reduces the dimension of a d x d - matrix to a (d-n)x(d-n) matrix,
keeping the largest eigenvalues unchanged.
"""
u, s, v = svd(fullmat)
return dot(u[:-n, :-n], dot(diag(s[:-n]), v[:-n, :-n]))
def homog3D(points2d, points3d):
"""
Compute a matrix relating homogeneous 3D points (4xN) to homogeneous
2D points (3xN)
Not sure why anyone would do this. Note that the returned transformation
*NOT* an isometry. But it's here... so deal with it.
"""
numPoints = points2d.shape[1]
assert numPoints >= 4
A = None
for i in range(0, numPoints):
xiPrime = points2d[:, i]
xi = points3d[:, i]
Ai_row0 = pl.concatenate((pl.zeros(4), -xiPrime[2] * xi, xiPrime[1] * xi))
Ai_row1 = pl.concatenate((xiPrime[2] * xi, pl.zeros(4), -xiPrime[0] * xi))
Ai = pl.row_stack((Ai_row0, Ai_row1))
if A is None:
A = Ai
else:
A = pl.vstack((A, Ai))
U, S, V = pl.svd(A)
V = V.T
h = V[:, -1]
P = pl.reshape(h, (3, 4))
return P
def homog2D (xPrime, x):
"""
Compute the 3x3 homography matrix mapping a set of N 2D homogeneous
points (3xN) to another set (3xN)
"""
numPoints = xPrime.shape[1]
assert (numPoints >= 4)
A = None
for i in range (0, numPoints):
xiPrime = xPrime[:,i]
xi = x[:,i]
Ai_row0 = pl.concatenate ((pl.zeros (3,), -xiPrime[2]*xi, xiPrime[1]*xi))
Ai_row1 = pl.concatenate ((xiPrime[2]*xi, pl.zeros (3,), -xiPrime[0]*xi))
Ai = pl.row_stack ((Ai_row0, Ai_row1))
if A is None:
A = Ai
else:
A = pl.vstack ((A, Ai))
U, S, V = pl.svd (A)
V = V.T
h = V[:,-1]
H = pl.reshape (h, (3, 3))
return H
def matDist(mat1, mat2,nidx = 100):
"""
returns the distance of two lists of matrices mat1 and mat2.
output: [d(mat1,mat1),d(mat2,mat2),d(mat1,mat2),d(mat2,mat1)]
d(mat1,mat2) and d(mat2,mat1) should be the same
up to random variance (when d(mat1,mat1) and d(mat2,mat2) have the same
width in "FWHM sense")
nidx: n matrices are compared to n matrices each, that is the result has
length n**2
"""
# pick up a random matrix from mat1
# compute distances from out-of-sample mat1
# compute distances from sample of same size in mat2
# repeat; for random matrix from mat2
d_11 = []
d_22 = []
d_12 = []
d_21 = []
nidx1 = nidx
nidx2 = nidx
# for d_11 and d_12
for nmat in randint(0,len(mat1),nidx1):
refmat = mat1[nmat]
for nmat_x in randint(0,len(mat1),nidx1):
if nmat_x == nmat:
nmat_x = (nmat - 1) if nmat > 0 else (nmat + 1)
d_11.append(svd(mat1[nmat_x] - refmat,False,False)[0])
# ... I could use a []-statement, but I do not want to reformat a list
# of lists ...
for nmat_x in randint(0,len(mat2),nidx1):
d_12.append(svd(mat2[nmat_x] - refmat,False,False)[0])
for nmat in randint(0,len(mat2),nidx2):
refmat = mat2[nmat]
for nmat_x in randint(0,len(mat2),nidx2):
if nmat_x == nmat:
nmat_x = (nmat - 1) if nmat > 0 else (nmat + 1)
d_22.append(svd(mat2[nmat_x] - refmat,False,False)[0])
# ... I could use a []-statement, but I do not want to reformat a list
# of lists ...
for nmat_x in randint(0,len(mat1),nidx2):
d_21.append(svd(mat1[nmat_x] - refmat,False,False)[0])
return (d_11,d_22,d_21,d_12)
def matDist(mat1, mat2, nidx=100):
"""
returns the distance of two lists of matrices mat1 and mat2.
output: [d(mat1,mat1),d(mat2,mat2),d(mat1,mat2),d(mat2,mat1)]
d(mat1,mat2) and d(mat2,mat1) should be the same
up to random variance (when d(mat1,mat1) and d(mat2,mat2) have the same
width in "FWHM sense")
nidx: n matrices are compared to n matrices each, that is the result has
length n**2
"""
# pick up a random matrix from mat1
# compute distances from out-of-sample mat1
# compute distances from sample of same size in mat2
# repeat; for random matrix from mat2
d_11 = []
d_22 = []
d_12 = []
d_21 = []
nidx1 = nidx
nidx2 = nidx
# for d_11 and d_12
for nmat in randint(0, len(mat1), nidx1):
refmat = mat1[nmat]
for nmat_x in randint(0, len(mat1), nidx1):
if nmat_x == nmat:
nmat_x = (nmat - 1) if nmat > 0 else (nmat + 1)
d_11.append(svd(mat1[nmat_x] - refmat, False, False)[0])
# ... I could use a []-statement, but I do not want to reformat a list
# of lists ...
for nmat_x in randint(0, len(mat2), nidx1):
d_12.append(svd(mat2[nmat_x] - refmat, False, False)[0])
for nmat in randint(0, len(mat2), nidx2):
refmat = mat2[nmat]
for nmat_x in randint(0, len(mat2), nidx2):
if nmat_x == nmat:
nmat_x = (nmat - 1) if nmat > 0 else (nmat + 1)
d_22.append(svd(mat2[nmat_x] - refmat, False, False)[0])
# ... I could use a []-statement, but I do not want to reformat a list
# of lists ...
for nmat_x in randint(0, len(mat1), nidx2):
d_21.append(svd(mat1[nmat_x] - refmat, False, False)[0])
return (d_11, d_22, d_21, d_12)
def singularValueDecomposition(self):
"""This method calculates the svd of the raw data.
"""
self.U = []
self.S = []
self.V = []
self.U, self.S, self.V = plab.svd(self.dockedOpt.TT)
def get_transformation_matrix(matrix):
# Get the vector p and the values that are in there by taking the SVD.
# Since D is diagonal with the eigenvalues sorted from large to small
# on the diagonal, the optimal q in min ||Dq|| is q = [[0]..[1]].
# Therefore, p = Vq means p is the last column in V.
U, D, V = svd(matrix)
p = V[8][:]
return inv(array([[p[0],p[1],p[2]], [p[3],p[4],p[5]], [p[6],p[7],p[8]]]))
def reduceDimDat(fulldat, n=1):
"""
reduces the dimension of a given data set by removing the
lowest principal component.
data must be given in D X N - format
(D: dimension, N: number of measurements)
"""
raise NotImplementedError, \
'Wait a minute - this function in raw form does not make much sense here ...'
u, s, v = svd(fulldat, full_matrices=False)
def reduceDimDat(fulldat,n=1):
"""
reduces the dimension of a given data set by removing the
lowest principal component.
data must be given in D X N - format
(D: dimension, N: number of measurements)
"""
raise NotImplementedError, \
'Wait a minute - this function in raw form does not make much sense here ...'
u,s,v = svd(fulldat, full_matrices = False)
def plotEllipse(pos,P,edge,face,transparency):
U, s , Vh = pl.svd(P)
orient = math.atan2(U[1,0],U[0,0])*180/math.pi
ellipsePlot = Ellipse(xy=pos, width=2.0*math.sqrt(s[0]),
height=2.0*math.sqrt(s[1]), angle=orient,
facecolor=face,edgecolor=edge,alpha=transparency,
zorder=0)
ax = pl.gca()
ax.add_patch(ellipsePlot)
return ellipsePlot
def fun(xp, threshold=1e-10, debug=False):
x = np.dot(projector, xp)
if np.linalg.norm(x - xp) > threshold:
if debug:
print "debug:", svd(projector, 0, 0)
print "threshold exceeded:", x, xp
print ("(norm of", x - xp, ":", np.linalg.norm(x - xp), " > ",
threshold, ") ")
return 0
return scale * np.exp(-0.5 * np.dot(x - x_ref, np.dot(cv_i, x - x_ref)))
def kern(mat, threshold = 1e-8):
"""
returns the kernel of the matrix
parameter:
mat: matrix to be analyzed
threshold: min. singular value to be considered as "different from 0"
"""
u,s,v = svd(mat,full_matrices = True)
dim_k = len(find(s < threshold)) + v.shape[0] - len(s)
if dim_k > 0:
return v[-dim_k:,:].T
else:
return None
def kern(mat, threshold=1e-8):
"""
returns the kernel of the matrix
parameter:
mat: matrix to be analyzed
threshold: min. singular value to be considered as "different from 0"
"""
u, s, v = svd(mat, full_matrices=True)
dim_k = len(find(s < threshold)) + v.shape[0] - len(s)
if dim_k > 0:
return v[-dim_k:, :].T
else:
return None
def perspectiveTransform(image, x1, y1, x2, y2, x3, y3, x4, y4, M, N):
# Construct the matrix M
x1_a, y1_a = 0, 0
x2_a, y2_a = M, 0
x3_a, y3_a = M, N
x4_a, y4_a = 0, N
mat_M = array([[x1, y1, 1, 0, 0, 0, -x1_a * x1, -x1_a * y1, -x1_a], \
[0, 0, 0, x1, y1, 1, -y1_a * x1, -y1_a * y1, -y1_a], \
[x2, y2, 1, 0, 0, 0, -x2_a * x2, -x2_a * y2, -x2_a], \
[0, 0, 0, x2, y2, 1, -y2_a * x2, -y2_a * y2, -y2_a], \
[x3, y3, 1, 0, 0, 0, -x3_a * x3, -x3_a * y3, -x3_a], \
[0, 0, 0, x3, y3, 1, -y3_a * x3, -y3_a * y3, -y3_a], \
[x4, y4, 1, 0, 0, 0, -x4_a * x4, -x4_a * y4, -x4_a], \
[0, 0, 0, x4, y4, 1, -y4_a * x4, -y4_a * y4, -y4_a]])
# Get the vector p and the values that are in there by taking the SVD.
# Since D is diagonal with the eigenvalues sorted from large to small on
# the diagonal, the optimal q in min ||Dq|| is q = [[0]..[1]]. Therefore,
# p = Vq means p is the last column in V.
U, D, V = svd(mat_M)
p = V[8][:]
a, b, c, d, e, f, g, h, i = p[0], \
p[1], \
p[2], \
p[3], \
p[4], \
p[5], \
p[6], \
p[7], \
p[8]
# P is the resulting matrix that describes the transformation
P = array([[a, b, c], \
[d, e, f], \
[g, h, i]])
# Create the new image
b = array([zeros(M, float)] * N)
for i in range(0, M):
for j in range(0, N):
or_coor = dot(inv(P),([[i],[j],[1]]))
or_coor_h = or_coor[1][0] / or_coor[2][0], \
or_coor[0][0] / or_coor[2][0]
b[j][i] = pV(image, or_coor_h[0], or_coor_h[1], 'linear')
return b
def generate(numDims, numClasses, k, numPatternsPerClass,
numPatterns, numTests, numSVDSamples, keep):
LOGGER.info('N dims=%s', numDims)
LOGGER.info('N classes=%s', numClasses)
LOGGER.info('k=%s', k)
LOGGER.info('N vectors per class=%s', numPatternsPerClass)
LOGGER.info('N training vectors=%s', numPatterns)
LOGGER.info('N test vectors=%s', numTests)
LOGGER.info('N SVD samples=%s', numSVDSamples)
LOGGER.info('N reduced dims=%s', int(keep*numDims))
LOGGER.info('Generating data')
numpy.random.seed(42)
data0 = numpy.zeros((numPatterns + numTests, numDims))
class0 = numpy.zeros((numPatterns + numTests), dtype='int')
c = 0
for i in range(numClasses):
pt = 5*i*numpy.ones((numDims))
for _j in range(numPatternsPerClass):
data0[c] = pt+5*numpy.random.random((numDims))
class0[c] = i
c += 1
if 0: # Change this to visualize the output
import pylab
pylab.ion()
pylab.figure()
_u, _s, vt = pylab.svd(data0[:numPatterns])
tmp = numpy.zeros((numPatterns, 2))
for i in range(numPatterns):
tmp[i] = numpy.dot(vt, data0[i])[:2]
pylab.scatter(tmp[:, 0], tmp[:, 1])
ind = numpy.random.permutation(numPatterns + numTests)
train_data = data0[ind[:numPatterns]]
train_class = class0[ind[:numPatterns]]
test_data = data0[ind[numPatterns:]]
test_class = class0[ind[numPatterns:]]
return train_data, train_class, test_data, test_class
def generate(numDims, numClasses, k, numPatternsPerClass,
numPatterns, numTests, numSVDSamples, keep):
LOGGER.info('N dims=%s', numDims)
LOGGER.info('N classes=%s', numClasses)
LOGGER.info('k=%s', k)
LOGGER.info('N vectors per class=%s', numPatternsPerClass)
LOGGER.info('N training vectors=%s', numPatterns)
LOGGER.info('N test vectors=%s', numTests)
LOGGER.info('N SVD samples=%s', numSVDSamples)
LOGGER.info('N reduced dims=%s', int(keep*numDims))
LOGGER.info('Generating data')
numpy.random.seed(42)
data0 = numpy.zeros((numPatterns + numTests, numDims))
class0 = numpy.zeros((numPatterns + numTests), dtype='int')
c = 0
for i in range(numClasses):
pt = 5*i*numpy.ones((numDims))
for _j in range(numPatternsPerClass):
data0[c] = pt+5*numpy.random.random((numDims))
class0[c] = i
c += 1
if 0: # Change this to visualize the output
import pylab
pylab.ion()
pylab.figure()
_u, _s, vt = pylab.svd(data0[:numPatterns])
tmp = numpy.zeros((numPatterns, 2))
for i in range(numPatterns):
tmp[i] = numpy.dot(vt, data0[i])[:2]
pylab.scatter(tmp[:, 0], tmp[:, 1])
ind = numpy.random.permutation(numPatterns + numTests)
train_data = data0[ind[:numPatterns]]
train_class = class0[ind[:numPatterns]]
test_data = data0[ind[numPatterns:]]
test_class = class0[ind[numPatterns:]]
return train_data, train_class, test_data, test_class
def triangulate(points2d, cameras):
"""
Compute the N-view triangulation of corresponding 2D image points, given calibrated camera poses
points2d: 2N-by-M matrix of 2D image points (N: number of cameras, M: number of image points)
cameras: length-N list of Camera objects
"""
N = len(cameras)
M = points2d.shape[1]
pointsNorm = pl.zeros(points2d.shape)
for camInd, camera in enumerate(cameras):
points = points2d[camInd * 2 : (camInd * 2) + 2, :]
pointsNorm[camInd * 2 : (camInd * 2) + 2, :] = camera.normalize(points)
X = pl.zeros((3, M))
for pointInd in range(M):
A = pl.zeros((3 * N, 4))
AStartRow = 0
skewsyms = pl.zeros((3, 3, N))
for camInd, camera in enumerate(cameras):
skewsyms[:, :, camInd] = skewsym(
homogeneous.homogenize(pointsNorm[camInd * 2 : (camInd * 2) + 2, pointInd])
)
A[AStartRow : AStartRow + 3, 0:4] = skewsyms[:, :, camInd].dot(camera.wHc[0:3, 0:4])
AStartRow += 3
U, S, VT = pl.svd(A)
V = VT.T
X[:, pointInd] = homogeneous.dehomogenize(V[:, -1])
return X
def _calculate_svd(pp, r_0, beta, N_piercepoints):
"""
Returns result (U) of svd for K-L vectors
Parameters
----------
pp : array
Array of piercepoint locations
r_0: float
Scale size of amp fluctuations (m)
beta: float
Power-law index for amp structure function (5/3 => pure Kolmogorov
turbulence)
N_piercepoints : int
Number of piercepoints
Returns
-------
C : array
C matrix
pinvC : array
Inv(C) matrix
U : array
Unitary matrix
"""
import numpy as np
from pylab import kron, concatenate, pinv, norm, newaxis, find, amin, svd, eye
D = np.resize(pp, (N_piercepoints, N_piercepoints, 3))
D = np.transpose(D, (1, 0, 2)) - D
D2 = np.sum(D**2, axis=2)
C = -(D2 / r_0**2)**(beta / 2.0) / 2.0
pinvC = pinv(C, rcond=1e-3)
U, S, V = svd(C)
return C, pinvC, U
def _calculate_svd(pp, r_0, beta, N_piercepoints):
"""
Returns result (U) of svd for K-L vectors
Parameters
----------
pp : array
Array of piercepoint locations
r_0: float
Scale size of amp fluctuations (m)
beta: float
Power-law index for amp structure function (5/3 => pure Kolmogorov
turbulence)
N_piercepoints : int
Number of piercepoints
Returns
-------
C : array
C matrix
pinvC : array
Inv(C) matrix
U : array
Unitary matrix
"""
import numpy as np
from pylab import kron, concatenate, pinv, norm, newaxis, find, amin, svd, eye
D = np.resize(pp, (N_piercepoints, N_piercepoints, 3))
D = np.transpose(D, (1, 0, 2)) - D
D2 = np.sum(D**2, axis=2)
C = -(D2 / r_0**2)**(beta / 2.0) / 2.0
pinvC = pinv(C, rcond=1e-3)
U, S, V = svd(C)
return C, pinvC, U
def triangulate (points2d, cameras):
"""
Compute the N-view triangulation of corresponding 2D image points, given calibrated camera poses
points2d: 2N-by-M matrix of 2D image points (N: number of cameras, M: number of image points)
cameras: length-N list of Camera objects
"""
N = len (cameras)
M = points2d.shape[1]
pointsNorm = pl.zeros (points2d.shape)
for camInd, camera in enumerate (cameras):
points = points2d[camInd*2:(camInd*2)+2,:]
pointsNorm[camInd*2:(camInd*2)+2,:] = camera.normalize (points)
X = pl.zeros ((3,M))
for pointInd in range (M):
A = pl.zeros ((3*N,4))
AStartRow = 0
skewsyms = pl.zeros ((3,3,N))
for camInd, camera in enumerate (cameras):
skewsyms[:,:,camInd] = skewsym (homogeneous.homogenize (pointsNorm[camInd*2:(camInd*2)+2,pointInd]))
A[AStartRow:AStartRow+3, 0:4] = skewsyms[:,:,camInd].dot (camera.wHc[0:3,0:4])
AStartRow += 3
U, S, VT = pl.svd (A)
V = VT.T
X[:,pointInd] = homogeneous.dehomogenize (V[:,-1])
return X
def ksvd(self, X, n_components, dictionary=None, max_err=0, max_iter=10, approx=False, preserve_dc=False):
(n_samples, n_features) = X.shape
# if we're not given a dictionary for starters, make our own
if dictionary is None:
dictionary = np.random.rand(n_samples, n_components)
# make the first dictionary element constant; remove the mean from the
# rest of the dictionary elements
if preserve_dc:
dictionary[:, 0] = 1
for i in range(1, n_components): dictionary[:, i] -= np.mean(dictionary[:, i])
# normalize the dictionary regardless
for i in range(n_components):
dictionary[:, i] /= np.linalg.norm(dictionary[:, i])
print("running ksvd on %d %d-dimensional vectors with K=%d" \
% (n_features, n_samples, n_components))
# algorithm stuff
code = np.zeros((n_components, n_samples))
err = np.inf
iter_num = 0
while iter_num < max_iter and err > max_err:
# batch omp, woo!
print("staring omp...")
# X = omp(dictionary, Y, T, max_err)
print("omp complete!")
print( \
'average l0 "norm" for ksvd iteration %d after omp was %f' \
% (iter_num, len(np.nonzero(code)[0]) / n_features))
# dictionary update -- protip: update dictionary columns in random
# order
atom_indices = range(n_components)
if preserve_dc: atom_indices = atom_indices[1:]
np.random.shuffle(atom_indices)
unused_atoms = []
for (i, j) in zip(atom_indices, xrange(n_components)):
if False:
if j % 25 == 0:
print("ksvd: iteration %d, updating atom %d of %d" \
% (iter_num + 1, j, n_components))
# find nonzero entries
x_using = np.nonzero(code[i, :])[0]
if len(x_using) == 0:
unused_atoms.append(i)
continue
if not approx:
# Non-approximate K-SVD, as described in the original K-SVD
# paper
# compute residual error ... here's a trick passing almost all the
# work to BLAS
code[i, x_using] = 0
Residual_err = X[:, x_using] - np.dot(dictionary, code[:, x_using])
# update dictionary and weights -- sparsity-restricted rank-1
# approximation
U, s, Vt = pl.svd(Residual_err)
dictionary[:, i] = U[:, 0]
code[i, x_using] = s[0] * Vt.T[:, 0]
else:
# Approximate K-SVD
dictionary[:, i] = 0
g = code[i, x_using]
d = np.dot(X[:, x_using], g) - np.dot(dictionary[:, x_using], g)
d = d / np.linalg.norm(d)
g = np.dot(X[:, x_using].T, d) - np.dot(dictionary[:, x_using].T, d)
dictionary[:, i] = d
code[i, x_using] = g
# fill in values for unused atoms
# unused column -> replace by signal in training data with worst
# representation
Repr_err = X - np.dot(dictionary, code)
Repr_err_norms = (np.linalg.norm(Repr_err[:, n]) for n in range(n_features))
err_indices = sorted(zip(Repr_err_norms, xrange(n_features)), reverse=True)
for (unused_index, err_tuple) in zip(unused_atoms, err_indices):
(err, err_idx) = err_tuple
d = X[:, err_idx].copy()
if preserve_dc: d -= np.mean(d)
d /= np.linalg.norm(d)
dictionary[:, unused_index] = d
# compute maximum representation error
Repr_err_norms = [np.linalg.norm(Repr_err[:, n]) for n in range(n_features)]
err = max(Repr_err_norms)
print("maximum representation error: %f" % (err))
def create_cm(dim,eigList1 = None, eigList2 = None):
"""
returns two real-valued commuting matrices of dimension dim x dim
the eigenvalues of each matrix can be given; single complex numbers will be
interpreted as pair of complex conjuates.
With this restriction, the (internally augmented) lists must have the length
of dim
"""
if eigList1 is None:
eigList1 = rand(dim)
if eigList2 is None:
eigList2 = rand(dim)
# order 1st array such that complex numbers are first
EL1 = array(eigList1)
imPos1 = find(iscomplex(EL1))
rePos1 = find(isreal(EL1)) # shorter than set comparisons :D
EL1 = hstack([EL1[imPos1],EL1[rePos1]])
# order 2nd array such that complex numbers are last
EL2 = array(eigList2)
imPos2 = find(iscomplex(EL2))
rePos2 = find(isreal(EL2)) # shorter than set comparisons :D
EL2 = hstack([EL2[rePos2],EL2[imPos2]])
# now: make eigenvalues of list #2, where a block is in list #1,
# pairwise equal, and other way round
EL2[1:2*len(imPos1):2] = EL2[0:2*len(imPos1):2]
EL1[-2*len(imPos2)+1::2] = EL1[-2*len(imPos2)::2]
if len(imPos2)*2 + len(imPos1)*2 > dim:
raise ValueError(
'too many complex eigenvalues - cannot create commuting matrices')
# augment lists
ev1 = []
nev1 = 0
for elem in EL1:
if elem.imag != 0.:
ev1.append( array( [[elem.real, -elem.imag],
[elem.imag, elem.real]]))
nev1 += 2
else:
ev1.append(elem)
nev1 += 1
if nev1 != dim:
raise ValueError(
'number of given eigenvalues #1 (complex: x2) does not match dim!')
ev2 = []
nev2 = 0
for elem in EL2:
if elem.imag != 0.:
ev2.append( array( [[elem.real, -elem.imag],
[elem.imag, elem.real]]))
nev2 += 2
else:
ev2.append(elem)
nev2 += 1
if nev2 != dim:
raise ValueError(
'number of given eigenvalues #2 (complex: x2) does not match dim!')
u,s,v = svd(randn(dim,dim))
# create a coordinate system v that is not orthogonal but not too skew
v = v + .2*rand(dim,dim) - .1
cm1 = dot(inv(v),dot(blockdiag(ev1),v))
cm2 = dot(inv(v),dot(blockdiag(ev2),v))
# create block diagonal matrices
return cm1, cm2
def svd(self):
self.U, self.s, self.Vh = pylab.svd(self.fish)
self.decomposed = True
return self.U , self.s , self.Vh
def svd(self):
self.U, self.s, self.Vh = pylab.svd(self.fish)
self.decomposed = True
return self.U, self.s, self.Vh
def _fit_tec_screen(station_names, source_names, pp, airmass, rr, weights,
times, height, order, r_0, beta, outQueue):
"""
Fits a screen to given TEC values using Karhunen-Lo`eve base vectors
Parameters
----------
station_names: array
Array of station names
source_names: array
Array of source names
pp: array
Array of piercepoint locations
airmass: array
Array of airmass values (note: not currently used)
rr: array
Array of TEC values to fit screen to
weights: array
Array of weights
times: array
Array of times
height: float
Height of screen (m)
order: int
Order of screen (i.e., number of KL base vectors to keep)
r_0: float
Scale size of phase fluctuations (m)
beta: float
Power-law index for phase structure function (5/3 => pure Kolmogorov
turbulence)
"""
import numpy as np
from pylab import kron, concatenate, pinv, norm, newaxis, find, amin, svd, eye
logging.info('Fitting screens...')
# Initialize arrays
N_stations = len(station_names)
N_sources = len(source_names)
N_times = len(times)
N_piercepoints = N_sources * N_stations
tec_fit_white_all = np.zeros((N_times, N_sources, N_stations))
tec_residual_all = np.zeros((N_times, N_sources, N_stations))
for k in range(N_times):
D = np.resize(pp[k, :, :], (N_piercepoints, N_piercepoints, 3))
D = np.transpose(D, (1, 0, 2)) - D
D2 = np.sum(D**2, axis=2)
C = -(D2 / r_0**2)**(beta / 2.0) / 2.0
pinvC = pinv(C, rcond=1e-3)
U, S, V = svd(C)
invU = pinv(np.dot(np.transpose(U[:, :order]),
np.dot(weights[:, :, k], U[:, :order])),
rcond=1e-3)
# Calculate screen
rr1 = np.dot(np.transpose(U[:, :order]),
np.dot(weights[:, :, k], rr[:, k]))
tec_fit = np.dot(pinvC, np.dot(U[:, :order], np.dot(invU, rr1)))
tec_fit_white_all[k, :, :] = tec_fit.reshape((N_sources, N_stations))
residual = rr - np.dot(C, tec_fit)[:, newaxis]
tec_residual_all[k, :, :] = residual.reshape((N_sources, N_stations))
outQueue.put([tec_fit_white_all, tec_residual_all, times])
# generate random data with random uncertainties in y
x = pylab.arange(0.5 / N, 1.0, 1.0 / N)
dy = pylab.array([0.1 * random.lognormvariate(0.0, 1.0) for i in range(N)])
y = [sum(a[j] * x[i]**j for j in range(M)) for i in range(N)]
y = pylab.array([random.gauss(y[i], dy[i]) for i in range(N)])
# construct vector b and design matrix X
b = pylab.zeros(N)
X = pylab.zeros((N, M))
for i in range(N):
b[i] = y[i] / dy[i]
for j in range(M):
X[i, j] = x[i]**j / dy[i]
# compute fit parameters ahat and covariance matrix Sigma
(U, w, VT) = pylab.svd(X)
wmax = max(w)
Winv = pylab.zeros((M, N))
Sigma = pylab.zeros((M, M))
eps = 1e-6
for j in range(M):
if w[j] > eps * wmax:
Winv[j, j] = 1.0 / w[j]
else:
Winv[j, j] = 0.0
Sigma[j, j] = Winv[j, j]**2
ahat = pylab.dot(VT.T, pylab.dot(Winv, pylab.dot(U.T, b)))
Sigma = pylab.dot(VT.T, pylab.dot(Sigma, VT))
# compute chi-square and p-value of the fit
chisq = pylab.norm(pylab.dot(X, ahat) - b)**2
def robust_combined_algo(y, u, f, p, s_tol, dt):
"""
Subspace Identification for stochastic systems with input
Robust combined algorithm from chapter 4 of (1)
assuming a system of the form:
x(k+1) = A x(k) + B u(k) + w(k)
y(k) = C x(k) + D u(k) + v(k)
E[(w_p; v_p) (w_q^T v_q^T)] = (Q S; S^T R) delta_pq
and given y and u.
Find the order of the system and A, B, C, D, Q, S, R
See page 131, and generally chapter 4, of (1)
A different implementation of the algorithm is presented in 6.1 of (1)
(1) Subspace Identification for Linear
Systems, by Van Overschee and Moor. 1996
"""
#pylint: disable=too-many-arguments, too-many-locals
# for this algorithm, we need future and past
# to be more than 1
assert f > 1
assert p > 1
# setup matrices
y = pl.matrix(y)
n_y = y.shape[0]
u = pl.matrix(u)
n_u = u.shape[0]
w = pl.vstack([y, u])
n_w = w.shape[0]
# make sure the input is column vectors
assert y.shape[0] < y.shape[1]
assert u.shape[0] < u.shape[1]
W = block_hankel(w, f + p)
U = block_hankel(u, f + p)
Y = block_hankel(y, f + p)
W_p = W[:n_w*p, :]
W_pp = W[:n_w*(p+1), :]
Y_f = Y[n_y*f:, :]
U_f = U[n_y*f:, :]
Y_fm = Y[n_y*(f+1):, :]
U_fm = U[n_u*(f+1):, :]
# step 1, calculate the oblique and orthogonal projections
#------------------------------------------
#TODO fix explanation
# Y_p = G_i Xd_p + Hd_i U_p
# After the oblique projection, U_p component is eliminated,
# without changing the Xd_p component:
# Proj_perp_(U_p) Y_p = W1 O_i W2 = G_i Xd_p
O_i = Y_f*project_oblique(U_f, W_p)
Z_i = Y_f*project(pl.vstack(W_p, U_f))
Z_ip = Y_fm*project(pl.vstack(W_pp, U_fm))
#TODO fix explanation
# step 2, calculate the SVD of the weighted oblique projection
#------------------------------------------
# given: W1 O_i W2 = G_i Xd_p
# want to solve for G_i, but know product, and not Xd_p
# so can only find Xd_p up to a similarity transformation
U0, s0, VT0 = pl.svd(O_i*project_perp(U_f)) #pylint: disable=unused-variable
# step 3, determine the order by inspecting the singular
#------------------------------------------
# values in S and partition the SVD accordingly to obtain U1, S1
#print s0
n_x = pl.find(s0/s0.max() > s_tol)[-1] + 1
U1 = U0[:, :n_x]
S1 = pl.matrix(pl.diag(s0[:n_x]))
# VT1 = VT0[:n_x, :n_x]
# step 4, determine Gi and Gim
#------------------------------------------
G_i = U1*pl.matrix(pl.diag(pl.sqrt(s1[:n_x])))
G_im = G_i[:-n_y, :]
# step 5, solve the linear equations for A and C
#------------------------------------------
# Recompute G_i and G_im from A and C
#TODO figure out what K (contains B and D) and the rhos (residuals) are in terms of knowns
AC_stack = (pl.vstack(G_im.I*Z_ip,Y_f(1,:))-K*U_f-pl.vstack(rho_w, rho_v))*(G_i.I*Z_i).I #TODO not done
def _fit_phase_screen(station_names, source_names, pp, airmass, rr, weights,
times, height, order, r_0, beta, outQueue):
"""
Fits a screen to given phase values using Karhunen-Lo`eve base vectors
Parameters
----------
station_names: array
Array of station names
source_names: array
Array of source names
pp: array
Array of piercepoint locations
airmass: array
Array of airmass values (note: not currently used)
rr: array
Array of phase values to fit screen to
weights: array
Array of weights
times: array
Array of times
height: float
Height of screen (m)
order: int
Order of screen (i.e., number of KL base vectors to keep)
r_0: float
Scale size of phase fluctuations (m)
beta: float
Power-law index for phase structure function (5/3 => pure Kolmogorov
turbulence)
"""
import numpy as np
from pylab import kron, concatenate, pinv, norm, newaxis, find, amin, svd, eye
logging.info('Fitting screens...')
# Initialize arrays
N_stations = len(station_names)
N_sources = len(source_names)
N_times = len(times)
N_piercepoints = N_sources * N_stations
real_fit_white_all = np.zeros((N_times, N_sources, N_stations))
imag_fit_white_all = np.zeros((N_times, N_sources, N_stations))
phase_fit_white_all = np.zeros((N_times, N_sources, N_stations))
real_residual_all = np.zeros((N_times, N_sources, N_stations))
imag_residual_all = np.zeros((N_times, N_sources, N_stations))
phase_residual_all = np.zeros((N_times, N_sources, N_stations))
# Change phase to real/imag
rr_real = np.cos(rr)
rr_imag = np.sin(rr)
for k in range(N_times):
try:
D = np.resize(pp[k, :, :], (N_piercepoints, N_piercepoints, 3))
D = np.transpose(D, (1, 0, 2)) - D
D2 = np.sum(D**2, axis=2)
C = -(D2 / r_0**2)**(beta / 2.0) / 2.0
pinvC = pinv(C, rcond=1e-3)
U, S, V = svd(C)
invU = pinv(np.dot(np.transpose(U[:, :order]),
np.dot(weights[:, :, k], U[:, :order])),
rcond=1e-3)
# Calculate real screen
rr1 = np.dot(np.transpose(U[:, :order]),
np.dot(weights[:, :, k], rr_real[:, k]))
real_fit = np.dot(pinvC, np.dot(U[:, :order], np.dot(invU, rr1)))
real_fit_white_all[k, :, :] = real_fit.reshape(
(N_sources, N_stations))
residual = rr_real - np.dot(C, real_fit)[:, newaxis]
real_residual_all[k, :, :] = residual.reshape(
(N_sources, N_stations))
# Calculate imag screen
rr1 = np.dot(np.transpose(U[:, :order]),
np.dot(weights[:, :, k], rr_imag[:, k]))
imag_fit = np.dot(pinvC, np.dot(U[:, :order], np.dot(invU, rr1)))
imag_fit_white_all[k, :, :] = imag_fit.reshape(
(N_sources, N_stations))
residual = rr_imag - np.dot(C, imag_fit)[:, newaxis]
imag_residual_all[k, :, :] = residual.reshape(
(N_sources, N_stations))
# Calculate phase screen
phase_fit = np.dot(
pinvC, np.arctan2(np.dot(C, imag_fit), np.dot(C, real_fit)))
phase_fit_white_all[k, :, :] = phase_fit.reshape(
(N_sources, N_stations))
residual = rr - np.dot(C, phase_fit)[:, newaxis]
phase_residual_all[k, :, :] = residual.reshape(
(N_sources, N_stations))
except:
# Set screen to zero if fit did not work
logging.debug('Screen fit failed for timeslot {}'.format(k))
real_fit_white_all[k, :, :] = np.zeros((N_sources, N_stations))
real_residual_all[k, :, :] = np.ones((N_sources, N_stations))
imag_fit_white_all[k, :, :] = np.zeros((N_sources, N_stations))
imag_residual_all[k, :, :] = np.ones((N_sources, N_stations))
phase_fit_white_all[k, :, :] = np.zeros((N_sources, N_stations))
phase_residual_all[k, :, :] = np.ones((N_sources, N_stations))
outQueue.put([
real_fit_white_all, real_residual_all, imag_fit_white_all,
imag_residual_all, phase_fit_white_all, phase_residual_all, times
])
def subspace_det_algo1(y, u, f, p, s_tol, dt):
"""
Subspace Identification for deterministic systems
algorithm 1 from (1)
assuming a system of the form:
x(k+1) = A x(k) + B u(k)
y(k) = C x(k) + D u(k)
and given y and u.
Find A, B, C, D
See page 52. of (1)
(1) Subspace Identification for Linear
Systems, by Van Overschee and Moor. 1996
"""
# pylint: disable=too-many-arguments, too-many-locals
# for this algorithm, we need future and past
# to be more than 1
assert f > 1
assert p > 1
# setup matrices
y = np.matrix(y)
n_y = y.shape[0]
u = np.matrix(u)
n_u = u.shape[0]
w = pl.vstack([y, u])
n_w = w.shape[0]
# make sure the input is column vectors
assert y.shape[0] < y.shape[1]
assert u.shape[0] < u.shape[1]
W = block_hankel(w, f + p)
U = block_hankel(u, f + p)
Y = block_hankel(y, f + p)
W_p = W[:n_w*p, :]
W_pp = W[:n_w*(p+1), :]
Y_f = Y[n_y*f:, :]
U_f = U[n_y*f:, :]
Y_fm = Y[n_y*(f+1):, :]
U_fm = U[n_u*(f+1):, :]
# step 1, calculate the oblique projections
# ------------------------------------------
# Y_p = G_i Xd_p + Hd_i U_p
# After the oblique projection, U_p component is eliminated,
# without changing the Xd_p component:
# Proj_perp_(U_p) Y_p = W1 O_i W2 = G_i Xd_p
O_i = Y_f*project_oblique(U_f, W_p)
O_im = Y_fm*project_oblique(U_fm, W_pp)
# step 2, calculate the SVD of the weighted oblique projection
# ------------------------------------------
# given: W1 O_i W2 = G_i Xd_p
# want to solve for G_i, but know product, and not Xd_p
# so can only find Xd_p up to a similarity transformation
W1 = np.matrix(pl.eye(O_i.shape[0]))
W2 = np.matrix(pl.eye(O_i.shape[1]))
U0, s0, VT0 = pl.svd(W1*O_i*W2) # pylint: disable=unused-variable
# step 3, determine the order by inspecting the singular
# ------------------------------------------
# values in S and partition the SVD accordingly to obtain U1, S1
# print s0
n_x = pl.where(s0/s0.max() > s_tol)[0][-1] + 1
U1 = U0[:, :n_x]
# S1 = np.matrix(pl.diag(s0[:n_x]))
# VT1 = VT0[:n_x, :n_x]
# step 4, determine Gi and Gim
# ------------------------------------------
G_i = W1.I*U1*np.matrix(pl.diag(pl.sqrt(s0[:n_x])))
G_im = G_i[:-n_y, :] # check
# step 5, determine Xd_ip and Xd_p
# ------------------------------------------
# only know Xd up to a similarity transformation
Xd_i = G_i.I*O_i
Xd_ip = G_im.I*O_im
# step 6, solve the set of linear eqs
# for A, B, C, D
# ------------------------------------------
Y_ii = Y[n_y*p:n_y*(p+1), :]
U_ii = U[n_u*p:n_u*(p+1), :]
a_mat = np.matrix(pl.vstack([Xd_ip, Y_ii]))
b_mat = np.matrix(pl.vstack([Xd_i, U_ii]))
ss_mat = a_mat*b_mat.I
A_id = ss_mat[:n_x, :n_x]
B_id = ss_mat[:n_x, n_x:]
assert B_id.shape[0] == n_x
assert B_id.shape[1] == n_u
C_id = ss_mat[n_x:, :n_x]
assert C_id.shape[0] == n_y
assert C_id.shape[1] == n_x
D_id = ss_mat[n_x:, n_x:]
assert D_id.shape[0] == n_y
assert D_id.shape[1] == n_u
if np.linalg.matrix_rank(C_id) == n_x:
T = C_id.I # try to make C identity, want it to look like state feedback
else:
T = np.matrix(pl.eye(n_x))
Q_id = pl.zeros((n_x, n_x))
R_id = pl.zeros((n_y, n_y))
sys = ss.StateSpaceDiscreteLinear(
A=T.I*A_id*T, B=T.I*B_id, C=C_id*T, D=D_id,
Q=Q_id, R=R_id, dt=dt)
return sys
def subspace_det_algo1(y, u, f, p, s_tol, dt):
"""
Subspace Identification for deterministic systems
deterministic algorithm 1 from (1)
assuming a system of the form:
x(k+1) = A x(k) + B u(k)
y(k) = C x(k) + D u(k)
and given y and u.
Find A, B, C, D
See page 52. of (1)
(1) Subspace Identification for Linear
Systems, by Van Overschee and Moor. 1996
"""
#pylint: disable=too-many-arguments, too-many-locals
# for this algorithm, we need future and past
# to be more than 1
assert f > 1
assert p > 1
# setup matrices
y = pl.matrix(y)
n_y = y.shape[0]
u = pl.matrix(u)
n_u = u.shape[0]
w = pl.vstack([y, u])
n_w = w.shape[0]
# make sure the input is column vectors
assert y.shape[0] < y.shape[1]
assert u.shape[0] < u.shape[1]
W = block_hankel(w, f + p)
U = block_hankel(u, f + p)
Y = block_hankel(y, f + p)
W_p = W[:n_w*p, :]
W_pp = W[:n_w*(p+1), :]
Y_f = Y[n_y*f:, :]
U_f = U[n_y*f:, :]
Y_fm = Y[n_y*(f+1):, :]
U_fm = U[n_u*(f+1):, :]
# step 1, calculate the oblique projections
#------------------------------------------
# Y_p = G_i Xd_p + Hd_i U_p
# After the oblique projection, U_p component is eliminated,
# without changing the Xd_p component:
# Proj_perp_(U_p) Y_p = W1 O_i W2 = G_i Xd_p
O_i = Y_f*project_oblique(U_f, W_p)
O_im = Y_fm*project_oblique(U_fm, W_pp)
# step 2, calculate the SVD of the weighted oblique projection
#------------------------------------------
# given: W1 O_i W2 = G_i Xd_p
# want to solve for G_i, but know product, and not Xd_p
# so can only find Xd_p up to a similarity transformation
W1 = pl.matrix(pl.eye(O_i.shape[0]))
W2 = pl.matrix(pl.eye(O_i.shape[1]))
U0, s0, VT0 = pl.svd(W1*O_i*W2) #pylint: disable=unused-variable
# step 3, determine the order by inspecting the singular
#------------------------------------------
# values in S and partition the SVD accordingly to obtain U1, S1
#print s0
n_x = pl.find(s0/s0.max() > s_tol)[-1] + 1
U1 = U0[:, :n_x]
# S1 = pl.matrix(pl.diag(s0[:n_x]))
# VT1 = VT0[:n_x, :n_x]
# step 4, determine Gi and Gim
#------------------------------------------
G_i = W1.I*U1*pl.matrix(pl.diag(pl.sqrt(s0[:n_x])))
G_im = G_i[:-n_y, :]
# step 5, determine Xd_ip and Xd_p
#------------------------------------------
# only know Xd up to a similarity transformation
Xd_i = G_i.I*O_i
Xd_ip = G_im.I*O_im
# step 6, solve the set of linear eqs
# for A, B, C, D
#------------------------------------------
Y_ii = Y[n_y*p:n_y*(p+1), :]
U_ii = U[n_u*p:n_u*(p+1), :]
a_mat = pl.matrix(pl.vstack([Xd_ip, Y_ii]))
b_mat = pl.matrix(pl.vstack([Xd_i, U_ii]))
ss_mat = a_mat*b_mat.I
A_id = ss_mat[:n_x, :n_x]
B_id = ss_mat[:n_x, n_x:]
assert B_id.shape[0] == n_x
assert B_id.shape[1] == n_u
C_id = ss_mat[n_x:, :n_x]
assert C_id.shape[0] == n_y
assert C_id.shape[1] == n_x
D_id = ss_mat[n_x:, n_x:]
assert D_id.shape[0] == n_y
assert D_id.shape[1] == n_u
if pl.matrix_rank(C_id) == n_x:
T = C_id.I # try to make C identity, want it to look like state feedback
else:
T = pl.matrix(pl.eye(n_x))
Q_id = pl.zeros((n_x, n_x))
R_id = pl.zeros((n_y, n_y))
sys = ss.StateSpaceDiscreteLinear(
A=T.I*A_id*T, B=T.I*B_id, C=C_id*T, D=D_id,
Q=Q_id, R=R_id, dt=dt)
return sys
def _fit_tec_screen(station_names, source_names, pp, airmass, rr, weights, times,
height, order, r_0, beta, outQueue):
"""
Fits a screen to given TEC values using Karhunen-Lo`eve base vectors
Parameters
----------
station_names: array
Array of station names
source_names: array
Array of source names
pp: array
Array of piercepoint locations
airmass: array
Array of airmass values (note: not currently used)
rr: array
Array of TEC values to fit screen to
weights: array
Array of weights
times: array
Array of times
height: float
Height of screen (m)
order: int
Order of screen (i.e., number of KL base vectors to keep)
r_0: float
Scale size of phase fluctuations (m)
beta: float
Power-law index for phase structure function (5/3 => pure Kolmogorov
turbulence)
"""
import numpy as np
from pylab import kron, concatenate, pinv, norm, newaxis, find, amin, svd, eye
logging.info('Fitting screens...')
# Initialize arrays
N_stations = len(station_names)
N_sources = len(source_names)
N_times = len(times)
N_piercepoints = N_sources * N_stations
tec_fit_white_all = np.zeros((N_times, N_sources, N_stations))
tec_residual_all = np.zeros((N_times, N_sources, N_stations))
for k in range(N_times):
D = np.resize(pp[k, :, :], (N_piercepoints, N_piercepoints, 3))
D = np.transpose(D, (1, 0, 2)) - D
D2 = np.sum(D**2, axis=2)
C = -(D2 / r_0**2)**(beta / 2.0) / 2.0
pinvC = pinv(C, rcond=1e-3)
U, S, V = svd(C)
invU = pinv(np.dot(np.transpose(U[:, :order]), np.dot(weights[:, :, k], U[:, :order])), rcond=1e-3)
# Calculate screen
rr1 = np.dot(np.transpose(U[:, :order]), np.dot(weights[:, :, k], rr[:, k]))
tec_fit = np.dot(pinvC, np.dot(U[:, :order], np.dot(invU, rr1)))
tec_fit_white_all[k, :, :] = tec_fit.reshape((N_sources, N_stations))
residual = rr - np.dot(C, tec_fit)[:, newaxis]
tec_residual_all[k, :, :] = residual.reshape((N_sources, N_stations))
outQueue.put([tec_fit_white_all, tec_residual_all, times])
def _fit_phase_screen(station_names, source_names, pp, airmass, rr, weights, times,
height, order, r_0, beta, outQueue):
"""
Fits a screen to given phase values using Karhunen-Lo`eve base vectors
Parameters
----------
station_names: array
Array of station names
source_names: array
Array of source names
pp: array
Array of piercepoint locations
airmass: array
Array of airmass values (note: not currently used)
rr: array
Array of phase values to fit screen to
weights: array
Array of weights
times: array
Array of times
height: float
Height of screen (m)
order: int
Order of screen (i.e., number of KL base vectors to keep)
r_0: float
Scale size of phase fluctuations (m)
beta: float
Power-law index for phase structure function (5/3 => pure Kolmogorov
turbulence)
"""
import numpy as np
from pylab import kron, concatenate, pinv, norm, newaxis, find, amin, svd, eye
logging.info('Fitting screens...')
# Initialize arrays
N_stations = len(station_names)
N_sources = len(source_names)
N_times = len(times)
N_piercepoints = N_sources * N_stations
real_fit_white_all = np.zeros((N_times, N_sources, N_stations))
imag_fit_white_all = np.zeros((N_times, N_sources, N_stations))
phase_fit_white_all = np.zeros((N_times, N_sources, N_stations))
real_residual_all = np.zeros((N_times, N_sources, N_stations))
imag_residual_all = np.zeros((N_times, N_sources, N_stations))
phase_residual_all = np.zeros((N_times, N_sources, N_stations))
# Change phase to real/imag
rr_real = np.cos(rr)
rr_imag = np.sin(rr)
for k in range(N_times):
try:
D = np.resize(pp[k, :, :], (N_piercepoints, N_piercepoints, 3))
D = np.transpose(D, (1, 0, 2)) - D
D2 = np.sum(D**2, axis=2)
C = -(D2 / r_0**2)**(beta / 2.0) / 2.0
pinvC = pinv(C, rcond=1e-3)
U, S, V = svd(C)
invU = pinv(np.dot(np.transpose(U[:, :order]), np.dot(weights[:, :, k], U[:, :order])), rcond=1e-3)
# Calculate real screen
rr1 = np.dot(np.transpose(U[:, :order]), np.dot(weights[:, :, k], rr_real[:, k]))
real_fit = np.dot(pinvC, np.dot(U[:, :order], np.dot(invU, rr1)))
real_fit_white_all[k, :, :] = real_fit.reshape((N_sources, N_stations))
residual = rr_real - np.dot(C, real_fit)[:, newaxis]
real_residual_all[k, :, :] = residual.reshape((N_sources, N_stations))
# Calculate imag screen
rr1 = np.dot(np.transpose(U[:, :order]), np.dot(weights[:, :, k], rr_imag[:, k]))
imag_fit = np.dot(pinvC, np.dot(U[:, :order], np.dot(invU, rr1)))
imag_fit_white_all[k, :, :] = imag_fit.reshape((N_sources, N_stations))
residual = rr_imag - np.dot(C, imag_fit)[:, newaxis]
imag_residual_all[k, :, :] = residual.reshape((N_sources, N_stations))
# Calculate phase screen
phase_fit = np.dot(pinvC, np.arctan2(np.dot(C, imag_fit), np.dot(C, real_fit)))
phase_fit_white_all[k, :, :] = phase_fit.reshape((N_sources, N_stations))
residual = rr - np.dot(C, phase_fit)[:, newaxis]
phase_residual_all[k, :, :] = residual.reshape((N_sources, N_stations))
except:
# Set screen to zero if fit did not work
logging.debug('Screen fit failed for timeslot {}'.format(k))
real_fit_white_all[k, :, :] = np.zeros((N_sources, N_stations))
real_residual_all[k, :, :] = np.ones((N_sources, N_stations))
imag_fit_white_all[k, :, :] = np.zeros((N_sources, N_stations))
imag_residual_all[k, :, :] = np.ones((N_sources, N_stations))
phase_fit_white_all[k, :, :] = np.zeros((N_sources, N_stations))
phase_residual_all[k, :, :] = np.ones((N_sources, N_stations))
outQueue.put([real_fit_white_all, real_residual_all,
imag_fit_white_all, imag_residual_all,
phase_fit_white_all, phase_residual_all,
times])
def fit_screen_to_tec(station_names, source_names, pp, airmass, rr, times,
height, order, r_0, beta):
"""
Fits a screen to given TEC values using Karhunen-Lo`eve base vectors
Keyword arguments:
station_names -- array of station names
source_names -- array of source names
pp -- array of piercepoint locations
airmass -- array of airmass values
rr -- array of TEC solutions
times -- array of times
height -- height of screen (m)
order -- order of screen (i.e., number of KL base vectors to keep)
r_0 -- scale size of phase fluctuations (m)
beta -- power-law index for phase structure function (5/3 =>
pure Kolmogorov turbulence)
"""
import numpy as np
from pylab import kron, concatenate, pinv, norm, newaxis, find, amin, svd, eye
try:
import progressbar
except ImportError:
import losoto.progressbar as progressbar
logging.info('Fitting screens to TEC values...')
N_stations = len(station_names)
N_sources = len(source_names)
N_times = len(times)
tec_fit_all = np.zeros((N_times, N_sources, N_stations))
residual_all = np.zeros((N_times, N_sources, N_stations))
A = concatenate([
kron(eye(N_sources), np.ones((N_stations, 1))),
kron(np.ones((N_sources, 1)), eye(N_stations))
],
axis=1)
N_piercepoints = N_sources * N_stations
P = eye(N_piercepoints) - np.dot(np.dot(A, pinv(np.dot(A.T, A))), A.T)
pbar = progressbar.ProgressBar(maxval=N_times).start()
ipbar = 0
for k in range(N_times):
try:
D = np.resize(pp[k, :, :], (N_piercepoints, N_piercepoints, 3))
D = np.transpose(D, (1, 0, 2)) - D
D2 = np.sum(D**2, axis=2)
C = -(D2 / r_0**2)**(beta / 2.0) / 2.0
P1 = eye(N_piercepoints) - np.ones(
(N_piercepoints, N_piercepoints)) / N_piercepoints
C1 = np.dot(np.dot(P1, C), P1)
U, S, V = svd(C1)
B = np.dot(P, np.dot(np.diag(airmass[k, :]), U[:, :order]))
pinvB = pinv(B, rcond=1e-3)
rr1 = np.dot(P, rr[:, k])
tec_fit = np.dot(U[:, :order], np.dot(pinvB, rr1))
tec_fit_all[k, :, :] = tec_fit.reshape((N_sources, N_stations))
residual = rr1 - np.dot(P, tec_fit)
residual_all[k, :, :] = residual.reshape((N_sources, N_stations))
except:
# Set screen to zero if fit did not work
logging.debug('Tecscreen fit failed for timeslot {0}'.format(k))
tec_fit_all[k, :, :] = np.zeros((N_sources, N_stations))
residual_all[k, :, :] = np.ones((N_sources, N_stations))
pbar.update(ipbar)
ipbar += 1
pbar.finish()
return tec_fit_all, residual_all
def create_cm(dim, eigList1=None, eigList2=None):
"""
returns two real-valued commuting matrices of dimension dim x dim
the eigenvalues of each matrix can be given; single complex numbers will be
interpreted as pair of complex conjuates.
With this restriction, the (internally augmented) lists must have the length
of dim
"""
if eigList1 is None:
eigList1 = rand(dim)
if eigList2 is None:
eigList2 = rand(dim)
# order 1st array such that complex numbers are first
EL1 = array(eigList1)
imPos1 = find(iscomplex(EL1))
rePos1 = find(isreal(EL1)) # shorter than set comparisons :D
EL1 = hstack([EL1[imPos1], EL1[rePos1]])
# order 2nd array such that complex numbers are last
EL2 = array(eigList2)
imPos2 = find(iscomplex(EL2))
rePos2 = find(isreal(EL2)) # shorter than set comparisons :D
EL2 = hstack([EL2[rePos2], EL2[imPos2]])
# now: make eigenvalues of list #2, where a block is in list #1,
# pairwise equal, and other way round
EL2[1:2 * len(imPos1):2] = EL2[0:2 * len(imPos1):2]
EL1[-2 * len(imPos2) + 1::2] = EL1[-2 * len(imPos2)::2]
if len(imPos2) * 2 + len(imPos1) * 2 > dim:
raise ValueError(
'too many complex eigenvalues - cannot create commuting matrices')
# augment lists
ev1 = []
nev1 = 0
for elem in EL1:
if elem.imag != 0.:
ev1.append(array([[elem.real, -elem.imag], [elem.imag,
elem.real]]))
nev1 += 2
else:
ev1.append(elem)
nev1 += 1
if nev1 != dim:
raise ValueError(
'number of given eigenvalues #1 (complex: x2) does not match dim!')
ev2 = []
nev2 = 0
for elem in EL2:
if elem.imag != 0.:
ev2.append(array([[elem.real, -elem.imag], [elem.imag,
elem.real]]))
nev2 += 2
else:
ev2.append(elem)
nev2 += 1
if nev2 != dim:
raise ValueError(
'number of given eigenvalues #2 (complex: x2) does not match dim!')
u, s, v = svd(randn(dim, dim))
# create a coordinate system v that is not orthogonal but not too skew
v = v + .2 * rand(dim, dim) - .1
cm1 = dot(inv(v), dot(blockdiag(ev1), v))
cm2 = dot(inv(v), dot(blockdiag(ev2), v))
# create block diagonal matrices
return cm1, cm2
def fit_screen_to_tec(station_names, source_names, pp, airmass, rr, times,
height, order, r_0, beta):
"""
Fits a screen to given TEC values using Karhunen-Lo`eve base vectors
Keyword arguments:
station_names -- array of station names
source_names -- array of source names
pp -- array of piercepoint locations
airmass -- array of airmass values
rr -- array of TEC solutions
times -- array of times
height -- height of screen (m)
order -- order of screen (i.e., number of KL base vectors to keep)
r_0 -- scale size of phase fluctuations (m)
beta -- power-law index for phase structure function (5/3 =>
pure Kolmogorov turbulence)
"""
import numpy as np
from pylab import kron, concatenate, pinv, norm, newaxis, find, amin, svd, eye
try:
import progressbar
except ImportError:
import losoto.progressbar as progressbar
logging.info('Fitting screens to TEC values...')
N_stations = len(station_names)
N_sources = len(source_names)
N_times = len(times)
tec_fit_all = np.zeros((N_times, N_sources, N_stations))
residual_all = np.zeros((N_times, N_sources, N_stations))
A = concatenate([kron(eye(N_sources), np.ones((N_stations, 1))),
kron(np.ones((N_sources, 1)), eye(N_stations))], axis=1)
N_piercepoints = N_sources * N_stations
P = eye(N_piercepoints) - np.dot(np.dot(A, pinv(np.dot(A.T, A))), A.T)
pbar = progressbar.ProgressBar(maxval=N_times).start()
ipbar = 0
for k in range(N_times):
try:
D = np.resize(pp[k, :, :], (N_piercepoints, N_piercepoints, 3))
D = np.transpose(D, (1, 0, 2)) - D
D2 = np.sum(D**2, axis=2)
C = -(D2 / r_0**2)**(beta / 2.0) / 2.0
P1 = eye(N_piercepoints) - np.ones((N_piercepoints, N_piercepoints)) / N_piercepoints
C1 = np.dot(np.dot(P1, C), P1)
U, S, V = svd(C1)
B = np.dot(P, np.dot(np.diag(airmass[k, :]), U[:, :order]))
pinvB = pinv(B, rcond=1e-3)
rr1 = np.dot(P, rr[:, k])
tec_fit = np.dot(U[:, :order], np.dot(pinvB, rr1))
tec_fit_all[k, :, :] = tec_fit.reshape((N_sources, N_stations))
residual = rr1 - np.dot(P, tec_fit)
residual_all[k, :, :] = residual.reshape((N_sources, N_stations))
except:
# Set screen to zero if fit did not work
logging.debug('Tecscreen fit failed for timeslot {0}'.format(k))
tec_fit_all[k, :, :] = np.zeros((N_sources, N_stations))
residual_all[k, :, :] = np.ones((N_sources, N_stations))
pbar.update(ipbar)
ipbar += 1
pbar.finish()
return tec_fit_all, residual_all
