def tridiag_eigs():
# Most of this code is just constructing
# tridiagonal matrices and calling functions
# they have already written.
m = 1000
k = 100
A = np.zeros((m, m))
a = rand(m)
b = rand(m-1)
np.fill_diagonal(A, a)
np.fill_diagonal(A[1:], b)
np.fill_diagonal(A[:,1:], b)
Amul = lambda u: tri_mul(a, b, u)
alpha, beta = lanczos(rand(m), Amul, k)
H = np.zeros((alpha.size, alpha.size))
np.fill_diagonal(H, alpha)
np.fill_diagonal(H[1:], beta)
np.fill_diagonal(H[:,1:], beta)
H_eigs = eig(H, right=False)
H_eigs.sort()
H_eigs = H_eigs[::-1]
print H_eigs[:10]
A = np.zeros((m, m))
np.fill_diagonal(A, a)
np.fill_diagonal(A[1:], b)
np.fill_diagonal(A[:,1:], b)
A_eigs = eig(A, right=False)
A_eigs.sort()
A_eigs = A_eigs[::-1]
print A_eigs[:10]
def algorithm3(self, e0, M):
# Compute global maximum expansion rate
Vtest = linspace(0,0.5,1000)
muplot = []
for Vi in Vtest:
J = self.Jac(0,[Vi,0])
muplot.append( 0.5*max(real(eig(J+J.T)[0])) )
index = argmax(muplot)
mustar = muplot[index]
Vstar = Vtest[index]
self.d1 = [e0]
self.d2 = [e0]
self.theta = [0]
c = []
for i in range(len(self.T)-1):
# compute maximal expansion rate c_i in a neigbourhood of V[i] using global vector field bound M
if abs(self.V[i] - Vstar) <= self.d1[i] + M*(self.T[i+1]-self.T[i]):
c.append(mustar)
elif self.V[i] + self.d1[i] + M*(self.T[i+1]-self.T[i]) < Vstar:
J = Jac(0,[self.V[i] + self.d1[i] + M*(self.T[i+1]-self.T[i]),0])
c.append(0.5*max(real(eig(J+J.T)[0])))
else:
J = Jac(0,[self.V[i] - self.d1[i] - M*(self.T[i+1]-self.T[i]),0])
c.append(0.5*max(real(eig(J+J.T)[0])))
# compute diameter of ball based on bound on expansion rate in neighbourhood of current state
self.d1.append(exp(c[i]*(self.T[i+1]-self.T[i]))*self.d1[i]+self.tolerance)
self.d2.append(exp(c[i]*(self.T[i+1]-self.T[i]))*self.d2[i]+self.tolerance)
self.theta.append(0)
def plot_ritz(A, n, iters):
''' Plot the relative error of the Ritz values of `A'.
'''
Amul = A.dot
b = np.random.rand(A.shape[0])
Q = np.empty((len(b), iters+1), dtype = np.complex128)
H = np.zeros((iters+1, iters), dtype = np.complex128)
Q[:, 0] = b / la.norm(b)
eigvals = np.sort(abs(la.eig(A)[0]))[::-1]
eigvals = eigvals[:n]
abs_err = np.zeros((iters,n))
for j in xrange(iters):
Q[:, j+1] = Amul(Q[:, j])
for i in xrange(j+1):
H[i,j] = np.vdot(Q[:,i].conjugate(), (Q[:, j+1]))
Q[:,j+1] = Q[:,j+1] - H[i,j] * (Q[:,i])
H[j+1, j] = np.sqrt(np.vdot(Q[:, j+1], Q[:, j+1].conjugate()))
Q[:,j+1] = Q[:,j+1] / H[j+1, j]
if j < n:
rit = np.zeros(n, dtype = np.complex128)
rit[:j+1] = np.sort(la.eig(H[:j+1, :j+1])[0])[::-1]
abs_err[j,:] = abs(eigvals - rit) / abs(eigvals)
else:
rit = np.sort(la.eig(H[:j+1,:j+1])[0])[::-1]
rit = rit[:n]
abs_err[j,:] = abs(eigvals - rit) / abs(eigvals)
for i in xrange(n):
plt.semilogy(abs_err[:,i])
plt.show()
def SlowDownFactor(temporalnet):
"""Returns a factor S that indicates how much slower (S>1) or faster (S<1)
a diffusion process in the temporal network evolves on a second-order model
compared to a first-order model. This value captures the effect of order
correlations on dynamical processes.
"""
g2 = temporalnet.iGraphSecondOrder().components(mode="STRONG").giant()
g2n = temporalnet.iGraphSecondOrderNull().components(mode="STRONG").giant()
A2 = np.matrix(list(g2.get_adjacency()))
T2 = np.zeros(shape=(len(g2.vs), len(g2.vs)))
D2 = np.diag(g2.strength(mode='out', weights=g2.es["weight"]))
for i in range(len(g2.vs)):
for j in range(len(g2.vs)):
T2[i,j] = A2[i,j]/D2[i,i]
A2n = np.matrix(list(g2n.get_adjacency()))
T2n = np.zeros(shape=(len(g2n.vs), len(g2n.vs)))
D2n = np.diag(g2n.strength(mode='out', weights=g2n.es["weight"]))
for i in range(len(g2n.vs)):
for j in range(len(g2n.vs)):
T2n[i,j] = A2n[i,j]/D2n[i,i]
w2, v2 = spl.eig(T2, left=True, right=False)
w2n, v2n = spl.eig(T2n, left=True, right=False)
return np.log(np.abs(w2n[1]))/np.log(np.abs(w2[1]))
def eig_linearised(Z, modes):
"""Solves a linearised approximation to the eigenvalue problem from
the impedance calculated at some fixed frequency.
The equation :math:`L = -s^2 S` is solved for `s`
Parameters
----------
Z : EfieImpedanceMatrixLoopStar
The impedance matrix calculated in a loop-star basis
modes : ndarray (int)
A list or array of the mode numbers required
Returns
-------
s_mode : ndarray, complex
The resonant frequencies of the modes (in Hz)
The complex pole `s` corresponding to the mode's eigenfrequency
j_mode : ndarray, complex
Columns of this matrix contain the corresponding modal currents
"""
modes = np.asarray(modes)
L = Z.matrices['L']
S = Z.matrices['S']
try:
# Try to find the loop and star parts of the matrix (all relevant
# matrices and vectors follow the same decomposition)
loop, star = loop_star_indices(L)
except AttributeError:
loop = [[], []]
star = [slice(None), slice(None)]
if len(loop[0]) > 0 and len(loop[1]) > 0:
L_conv = la.solve(L[loop[0], loop[1]],
L[loop[0], star[1]])
L_red = (L[star[0], star[1]] -
np.dot(L[star[0], loop[1]], L_conv))
# find eigenvalues, and star part of eigenvectors
w, v_s = la.eig(S[star[0], star[1]], -L_red)
vr = np.empty((L.shape[0], len(w)), np.complex128)
vr[star[1]] = v_s
vr[loop[1]] = -np.dot(L_conv, v_s)
else:
# Matrix does not have loop-star decomposition, so use the whole thing
# TODO: implement some filtering to eliminate null-space solutions?
w, vr = la.eig(S, -L)
w_freq = np.sqrt(w)
# make sure real part is negative
w_freq = np.where(w_freq.real > 0, -w_freq, w_freq)
w_selected = np.ma.masked_array(w_freq, abs(w_freq.real) > abs(w_freq.imag))
which_modes = np.argsort(abs(w_selected.imag))[modes]
return w_freq[which_modes], vr[:, which_modes]
def coupled_modes(self, w, ignore_damping=False, **kwargs):
M, B, C = self.linearised_matrices(w, **kwargs)
if ignore_damping:
wn, vn = linalg.eig(C, M)
order = np.argsort(abs(wn))
wn = np.sqrt(abs(wn[order]))
vn = vn[:, order]
else:
AA = r_[c_[zeros_like(C), C], c_[C, B]]
BB = r_[c_[C, zeros_like(C)], c_[zeros_like(C), -M]]
wn, vn = linalg.eig(AA, BB)
order = np.argsort(abs(wn))
wn = abs(wn[order])
# Mode shapes are the first half of the rows; the second
# half of the rows should be same multiplied by eigenvalues.
vn = vn[:M.shape[0], order]
# We expect all the modes to be complex conjugate; return
# every other one.
# First: make sure all are the same sign
norm_vn = vn / vn[np.argmax(abs(vn), axis=0), range(vn.shape[1])]
assert (np.allclose(wn[::2], wn[1::2], rtol=1e-4) and
np.allclose(norm_vn[:, ::2], norm_vn[:, 1::2].conj(), atol=1e-2)), \
"Expect conjugate modes"
wn = wn[::2]
vn = norm_vn[:, ::2]
return wn, vn
def spatialFilter(Ra,Rb): R = Ra + Rb E,U = la.eig(R) # CSP requires the eigenvalues E and eigenvector U be sorted in descending order ord = np.argsort(E) ord = ord[::-1] # argsort gives ascending order, flip to get descending E = E[ord] U = U[:,ord] # Find the whitening transformation matrix P = np.dot(np.sqrt(la.inv(np.diag(E))),np.transpose(U)) # The mean covariance matrices may now be transformed Sa = np.dot(P,np.dot(Ra,np.transpose(P))) Sb = np.dot(P,np.dot(Rb,np.transpose(P))) # Find and sort the generalized eigenvalues and eigenvector # Find and sort the generalized eigenvalues and eigenvector E1,U1 = la.eig(Sa,Sb) ord1 = np.argsort(E1) ord1 = ord1[::-1] E1 = E1[ord1] U1 = U1[:,ord1] # The projection matrix (the spatial filter) may now be obtained SFa = np.dot(np.transpose(U1),P) return SFa.astype(np.float32)
def FLQTEB(engine,app):
nmatrix=len(engine.generators['h'].table)
if app.path!=None:
result=zeros((app.path.rank,nmatrix+1))
key=app.path.mesh.keys()[0]
if len(app.path.mesh[key].shape)==1:
result[:,0]=app.path.mesh[key]
else:
result[:,0]=array(xrange(app.path.rank[key]))
for i,parameter in enumerate(list(app.path.mesh[key])):
result[i,1:]=phase(eig(engine.evolution(t=app.ts.mesh['t'],**{key:parameter}))[0])/app.ts.volume['t']
else:
result=zeros((2,nmatrix+1))
result[:,0]=array(xrange(2))
result[0,1:]=angle(eig(engine.evolution(t=app.ts.mesh['t']))[0])/app.ts.volume['t']
result[1,1:]=result[0,1:]
if app.save_data:
savetxt(engine.dout+'/'+engine.name.full+'_EB.dat',result)
if app.plot:
plt.title(engine.name.full+'_EB')
plt.plot(result[:,0],result[:,1:])
if app.show:
plt.show()
else:
plt.savefig(engine.dout+'/'+engine.name.full+'_EB.png')
def test_streamline_tensors():
# Small streamline
streamline = [[1, 2, 3], [4, 5, 3], [5, 6, 3]]
# Non-default eigenvalues:
evals = [0.0012, 0.0006, 0.0004]
streamline_tensors = life.streamline_tensors(streamline, evals=evals)
npt.assert_array_almost_equal(streamline_tensors[0],
np.array([[0.0009, 0.0003, 0.],
[0.0003, 0.0009, 0.],
[0., 0., 0.0004]]))
# Get the eigenvalues/eigenvectors:
eigvals, eigvecs = la.eig(streamline_tensors[0])
eigvecs = eigvecs[np.argsort(eigvals)[::-1]]
eigvals = eigvals[np.argsort(eigvals)[::-1]]
npt.assert_array_almost_equal(eigvals,
np.array([0.0012, 0.0006, 0.0004]))
npt.assert_array_almost_equal(eigvecs[0],
np.array([0.70710678, -0.70710678, 0.]))
# Another small streamline
streamline = [[1, 0, 0], [2, 0, 0], [3, 0, 0]]
streamline_tensors = life.streamline_tensors(streamline, evals=evals)
for t in streamline_tensors:
eigvals, eigvecs = la.eig(t)
eigvecs = eigvecs[np.argsort(eigvals)[::-1]]
# This one has no rotations - all tensors are simply the canonical:
npt.assert_almost_equal(np.rad2deg(np.arccos(
np.dot(eigvecs[0], [1, 0, 0]))), 0)
npt.assert_almost_equal(np.rad2deg(np.arccos(
np.dot(eigvecs[1], [0, 1, 0]))), 0)
npt.assert_almost_equal(np.rad2deg(np.arccos(
np.dot(eigvecs[2], [0, 0, 1]))), 0)
def regular_svd_by_pca(K, k=0):
K_size = K.shape;
if( K_size[0] < K_size[1] ):
K_squared = np.dot(K, K.T);
tsp, tUp = la.eig(K_squared);
else:
K_squared = np.dot(K.T, K);
tsp, tVp = la.eig(K_squared);
# As la.eig returns complex number, use its absolute value.
tsp = abs(tsp);
tsp = np.sqrt(tsp);
n_pos_sigs = sum(tsp > 0);
tSp = np.diag(map(lambda s: 1.0/s, tsp[0:n_pos_sigs]));
if( K_size[0] < K_size[1] ):
tVp = np.dot(K.T, tUp);
tVp[:, 0:n_pos_sigs] = np.dot(tVp[:, 0:n_pos_sigs], tSp);
else:
tUp = np.dot(K, tVp);
tUp[:, 0:n_pos_sigs] = np.dot(tUp[:, 0:n_pos_sigs], tSp);
if( 0 < k and k < min(K_size) ):
tUp = tUp[:, 0:k];
tVp = tVp[:, 0:k];
tsp = tsp[0:k];
return tUp, tsp, tVp;
def FindMaximumQAQ(A, vertices, tetra):
lambdas = []
Q = np.zeros((4,4))
for i in range(4):
Q[:,i] = vertices[tetra[i],:]
print "Q", Q
# Full problem:
A_ = Q.T.dot(A).dot(Q)
B_ = Q.T.dot(Q)
e, V = eig(A_, B_)
alpha = np.real(V[:,np.argmax(e)])
if np.all(alpha >= 0.) or np.all(alpha <= 0.):
lambdas.append(np.max(np.real(e)))
# Only three qs:
for comb in combinations(range(4), 3):
A__ = np.array([[A_[i,j] for j in comb] for i in comb])
B__ = np.array([[B_[i,j] for j in comb] for i in comb])
e, V = eig(A__, B__)
alpha = np.real(V[:,np.argmax(e)])
if np.all(alpha >= 0.) or np.all(alpha <= 0.):
lambdas.append(np.max(np.real(e)))
# Only two qs:
for comb in combinations(range(4), 2):
A__ = np.array([[A_[i,j] for j in comb] for i in comb])
B__ = np.array([[B_[i,j] for j in comb] for i in comb])
e, V = eig(A__, B__)
alpha = np.real(V[:,np.argmax(e)])
if np.all(alpha >= 0.) or np.all(alpha <= 0.):
lambdas.append(np.max(np.real(e)))
# Only one q:
for i in range(4):
lambdas.append((Q[:,i]).T.dot(A).dot(Q[:,i]))
print lambdas
return np.max(np.array(lambdas))
def pca(data, base_num=1):
N, dim = data.shape
data_m = data.mean(0)
data_new = data - data_m
# データ数 > 次元数
if N > dim:
# データ行列の共分散行列
cov_mat = sp.dot(data_new.T, data_new) / float(N)
# 固有値・固有ベクトルを計算
l, vm = linalg.eig(cov_mat)
# 固有値が大きい順に並び替え
axis = vm[:, l.argsort()[-min(base_num, dim) :][::-1]].T
# 次元数 > データ数
else:
base_num = min(base_num, N)
cov_mat = sp.dot(data_new, data_new.T) / float(N)
l, v = linalg.eig(cov_mat)
# 固有値と固有ベクトルを並び替え
idx = l.argsort()[::-1]
l = l[idx]
v = vm[:, idx]
# 固有ベクトルを変換
vm = sp.dot(data_m.T, v[:, :base_num])
# （主成分の）基底を計算
axis = sp.zeros([base_num, dim], dtype=sp.float64)
for ii in range(base_num):
if l[ii] <= 0:
break
axis[ii] = vm[:, ii] / linalg.norm(vm[:, ii])
return axis
def test_create(self):
basis_set = SphericalGTOSet()
xyz = (0.0, 0.0, 0.0)
r0 = 10.0
for n in range(-10, 10):
z = 2.0**n
basis_set.add_one_basis(0, 0, xyz, z)
for L in [0]:
basis_set.add_basis(L, (0.0, 0.0, +r0), z)
basis_set.add_basis(L, (0.0, 0.0, -r0), z)
basis_set.add_basis(L, (0.0, +r0, 0.0), z)
basis_set.add_basis(L, (0.0, -r0, 0.0), z)
basis_set.add_basis(L, (+r0, 0.0, 0.0), z)
basis_set.add_basis(L, (-r0, 0.0, 0.0), z)
smat = basis_set.s_mat()
for e in sorted(abs(la.eig(smat)[0]))[0:10]:
print e
print "-----"
hmat = basis_set.t_mat() + basis_set.v_mat(1.0, xyz)
zmat = basis_set.xyz_mat((0, 0, 1))
for e in sorted(la.eig(hmat, smat)[0].real)[0:10]:
print e
"""
def eig(A,B): """ To ensure matlab compatibility, we need to swap matrices A and B around !! """ (XX1,XX2) = LIN.eig(A,B) (XX3,XX4) = LIN.eig(B,A) return (mat(XX4),mat(XX1))
def __init__(self,matrix,error,overlap=None,overlap_err=None):
if overlap_err is None:
self.func = lambda mat:lin.eig(mat,overlap)[0]
self.resample = lambda: gaussian_matrix_resample(matrix,error)
else:
self.func = lambda mats:lin.eig(mats[0],mats[1])[0]
self.resample = lambda: (gaussian_matrix_resample(matrix,error),
gaussian_matrix_resample(overlap,overlap_err))
def compute_method(self, parametro= None):
''' eigenvalues and eigen vectors '''
#first era method
# self.__eraresOut.A, self.__eraresOut.B, self.__eraresOut.C = mr.compute_ERA_model(np.array(self._signalOut), 2)
# second, eigenvalues and eigenvectors
self.__eraresOut.lambdaValues, self.__eraresOut.lambdaVector = linalg.eig(self.__eraresOut.A)
# self.__eraresRef.A, self.__eraresRefB, self.__eraresRefC = mr.compute_ERA_model(np.array(self._signalRef), 2)
# second, eigenvalues and eigenvectors
self.__eraresRef.lambdaValues, self.__eraresRef.lamdaVector = linalg.eig(self.__eraresRef.A)
def designCSP(dataA, dataB, nb):
# return v, a, d
n_channels = dataA.shape[0]
q = dataA.shape[1]
cA = np.zeros([dataA.shape[0], n_channels, n_channels])
cB = np.zeros([dataB.shape[0], n_channels, n_channels])
# Compute the covariance matrix of each epoch of the same class (A and B)
for i in range(dataA.shape[0]):
# cA[i,...] = np.cov(dataA[i,:,:])
c = np.dot(dataA[i, :, :], dataA[i, :, :].transpose())
cA[i, ...] = c / (np.trace(c) * q)
# cA[i,...] = c
# compute the mean of the covariance matrices of each epoch
cA_mean = cA.mean(0)
for i in range(dataB.shape[0]):
# cB[i,...] = np.cov(dataB[i,:,:])
c = np.dot(dataB[i, :, :], dataB[i, :, :].transpose())
cB[i, ...] = c / (np.trace(c) * q)
# cB[i,...] = c
# compute the mean of the covariance matrices of each epoch
cB_mean = cB.mean(0)
lamb, v = lg.eig(cA_mean + cB_mean) # eigvalue and eigvector decomposition
lamb = lamb.real # return only real part of eigen vector
# returns the index of array lamb in crescent order
index = np.argsort(lamb)
# reverse the order, now index has the positon of lamb in descendent order
index = index[::-1]
lamb = lamb[index] # sort the eingenvalues in descendent order
# the same goes for the eigenvectors along axis y
v = v.take(index, axis=1)
# whitening matrix computation
Q = np.dot(np.diag(1 / np.sqrt(lamb)), v.transpose())
# eig decomposition of whiten cov matrix
D, V = lg.eig(np.dot(Q, np.dot(cA_mean, Q.transpose())))
W_full = np.dot(V.transpose(), Q)
# select only the neighbours defined in NB; get the first 3 eigenvectors
W = W_full[:nb, :]
W = np.vstack((W, W_full[-nb:, :])) # get the three last eigenvectors
return W
def mysqrtm(m):
m = 0.5 * (m.H + m)
ls, vs = la.eigh(m)
vs = np.matrix(vs)
try:
ls = [math.sqrt(max(l.real, 0)) for l in ls]
except ValueError:
print m.H - m
print la.eig(m)
raise ValueError
return vs * np.diag(ls) * vs.H
def printEigen(A, F):
print 'Pole Locations:'
(w_A, v_A) = la.eig(A)
(w_F, v_F) = la.eig(F)
for i in range(0, len(w_A)):
print 'w_A = ', w_A[i]
print 'v_A = \n', v_A[:,i]
for i in range(0, len(w_F)):
print 'w_F = ', w_F[i]
print 'v_F = \n', v_F[:,i]
return
def main() :
print power_iteration(Q, A)
w, pi = ln.eig(A, left=True, right=False)
print w
print pi
print power_iteration(Q, B)
w, pi = ln.eig(B, left=True, right=False)
print w
print pi
def test_aligned_mem():
"""Check linalg works with non-aligned memory"""
# Allocate 804 bytes of memory (allocated on boundary)
a = arange(804, dtype=np.uint8)
# Create an array with boundary offset 4
z = np.frombuffer(a.data, offset=4, count=100, dtype=float)
z.shape = 10, 10
eig(z, overwrite_a=True)
eig(z.T, overwrite_a=True)
def eigen(A, B=None):
"""
This function is used to sort eigenvalues and eigenvectors
e.g. for a given system linalg.eig will return eingenvalues as:
(array([ 0. +89.4j, 0. -89.4j, 0. +89.4j, 0. -89.4j, 0.+983.2j, 0.-983.2j, 0. +40.7j, 0. -40.7j])
This function will sort this eigenvalues as:
(array([ 0. +40.7j, 0. +89.4j, 0. +89.4j, 0.+983.2j, 0. -40.7j, 0. -89.4j, 0. -89.4j, 0.-983.2j])
Correspondent eigenvectors will follow the same order.
Parameters
----------
A: array
A complex or real matrix whose eigenvalues and eigenvectors
will be computed.
B: float or str
Right-hand side matrix in a generalized eigenvalue problem.
Default is None, identity matrix is assumed.
Returns
----------
evalues: array
Sorted eigenvalues
evectors: array
Sorted eigenvalues
Examples:
>>> L = sp.array([[2, -1, 0],
... [-4, 8, -4],
... [0, -4, 4]])
>>> lam, P = eigen(L)
>>> lam
array([ 0.56258062+0.j, 2.63206172+0.j, 10.80535766+0.j])
"""
if B is None:
evalues, evectors = la.eig(A)
else:
evalues, evectors = la.eig(A, B)
if all(eigs == 0 for eigs in evalues.imag):
if all(eigs > 0 for eigs in evalues.real):
idxp = evalues.real.argsort() # positive in increasing order
idxn = sp.array([], dtype=int)
else:
idxp = evalues.real.argsort()[int(len(evalues)/2):] # positive in increasing order
idxn = evalues.real.argsort()[int(len(evalues)/2) - 1::-1] # negative in decreasing order
else:
idxp = evalues.imag.argsort()[int(len(evalues)/2):] # positive in increasing order
idxn = evalues.imag.argsort()[int(len(evalues)/2) - 1::-1] # negative in decreasing order
idx = sp.hstack([idxp, idxn])
return evalues[idx], evectors[:, idx]
def test_aligned_mem_complex():
"""Check that complex objects don't need to be completely aligned"""
# Allocate 1608 bytes of memory (allocated on boundary)
a = zeros(1608, dtype=np.uint8)
# Create an array with boundary offset 8
z = np.frombuffer(a.data, offset=8, count=100, dtype=complex)
z.shape = 10, 10
eig(z, overwrite_a=True)
# This does not need special handling
eig(z.T, overwrite_a=True)
def _poles_and_tangential_directions(rom):
"""Compute the poles and tangential directions of a reduced order model."""
if isinstance(rom.E, IdentityOperator):
poles, Y, X = spla.eig(to_matrix(rom.A, format='dense'),
left=True, right=True)
else:
poles, Y, X = spla.eig(to_matrix(rom.A, format='dense'), to_matrix(rom.E, format='dense'),
left=True, right=True)
Y = rom.B.range.make_array(Y.conj().T)
X = rom.C.source.make_array(X.T)
b = rom.B.apply_adjoint(Y)
c = rom.C.apply(X)
return poles, b, c
def transitionMatrix(cg, minstrength=0.1):
A = gk.CG2adj(cg)
edges = scipy.where(A == 1)
A[edges] = randweights(edges[0].shape[0], c=minstrength)
l = linalg.eig(A)[0]
c = 0
pbar = ProgressBar(widgets=['Searching for weights: ', Percentage(), ' '], maxval=10000).start()
while max(l*scipy.conj(l)) > 1:
A[edges] = randweights(edges[0].shape[0], c=c)
c += 1
l = linalg.eig(A)[0]
pbar.update(c)
pbar.finish()
return A
def ps_contour_plot(A, m = 20,epsilon_vals=None):
'''Plots the pseudospectrum of the matrix A as a contour plot. Also,
plots the eigenvalues.
Parameters:
A : square, 2D ndarray
The matrix whose pseudospectrum is to be plotted
m : int
accuracy
epsilon_vals : list of floats
If k is in epsilon_vals, then the epsilon-pseudospectrum
is plotted for epsilon=10**-k
If epsilon_vals=None, the defaults of plt.contour() are used
instead of any specified values.
'''
n = A.shape[0]
T = la.schur(A)[0]
eigsA = np.diagonal(T)
xvals, yvals = ps_grid(eigsA, m)
sigmin = np.zeros((m, m))
for k in xrange(m):
for j in xrange(m):
T1 = (xvals[k] + 1j*yvals[j]) * np.eye(n) - T
T2 = T1.T.conjugate()
sigold = 0
qold = np.zeros((n, 1))
beta = 0
H = np.zeros((n, n))
q = np.random.normal(size=(n, 1)) + 1j * np.random.normal(size=(n, 1))
q = q/la.norm(q, ord=2)
for p in xrange(n-1):
b1 = la.solve(T2, q)
b2 = la.solve(T1, b1)
v = b2 - beta * qold
alpha = np.real(np.vdot(q,v))
v = v - alpha * q
beta = la.norm(v)
qold = q
q = v/beta
H[p+1, p] = beta
H[p, p+1] = beta
H[p, p] = alpha
sig = np.abs(np.max(la.eig(H[:p+1,:p+1])[0]))
if np.abs(sigold/sig - 1) < .001:
break
sigold = sig
sigmin[j, k] = np.sqrt(sig)
plt.contour(xvals,yvals,np.log10(sigmin), levels=epsilon_vals)
plt.scatter(la.eig(A)[0].real, la.eig(A)[0].imag)
plt.show()
def test_approximate_spectral_radius(self):
cases = []
cases.append( matrix([[-4-4.0j]]) )
cases.append( matrix([[-4+8.2j]]) )
cases.append( matrix([[2.0-2.9j,0],[0,1.5]]) )
cases.append( matrix([[-2.0-2.4j,0],[0,1.21]]) )
cases.append( matrix([[100+1.0j,0,0],[0,101-1.0j,0],[0,0,99+9.9j]]) )
for i in range(1,6):
cases.append( matrix(rand(i,i)+1.0j*rand(i,i)) )
# method should be almost exact for small matrices
for A in cases:
Asp = csr_matrix(A)
[E,V] = linalg.eig(A)
E = abs(E)
largest_eig = (E == E.max()).nonzero()[0]
expected_eig = E[largest_eig]
expected_vec = V[:,largest_eig]
assert_almost_equal( approximate_spectral_radius(A), expected_eig )
assert_almost_equal( approximate_spectral_radius(Asp), expected_eig )
vec = approximate_spectral_radius(A, return_vector=True)[1]
rayleigh = abs( dot(ravel(A*vec), ravel(vec)) / dot(ravel(vec), ravel(vec)) )
assert_almost_equal(rayleigh, expected_eig, decimal=4 )
vec = approximate_spectral_radius(Asp, return_vector=True)[1]
rayleigh = abs( dot(ravel(Asp*vec), ravel(vec)) / dot(ravel(vec), ravel(vec)) )
assert_almost_equal(rayleigh, expected_eig, decimal=4 )
AA = mat(A).H*mat(A)
AAsp = csr_matrix(AA)
[E,V] = linalg.eig(AA)
E = abs(E)
largest_eig = (E == E.max()).nonzero()[0]
expected_eig = E[largest_eig]
expected_vec = V[:,largest_eig]
assert_almost_equal( approximate_spectral_radius(AA), expected_eig )
assert_almost_equal( approximate_spectral_radius(AAsp), expected_eig )
vec = approximate_spectral_radius(AA, return_vector=True)[1]
rayleigh = abs( dot(ravel(AA*vec), ravel(vec)) / dot(ravel(vec), ravel(vec)) )
assert_almost_equal(rayleigh, expected_eig, decimal=4 )
vec = approximate_spectral_radius(AAsp, return_vector=True)[1]
rayleigh = abs( dot(ravel(AAsp*vec), ravel(vec)) / dot(ravel(vec), ravel(vec)) )
assert_almost_equal(rayleigh, expected_eig, decimal=4 )
def process(self, X, V, C):
BifPoint.process(self, X, V, C)
J_coords = C.sysfunc.jac(X, C.coords)
eigs, VL, VR = linalg.eig(J_coords, left=1, right=1)
# Check for nonreal multipliers
found = False
for i in range(len(eigs)):
for j in range(i+1,len(eigs)):
if abs(imag(eigs[i])) > 1e-10 and \
abs(imag(eigs[j])) > 1e-10 and \
abs(eigs[i]*eigs[j] - 1) < 1e-5:
found = True
if not found:
del self.found[-1]
return False
self.found[-1].eigs = eigs
self.info(C, -1)
return True
def get_eq_from_eig(m):
''' get the equilibrium frequencies from the matrix. the eq freqs are the left eigenvector corresponding to eigenvalue of 0.
Code here is largely taken from Bloom. See here - https://github.com/jbloom/phyloExpCM/blob/master/src/submatrix.py, specifically in the fxn StationaryStates
'''
(w, v) = linalg.eig(m, left=True, right=False)
max_i = 0
max_w = w[max_i]
for i in range(1, len(w)):
if w[i] > max_w:
max_w = w[i]
max_i = i
assert( abs(max_w) < ZERO ), "Maximum eigenvalue is not close to zero."
max_v = v[:,max_i]
max_v /= np.sum(max_v)
eq_freqs = max_v.real # these are the stationary frequencies
# SOME SANITY CHECKS
assert np.allclose(np.zeros(61), np.dot(eq_freqs, m)) # should be true since eigenvalue of zero
pi_inv = np.diag(1.0 / eq_freqs)
s = np.dot(m, pi_inv)
assert np.allclose(m, np.dot(s, np.diag(eq_freqs)), atol=ZERO, rtol=1e-5), "exchangeability and equilibrium does not recover matrix"
# And for some impressive overkill, double check pi_i*q_ij = pi_j*q_ji
for i in range(61):
pi_i = eq_freqs[i]
for j in range(61):
pi_j = eq_freqs[j]
forward = pi_i * m[i][j]
backward = pi_j * m[j][i]
assert(abs(forward - backward) < ZERO), "Detailed balance violated."
return eq_freqs
def propagator_steadystate(U):
"""Find the steady state for successive applications of the propagator
:math:`U`.
Parameters
----------
U : qobj
Operator representing the propagator.
Returns
-------
a : qobj
Instance representing the steady-state density matrix.
"""
evals, evecs = la.eig(U.full())
ev_min, ev_idx = _get_min_and_index(abs(evals - 1.0))
evecs = evecs.T
rho = Qobj(vec2mat(evecs[ev_idx]), dims=U.dims[0])
rho = rho * (1.0 / rho.tr())
rho = 0.5 * (rho + rho.dag()) # make sure rho is herm
return rho
def Checkvalid(Object,Order,alpha,inorout,mur,sig):
Object = Object[:-4]+".vol"
#Set order Ordercheck to be of low order to speed up computation.
Ordercheck = 1
#Accuracy is increased by increaing noutput, but at greater cost
noutput=20
#Set up the Solver Parameters
Solver,epsi,Maxsteps,Tolerance = SolverParameters()
#Loading the object file
ngmesh = ngmeshing.Mesh(dim=3)
ngmesh.Load("VolFiles/"+Object)
#Creating the mesh and defining the element types
mesh = Mesh("VolFiles/"+Object)
mesh.Curve(5)#This can be used to refine the mesh
#Set materials
mu_coef = [ mur[mat] for mat in mesh.GetMaterials() ]
mu = CoefficientFunction(mu_coef)
inout_coef = [inorout[mat] for mat in mesh.GetMaterials() ]
inout = CoefficientFunction(inout_coef)
sigma_coef = [sig[mat] for mat in mesh.GetMaterials() ]
sigma = CoefficientFunction(sigma_coef)
#Scalars
Mu0 = 4*np.pi*10**(-7)
#Setup the finite element space
dom_nrs_metal = [0 if mat == "air" else 1 for mat in mesh.GetMaterials()]
femfull = H1(mesh, order=Ordercheck,dirichlet="default|outside")
freedofs = femfull.FreeDofs()
ndof=femfull.ndof
Output = np.zeros([ndof,noutput],dtype=float)
Averg = np.zeros([1,3],dtype=float)
# we want to create a list of coordinates where we would like to apply BCs
list=np.zeros([noutput,3],dtype=float)
npp=0
for el in mesh.Elements(BND):
if el.mat == "default":
Averg[0,:]=0
#determine the average coordinate
for v in el.vertices:
Averg[0,:]+=mesh[v].point[:]
Averg=Averg/3
if npp < noutput:
list[npp,:]=Averg[0,:]
npp+=1
print(" solving problems", end='\r')
sval=(Integrate(inout, mesh))**(1/3)
for i in range(noutput):
sol=GridFunction(femfull)
sol.Set(exp(-((x-list[i,0])**2 + (y-list[i,1])**2 + (z-list[i,2])**2)/sval**2),definedon=mesh.Boundaries("default"))
u = femfull.TrialFunction()
v = femfull.TestFunction()
# the bilinear-form
a = BilinearForm(femfull, symmetric=True, condense=True)
a += 1/alpha**2*grad(u)*grad(v)*dx
a += u*v*dx
# the right hand side
f = LinearForm(femfull)
f += 0 * v * dx
if Solver=="bddc":
c = Preconditioner(a,"bddc")#Apply the bddc preconditioner
a.Assemble()
f.Assemble()
if Solver=="local":
c = Preconditioner(a,"local")#Apply the local preconditioner
c.Update()
#Solve the problem
f.vec.data += a.harmonic_extension_trans * f.vec
res = f.vec.CreateVector()
res.data = f.vec - a.mat * sol.vec
inverse= CGSolver(a.mat, c.mat, precision=Tolerance, maxsteps=Maxsteps)
sol.vec.data += inverse * res
sol.vec.data += a.inner_solve * f.vec
sol.vec.data += a.harmonic_extension * sol.vec
Output[:,i] = sol.vec.FV().NumPy()
print(" problems solved ")
Mc = np.zeros([noutput,noutput],dtype=float)
M0 = np.zeros([noutput,noutput],dtype=float)
print(" computing matrices", end='\r')
# create numpy arrays by passing solutions back to NG Solve
Soli=GridFunction(femfull)
Solj=GridFunction(femfull)
for i in range(noutput):
Soli.Set(exp(-((x-list[i,0])**2 + (y-list[i,1])**2 + (z-list[i,2])**2)/sval**2),definedon=mesh.Boundaries("default"))
Soli.vec.FV().NumPy()[:]=Output[:,i]
for j in range(i,noutput):
Solj.Set(exp(-((x-list[j,0])**2 + (y-list[j,1])**2 + (z-list[j,2])**2)/sval**2),definedon=mesh.Boundaries("default"))
Solj.vec.FV().NumPy()[:]=Output[:,j]
Mc[i,j] = Integrate(inout * (InnerProduct(grad(Soli),grad(Solj))/alpha**2+ InnerProduct(Soli,Solj)),mesh)
Mc[j,i] = Mc[i,j]
M0[i,j] = Integrate((1-inout) * (InnerProduct(grad(Soli),grad(Solj))/alpha**2+ InnerProduct(Soli,Solj)),mesh)
M0[j,i] = M0[i,j]
print(" matrices computed ")
# solve the eigenvalue problem
print(" solving eigenvalue problem", end='\r')
out=slin.eig(Mc+M0,Mc,left=False, right=False)
print(" eigenvalue problem solved ")
# compute contants
etasq = np.max((out.real))
C = 1 # It is not clear what this value is.
C1 = C * ( 1 + np.sqrt(etasq) )**2
C2 = 2 * etasq
epsilon = 8.854*10**-12
sigmamin = Integrate(inout * sigma, mesh)/Integrate(inout, mesh)
mumax = Integrate(inout * mu * Mu0, mesh)/Integrate(inout, mesh)
volume = Integrate(inout, mesh)
D = (volume * alpha**3)**(1/3)
cond1 = np.sqrt(1/epsilon/mumax/D**2/C1)
cond2 = 1/epsilon*sigmamin/C2
cond = min(cond1,cond2)
print(" maximum recomeneded frequency is ",str(round(cond/100)))
return cond/100
#!/usr/bin/env python3
# -*- coding: utf-8 -*-
"""
Created on Wed Oct 18 17:14:57 2017
@author: justinwu
"""
import numpy as np
from scipy import linalg as la
A = np.array([[1, 5, 2], [2, 4, 1], [3, 6, 2]])
lna, v = la.eig(A)
l1, l2, l3 = lna
print(l1, l2, l3)
print(v)
print(v[:, 0])
print(v[:, 1])
print(v[:, 2])
print('--------------')
v1 = np.array(v[:, 0]).T
print(v1)
print(linalg.norm(A.dot(v1) - l1 * v1))
n=36
# Legendre Gauss-Lobatto nodes
x,vn=zelegl(n)
# Derivative matrix for Legendre at Gauss-Lobatto nodes
D=dmlegl(n,x,vn,n)
D2=np.dot(D,D)
D2=D2[1:n,1:n]
# Get eigenvalues and right-eigenvectors using scipy.linalg.eig
# Notice: lam,V in reverse order that in Matlab where it's [V,Lam]=eig(D2)
# Also notice the difference In Matlab we get diagonal matrix lam, and here an numpy array lam!!!
lam,V=eig(D2)
ii = np.argsort(-lam) # sort eigenvalues
lam=lam[ii]
V=V[:,ii]
fig = plt.figure()
#fig, axes = plt.subplots(nrows=6, ncols=1)
#fig.tight_layout()
eigs=np.linspace(5,30,6) # plot 6 eigenmodes
for j in eigs:
lv = np.shape(V)[0]+2
u = np.zeros(lv)
u[1:lv-1] = V[:,int(j)]
ax1 = fig.add_subplot(6,1,j/5)
ax1.plot(x,u,'bo')
plt.subplots_adjust(hspace = 0.9)
H = knn(M,k) for u,vec in H.iteritems(): for v in vec: d = ker(M[u],M[v]) A[u,v] = d A = A + A.T D = np.diag( [sum(v) for v in A] ) #print A #return A return D-A L_EN = L(EN,0) L_ES = L(ES,1) evals1, enpvecs = eig(L_EN) evals2, eshvecs = eig(L_ES) New_EN = [] New_ES = [] for e,n in seeded: print e,n,evals1[en[e]].real, n,evals2[es[n]] b_en = enpvecs[en[e]] b_es = eshvecs[es[n]] New_EN.append(b_en) New_ES.append(b_es) EN2 = np.array(New_EN).T ES2 = np.array(New_ES).T print EN2.shape, ES2.shape
iArr = slg.inv(arr)
print('iArr:', iArr)
# 奇异值分解
arr = np.arange(9).reshape((3, 3)) + np.diag([1, 0, 1])
u, sigma, v = slg.svd(arr)
print(u, sigma, v)
# sig3 = np.mat([[sigma[0], 0, 0], [0, sigma[1], 0], [0, 0, sigma[2]]]) # 重构原始矩阵
# newArr = u*sig3*v
# print('arr:', arr)
# print('newArr', newArr)
sig3 = np.diag(sigma) # 重构原始矩阵
newArr = u.dot(sig3).dot(v)
print('arr:', arr)
print('newArr', newArr)
# QR
q, r = slg.qr(arr)
print('q:', q)
print('r:', r)
# 求解
b = np.array([6, 14])
arr = np.array([[1, 2], [3, 4]])
print('Solve:', slg.solve(arr, b))
# 特征值和特征向量
# arr = np.array([[1, 0, 0], [0, 2, 0], [0, 0, 3]])
print('eig:', slg.eig(arr))
def olf_bulb_10(Nmitral,H_in,W_in,P_odor_in,dam):
# Nmitral = 10 #number of mitral cells
Ngranule = np.copy(Nmitral) #number of granule cells pg. 383 of Li/Hop
Ndim = Nmitral+Ngranule #total number of cells
t_inh = 25 ; # time when inhalation starts
t_exh = 205; #time when exhalation starts
finalt = 395; # end time of the cycle
#y = zeros(ndim,1);
Sx = 1.43 #Sx,Sx2,Sy,Sy2 are parameters for the activation functions
Sx2 = 0.143
Sy = 2.86 #These are given in Li/Hopfield pg 382, slightly diff in her thesis
Sy2 = 0.286
th = 1 #threshold for the activation function
tau_exh = 33.3333; #Exhale time constant, pg. 382 of Li/Hop
exh_rate = 1/tau_exh
alpha = .15 #decay rate for the neurons
#Li/Hop have it as 1/7 or .142 on pg 383
P_odor0=np.zeros(Nmitral) #odor pattern, no odor
H0 = H_in #weight matrix: to mitral from granule
W0 = W_in #weights: to granule from mitral
Ib = np.ones((Nmitral,1))*.243 #initial external input to mitral cells
Ic = np.ones((Ngranule,1))*.1 #initial input to granule cells, these values are
#given on pg 382 of Li/Hop
signalflag = 1 # 0 for linear output, 1 for activation function
noise = np.zeros((Ndim,1)) #noise in inputs
noiselevel = .00143
noisewidth = 7 #noise correlation time, given pg 383 Li/Hop as 9, but 7 in thesis
lastnoise = np.zeros((Ndim,1)) #initial time of last noise pule
#******************************************************************************
#CALCULATE FIXED POINTS
#Calculating equilibrium value with no input
rest0 = np.zeros((Ndim,1))
restequi = fsolve(lambda x: equi(x,Ndim,Nmitral,Sx,Sx2,Sy,Sy2,th,alpha,\
t_inh,H0,W0,P_odor0,Ib,Ic,dam),rest0) #about 20 ms to run this
np.random.seed(seed=23)
#init0 = restequi+np.random.rand(Ndim)*.00143 #initial conditions plus some noise
#for no odor input
init0 = restequi+np.random.rand(Ndim)*.00143 #initial conditions plus some noise
#for no odor input
np.random.seed()
#Now calculate equilibrium value with odor input
lastnoise = lastnoise + t_inh - noisewidth #initialize lastnoise value
#But what is it for? to have some
#kind of correlation in the noise
#find eigenvalues of A to see if input produces oscillating signal
xequi = fsolve(lambda x: equi(x,Ndim,Nmitral,Sx,Sx2,Sy,Sy2,th,alpha,\
t_inh,H0,W0,P_odor_in,Ib,Ic,dam),rest0)
#equilibrium values with some input, about 20 ms to run
#******************************************************************************
#CALCULATE A AND DETERMINE EXISTENCE OF OSCILLATIONS
diffgy = celldiff(xequi[Nmitral:],Sy,Sy2,th)
diffgx = celldiff(xequi[0:Nmitral],Sx,Sx2,th)
H1 = np.dot(H0,diffgy)
W1 = np.dot(W0,diffgx) #intermediate step in constructing A
A = np.dot(H1,W1) #Construct A
dA,vA = lin.eig(A) #about 20 ms to run this
#Find eigenvalues of A
diff = (1j)*(dA)**.5 - alpha #criteria for a growing oscillation
negsum = -(1j)*(dA)**.5 - alpha #Same
diff_re = np.real(diff)
#Take the real part
negsum_re = np.real(negsum)
#do an argmax to return the eigenvalue that will cause the fastest growing oscillations
#Then do a spectrograph to track the growth of the associated freq through time
indices = np.where(diff_re>0) #Find the indices where the criteria is met
indices2 = np.where(negsum_re>0)
#eigenvalues that could lead to growing oscillations
# candidates = np.append(np.real((dA[indices])**.5),np.real((dA[indices2])**.5))
largest = np.argmax(diff_re)
check = np.size(indices)
check2 = np.size(indices2)
if check==0 and check2==0:
# print("No Odor Recognized")
dominant_freq = 0
else:
dominant_freq = np.real((dA[largest])**.5)/(2*np.pi) #find frequency of the dominant mode
#Divide by 2pi to get to cycles/ms
# print("Odor detected. Eigenvalues:",dA[indices],dA[indices2],\
# "\nEigenvectors:",vA[indices],vA[indices2],\
# "\nDominant Frequency:",dominant_freq)
#*************************************************************************
#SOLVE DIFFERENTIAL EQUATIONS TO GET INPUT AND OUTPUTS AS FN'S OF t
#differential equation to solve
teval = np.r_[0:finalt]
#solve the differential equation
sol = solve_ivp(lambda t,y: diffeq(t,y,Nmitral,Ngranule,Ndim,lastnoise,\
noise,noisewidth,noiselevel, t_inh,t_exh,exh_rate,alpha,Sy,\
Sy2,Sx,Sx2,th,H0,W0,P_odor_in,Ic,Ib,dam),\
[0,395],init0,t_eval = teval,method = 'RK45')
t = sol.t
y = sol.y
y = np.transpose(y)
yout = np.copy(y)
#convert signal into output signal given by the activation fn
if signalflag ==1:
for i in np.arange(np.size(t)):
yout[i,:Nmitral] = cellout(y[i,:Nmitral],Sx,Sx2,th)
yout[i,Nmitral:] = cellout(y[i,Nmitral:],Sy,Sy2,th)
#solve diffeq for P_odor = 0
#first, reinitialize lastnoise & noise
noise = np.zeros((Ndim,1))
lastnoise = np.zeros((Ndim,1))
lastnoise = lastnoise + t_inh - noisewidth
sol0 = sol = solve_ivp(lambda t,y: diffeq(t,y,Nmitral,Ngranule,Ndim,lastnoise,\
noise,noisewidth,noiselevel, t_inh,t_exh,exh_rate,alpha,Sy,\
Sy2,Sx,Sx2,th,H0,W0,P_odor0,Ic,Ib,dam),\
[0,395],init0,t_eval = teval,method = 'RK45')
y0 = sol0.y
y0 = np.transpose(y0)
y0out = np.copy(y0)
#convert signal into output signal given by the activation fn
if signalflag ==1:
for i in np.arange(np.size(t)):
y0out[i,:Nmitral] = cellout(y0[i,:Nmitral],Sx,Sx2,th)
y0out[i,Nmitral:] = cellout(y0[i,Nmitral:],Sy,Sy2,th)
#*****************************************************************************
#SIGNAL PROCESSING
#Filtering the signal - O_mean: Lowpass fitered signal, under 20 Hz
#S_h: Highpass filtered signal, over 20 Hz
fs = 1/(.001*(t[1]-t[0])) #sampling freq, converting from ms to sec
f_c = 15/fs # Cutoff freq at 20 Hz, written as a ratio of fc to sample freq
flter = np.sinc(2*f_c*(t - (finalt-1)/2))*np.blackman(finalt) #creating the
#windowed sinc filter
#centered at the middle
#of the time data
flter = flter/np.sum(flter) #normalize
hpflter = -np.copy(flter)
hpflter[int((finalt-1)/2)] += 1 #convert the LP filter into a HP filter
Sh = np.zeros(np.shape(yout))
Sl = np.copy(Sh)
Sl0 = np.copy(Sh)
Sbp = np.copy(Sh)
for i in np.arange(Ndim):
Sh[:,i] = np.convolve(yout[:,i],hpflter,mode='same')
Sl[:,i] = np.convolve(yout[:,i],flter,mode='same')
Sl0[:,i] = np.convolve(y0out[:,i],flter,mode='same')
#find the oscillation period Tosc (Tosc must be greater than 5 ms to exclude noise)
Tosc0 = np.zeros(np.size(np.arange(5,50)))
for i in np.arange(5,50):
Sh_shifted=np.roll(Sh,i,axis=0)
Tosc0[i-5] = np.sum(np.diagonal(np.dot(np.transpose(Sh[:,:Nmitral]),Sh_shifted[:,:Nmitral])))
#That is, do the correlation matrix (time correlation), take the diagonal to
#get the autocorrelations, and find the max
Tosc = np.argmax(Tosc0)
Tosc = Tosc + 5
f_c2 = 1000*(1.3/Tosc)/fs #Filter out components with frequencies higher than this
#to get rid of noise effects in cross-correlation
#times 1000 to get units right
flter2 = np.sinc(2*f_c2*(t - (finalt-1)/2))*np.blackman(finalt)
flter2 = flter2/np.sum(flter2)
for i in np.arange(Ndim):
Sbp[:,i] = np.convolve(Sh[:,i],flter2,mode='same')
#CALCULATE THE DISTANCE MEASURES
#calculate phase via cross-correlation with each cell
phase = np.zeros(Nmitral)
for i in np.arange(1,Nmitral):
crosscor = signal.correlate(Sbp[:,0],Sbp[:,i])
tdiff = np.argmax(crosscor)-(finalt-1)
phase[i] = tdiff/Tosc * 2*np.pi
#Problem with the method below is that it will only give values from 0 to pi
#for i in np.arange(1,Nmitral):
# phase[i]=np.arccos(np.dot(Sbp[:,0],Sbp[:,i])/(lin.norm(Sbp[:,0])*lin.norm(Sbp[:,i])))
OsciAmp = np.zeros(Nmitral)
Oosci = np.copy(OsciAmp)*0j
Omean = np.zeros(Nmitral)
for i in np.arange(Nmitral):
OsciAmp[i] = np.sqrt(np.sum(Sh[125:250,i]**2)/np.size(Sh[125:250,i]))
Oosci[i] = OsciAmp[i]*np.exp(1j*phase[i])
Omean[i] = np.average(Sl[:,i] - Sl0[:,i])
Omean = np.maximum(Omean,0)
Ooscibar = np.sqrt(np.dot(Oosci,np.conjugate(Oosci)))/Nmitral #can't just square b/c it's complex
Omeanbar = np.sqrt(np.dot(Omean,Omean))/Nmitral
maxlam = np.max(np.abs(np.imag(np.sqrt(dA))))
return yout,y0out,Sh,t,OsciAmp,Omean,Oosci,Omeanbar,Ooscibar,dominant_freq,maxlam
def pagerank_weighted_scipy(graph):
matrix = build_matrix(graph)
vals, vecs = eig(matrix.todense(), left=True, right=False)
return process_results(graph, vecs)
def rsw_rect(grid):
H = 5e2 # Fluid Depth
beta = 2e-11 # beta parameter
f0 = 1e-4 # Mean Coriolis parameter
g = 9.81 # gravity
Lx = np.sqrt(2) * 1e5 # Zonal qidth
Ly = 1e5 # Meridional width
Nx = grid[0]
Ny = grid[1]
# # Using Finite Difference
# [Dx,Dx2,x] = fd2(Nx); [Dy,Dy2,y] = fd2(Ny)
# x = Lx/2*x; y = Ly/2*y
# Dx = 2/Lx*Dx; Dy = 2/Ly*Dy
# Dx2 = (2/Lx)**2*Dx2; Dy2 = (2/Ly)**2*Dy2
# Using cheb
# x derivative
Dx, x = cheb(Nx)
x = Ly / 2 * x
Dx = 2 / Lx * Dx
# y derivative
Dy, y = cheb(Ny)
y = Ly / 2 * y
Dy = 2 / Ly * Dy
# Define Differentiation Matrices using kronecker product
F = np.kron(np.diag(np.ravel(f0 + beta * y)), np.eye(Nx + 1))
Z = np.zeros([(Nx + 1) * (Ny + 1), (Nx + 1) * (Ny + 1)])
DX = np.kron(np.eye(Ny + 1), Dx)
DY = np.kron(Dy, np.eye(Nx + 1))
# Sx and Sy are used to select which rows/columns need to be
# deleted for the boundary conditions.
Sy = np.ones((Nx + 1) * (Ny + 1), dtype=bool)
Sy[0:Nx + 1] = 0
Sy[Ny * (Nx + 1):] = 0
Sx = np.ones((Nx + 1) * (Ny + 1), dtype=bool)
Sx[0:((Nx + 1) * (Ny + 1)):(Nx + 1)] = 0
Sx[Nx:((Nx + 1) * (Ny + 1)):(Nx + 1)] = 0
# Define Matrices
Zx = Z[Sx, :]
Fx = F[Sx, :]
Zxx = Zx[:, Sx]
Fxy = Fx[:, Sy]
Fy = F[Sy, :]
Zy = Z[Sy, :]
Fyx = Fy[:, Sx]
Zyy = Zy[:, Sy]
A0 = np.hstack([Zxx, Fxy, -g * DX[Sx, :]])
A1 = np.hstack([-Fyx, Zyy, -g * DY[Sy, :]])
A2 = np.hstack([-H * DX[:, Sx], -H * DY[:, Sy], Z])
#size = (Nx-1)(Ny+1)+(Nx+1)(Ny-1)+(Nx+1)(Ny+1) = 3*Nx*Ny + Ny + Nx + 1 ^2
A = np.vstack([A0, A1, A2])
# B = np.eye(A.shape[0])
# Using eig
eigVals, eigVecs = spalg.eig(1j * A)
ind = (np.real(eigVals)).argsort() #get indices in ascending order
eigVecs = eigVecs[:, ind]
eigVals = eigVals[ind]
omega = eigVals
# # Using eigs
# evals_all, evecs_all = eigs(1j*A,80,which='SR',maxiter=500)
# print evals_all[0:5]
# omega = eigVals
evals = len(eigVals) # how many eigenvalues we have
# for i in range(evals):
# plt.plot(np.arange(0,evals_all.shape[0]),evals_all[:].real, 'o')
# plt.title("Plot of Real Part of Eigenvalues Using eigs")
# plt.show()
# for i in range(evals):
# plt.plot(np.arange(0,eigVals.shape[0]),eigVals[:].real, 'o')
# plt.title("Plot of Real Part of Eigenvalues Using eig")
# plt.show()
print "First 5 eigenvalues:"
posReal = eigVals[eigVals.real > 1e-10]
print posReal[0:5]
omega = eigVals.real
fieldNames = ["u_x", "u_y", "eta"]
nsol = eigVecs.shape[1]
fields = np.empty(3, dtype='object')
fields = [np.reshape(eigVecs[0:(Nx-1)*(Ny+1),:], [Ny+1,Nx-1,nsol]), \
np.reshape(eigVecs[(Nx-1)*(Ny+1):2*Nx*Ny-2,:], [Ny-1, Nx+1, nsol]), \
np.reshape(eigVecs[2*Nx*Ny-2:,:], [Nx+1, Ny+1, nsol])]
om = omega.real
om[om <= f0] = np.Inf
ii = (abs(om.real)).argmin(0)
for i in range(ii - 2, ii + 9):
uf = fields[0]
u = np.squeeze(uf[:, :, i])
vf = fields[1]
v = np.squeeze(vf[:, :, i])
hf = fields[2]
h = np.squeeze(hf[:, :, i])
v = np.vstack([np.zeros([1, Nx + 1]), v, np.zeros([1, Nx + 1])])
u = np.hstack([np.zeros([Ny + 1, 1]), u, np.zeros([Ny + 1, 1])])
X, Y = np.meshgrid(x, y)
plt.subplot(3, 2, 1)
plt.contourf(X / 1e3, Y / 1e3, (u.real).conj().transpose(), 20)
plt.colorbar()
plt.subplot(3, 2, 2)
plt.contourf(X / 1e3, Y / 1e3, (u.imag).conj().transpose(), 20)
plt.colorbar()
plt.subplot(3, 2, 3)
plt.contourf(X / 1e3, Y / 1e3, (v.real).conj().transpose(), 20)
plt.colorbar()
plt.subplot(3, 2, 4)
plt.contourf(X / 1e3, Y / 1e3, (v.imag).conj().transpose(), 20)
plt.colorbar()
plt.subplot(3, 2, 5)
plt.contourf(X / 1e3, Y / 1e3, (h.real).conj().transpose(), 20)
plt.colorbar()
plt.subplot(3, 2, 6)
plt.contourf(X / 1e3, Y / 1e3, (h.imag).conj().transpose(), 20)
plt.colorbar()
plt.show()
def CalAbsorb(self):
Energyp = []
Epsreal = []
Absorpv = []
Epsimag = []
Reflexd = []
Energyp, Epsreal, Epsimag = self.GetEpsilon()
for i in range(len(Epsimag)):
Energy = 0.
ImagXX = 0.
ImagYY = 0.
ImagZZ = 0.
RealXX = 0.
RealYY = 0.
RealZZ = 0.
ImagXY = 0.
ImagYZ = 0.
ImagZX = 0.
RealXY = 0.
RealYZ = 0.
RealZX = 0.
Energy = float(Energyp[i])
ImagXX = float(Epsimag[i][1])
ImagYY = float(Epsimag[i][2])
ImagZZ = float(Epsimag[i][3])
RealXX = float(Epsreal[i][1])
RealYY = float(Epsreal[i][2])
RealZZ = float(Epsreal[i][3])
ImagXY = float(Epsimag[i][4])
ImagYZ = float(Epsimag[i][5])
ImagZX = float(Epsimag[i][6])
RealXY = float(Epsreal[i][4])
RealYZ = float(Epsreal[i][5])
RealZX = float(Epsreal[i][6])
Cxx = complex(RealXX, ImagXX)
Cyy = complex(RealYY, ImagYY)
Czz = complex(RealZZ, ImagZZ)
Cxy = complex(RealXY, ImagXY)
Cyz = complex(RealYZ, ImagYZ)
Czx = complex(RealZX, ImagZX)
C_eps = mat([[Cxx, Cxy, conj(Czx)], [conj(Cxy), Cyy, Cyz],
[Czx, conj(Cyz), Czz]])
eps_eig, eps_v = linalg.eig(C_eps)
hv = Energy
alpha_a1 = hv * 71618.96076 * sqrt(
abs(eps_eig[0]) - real(eps_eig[0]))
alpha_a2 = hv * 71618.96076 * sqrt(
abs(eps_eig[1]) - real(eps_eig[1]))
alpha_a3 = hv * 71618.96076 * sqrt(
abs(eps_eig[2]) - real(eps_eig[2]))
alpha_av = (alpha_a1 + alpha_a2 + alpha_a3) / 3
Absorpv.append(alpha_av)
n1 = sqrt(0.5 * (abs(eps_eig[0]) + real(eps_eig[0])))
n2 = sqrt(0.5 * (abs(eps_eig[1]) + real(eps_eig[1])))
n3 = sqrt(0.5 * (abs(eps_eig[2]) + real(eps_eig[2])))
n_av = (n1 + n2 + n3) / 3.0
Reflexd.append(n_av)
return Energyp, Absorpv, Reflexd
def matrix_eigenvalues(a):
return la.eig(a)
trans_m = np.delete(trans_m, row, axis=1)
for icbin, cbin in enumerate(CBINS):
if cbin > row:
CBINS[icbin] -= 1
elif cbin == row:
del CBINS[icbin]
if INIT_BINS[0] > row:
INIT_BINS[0] -= 1
if TARGET_BINS[0] > row:
TARGET_BINS[0] -= 1
print(CBINS)
print(trans_m)
K = np.nan_to_num(trans_m)
print(K)
eigvals, eigvecs = LA.eig(K.T)
unity = (np.abs(np.real(eigvals) - 1)).argmin()
print(eigvals, eigvecs)
eq_pop = np.abs(np.real(eigvecs)[unity])
eq_pop /= eq_pop.sum()
# probability of starting in init bin A.
distr_prob = np.random.rand(len(INIT_BINS))
paths = []
# t_bins: all bins which are not target bins.
t_bins = list(x for x in range(0, NBINS) if x not in TARGET_BINS)
# lower_bound = mfpt - error
# lower_bound = 121.8
lower_bound = 116
def numpy_eigen():
A = np.array([[1, 3, 5], [3, 5, 3], [5, 3, 9]])
evals, evecs = la.eig(A)
return la.eigvalsh(A)
r+=t+" NO es normal "+str(ks)+" \n"
elif ks>0.01 :
r+=t+" es normal\n"
print("-------------------------------\n")
if not "no " in r: print("¡Todo parece ser normal!\n")
else: print(r)
print("--------------------------------\n")
#------------------- Cálculo de Matriz de componentes principales -----------------------------#
import scipy.linalg as la
from math import log
CV = Trait_matrix.cov()
vals,vects = la.eig(CV)
perct = [] #Porcentaje de varianza explicada
for i in vals:
perct.append((i*100)/vals.sum())
p=len(traits) #número de variames
N=Trait_matrix[[traits[0]]].size #Tamaño muestral
def eme (m,lg=False): #la sumatoria de los valores propios hasta m
s=0
if lg: #si es la suma de los logaritmos
for i in range(m,len(vals)+1): s+=log(vals[i-1].real,10)
else:
for i in range(m,len(vals)+1): s+=vals[i-1].real
return s
x_2 = 0
gl = 0
# File: Hello.py
import math
import scipy.linalg as la
import numpy as np
import matplotlib.pyplot as plt
A = [[1,4],[-4,1]]
eigvals,eigvecs = la.eig(A)
new = []
old = []
norm1 = [0,0]
norm1dup = []
norm2 = [0,0]
norm2dup = []
norminf = [0,0]
norminfdup = []
x, y = -1.001, 0
for k in range(2000):
x += 0.001
y = 1
old.append([x,y])
old.append([x,-y])
a, a_inv = np.matmul(A, [x, y]), np.matmul(A, [x, -y])
new.append(a)
new.append(a_inv)
if abs(a[0]) + abs(a[1]) > abs(norm1[0]) + abs(norm1[1]):
norm1 = a
# -------------------------------------------------------------------------- #
# Main
# -------------------------------------------------------------------------- #
# == Compute the optimal rule == #
optimal_lq = qe.lqcontrol.LQ(Q, R, A, B, C, beta)
Po, Fo, do = optimal_lq.stationary_values()
# == Compute a robust rule given theta == #
baseline_robust = qe.robustlq.RBLQ(Q, R, A, B, C, beta, theta)
Fb, Kb, Pb = baseline_robust.robust_rule()
# == Check the positive definiteness of worst-case covariance matrix to == #
# == ensure that theta exceeds the breakdown point == #
test_matrix = np.identity(Pb.shape[0]) - np.dot(C.T, Pb.dot(C)) / theta
eigenvals, eigenvecs = eig(test_matrix)
assert (eigenvals >= 0).all(), 'theta below breakdown point.'
emax = 1.6e6
optimal_best_case = value_and_entropy(emax, Fo, 'best')
robust_best_case = value_and_entropy(emax, Fb, 'best')
optimal_worst_case = value_and_entropy(emax, Fo, 'worst')
robust_worst_case = value_and_entropy(emax, Fb, 'worst')
fig, ax = plt.subplots()
ax.set_xlim(0, emax)
ax.set_ylabel("Value")
ax.set_xlabel("Entropy")
ax.grid()
#import matplotlib.pyplot as plt
Nset = [100, 200, 400, 800]
nset = [2, 5, 10, 20]
m = 2
p = 8
MC = 100
tim_res = np.zeros((4, 4))
nidx = 0
for n in nset:
Nidx = 0
for N in Nset:
A = np.random.randn(n, n)
lam = linalg.eig(A)[0]
rho = np.max(np.abs(lam))
## Here we create a random stable DT system
A = A / rho / 1.01
B = np.random.randn(n, m)
C = np.random.randn(p, n)
D = np.random.randn(p, m)
fset = np.arange(0, N, dtype='float') / N
u = np.random.randn(N, m)
y = fsid.lsim((A, B, C, D), u, dtype='float')
yf = np.fft.fft(y, axis=0)
uf = np.fft.fft(u, axis=0)
fset = np.arange(0, N, dtype='float') / N
wexp = np.exp(1j * 2 * np.pi * fset)
W = np.zeros((N, p, p))
def process(self):
after = self.after
before = self.before
(rows, cols, bands) = after.shape
after = np.transpose(np.reshape(after, (rows * cols, bands)), (1, 0))
before = np.transpose(np.reshape(before, (rows * cols, bands)), (1, 0))
after_mean = np.mean(after, axis=1)
after_var = np.std(after, axis=1)
before_mean = np.mean(before, axis=1)
before_var = np.std(before, axis=1)
# for i in range(bands):
# #test = after[:, i] - after_mean[i]
# after[i,:] = (after[i,:]-after_mean[i])/after_var[i]
# before[i,:] = (before[i,:]-before_mean[i])/before_var[i]
cov_aa_mari = np.cov(after)
cov_aa_mat_i = np.linalg.inv(cov_aa_mari)
con_cov = np.cov(after, before)
cov_xx = con_cov[0:bands, 0:bands]
cov_xy = con_cov[0:bands, bands:]
cov_yx = con_cov[bands:, 0:bands]
cov_yy = con_cov[bands:, bands:]
# yy_cov = np.cov(before)
A = inv(cov_xx) @ cov_xy @ inv(cov_yy) @ cov_yx
B = inv(cov_yy) @ cov_yx @ inv(cov_xx) @ cov_xy # 与A特征值相同，但特征向量不同
# A的特征值与特征向量 av 特征值， ad 特征向量
[av, ad] = eig(A)
# 对特征值从小到大排列 与 CCA相反
swap_av_index = np.argsort(av)
swap_av = av[swap_av_index[:av.size:1]]
swap_ad = ad[swap_av_index[:av.size:1], :]
# 满足st 条件
ma = inv(sqrtm(swap_ad.T @ cov_xx @ swap_ad)) # 条件一
swap_ad = swap_ad @ ma
# 对应b的值
[bv, bd] = eig(B)
swap_bv = bv[swap_av_index[:bv.size:1]]
swap_bd = bd[swap_av_index[:bd.size:1]]
mb = inv(sqrtm(swap_bd.T @ cov_yy @ swap_bd)) # 条件二
swap_bd = swap_bd @ mb
# ab = np.linalg.inv(cov_yy) @ cov_yx @ swap_ad
# bb = np.linalg.inv()
MAD = swap_ad.T @ after - (swap_bd.T @ before)
[i, j] = MAD.shape
var_mad = np.zeros(i)
for k in range(i):
var_mad[k] = np.var(MAD[k])
var_mad = np.transpose(np.matlib.repmat(var_mad, j, 1), (1, 0))
res = MAD * MAD / var_mad
T = res.sum(axis=0)
# T = np.zeros(j)
# #for row in range(j):
# sum = 0.
# for col in range(i):
# sum = np.sum(np.square(MAD[col,:] / np.var(MAD[col])))
# T[i] = sum
# Kmeans 聚类
re = np.reshape(T, (j, 1))
kmeans = KMeans(n_clusters=2, random_state=0).fit(re)
img = np.reshape(kmeans.labels_, (
rows,
cols,
))
center = kmeans.cluster_centers_
pyplot.imshow(np.uint8(img))
pyplot.show()
# scipy.misc.imsave('c.jpg', img)
print(center)
JAC[M - 1, :] = concatenate((BBOT, zeros(2 * M)))
# V
JAC[2 * M - 2, :] = concatenate((zeros(M), BTOP, zeros(M)))
JAC[2 * M - 1, :] = concatenate((zeros(M), BBOT, zeros(M)))
B = zeros((3 * M, 3 * M), dtype='complex')
B[:2 * M, :2 * M] = eye(2 * M, 2 * M)
### BCs on RHS matrix
B[M - 2, :] = 0
B[M - 1, :] = 0
B[2 * M - 2, :] = 0
B[2 * M - 1, :] = 0
eigenvalues = linalg.eig(JAC, B, left=False, right=False)
eigenvalues = eigenvalues[~isnan(eigenvalues * conj(eigenvalues))]
eigenvalues = eigenvalues[~isinf(eigenvalues * conj(eigenvalues))]
eigarr = zeros((len(eigenvalues), 2))
eigarr[:, 0] = real(eigenvalues)
eigarr[:, 1] = imag(eigenvalues)
#savetxt("eigs.txt", eigarr)
actualDecayRate = amax(eigarr[:, 0])
while actualDecayRate > 100:
eigarr[argmax(eigarr[:, 0]), 0] = -10.0
actualDecayRate = amax(eigarr[:, 0])
evals.append([kx, actualDecayRate])
I_eps_zz = I_eps_zz * dE / sqrt(2.0 * pi) / sigma
I_eps_xy = I_eps_xy * dE / sqrt(2.0 * pi) / sigma
I_eps_yz = I_eps_yz * dE / sqrt(2.0 * pi) / sigma
I_eps_zx = I_eps_zx * dE / sqrt(2.0 * pi) / sigma
Cxx = complex(R_eps_xx, I_eps_xx)
Cyy = complex(R_eps_yy, I_eps_yy)
Czz = complex(R_eps_zz, I_eps_zz)
Cxy = complex(R_eps_xy, I_eps_xy)
Cyz = complex(R_eps_yz, I_eps_yz)
Czx = complex(R_eps_zx, I_eps_zx)
C_eps = mat([[Cxx, Cxy, conj(Czx)], [conj(Cxy), Cyy, Cyz],
[Czx, conj(Cyz), Czz]])
eps_eig, eps_v = linalg.eig(C_eps)
# print hv, imag(eps_eig[0]),imag(eps_eig[1]),imag(eps_eig[2])
alpha_a1 = hv * 71618.96076 * sqrt(abs(eps_eig[0]) - real(eps_eig[0]))
alpha_a2 = hv * 71618.96076 * sqrt(abs(eps_eig[1]) - real(eps_eig[1]))
alpha_a3 = hv * 71618.96076 * sqrt(abs(eps_eig[2]) - real(eps_eig[2]))
alpha_av = (alpha_a1 + alpha_a2 + alpha_a3) / 3
n1 = sqrt(0.5 * (abs(eps_eig[0]) + real(eps_eig[0])))
n2 = sqrt(0.5 * (abs(eps_eig[1]) + real(eps_eig[1])))
n3 = sqrt(0.5 * (abs(eps_eig[2]) + real(eps_eig[2])))
n_av = (n1 + n2 + n3) / 3.0
f.write("%9.4f %13.6E %13.6E %13.6E %13.6E %9.4f\n" %
(hv, alpha_a1, alpha_a2, alpha_a3, alpha_av, n_av**2))
alpha.append([hv, alpha_av, n_av**2])
def lda_train2(data=None, label=None):
"""
Perform Linear Discriminant Analysis on input data.
Input:
data (array): input data array of shape (number samples x number features).
label (array): input data label array of shape (number of samples x 1). Must start in 0 (e.g., label = [0,0,0,1,2] for 5 samples).
Output:
success (boolean): indicates whether training was successful (True) or not (False).
Configurable fields:{"name": "dimreduction.lda.train", "config": {"": ""}, "inputs": ["data", "label"], "outputs": ["success"]}
See Also:
Notes:
Example:
References:
.. [1] ...
.. [2] ...
.. [3] ...
"""
# Check inputs
if data is None:
raise TypeError, "Please provide input data."
if label is None:
raise TypeError, "Please provide input data label."
if 0 not in label:
raise TypeError, "Label must start in 0 (e.g., label = [0,0,0,1,2] for 5 samples)."
# success = False
try:
# Compute mean of each set (mi)
m = []
for c in set(label):
m.append(scipy.mean(data[label == c], axis=0))
m = scipy.array(m)
# Compute Scatter Matrix of eah set (Si)
S = []
for c in set(label):
S.append(scipy.cov(scipy.array(data[label == c]).T))
# Compute Within Scatter Matrix (SW)
SW = 0
for s in S:
SW += s
# Compute Total Mean (mt)
mt = scipy.mean(data, axis=0)
# Compute Total Scatter Matrix (ST)
ST = 0
for xi in data:
aux = scipy.matrix(xi - mt)
ST += aux.T * aux
# Compute Between Scatter Matrix (SB)
SB = 0
for c in set(label):
aux = scipy.matrix(m[c, :] - mt)
SB += len(pylab.find(label == c)) * aux.T * aux
# Solve (Sb - li*Sw)Wi = 0 for the eigenvectors wi
eigenvalues, v = linalg.eig(SB, SW)
# Get real part and sort eigenvalues
real_sorted_eigenvalues = []
for i in xrange(len(eigenvalues)):
real_sorted_eigenvalues.append([scipy.real(eigenvalues[i]), i])
real_sorted_eigenvalues.sort()
# Get the (nclasses-1) main eigenvectors
# Assures eigenvalue is not NaN
nclasses = len(set(label)) - 1
# nclasses = 5
eigenvectors = []
for i in xrange(-1, -len(real_sorted_eigenvalues) - 1, -1):
if not scipy.isnan(real_sorted_eigenvalues[i][0]):
eigenvectors.append(v[real_sorted_eigenvalues[i][1]])
if len(eigenvectors) == nclasses: break
# Updates variables
# self.eigen_values = real_sorted_eigenvalues
# self.eigen_vectors = eigenvectors
# self.transform_matrix = scipy.matrix(eigenvectors)
transform_matrix = scipy.matrix(eigenvectors)
# success = True
# self.is_trained = True
except Exception as e:
print e
print traceback.format_exc()
# return success
return transform_matrix
def my_LDA(X, Y):
"""
Train a LDA classifier from the training set
X: training data
Y: class labels of training data
"""
classLabels = np.unique(Y) # different class labels on the dataset
classNum = len(classLabels)
datanum, dim = X.shape # dimensions of the dataset
totalMean = np.mean(X, 0) # total mean of the data
# ====================== YOUR CODE HERE ======================
# Instructions: Implement the LDA technique, following the
# steps given in the pseudocode on the assignment.
# The function should return the projection matrix W,
# the centroid vector for each class projected to the new
# space defined by W and the projected data X_lda.
# =============================================================
# partition class labels per label - list of arrays per label
partition = [np.where(Y == label)[0] for label in classLabels]
# find mean value per class (per attribute) - list of arrays per label
classMean = [(np.mean(X[idx], 0), len(idx)) for idx in partition]
# Compute the within-class scatter matrix
Sw = np.zeros((dim, dim))
for idx in partition:
Sw += np.cov(X[idx], rowvar=0) * len(
idx) # covariance matrix * fraction of instances per class
# Compute the between-class scatter matrix
Sb = np.zeros((dim, dim))
for mu, class_size in classMean:
mu = mu.reshape(dim, 1) #make column vector
Sb += class_size * np.dot(
(mu - totalMean), np.transpose((mu - totalMean)))
# Solve the eigenvalue problem for discriminant directions to maximize class seperability while simultaneously minimizing
# the variance within each class
# The exception code can be ignored for the example dataset
try:
S = np.dot(linalg.inv(Sw), Sb)
eigval, eigvec = linalg.eig(S)
except: #SingularMatrix
print "Singular matrix"
eigval, eigvec = linalg.eig(Sb, Sw + Sb)
idx = eigval.argsort()[::-1] # Sort eigenvalues
eigvec = eigvec[:, idx] # Sort eigenvectors according to eigenvalues
W = np.real(
eigvec[:, :classNum -
1]) # eigenvectors correspond to k-1 largest eigenvalues
# Project data onto the new LDA space
X_lda = np.real(np.dot(X, np.real(W)))
# project the mean vectors of each class onto the LDA space
projected_centroid = [
np.dot(mu, np.real(W)) for mu, class_size in classMean
]
return W, projected_centroid, X_lda
slice_height, slice_width = det_pixels.shape
# The '300' is the width/height of the image
if slice_height > 300 / np.sqrt(2.) or slice_width > 300 / np.sqrt(2.):
det_pixels = (det_pixels == i + 1)
x = np.arange(slice_width)
y = np.arange(slice_height)
flux = np.sum(det_pixels)
x_cen = np.sum(x[np.newaxis, :] * det_pixels) / flux
y_cen = np.sum(y[:, np.newaxis] * det_pixels) / flux
c_xx = np.sum(((x - x_cen)**2)[np.newaxis, :] * det_pixels) / flux
c_yy = np.sum(((y - y_cen)**2)[:, np.newaxis] * det_pixels) / flux
c_xy = np.sum((x - x_cen)[np.newaxis, :] *
(y - y_cen)[:, np.newaxis] * det_pixels) / flux
C = np.array([[c_xx, c_xy], [c_xy, c_yy]])
w, v = sl.eig(C)
a, b = np.max(2.0 * np.sqrt(np.real(w))), \
np.min(2.0 * np.sqrt(np.real(w)))
# The final check whether we have a track:
if a > (300 / (2. * np.sqrt(2.))) and b / a < 0.1:
track_pixels[slc] = det_pixels
# make a plot from original image and track pixels if track(s) were found:
if np.sum(track_pixels) > 0:
print('Track(s) detected! Creating plot %s' % (out_plot))
fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(8, 4))
ax1.imshow(data, origin='lower', cmap='gray_r', vmin=min_val, vmax=max_val)
ax1.set_xlabel('x [pix]')
ax1.set_ylabel('y [pix]')
ax1.set_title('original data')
import datetime
starttime = datetime.datetime.now()
import numpy as np
import scipy.linalg as la
A = np.array([[17.125, -8, 0, -8], [-8, 16.2815, -8, 0], [0, -8, 16, -8],
[-8, 0, -8, 16.28125]])
print(la.eig(A)[0])
endtime = datetime.datetime.now()
print("total time used", endtime - starttime)
def ex17(exclude=sc.array([3]),
plotfilename='ex17.png',
nburn=5000,
nsamples=200000,
parsigma=[1, m.pi / 200., .1],
bovyprintargs={}):
"""ex17: solve exercise 17 by MCMC
Input:
exclude - ID numbers to exclude from the analysis
plotfilename - filename for the output plot
nburn - number of burn-in samples
nsamples - number of samples to take after burn-in
parsigma - proposal distribution width (Gaussian)
Output:
plot
History:
2010-05-07 - Written - Bovy (NYU)
"""
#Read the data
data = read_data('data_allerr.dat', allerr=True)
ndata = len(data)
nsample = ndata - len(exclude)
#First find the chi-squared solution, which we will use as an
#initial gues
#Put the dat in the appropriate arrays and matrices
Y = sc.zeros(nsample)
X = sc.zeros(nsample)
A = sc.ones((nsample, 2))
C = sc.zeros((nsample, nsample))
Z = sc.zeros((nsample, 2))
yerr = sc.zeros(nsample)
ycovar = sc.zeros((2, nsample, 2)) #Makes the sc.dot easier
jj = 0
for ii in range(ndata):
if sc.any(exclude == data[ii][0]):
pass
else:
Y[jj] = data[ii][1][1]
X[jj] = data[ii][1][0]
Z[jj, 0] = X[jj]
Z[jj, 1] = Y[jj]
A[jj, 1] = data[ii][1][0]
C[jj, jj] = data[ii][2]**2.
yerr[jj] = data[ii][2]
ycovar[0, jj, 0] = data[ii][3]**2.
ycovar[1, jj, 1] = data[ii][2]**2.
ycovar[0, jj, 1] = data[ii][4] * m.sqrt(
ycovar[0, jj, 0] * ycovar[1, jj, 1])
ycovar[1, jj, 0] = ycovar[0, jj, 1]
jj = jj + 1
#Now compute the best fit and the uncertainties
bestfit = sc.dot(linalg.inv(C), Y.T)
bestfit = sc.dot(A.T, bestfit)
bestfitvar = sc.dot(linalg.inv(C), A)
bestfitvar = sc.dot(A.T, bestfitvar)
bestfitvar = linalg.inv(bestfitvar)
bestfit = sc.dot(bestfitvar, bestfit)
#Now sample
inittheta = m.acos(1. / m.sqrt(1. + bestfit[1]**2.))
if bestfit[1] < 0.:
inittheta = m.pi - inittheta
initialguess = sc.array(
[bestfit[0] * m.cos(inittheta), inittheta,
sc.log(1.)]) #(m,b,logV)
#With this initial guess start off the sampling procedure
initialX = objective(initialguess, Z, ycovar)
currentX = initialX
bestX = initialX
bestfit = initialguess
currentguess = initialguess
naccept = 0
samples = []
samples.append(currentguess)
for jj in range(nburn + nsamples):
#Draw a sample from the proposal distribution
newsample = sc.zeros(3)
newsample[0] = currentguess[0] + stats.norm.rvs() * parsigma[0]
newsample[1] = currentguess[1] + stats.norm.rvs() * parsigma[1]
newsample[2] = currentguess[2] + stats.norm.rvs() * parsigma[2]
#Calculate the objective function for the newsample
newX = objective(newsample, Z, ycovar)
#Accept or reject
#Reject with the appropriate probability
u = stats.uniform.rvs()
try:
test = m.exp(newX - currentX)
except OverflowError:
test = 2.
if u < test:
#Accept
currentX = newX
currentguess = newsample
naccept = naccept + 1
if currentX > bestX:
bestfit = currentguess
bestX = currentX
samples.append(currentguess)
if double(naccept) / (nburn + nsamples) < .5 or double(naccept) / (
nburn + nsamples) > .8:
print "Acceptance ratio was " + str(
double(naccept) / (nburn + nsamples))
samples = sc.array(samples).T[:, nburn:-1]
print "Best-fit, overall"
print bestfit, sc.mean(samples[2, :]), sc.median(samples[2, :])
histmb, edges = sc.histogramdd(samples.T[:, 0:2],
bins=round(sc.sqrt(nsamples) / 2.))
indxi = sc.argmax(sc.amax(histmb, axis=1))
indxj = sc.argmax(sc.amax(histmb, axis=0))
print "Best-fit, marginalized"
print edges[0][indxi - 1], edges[1][indxj - 1]
print edges[0][indxi], edges[1][indxj]
print edges[0][indxi + 1], edges[1][indxj + 1]
t = edges[1][indxj]
bcost = edges[0][indxi]
mf = m.sqrt(1. / m.cos(t)**2. - 1.)
b = bcost / m.cos(t)
print b, mf
#Plot
plot.bovy_print(**bovyprintargs)
hist, bins, patchess = plot.bovy_hist(sc.exp(samples.T[:, 2] / 2.),
edgecolor='k',
bins=round(sc.sqrt(nsamples) / 2.),
xlabel=r'$\sqrt{V}$',
normed=True,
histtype='step')
cumhist = sc.cumsum(hist) / sc.sum(hist) / (bins[1] - bins[0])
ninefive = 0.
ninenine = 0.
foundfive = False
foundnine = False
for ii in range(len(cumhist)):
if cumhist[ii] * (bins[1] - bins[0]) > 0.95 and not foundfive:
ninefive = bins[ii]
foundfive = True
if cumhist[ii] * (bins[1] - bins[0]) > 0.99 and not foundnine:
ninenine = bins[ii]
foundnine = True
print ninefive, ninenine
axvline(ninefive, color='0.5', lw=2.)
axvline(ninenine, color='0.5', lw=2.)
plot.bovy_end_print(plotfilename)
return
#Plot result
plot.bovy_print()
xrange = [0, 300]
yrange = [0, 700]
plot.bovy_plot(sc.array(xrange),
mf * sc.array(xrange) + b,
'k--',
xrange=xrange,
yrange=yrange,
xlabel=r'$x$',
ylabel=r'$y$',
zorder=2)
for ii in range(10):
#Random sample
ransample = sc.floor((stats.uniform.rvs() * nsamples))
ransample = samples.T[ransample, 0:2]
mf = m.sqrt(1. / m.cos(ransample[1])**2. - 1.)
b = ransample[0] / m.cos(ransample[1])
bestb = b
bestm = mf
plot.bovy_plot(sc.array(xrange),
bestm * sc.array(xrange) + bestb,
overplot=True,
color='0.75',
zorder=0)
#Add labels
nsamples = samples.shape[1]
for ii in range(nsample):
Pb = 0.
for jj in range(nsamples):
Pb += Pbad(samples[:, jj], Z[ii, :], ycovar[:, ii, :])
Pb /= nsamples
text(Z[ii, 0] + 5, Z[ii, 1] + 5, '%.1f' % Pb, color='0.5', zorder=3)
#Plot the data OMG straight from plot_data.py
data = read_data('data_allerr.dat', True)
ndata = len(data)
#Create the ellipses and the data points
id = sc.zeros(nsample)
x = sc.zeros(nsample)
y = sc.zeros(nsample)
ellipses = []
ymin, ymax = 0, 0
xmin, xmax = 0, 0
jj = 0
for ii in range(ndata):
if sc.any(exclude == data[ii][0]):
continue
id[jj] = data[ii][0]
x[jj] = data[ii][1][0]
y[jj] = data[ii][1][1]
#Calculate the eigenvalues and the rotation angle
ycovar = sc.zeros((2, 2))
ycovar[0, 0] = data[ii][3]**2.
ycovar[1, 1] = data[ii][2]**2.
ycovar[0, 1] = data[ii][4] * m.sqrt(ycovar[0, 0] * ycovar[1, 1])
ycovar[1, 0] = ycovar[0, 1]
eigs = linalg.eig(ycovar)
angle = m.atan(-eigs[1][0, 1] / eigs[1][1, 1]) / m.pi * 180.
thisellipse = Ellipse(sc.array([x[jj], y[jj]]), 2 * m.sqrt(eigs[0][0]),
2 * m.sqrt(eigs[0][1]), angle)
ellipses.append(thisellipse)
if (x[jj] + m.sqrt(ycovar[0, 0])) > xmax:
xmax = (x[jj] + m.sqrt(ycovar[0, 0]))
if (x[jj] - m.sqrt(ycovar[0, 0])) < xmin:
xmin = (x[jj] - m.sqrt(ycovar[0, 0]))
if (y[jj] + m.sqrt(ycovar[1, 1])) > ymax:
ymax = (y[jj] + m.sqrt(ycovar[1, 1]))
if (y[jj] - m.sqrt(ycovar[1, 1])) < ymin:
ymin = (y[jj] - m.sqrt(ycovar[1, 1]))
jj = jj + 1
#Add the error ellipses
ax = gca()
for e in ellipses:
ax.add_artist(e)
e.set_facecolor('none')
ax.plot(x, y, color='k', marker='o', linestyle='None')
plot.bovy_end_print(plotfilename)
def dmd(A,
rank=None,
dt=1,
modes='exact',
return_amplitudes=False,
return_vandermonde=False,
order=True):
"""Dynamic Mode Decomposition.
Dynamic Mode Decomposition (DMD) is a data processing algorithm which
allows to decompose a matrix `A` in space and time. The matrix `A` is
decomposed as `A = F * B * V`, where the columns of `F` contain the dynamic modes.
The modes are ordered corresponding to the amplitudes stored in the diagonal
matrix `B`. `V` is a Vandermonde matrix describing the temporal evolution.
Parameters
----------
A : array_like, shape `(m, n)`.
Input array.
rank : int
If `rank < (n-1)` low-rank Dynamic Mode Decomposition is computed.
dt : scalar or array_like, optional (default: 1)
Factor specifying the time difference between the observations.
modes : str `{'standard', 'exact', 'exact_scaled'}`
- 'standard' : uses the standard definition to compute the dynamic modes, `F = U * W`.
- 'exact' : computes the exact dynamic modes, `F = Y * V * (S**-1) * W`.
- 'exact_scaled' : computes the exact dynamic modes, `F = (1/l) * Y * V * (S**-1) * W`.
return_amplitudes : bool `{True, False}`
True: return amplitudes in addition to dynamic modes.
return_vandermonde : bool `{True, False}`
True: return Vandermonde matrix in addition to dynamic modes and amplitudes.
order : bool `{True, False}`
True: return modes sorted.
Returns
-------
F : array_like
Matrix containing the dynamic modes of shape `(m, n-1)` or `(m, k)`.
b : array_like, if `return_amplitudes=True`
1-D array containing the amplitudes of length `min(n-1, rank)`.
V : array_like, if `return_vandermonde=True`
Vandermonde matrix of shape `(n-1, n-1)` or `(rank, n-1)`.
omega : array_like
Time scaled eigenvalues: `ln(l)/dt`.
References
----------
J. H. Tu, et al.
"On Dynamic Mode Decomposition: Theory and Applications" (2013).
(available at `arXiv <http://arxiv.org/abs/1312.0041>`_).
N. B. Erichson and C. Donovan.
"Randomized Low-Rank Dynamic Mode Decomposition for Motion Detection" (2015).
Under Review.
"""
# converts A to array, raise ValueError if A has inf or nan
A = np.asarray_chkfinite(A)
m, n = A.shape
if modes not in _VALID_MODES:
raise ValueError('modes must be one of %s, not %s' %
(' '.join(_VALID_MODES), modes))
if A.dtype not in _VALID_DTYPES:
raise ValueError('A.dtype must be one of %s, not %s' %
(' '.join(_VALID_DTYPES), A.dtype))
if rank is not None and (rank < 1 or rank > n):
raise ValueError('rank must be > 1 and less than n')
#Split data into lef and right snapshot sequence
X = A[:, :(n - 1)] #pointer
Y = A[:, 1:n] #pointer
#Singular Value Decomposition
U, s, Vh = linalg.svd(X,
compute_uv=True,
full_matrices=False,
overwrite_a=False,
check_finite=True)
if rank is not None:
U = U[:, :rank]
s = s[:rank]
Vh = Vh[:rank, :]
#~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#Solve the LS problem to find estimate for M using the pseudo-inverse
#~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#real: M = U.T * Y * Vt.T * S**-1
#complex: M = U.H * Y * Vt.H * S**-1
#Let G = Y * Vt.H * S**-1, hence M = M * G
G = np.dot(Y, conjugate_transpose(Vh)) / s
M = np.dot(conjugate_transpose(U), G)
#Eigen Decomposition
l, W = linalg.eig(M, right=True, overwrite_a=True)
omega = np.log(l) / dt
if order:
# return ordered result
sort_idx = np.argsort(np.abs(omega))
W = W[:, sort_idx]
l = l[sort_idx]
omega = omega[sort_idx]
#Compute DMD Modes
if modes == 'standard':
F = np.dot(U, W)
else:
F = np.dot(G, W)
if modes == 'exact_scaled':
F /= l
result = [F]
if return_amplitudes:
#Compute amplitueds b using least-squares: Fb=x1
b = _get_amplitudes(F, A)
result.append(b)
if return_vandermonde:
#Compute Vandermonde matrix
V = np.fliplr(np.vander(l, N=n))
result.append(V)
result.append(omega)
return result
def __init__(self,
x,
y,
z,
dim,
file,
percentage_max_n=0.1,
typ="",
motion=""):
"""
:param x: list, x coordinates
:param y: list, y coordinates
:param y: list, z coordinates
:param dim: int, dimension
:param file: str, path to trajectory
:param percentage_max_n: float, percentage of length of the trajectory for msd generating
:param typ: str, type of diffusion i.e sub, super, rand
:param motion: str, mode of diffusion eg. normal, directed
"""
CharacteristicFour.__init__(self, x, y, z, dim, file, percentage_max_n,
typ, motion)
self.velocity_autocorrelation, self.velocity_autocorrelation_names = self.get_velocity_autocorrelation(
[1, 2])
self.ksstat = self.get_ksstat()
self.dagostino_stats, self.dagostino_stats_names = self.get_dagostino_stats(
)
self.mv_features, self.mv_features_names = self.moving_window(
windows=[10, 20])
self.eM, self.eL, self.eJ = self.get_exponents()
self.maximum_ts = self.get_maximum_test_statistic()
self.radius_gyration_tensor = self.get_tensor()
self.trappedness = self.get_trappedness()
self.eigenvalues, self.eigenvectors = LA.eig(
self.radius_gyration_tensor)
self.asymmetry = self.get_asymmetry()
self.diff_kurtosis = self.get_kurtosis_corrected()
self.efficiency = self.get_efficiency()
self.max_std_x, self.max_std_y = self.max_min_std()
self.max_std_change_x, self.max_std_change_y = self.max_std_change()
self.velocity_autocorrelation, self.velocity_autocorrelation_names = self.get_velocity_autocorrelation(
[1, 2])
self.dma, self.dma_names = self.get_dma([1, 2])
self.values = [self.file, self.type, self.motion, self.D, self.alpha,
self.alpha_n_1, self.alpha_n_2, self.alpha_n_3,
self.fractal_dimension, self.mean_gaussianity,
self.mean_squared_displacement_ratio, self.straightness,
self.p_variation, self.max_excursion_normalised, self.ksstat,
self.eM, self.eL, self.eJ, self.maximum_ts, self.trappedness,
self.asymmetry, self.diff_kurtosis, self.efficiency,
self.max_std_x, self.max_std_y, self.max_std_change_x, self.max_std_change_y] \
+ list(self.velocity_autocorrelation) \
+ list(self.p_variations) + self.dagostino_stats + self.mv_features + list(self.dma)
self.columns = ["file", "Alpha", "motion", "D", "alpha",
"alpha_n_1", "alpha_n_2", "alpha_n_3",
"fractal_dimension", "mean_gaussianity",
"mean_squared_displacement_ratio", "straightness",
"p-variation", "max_excursion_normalised", "ksstat_chi2",
"M", "L", "J", "max_ts", "trappedness", 'asymmetry',
"diff_kurtosis", "efficiency", 'max_std_x', 'max_std_y',
'max_std_change_x', 'max_std_change_y'] + self.velocity_autocorrelation_names \
+ self.p_variation_names + self.dagostino_stats_names + self.mv_features_names \
+ list(self.dma_names)
self.data = pd.DataFrame([self.values], columns=self.columns)
def compute_cca(views, k=300, eps=1e-12):
"""
views: list of views, each N x v_i_emb where N is the number of observations and v_i_emb is the embedding dimensionality of that view
k: integer for the dimensionality of the joint projection space
eps: float added to diagonals of matrices A and B for stability
"""
m = views[0].size(0)
t = views[0].type()
o = [v.size(1) for v in views]
os = sum(o)
A = torch.zeros(os, os).type(t)
B = torch.zeros(os, os).type(t)
print('doing generalised eigendecomposition...')
row_i = 0
for i, V_i in enumerate(views):
V_i = V_i.t()
o_i = V_i.size(0)
mu_i = V_i.mean(dim=1, keepdim=True)
# mean center view i
V_i_bar = V_i - mu_i.expand_as(V_i) # o_i x N
col_i = 0
for j, V_j in enumerate(views):
V_j = V_j.t()
o_j = V_j.size(0)
if i > j:
col_i += o_j
continue
mu_j = V_j.mean(dim=1, keepdim=True)
# mean center view j
V_j_bar = V_j - mu_j.expand_as(V_j) # o_j x N
C_ij = (1.0 /
(m - 1)) * torch.mm(V_i_bar, V_j_bar.t()) # o_i x o_j
A[row_i:row_i + o_i, col_i:col_i + o_j] = C_ij
A[col_i:col_i + o_j, row_i:row_i + o_i] = C_ij.t()
if i == j:
B[row_i:row_i + o_i, col_i:col_i + o_j] = C_ij.clone()
col_i += o_j
row_i += o_i
diagonal(A).add_(eps)
diagonal(B).add_(eps)
A = A.cpu().numpy()
B = B.cpu().numpy()
l, v = la.eig(A, B)
idx = l.argsort()[-k:][::-1]
l = l[idx] # eigenvalues
v = v[:, idx] # eigenvectors
l = torch.from_numpy(l.real)
v = torch.from_numpy(v.real)
# extracting projection matrices
proj_matrices = [
v[sum(o[:i]):sum(o[:i]) + views[i].size(1)].type(t)
for i in range(len(views))
]
return l.type(t), proj_matrices
def compute_eigvals(self):
self.eigvals = []
for i in range(len(self.J)):
w, _ = eig(self.J[i])
self.eigvals.append(w)
ans = integral(this_func, 0, 1)
print("integral:", ans)
# scipy integrate!!
value, error = integrate.quad(this_func, 0, 1)
print("scipy integrate value", value)
print("scipy integrate error", error)
# scipy linear algebra
A = np.random.rand(3, 3)
b = np.random.rand(3)
x = linalg.solve(A, b) # solve A x = b
print(x)
eigen = linalg.eig(A) # eigens
print(eigen)
det = linalg.det(A) # determinant
print(det)
U, s, Vh = linalg.svd(A) # singular value decomposition
# statistics in scipy
y = stats.norm.cdf(0.311) # norm CDF
print(y)
# data fitting
def func(x, a, b, c):
return a * np.exp(-b * x) + c
Xt.append(xt)
Yt.append(y)
# labelled source
Xs = np.row_stack(Xs)
Xs = Xs / Xs.std(0) # scale
Ys = np.concatenate(Ys)
# unlabelled target
Xt = np.row_stack(Xt)
Xt = Xt / Xt.std(0) # scale
Yt = np.concatenate(Yt)
# PCA
Zd = 2 # dimensionality of the subspace
# weights
_, w = eig(np.cov(np.row_stack([Xs[0:-1:2], Xt[0:-1:2]]).T)) # train with even
# project
pcaZs = Xs @ w[:, :Zd]
pcaZt = Xt @ w[:, :Zd]
# -- plot
# train (even idxs)
plt.scatter(pcaZs[0:-1:2, 0],
pcaZs[0:-1:2, 1],
s=4,
c=np.take(clr, Ys[0:-1:2] - 1),
label='$D_s$')
plt.scatter(pcaZt[0:-1:2, 0],
pcaZt[0:-1:2, 1],
s=8,
ec=np.take(clr, Yt[0:-1:2] - 1),