def calculateKernelBasisFunctionsMC(kernel, numBasis, mcPoints):
"""
Calculates basis functions using Monte Carlo and Nystrom method.
Parameters
----------
numBasis : int
calculate numBasis eigenvectors
mcPoints : ndarray
Monte Carlo points
Returns
-------
eigv : 1darray
eigenvalues numBasisx1
eigve : ndarray
eigenvectors mcPoints x numBasis
Notes
-----
"""
linearOpVersion = 1
nMC = mcPoints.shape[0]
numberEigenVectors = int(min(numBasis, nMC))
#print "looking for ", numberEigenVectors, "eigenvalues"e
if linearOpVersion:
#covm = calculateCovarianceMatrix(kernel, mcPoints)
covop = lambda v, k=kernel, p=mcPoints: covTimesV(v, k, p)
A = LinearOperator( (nMC, nMC), matvec=covop, \
dtype=float)
eigv, eigve = sparpack.eigsh(A,
k=numberEigenVectors,
maxiter=10 * numberEigenVectors)
else:
#print "calculateCovariance Matrix "
covm = calculateCovarianceMatrix(kernel, mcPoints)
eigv, eigve = sparpack.eigsh(covm,
k=numberEigenVectors,
maxiter=50 * numberEigenVectors)
#print "completed eigenvalue decomposition "
#sort in decending order
eigv = eigv[::-1]
eigve = eigve[:, ::-1]
# perform nystrom
eigve = eigve * np.sqrt(float(nMC))
eigv = eigv / float(nMC)
return eigv, eigve
def calculateKernelBasisFunctionsMC(kernel, numBasis, mcPoints):
"""
Calculates basis functions using Monte Carlo and Nystrom method.
Parameters
----------
numBasis : int
calculate numBasis eigenvectors
mcPoints : ndarray
Monte Carlo points
Returns
-------
eigv : 1darray
eigenvalues numBasisx1
eigve : ndarray
eigenvectors mcPoints x numBasis
Notes
-----
"""
linearOpVersion = 1
nMC = mcPoints.shape[0]
numberEigenVectors = int(min(numBasis,nMC))
#print "looking for ", numberEigenVectors, "eigenvalues"e
if linearOpVersion:
#covm = calculateCovarianceMatrix(kernel, mcPoints)
covop = lambda v, k=kernel, p=mcPoints: covTimesV(v, k, p)
A = LinearOperator( (nMC, nMC), matvec=covop, \
dtype=float)
eigv, eigve= sparpack.eigsh(A,
k=numberEigenVectors, maxiter=10*numberEigenVectors)
else:
#print "calculateCovariance Matrix "
covm= calculateCovarianceMatrix(kernel, mcPoints)
eigv, eigve= sparpack.eigsh(covm,
k=numberEigenVectors, maxiter = 50*numberEigenVectors)
#print "completed eigenvalue decomposition "
#sort in decending order
eigv = eigv[::-1]
eigve = eigve[:, ::-1]
# perform nystrom
eigve = eigve*np.sqrt(float(nMC))
eigv = eigv/float(nMC)
return eigv, eigve
def _sparse_spectral(A, dim=2):
# Input adjacency matrix A
# Uses sparse eigenvalue solver from scipy
# Could use multilevel methods here, see Koren "On spectral graph drawing"
import numpy as np
from scipy.sparse import spdiags
from scipy.sparse.linalg.eigen import eigsh
try:
nnodes, _ = A.shape
except AttributeError:
msg = "sparse_spectral() takes an adjacency matrix as input"
raise nx.NetworkXError(msg)
# form Laplacian matrix
data = np.asarray(A.sum(axis=1).T)
D = spdiags(data, 0, nnodes, nnodes)
L = D - A
k = dim + 1
# number of Lanczos vectors for ARPACK solver.What is the right scaling?
ncv = max(2 * k + 1, int(np.sqrt(nnodes)))
# return smallest k eigenvalues and eigenvectors
eigenvalues, eigenvectors = eigsh(L, k, which='SM', ncv=ncv)
index = np.argsort(eigenvalues)[1:k] # 0 index is zero eigenvalue
return np.real(eigenvectors[:, index])
def _sparse_spectral(A, dim=2):
# Input adjacency matrix A
# Uses sparse eigenvalue solver from scipy
# Could use multilevel methods here, see Koren "On spectral graph drawing"
try:
import numpy as np
from scipy.sparse import spdiags
except ImportError:
raise ImportError("_sparse_spectral() requires scipy & numpy: http://scipy.org/ ")
try:
from scipy.sparse.linalg.eigen import eigsh
except ImportError:
# scipy <0.9.0 names eigsh differently
from scipy.sparse.linalg import eigen_symmetric as eigsh
try:
nnodes, _ = A.shape
except AttributeError:
raise nx.NetworkXError("sparse_spectral() takes an adjacency matrix as input")
# form Laplacian matrix
data = np.asarray(A.sum(axis=1).T)
D = spdiags(data, 0, nnodes, nnodes)
L = D - A
k = dim + 1
# number of Lanczos vectors for ARPACK solver.What is the right scaling?
ncv = max(2 * k + 1, int(np.sqrt(nnodes)))
# return smallest k eigenvalues and eigenvectors
eigenvalues, eigenvectors = eigsh(L, k, which="SM", ncv=ncv)
index = np.argsort(eigenvalues)[1:k] # 0 index is zero eigenvalue
return np.real(eigenvectors[:, index])
def get_eigenValsVecs(self):
nDOF = self.space.dim() # number of dof
d = self.mesh.geometry().dim()
c4dof = self.space.tabulate_dof_coordinates().reshape(nDOF, d)
u = TrialFunction(self.space)
v = TestFunction(self.space)
MassM = assemble(u * v * dx)
M = MassM.array()
c0 = np.repeat(c4dof, nDOF, axis=0)
c1 = np.tile(c4dof, [nDOF, 1])
r = np.linalg.norm(c0 - c1, axis=1)
C = self.cov_scal * np.exp(-r ** 2 / self.cov_lenght ** 2) # covariance
C.shape = [nDOF, nDOF]
A = np.dot(M, np.dot(C, M))
w, v = eigsh(A, self.nModes, M)
dof2vert = self.space.dofmap().dofs()
eigenVectors = np.array([z[dof2vert] for z in v.T])
eigenVectors = list(eigenVectors)
eigenValues = list(w)
return eigenValues[::-1], eigenVectors[::-1]
This can be done by projection with inner products:
y_1*y_4 = a_1 y_1*y_1 + a_2 y_1*y_2 + a_3 y_1*y_3
y_2*y_4 = a_1 y_2*y_1 + a_2 y_2*y_2 + a_3 y_2*y_3
y_3*y_4 = a_1 y_3*y_1 + a_2 y_3*y_2 + a_3 y_3*y_3
We see that we have to solve a linear system with the
covariance matrix M_ij = y_i*y_j . To find the most relevant features
and reduce dimensionality, we use a PCA with only the highest eigenvalues.
We center the covariance around the mean, i.e. subtract ymean before.
"""
ymean = np.mean(ytrain, 0)
dytrain = ytrain - ymean
M = dytrain @ dytrain.T
w, Q = eigsh(M, 10)
plt.figure()
plt.plot(w)
plt.figure()
for i in range(len(w)):
plt.plot(w[i]*Q[:,i] @ dytrain)
plt.title('10 most significant eigenvectors')
plt.xlabel('Independent variable x')
plt.ylabel('Eigenvectors scaled by eigenvalue')
#%% Testing
utest = 0.7
ytest = f(utest, x)
b = np.empty(ntrain)
for i in range(ntrain):
plt.plot(atrain[:, 0], atrain[:, 1], 'x')
plt.axis('equal')
plt.title('Weights')
# %%
from scipy.sparse.linalg.eigen import eigsh
neig = 5
M = np.empty([2 * ntrain, 2 * ntrain])
M[:ntrain, :ntrain] = dqtrain @ dqtrain.T
M[:ntrain, ntrain:] = dqtrain @ dptrain.T
M[ntrain:, :ntrain] = dptrain @ dqtrain.T
M[ntrain:, ntrain:] = dptrain @ dptrain.T
w, Q = eigsh(M, neig)
Q = Q[:, :-neig - 1:-1]
w = w[:-neig - 1:-1]
plt.figure()
plt.semilogy(w)
for i in range(len(w)):
plt.figure()
qeig = Q[:ntrain, i] @ dqtrain
peig = Q[ntrain:, i] @ dptrain
plt.figure()
plt.plot(qeig, peig)
plt.xlabel('qeig')
plt.ylabel('peig')
plt.xlabel('Independent variable t')
plt.ylabel('Output f(t;u)')
plt.title(f'Output f(t;u) for different u')
#%%
utrain, utest, ytrain, ytest = train_test_split(indata,
outdata,
test_size=0.95)
ntrain = utrain.shape[0]
ntest = utest.shape[1]
ymean = np.mean(ytrain, 0)
dytrain = ytrain - ymean
M = dytrain @ dytrain.T
w, Q = eigsh(M, 22)
plt.figure()
plt.semilogy(w[::-1] / w[-1])
plt.xlabel('Number')
plt.ylabel('Relative size')
plt.title('Eigenvalue decay')
neig = 5
Q = Q[:, :-neig - 1:-1]
w = w[:-neig - 1:-1]
plt.figure()
for i in range(len(w)):
efunc = Q[:, i] @ dytrain
plt.plot(-efunc * np.sign(efunc[0])) # Sign-flipped eigenfunctions
#plt.plot(efunc) # Eigenfunctions