print('forward gradient g(x) = \n', grad1)
# Now we want to compute the same Jacobian in the reverse mode of AD
# before we do that we have a look what the computational graph looks like:
# print 'Computational graph is', cg
# the reverse mode is called by cg.pullback([ybar])
# it is a little hard to explain what's going on here. Suffice to say that we
# now compute one row of the Jacobian instead of one column as in the forward mode
zbar = z.x.zeros_like()
# compute gradient in the reverse mode
zbar.data[0, :, 0, 0] = 1
cg.pullback([zbar])
grad2_x = x.xbar.data[0, 0]
grad2_y = y.xbar.data[0, 0]
grad2 = numpy.concatenate([grad2_x, grad2_y])
print('reverse gradient g(x) = \n', grad2)
#check that the forward computed gradient equals the reverse gradient
print('difference forward/reverse gradient=\n', grad1 - grad2)
# one can also easiliy extract the Hessian
H = numpy.zeros((2 * N, 2 * N))
H[:, :N] = x.xbar.data[1, :, :, 0]
H[:, N:] = y.xbar.data[1, :, :, 0]
print('Hessian = \n', H)
# 6. + 7.*t + 8.*t**2 + 9.*t**3 + 10.*t**5 )
#
# where
#
# Jx = dg/dx
# Jy = dg/dy
# setup input Taylor polynomials
D,P = 5, 3 # order D=5, number of directions P
ax = UTPM(numpy.zeros((D, P)))
ay = UTPM(numpy.zeros((D, P)))
ax.data[:, :] = numpy.array([1., 2. ,3., 4. ,5.]).reshape((5,1)) # input Taylor polynomial
ay.data[:, :] = numpy.array([6., 7. ,8., 9. ,10.]).reshape((5,1)) # input Taylor polynomial
# forward sweep
cg.pushforward([ax, ay])
azbar = UTPM(numpy.zeros((D, P, 3)))
azbar.data[0, ...] = numpy.eye(3)
# reverse sweep
cg.pullback([azbar])
# get results
Jx = cg.independentFunctionList[0].xbar
Jy = cg.independentFunctionList[1].xbar
print('Taylor series of Jx =\n', Jx)
print('Taylor series of Jy =\n', Jy)
# # checking against the analytical result
print('J - A =\n', J - A.x.data[0, 0])
# Now we want to compute the same Jacobian in the reverse mode of AD
# before we do that we have a look what the computational graph looks like:
# print 'Computational graph is', cg
# the reverse mode is called by cg.pullback([ybar])
# it is a little hard to explain what's going on here. Suffice to say that we
# now compute one row of the Jacobian instead of one column as in the forward mode
ybar = y.x.zeros_like()
# compute first row of J
ybar.data[0, 0, 0, 0] = 1
cg.pullback([ybar])
J_row1 = x.xbar.data[0, 0]
# compute second row of J
ybar.data[...] = 0
ybar.data[0, 0, 1, 0] = 1
cg.pullback([ybar])
J_row2 = x.xbar.data[0, 0]
# build Jacobian
J2 = numpy.vstack([J_row1.T, J_row2.T])
print('J - J2 =\n', J - J2)
# one can also easiliy extract the Hessian which is here a (M,N,N)-tensor
# e.g. the hessian of y[1] is zero since y[1] is linear in x
print('Hessian of y[1] w.r.t. x = \n', x.xbar.data[1, :, :, 0])
# METHOD 2: image space method (potentially numerically unstable)
cg2 = CGraph()
J1 = Function(J1.x)
J2 = Function(J2.x)
M = Function(UTPM(numpy.zeros((D, P, N + K, N + K))))
M[:N, :N] = dot(J1.T, J1)
M[:N, N:] = J2.T
M[N:, :N] = J2
C2 = inv(M)[:N, :N]
cg2.trace_off()
cg2.independentFunctionList = [J1, J2]
cg2.dependentFunctionList = [C2]
print('covariance matrix: C =\n', C2)
print('difference between image and nullspace method:\n', C - C2)
Cbar = UTPM(numpy.random.rand(D, P, N, N))
cg1.pullback([Cbar])
cg2.pullback([Cbar])
print(
'J1\n',
cg2.independentFunctionList[0].xbar - cg1.independentFunctionList[0].xbar)
print(
'J2\n',
cg2.independentFunctionList[1].xbar - cg1.independentFunctionList[1].xbar)
# # checking against the analytical result
print('J - A =\n', J - A.x.data[0,0])
# Now we want to compute the same Jacobian in the reverse mode of AD
# before we do that we have a look what the computational graph looks like:
# print 'Computational graph is', cg
# the reverse mode is called by cg.pullback([ybar])
# it is a little hard to explain what's going on here. Suffice to say that we
# now compute one row of the Jacobian instead of one column as in the forward mode
ybar = y.x.zeros_like()
# compute first row of J
ybar.data[0,0,0,0] = 1
cg.pullback([ybar])
J_row1 = x.xbar.data[0,0]
# compute second row of J
ybar.data[...] = 0
ybar.data[0,0,1,0] = 1
cg.pullback([ybar])
J_row2 = x.xbar.data[0,0]
# build Jacobian
J2 = numpy.vstack([J_row1.T, J_row2.T])
print('J - J2 =\n', J - J2)
# one can also easiliy extract the Hessian which is here a (M,N,N)-tensor
# e.g. the hessian of y[1] is zero since y[1] is linear in x
print('Hessian of y[1] w.r.t. x = \n',x.xbar.data[1,:,:,0])
# forward mode with ALGOPY
utp = logp(x, mu, sigma).data[:, 0]
print(
'function evaluation = %f\n1st directional derivative = %f\n2nd directional derivative = %f'
% (utp[0], 1. * utp[1], 2. * utp[2]))
# finite differences solution:
print('finite differences derivative =\n',
(logp(x, 3.5 + 10**-8, sigma) - logp(x, 3.5, sigma)) / 10**-8)
# trace function evaluation
cg = CGraph()
mu = Function(UTPM([[3.5], [1], [0]])) #unknown variable
out = logp(x, mu, sigma)
cg.trace_off()
cg.independentFunctionList = [mu]
cg.dependentFunctionList = [out]
cg.plot(
os.path.join(os.path.dirname(os.path.realpath(__file__)),
'posterior_log_probability_cgraph.png'))
# reverse mode with ALGOPY
outbar = UTPM([[1.], [0], [0]])
cg.pullback([outbar])
gradient = mu.xbar.data[0, 0]
Hess_vec = mu.xbar.data[1, 0]
print('gradient = ', gradient)
print('Hessian vector product = ', Hess_vec)
print 'function evaluation =\n',logp(x,3.5,sigma)
# forward mode with ALGOPY
utp = logp(x, mu, sigma).data[:,0]
print 'function evaluation = %f\n1st directional derivative = %f\n2nd directional derivative = %f'%(utp[0], 1.*utp[1], 2.*utp[2])
# finite differences solution:
print 'finite differences derivative =\n',(logp(x,3.5+10**-8,sigma) - logp(x, 3.5, sigma))/10**-8
# trace function evaluation
cg = CGraph()
mu = Function(UTPM([[3.5],[1],[0]])) #unknown variable
out = logp(x, mu, sigma)
cg.trace_off()
cg.independentFunctionList = [mu]
cg.dependentFunctionList = [out]
cg.plot(os.path.join(os.path.dirname(os.path.realpath(__file__)),'posterior_log_probability_cgraph.png'))
# reverse mode with ALGOPY
outbar = UTPM([[1.],[0],[0]])
cg.pullback([outbar])
gradient = mu.xbar.data[0,0]
Hess_vec = mu.xbar.data[1,0]
print 'gradient = ', gradient
print 'Hessian vector product = ', Hess_vec
M = Function(UTPM(numpy.zeros((D,P,N+K,N+K))))
M[:N,:N] = dot(J1.T,J1)
M[:N,N:] = J2.T
M[N:,:N] = J2
C2 = inv(M)[:N,:N]
cg2.trace_off()
cg2.independentFunctionList = [J1, J2]
cg2.dependentFunctionList = [C2]
print('covariance matrix: C =\n',C2)
print('difference between image and nullspace method:\n',C - C2)
Cbar = UTPM(numpy.random.rand(D,P,N,N))
cg1.pullback([Cbar])
cg2.pullback([Cbar])
print('J1\n',cg2.independentFunctionList[0].xbar - cg1.independentFunctionList[0].xbar)
print('J2\n',cg2.independentFunctionList[1].xbar - cg1.independentFunctionList[1].xbar)
print 'forward gradient g(x) = \n', grad1 # Now we want to compute the same Jacobian in the reverse mode of AD # before we do that we have a look what the computational graph looks like: # print 'Computational graph is', cg # the reverse mode is called by cg.pullback([ybar]) # it is a little hard to explain what's going on here. Suffice to say that we # now compute one row of the Jacobian instead of one column as in the forward mode zbar = z.x.zeros_like() # compute gradient in the reverse mode zbar.data[0,:,0,0] = 1 cg.pullback([zbar]) grad2_x = x.xbar.data[0,0] grad2_y = y.xbar.data[0,0] grad2 = numpy.concatenate([grad2_x, grad2_y]) print 'reverse gradient g(x) = \n', grad2 #check that the forward computed gradient equals the reverse gradient print 'difference forward/reverse gradient=\n',grad1 - grad2 # one can also easiliy extract the Hessian H = numpy.zeros((2*N,2*N)) H[:,:N] = x.xbar.data[1,:,:,0] H[:,N:] = y.xbar.data[1,:,:,0] print 'Hessian = \n', H
#
# where
#
# Jx = dg/dx
# Jy = dg/dy
# setup input Taylor polynomials
D, P = 5, 3 # order D=5, number of directions P
ax = UTPM(numpy.zeros((D, P)))
ay = UTPM(numpy.zeros((D, P)))
ax.data[:, :] = numpy.array([1., 2., 3., 4., 5.]).reshape(
(5, 1)) # input Taylor polynomial
ay.data[:, :] = numpy.array([6., 7., 8., 9., 10.]).reshape(
(5, 1)) # input Taylor polynomial
# forward sweep
cg.pushforward([ax, ay])
azbar = UTPM(numpy.zeros((D, P, 3)))
azbar.data[0, ...] = numpy.eye(3)
# reverse sweep
cg.pullback([azbar])
# get results
Jx = cg.independentFunctionList[0].xbar
Jy = cg.independentFunctionList[1].xbar
print('Taylor series of Jx =\n', Jx)
print('Taylor series of Jy =\n', Jy)
