def lnprob(theta, Deltam, nsne, xi, redshiftterm):
A, M, sigma, pv = theta
if (A <= 0 or sigma < 0 or pv < 0):
return -numpy.inf
C = A * numpy.array(xi)
numpy.fill_diagonal(
C,
C.diagonal() + sigma**2 / nsne + (pv * redshiftterm)**2)
mterm = Deltam - M
C = matrix(C)
W = matrix(mterm)
try:
lapack.posv(C, W, uplo='U')
except ArithmeticError:
return -np.inf
logdetC = 2 * numpy.log(numpy.array(C).diagonal()).sum()
lp = -0.5 * (logdetC + blas.dot(matrix(mterm), W)) + cauchy.logpdf(
sigma, loc=0.08, scale=0.5) + cauchy.logpdf(
pv, loc=0, scale=600 / 3e5)
if not numpy.isfinite(lp):
return -np.inf
return lp
def lapack_posv(a, b):
"""
Inverse using LaPaCK posv (definite positive) algorithm.
:param a:
:param b:
:return:
"""
b_posv = matrix(b)
lapack.posv(matrix(a), b_posv)
return b_posv
def PLS2D_getBeta(theta, ZtX, ZtY, XtX, ZtZ, XtY, YtX, YtZ, XtZ, YtY, n, P,
tinds, rinds, cinds):
# Obtain Lambda
Lambda = mapping2D(theta, tinds, rinds, cinds)
# Obtain Lambda'
Lambdat = spmatrix.trans(Lambda)
# Obtain Lambda'Z'Y and Lambda'Z'X
LambdatZtY = Lambdat * ZtY
LambdatZtX = Lambdat * ZtX
# Set the factorisation to use LL' instead of LDL'
cholmod.options['supernodal'] = 2
# Obtain the cholesky decomposition
LambdatZtZLambda = Lambdat * ZtZ * Lambda
I = spmatrix(1.0, range(Lambda.size[0]), range(Lambda.size[0]))
chol_dict = sparse_chol2D(LambdatZtZLambda + I,
perm=P,
retF=True,
retP=False,
retL=False)
F = chol_dict['F']
# Obtain C_u (annoyingly solve writes over the second argument,
# whereas spsolve outputs)
Cu = LambdatZtY[P, :]
cholmod.solve(F, Cu, sys=4)
# Obtain RZX
RZX = LambdatZtX[P, :]
cholmod.solve(F, RZX, sys=4)
# Obtain RXtRX
RXtRX = XtX - matrix.trans(RZX) * RZX
# Obtain beta estimates (note: gesv also replaces the second
# argument)
betahat = XtY - matrix.trans(RZX) * Cu
try:
lapack.posv(RXtRX, betahat)
except:
lapack.gesv(RXtRX, betahat)
return (betahat)
def barrier():
# variables kept same from cvxopt example
MAXITERS = 100
ALPHA = 0.01
BETA = 0.5
x = matrix(0.0, (n, 1))
H = matrix(0.0, (n, n)) # Symmetrix matrix
for iter in range(MAXITERS):
# get the gradient of the function
d = (b - A * x)**-1
g = A.T * d
# print(d[:, n*[0]].size)
# print(A.size)
"""bug here: won't multiply of two matrix of same dimension. code is looking into another dimension?"""
# get Hessian
# lower diaganol multiplied to constraint matrix, n*[0] is first center x^t(0)
h = np.zeros(shape=(m, n))
np.matmul(d[:, n * [0]], A, h)
# use the BLAS solver to get the symmetric matrix and get roots
blas.syrk(h, H, trans='T')
# do Newton's step
v = -g # g is our gradient
# LAPACK solves the matrix and gives us the tep value to transverse with
lapack.posv(H, v)
# Stop condition if exceeding tolerance
lam = blas.dot(g, v)
if sqrt(-lam) < mu:
return x # return the orignal value if we're above tolerance
# Line search to go to optimal using ALPHA and BETA
y = mul(A * v, d)
step = 1.0
while 1 - step * max(y) < 0:
step *= BETA
while True:
if -sum(log(1 - step * y)) < (ALPHA * step * lam):
break
step *= BETA
# increment x by the step times the negative gradient otherwise
x += step * v
def acent(A, b):
"""
Computes analytic center of A*x <= b with A m by n of rank n.
We assume that b > 0 and the feasible set is bounded.
"""
MAXITERS = 100
ALPHA = 0.01
BETA = 0.5
TOL = 1e-8
ntdecrs = []
m, n = A.size
x = matrix(0.0, (n, 1))
H = matrix(0.0, (n, n))
for iter in range(MAXITERS):
# Gradient is g = A^T * (1./(b-A*x)).
d = (b - A * x) ** -1
g = A.T * d
# Hessian is H = A^T * diag(1./(b-A*x))^2 * A.
Asc = mul(d[:, n * [0]], A)
blas.syrk(Asc, H, trans="T")
# Newton step is v = H^-1 * g.
v = -g
lapack.posv(H, v)
# Directional derivative and Newton decrement.
lam = blas.dot(g, v)
ntdecrs += [sqrt(-lam)]
print("%2d. Newton decr. = %3.3e" % (iter, ntdecrs[-1]))
if ntdecrs[-1] < TOL:
return x, ntdecrs
# Backtracking line search.
y = mul(A * v, d)
step = 1.0
while 1 - step * max(y) < 0:
step *= BETA
while True:
if -sum(log(1 - step * y)) < ALPHA * step * lam:
break
step *= BETA
x += step * v
def acent(A, b):
"""
Computes analytic center of A*x <= b with A m by n of rank n.
We assume that b > 0 and the feasible set is bounded.
"""
MAXITERS = 100
ALPHA = 0.01
BETA = 0.5
TOL = 1e-8
ntdecrs = []
m, n = A.size
x = matrix(0.0, (n, 1))
H = matrix(0.0, (n, n))
for iter in range(MAXITERS):
# Gradient is g = A^T * (1./(b-A*x)).
d = (b - A * x)**-1
g = A.T * d
# Hessian is H = A^T * diag(1./(b-A*x))^2 * A.
Asc = mul(d[:, n * [0]], A)
blas.syrk(Asc, H, trans='T')
# Newton step is v = H^-1 * g.
v = -g
lapack.posv(H, v)
# Directional derivative and Newton decrement.
lam = blas.dot(g, v)
ntdecrs += [sqrt(-lam)]
print("%2d. Newton decr. = %3.3e" % (iter, ntdecrs[-1]))
if ntdecrs[-1] < TOL: return x, ntdecrs
# Backtracking line search.
y = mul(A * v, d)
step = 1.0
while 1 - step * max(y) < 0:
step *= BETA
while True:
if -sum(log(1 - step * y)) < ALPHA * step * lam: break
step *= BETA
x += step * v
# Nominal problem: minimize || A*x - b ||_2
xnom = +b
lapack.gels(+A, xnom)
xnom = xnom[:n]
# Stochastic problem.
#
# minimize E || (A+u*B) * x - b ||_2^2
# = || A*x - b||_2^2 + x'*P*x
#
# with P = E(u^2) * B'*B = (1/3) * B'*B
S = A.T * A + (1.0 / 3.0) * B.T * B
xstoch = A.T * b
lapack.posv(S, xstoch)
# Worst case approximation.
#
# minimize max_{-1 <= u <= 1} ||A*u - b||_2^2.
xwc = wcls(A, [B], b)
nopts = 500
us = -2.0 + (2.0 - (-2.0)) / (nopts - 1) * matrix(list(range(nopts)), tc='d')
rnom = [blas.nrm2((A + u * B) * xnom - b) for u in us]
rstoch = [blas.nrm2((A + u * B) * xstoch - b) for u in us]
rwc = [blas.nrm2((A + u * B) * xwc - b) for u in us]
if pylab_installed:
pylab.figure(1, facecolor='w')
def covsel(Y):
"""
Returns the solution of
minimize -log det K + Tr(KY)
subject to K_{ij}=0, (i,j) not in indices listed in I,J.
Y is a symmetric sparse matrix with nonzero diagonal elements.
I = Y.I, J = Y.J.
"""
I, J = Y.I, Y.J
n, m = Y.size[0], len(I)
N = I + J*n # non-zero positions for one-argument indexing
D = [k for k in range(m) if I[k]==J[k]] # position of diagonal elements
# starting point: symmetric identity with nonzero pattern I,J
K = spmatrix(0.0, I, J)
K[::n+1] = 1.0
# Kn is used in the line search
Kn = spmatrix(0.0, I, J)
# symbolic factorization of K
F = cholmod.symbolic(K)
# Kinv will be the inverse of K
Kinv = matrix(0.0, (n,n))
for iters in range(100):
# numeric factorization of K
cholmod.numeric(K, F)
d = cholmod.diag(F)
# compute Kinv by solving K*X = I
Kinv[:] = 0.0
Kinv[::n+1] = 1.0
cholmod.solve(F, Kinv)
# solve Newton system
grad = 2*(Y.V - Kinv[N])
hess = 2*(mul(Kinv[I,J],Kinv[J,I]) + mul(Kinv[I,I],Kinv[J,J]))
v = -grad
lapack.posv(hess,v)
# stopping criterion
sqntdecr = -blas.dot(grad,v)
print("Newton decrement squared:%- 7.5e" %sqntdecr)
if (sqntdecr < 1e-12):
print("number of iterations: ", iters+1)
break
# line search
dx = +v
dx[D] *= 2 # scale the diagonal elems
f = -2.0 * sum(log(d)) # f = -log det K
s = 1
for lsiter in range(50):
Kn.V = K.V + s*dx
try:
cholmod.numeric(Kn, F)
except ArithmeticError:
s *= 0.5
else:
d = cholmod.diag(F)
fn = -2.0 * sum(log(d)) + 2*s*blas.dot(v,Y.V)
if (fn < f - 0.01*s*sqntdecr):
break
s *= 0.5
K.V = Kn.V
return K
# Worst-case distribution from dual solution.
X = [Z[2, :2].T / Z[2, 2] for Z in sol['zs'] if Z[2, 2] > 1e-5]
return bound, P, q, r, X
# Compute bound for s0 with sigma = 1.0.
# Write ellipse {x | x'*P*x + 2*q'*x + r = 1} in the form
# {xc + L^{-T}*u | ||u||_2 = 1}
Sigma = matrix([1.0, 0.0, 0.0, 1.0], (2, 2))
bnd, P, q, r, X = cheb(A0, b0, Sigma)
xc = -q
L = +P
lapack.posv(L, xc)
L /= sqrt(1 - r - blas.dot(q, xc))
if pylab_installed:
def makefig1():
pylab.figure(1, facecolor='w', figsize=(6, 6))
pylab.plot(V[0, :].T, V[1, :].T, 'b-')
nopts = 1000
angles = matrix([a * 2.0 * pi / nopts for a in range(nopts)],
(1, nopts))
circle = matrix(0.0, (2, nopts))
circle[0, :], circle[1, :] = cos(angles), sin(angles)
for k in range(len(C)):
c = C[k]
pylab.plot([c[0]], [c[1]], 'ow')
# Nominal problem: minimize || A*x - b ||_2
xnom = +b
lapack.gels(+A, xnom)
xnom = xnom[:n]
# Stochastic problem.
#
# minimize E || (A+u*B) * x - b ||_2^2
# = || A*x - b||_2^2 + x'*P*x
#
# with P = E(u^2) * B'*B = (1/3) * B'*B
S = A.T * A + (1.0/3.0) * B.T * B
xstoch = A.T * b
lapack.posv(S, xstoch)
# Worst case approximation.
#
# minimize max_{-1 <= u <= 1} ||A*u - b||_2^2.
xwc = wcls(A, [B], b)
nopts = 500
us = -2.0 + (2.0 - (-2.0))/(nopts-1) * matrix(list(range(nopts)),tc='d')
rnom = [ blas.nrm2( (A+u*B)*xnom - b) for u in us ]
rstoch = [ blas.nrm2( (A+u*B)*xstoch - b) for u in us ]
rwc = [ blas.nrm2( (A+u*B)*xwc - b) for u in us ]
if pylab_installed:
# Worst-case distribution from dual solution.
X = [ Z[2,:2].T / Z[2,2] for Z in sol['zs'] if Z[2,2] > 1e-5 ]
return bound, P, q, r, X
# Compute bound for s0 with sigma = 1.0.
# Write ellipse {x | x'*P*x + 2*q'*x + r = 1} in the form
# {xc + L^{-T}*u | ||u||_2 = 1}
Sigma = matrix([1.0, 0.0, 0.0, 1.0], (2,2))
bnd, P, q, r, X = cheb(A0, b0, Sigma)
xc = -q
L = +P
lapack.posv(L, xc)
L /= sqrt(1 - r - blas.dot(q, xc))
if pylab_installed:
def makefig1():
pylab.figure(1, facecolor='w', figsize=(6,6))
pylab.plot(V[0,:].T, V[1,:].T, 'b-')
nopts = 1000
angles = matrix( [a*2.0*pi/nopts for a in range(nopts) ],
(1,nopts) )
circle = matrix(0.0, (2,nopts))
circle[0,:], circle[1,:] = cos(angles), sin(angles)
for k in range(len(C)):
c = C[k]
pylab.plot([c[0]], [c[1]], 'og')
pylab.text(c[0], c[1], "s%d" %k)
def PLS2D_getSigma2(theta, ZtX, ZtY, XtX, ZtZ, XtY, YtX, YtZ, XtZ, YtY, n, P,
I, tinds, rinds, cinds):
# Obtain Lambda
#t1 = time.time()
Lambda = mapping2D(theta, tinds, rinds, cinds)
#t2 = time.time()
#print(t2-t1)#3.170967102050781e-05 9
# Obtain Lambda'
#t1 = time.time()
Lambdat = spmatrix.trans(Lambda)
#t2 = time.time()
#print(t2-t1)# 3.5762786865234375e-06
# Obtain Lambda'Z'Y and Lambda'Z'X
#t1 = time.time()
LambdatZtY = Lambdat * ZtY
LambdatZtX = Lambdat * ZtX
#t2 = time.time()
#print(t2-t1)#1.049041748046875e-05 13
# Obtain the cholesky decomposition
#t1 = time.time()
LambdatZtZLambda = Lambdat * (ZtZ * Lambda)
#t2 = time.time()
#print(t2-t1)#3.790855407714844e-05 2
#t1 = time.time()
chol_dict = sparse_chol2D(LambdatZtZLambda + I,
perm=P,
retF=True,
retP=False,
retL=False)
F = chol_dict['F']
#t2 = time.time()
#print(t2-t1)#0.0001342296600341797 1
# Obtain C_u (annoyingly solve writes over the second argument,
# whereas spsolve outputs)
#t1 = time.time()
Cu = LambdatZtY[P, :]
cholmod.solve(F, Cu, sys=4)
#t2 = time.time()
#print(t2-t1)#1.5974044799804688e-05 5
# Obtain RZX
#t1 = time.time()
RZX = LambdatZtX[P, :]
cholmod.solve(F, RZX, sys=4)
#t2 = time.time()
#print(t2-t1)#1.2159347534179688e-05 7
# Obtain RXtRX
#t1 = time.time()
RXtRX = XtX - matrix.trans(RZX) * RZX
#t2 = time.time()
#print(t2-t1)#9.775161743164062e-06 11
# Obtain beta estimates (note: gesv also replaces the second
# argument)
#t1 = time.time()
betahat = XtY - matrix.trans(RZX) * Cu
try:
lapack.posv(RXtRX, betahat)
except:
lapack.gesv(RXtRX, betahat)
#t2 = time.time()
#print(t2-t1)#1.7404556274414062e-05 6
# Obtain u estimates
#t1 = time.time()
uhat = Cu - RZX * betahat
cholmod.solve(F, uhat, sys=5)
cholmod.solve(F, uhat, sys=8)
#t2 = time.time()
#print(t2-t1)#1.2874603271484375e-05 8
# Obtain b estimates
#t1 = time.time()
bhat = Lambda * uhat
#t2 = time.time()
#print(t2-t1)#2.86102294921875e-06 15
# Obtain residuals sum of squares
#t1 = time.time()
resss = YtY - 2 * YtX * betahat - 2 * YtZ * bhat + 2 * matrix.trans(
betahat) * XtZ * bhat + matrix.trans(
betahat) * XtX * betahat + matrix.trans(bhat) * ZtZ * bhat
#t2 = time.time()
#print(t2-t1)#3.409385681152344e-05 4
# Obtain penalised residual sum of squares
#t1 = time.time()
pss = resss + matrix.trans(uhat) * uhat
return (pss / n)
def covsel(Y):
"""
Returns the solution of
minimize -log det K + tr(KY)
subject to K_ij = 0 if (i,j) not in zip(I, J).
Y is a symmetric sparse matrix with nonzero diagonal elements.
I = Y.I, J = Y.J.
"""
cholmod.options['supernodal'] = 2
I, J = Y.I, Y.J
n, m = Y.size[0], len(I)
# non-zero positions for one-argument indexing
N = I + J*n
# position of diagonal elements
D = [ k for k in range(m) if I[k]==J[k] ]
# starting point: symmetric identity with nonzero pattern I,J
K = spmatrix(0.0, I, J)
K[::n+1] = 1.0
# Kn is used in the line search
Kn = spmatrix(0.0, I, J)
# symbolic factorization of K
F = cholmod.symbolic(K)
# Kinv will be the inverse of K
Kinv = matrix(0.0, (n,n))
for iters in range(100):
# numeric factorization of K
cholmod.numeric(K, F)
d = cholmod.diag(F)
# compute Kinv by solving K*X = I
Kinv[:] = 0.0
Kinv[::n+1] = 1.0
cholmod.solve(F, Kinv)
# solve Newton system
grad = 2 * (Y.V - Kinv[N])
hess = 2 * ( mul(Kinv[I,J], Kinv[J,I]) +
mul(Kinv[I,I], Kinv[J,J]) )
v = -grad
lapack.posv(hess,v)
# stopping criterion
sqntdecr = -blas.dot(grad,v)
print("Newton decrement squared:%- 7.5e" %sqntdecr)
if (sqntdecr < 1e-12):
print("number of iterations: %d" %(iters+1))
break
# line search
dx = +v
dx[D] *= 2
f = -2.0*sum(log(d)) # f = -log det K
s = 1
for lsiter in range(50):
Kn.V = K.V + s*dx
try:
cholmod.numeric(Kn, F)
except ArithmeticError:
s *= 0.5
else:
d = cholmod.diag(F)
fn = -2.0 * sum(log(d)) + 2*s*blas.dot(v,Y.V)
if (fn < f - 0.01*s*sqntdecr): break
else: s *= 0.5
K.V = Kn.V
return K
def PLS(theta, ZtX, ZtY, XtX, ZtZ, XtY, YtX, YtZ, XtZ, YtY, P, tinds, rinds,
cinds):
#t1 = time.time()
# Obtain Lambda from theta
Lambda = mapping(theta, tinds, rinds, cinds)
#t2 = time.time()
#print(t2-t1)
#t1 = time.time()
# Obtain Lambda'
Lambdat = spmatrix.trans(Lambda)
#t2 = time.time()
#print(t2-t1)
#t1 = time.time()
LambdatZtY = Lambdat * ZtY
#t2 = time.time()
#print(t2-t1)
#t1 = time.time()
LambdatZtX = Lambdat * ZtX
#t2 = time.time()
#print(t2-t1)
#t1 = time.time()
# Set the factorisation to use LL' instead of LDL'
cholmod.options['supernodal'] = 2
#t2 = time.time()
#print(t2-t1)
# Obtain L
#t1 = time.time()
LambdatZtZLambda = Lambdat * ZtZ * Lambda
#t2 = time.time()
#print(t2-t1)
#t1 = time.time()
I = spmatrix(1.0, range(Lambda.size[0]), range(Lambda.size[0]))
#t2 = time.time()
#print(t2-t1)
#t1 = time.time()
chol_dict = sparse_chol(LambdatZtZLambda + I,
perm=P,
retF=True,
retP=False,
retL=False)
#t2 = time.time()
#print(t2-t1)
#t1 = time.time()
F = chol_dict['F']
#t2 = time.time()
#print(t2-t1)
# Obtain C_u (annoyingly solve writes over the second argument,
# whereas spsolve outputs)
#t1 = time.time()
Cu = LambdatZtY[P, :]
#t2 = time.time()
#print(t2-t1)
#t1 = time.time()
cholmod.solve(F, Cu, sys=4)
#t2 = time.time()
#print(t2-t1)
# Obtain RZX
#t1 = time.time()
RZX = LambdatZtX[P, :]
#t2 = time.time()
#print(t2-t1)
#t1 = time.time()
cholmod.solve(F, RZX, sys=4)
#t2 = time.time()
#print(t2-t1)
# Obtain RXtRX
#t1 = time.time()
RXtRX = XtX - matrix.trans(RZX) * RZX
#t2 = time.time()
#print(t2-t1)
#print(RXtRX.size)
#print(X.size)
#print(Y.size)
#print(RZX.size)
#print(Cu.size)
# Obtain beta estimates (note: gesv also replaces the second
# argument)
#t1 = time.time()
betahat = XtY - matrix.trans(RZX) * Cu
#t2 = time.time()
#print(t2-t1)
#t1 = time.time()
lapack.posv(RXtRX, betahat)
#t2 = time.time()
#print(t2-t1)
# Obtain u estimates
#t1 = time.time()
uhat = Cu - RZX * betahat
#t2 = time.time()
#print(t2-t1)
#t1 = time.time()
cholmod.solve(F, uhat, sys=5)
#t2 = time.time()
#print(t2-t1)
#t1 = time.time()
cholmod.solve(F, uhat, sys=8)
#t2 = time.time()
#print(t2-t1)
# Obtain b estimates
#t1 = time.time()
bhat = Lambda * uhat
#t2 = time.time()
#print(t2-t1)
# Obtain residuals sum of squares
#t1 = time.time()
resss = YtY - 2 * YtX * betahat - 2 * YtZ * bhat + 2 * matrix.trans(
betahat) * XtZ * bhat + matrix.trans(
betahat) * XtX * betahat + matrix.trans(bhat) * ZtZ * bhat
#t2 = time.time()
#print(t2-t1)
# Obtain penalised residual sum of squares
#t1 = time.time()
pss = resss + matrix.trans(uhat) * uhat
#t2 = time.time()
#print(t2-t1)
# Obtain Log(|L|^2)
#t1 = time.time()
logdet = 2 * sum(cvxopt.log(
cholmod.diag(F))) # this method only works for symm decomps
# Need to do tr(R_X)^2 for rml
#t2 = time.time()
#print(t2-t1)
# Obtain log likelihood
logllh = -logdet / 2 - X.size[0] / 2 * (1 + np.log(2 * np.pi * pss) -
np.log(X.size[0]))
#print(L[::(L.size[0]+1)]) # gives diag
#print(logllh[0,0])
#print(theta)
return (-logllh[0, 0])
def PeLS2D(theta, ZtX, ZtY, XtX, ZtZ, XtY, YtX, YtZ, XtZ, YtY, n, P, I, tinds,
rinds, cinds):
# Obtain Lambda
Lambda = mapping2D(theta, tinds, rinds, cinds)
# Obtain Lambda'
Lambdat = spmatrix.trans(Lambda)
# Obtain Lambda'Z'Y and Lambda'Z'X
LambdatZtY = Lambdat * ZtY
LambdatZtX = Lambdat * ZtX
# Obtain the cholesky decomposition
LambdatZtZLambda = Lambdat * (ZtZ * Lambda)
chol_dict = sparse_chol2D(LambdatZtZLambda + I,
perm=P,
retF=True,
retP=False,
retL=False)
F = chol_dict['F']
# Obtain C_u (annoyingly solve writes over the second argument,
# whereas spsolve outputs)
Cu = LambdatZtY[P, :]
cholmod.solve(F, Cu, sys=4)
# Obtain RZX
RZX = LambdatZtX[P, :]
cholmod.solve(F, RZX, sys=4)
# Obtain RXtRX
RXtRX = XtX - matrix.trans(RZX) * RZX
# Obtain beta estimates (note: gesv also replaces the second
# argument)
betahat = XtY - matrix.trans(RZX) * Cu
try:
lapack.posv(RXtRX, betahat)
except:
lapack.gesv(RXtRX, betahat)
# Obtain u estimates
uhat = Cu - RZX * betahat
cholmod.solve(F, uhat, sys=5)
cholmod.solve(F, uhat, sys=8)
# Obtain b estimates
bhat = Lambda * uhat
# Obtain residuals sum of squares
resss = YtY - 2 * YtX * betahat - 2 * YtZ * bhat + 2 * matrix.trans(
betahat) * XtZ * bhat + matrix.trans(
betahat) * XtX * betahat + matrix.trans(bhat) * ZtZ * bhat
# Obtain penalised residual sum of squares
pss = resss + matrix.trans(uhat) * uhat
# Obtain Log(|L|^2)
logdet = 2 * sum(cvxopt.log(cholmod.diag(F)))
# Obtain log likelihood
logllh = -logdet / 2 - n / 2 * (1 + np.log(2 * np.pi * pss[0, 0]) -
np.log(n))
return (-logllh)