def assert_double_centering_identities(fs, M):
"""
Raise a Counterexample if one is found.
"""
A = M[:fs.nsmall, :fs.nsmall]
HMH = double_centered_slow(M)
HMH_A = HMH[:fs.nsmall, :fs.nsmall]
# check an identity between two ways to compute a centered submatrix
HAH_direct = double_centered_slow(A)
HAH_indirect = double_centered_slow(HMH_A)
assert_named_equation(
(HAH_direct, 'double centered submatrix'),
(HAH_indirect, 're-centered submatrix of doubly centered matrix'), M)
HAH = HAH_indirect
# This is not true:
# check that HAH <==inverse_in_H==> H A^-1 H
HAH_pinv = inverse_in_H(HAH)
H_Ainv_H = double_centered_slow(np.linalg.inv(A))
"""
assert_named_equation(
(HAH_pinv, 'inverse-in-H of HAH'),
(H_Ainv_H, 'double centered inverse of A'))
"""
# check a submatrix-schur-inverse commutativity in double centering
HMH_pinv = inverse_in_H(HMH)
schur_of_HMH_pinv = schur(HMH_pinv, fs.nsmall)
assert_named_equation((HAH_pinv, 'inverse-in-H of HAH'),
(schur_of_HMH_pinv, 'schur of pinv of HMH'), M)
def assert_double_centering_identities(fs, M):
"""
Raise a Counterexample if one is found.
"""
A = M[:fs.nsmall, :fs.nsmall]
HMH = double_centered_slow(M)
HMH_A = HMH[:fs.nsmall, :fs.nsmall]
# check an identity between two ways to compute a centered submatrix
HAH_direct = double_centered_slow(A)
HAH_indirect = double_centered_slow(HMH_A)
assert_named_equation(
(HAH_direct, 'double centered submatrix'),
(HAH_indirect, 're-centered submatrix of doubly centered matrix'))
HAH = HAH_indirect
# This is not true:
# check that HAH <==inverse_in_H==> H A^-1 H
HAH_pinv = inverse_in_H(HAH)
H_Ainv_H = double_centered_slow(np.linalg.inv(A))
assert_named_equation(
(HAH_pinv, 'inverse-in-H of HAH'),
(H_Ainv_H, 'double centered inverse of A'))
# check a submatrix-schur-inverse commutativity in double centering
HMH_pinv = inverse_in_H(HMH)
schur_of_HMH_pinv = schur(HMH_pinv, fs.nsmall)
assert_named_equation(
(HAH_pinv, 'inverse-in-H of HAH'),
(schur_of_HMH_pinv, 'schur of pinv of HMH'))
def assert_pinvproj(fs, M):
"""
Raise a Counterexample if one is found.
"""
M_sub = M[:fs.nsmall, :fs.nsmall]
pinvproj_of_sub = pinvproj(M_sub)
schur_of_pinvproj = schur(pinvproj(M), fs.nsmall)
bottduff_of_sub = numpyutils.bott_duffin_const(M_sub)
schur_of_bottduff = schur(numpyutils.bott_duffin_const(M), fs.nsmall)
"""
assert_named_equation(
(np.array([[1]]), 'one'),
(np.array([[2]]), 'two'),
MatrixUtil.double_centered(M))
"""
assert_named_equation(
(pinvproj_of_sub, 'pinvproj of sub'),
(schur_of_pinvproj, 'schur of pinvproj'),
double_centered_slow(M))
assert_named_equation(
(pinvproj_of_sub, 'pinvproj of sub'),
(bottduff_of_sub, 'bottduff of sub'),
double_centered_slow(M))
assert_named_equation(
(schur_of_pinvproj, 'schur of pinvproj'),
(schur_of_bottduff, 'schur of bottduff'),
double_centered_slow(M))
def edm_to_gower_matrix(D):
"""
@param D: an EDM
@return: Gower's double centered matrix
"""
G = -0.5 * double_centered_slow(D)
return G
def get_response_content(fs):
nbig = 10
nsmall = 4
e = np.ones(nbig)
I = np.eye(nbig)
Q = np.outer(e, e) / np.inner(e, e)
H = I - Q
M = sample_sym(nbig)
W = double_centered_slow(M)
WpQ = W + Q
# get eigendecomposition of schur complement
W_SC = schur(W, nsmall)
W_SC_w, W_SC_vt = scipy.linalg.eigh(W_SC)
# get eigendecomposition of schur complement
WpQ_SC = schur(WpQ, nsmall)
WpQ_SC_w, WpQ_SC_vt = scipy.linalg.eigh(WpQ_SC)
# show the results
out = StringIO()
np.set_printoptions(linewidth=200)
print >> out, 'random symmetric matrix:'
print >> out, M
print >> out, 'after centering:'
print >> out, W
print >> out, 'after adding a constant eigenvector with unit eigenvalue:'
print >> out, WpQ
print >> out
print >> out, 'eigendecomposition of Schur complement of centered matrix:'
print >> out, W_SC_w
print >> out, W_SC_vt
print >> out
print >> out, 'eigendecomposition of Schur complement of decentered matrix:'
print >> out, WpQ_SC_w
print >> out, WpQ_SC_vt
print >> out
return out.getvalue()
def edm_to_gower_matrix(D):
"""
@param D: an EDM
@return: Gower's double centered matrix
"""
G = -0.5 * double_centered_slow(D)
return G
def get_response_content(fs):
nbig = 10
nsmall = 4
e = np.ones(nbig)
I = np.eye(nbig)
Q = np.outer(e, e) / np.inner(e, e)
H = I - Q
M = sample_sym(nbig)
W = double_centered_slow(M)
WpQ = W + Q
# get eigendecomposition of schur complement
W_SC = schur(W, nsmall)
W_SC_w, W_SC_vt = scipy.linalg.eigh(W_SC)
# get eigendecomposition of schur complement
WpQ_SC = schur(WpQ, nsmall)
WpQ_SC_w, WpQ_SC_vt = scipy.linalg.eigh(WpQ_SC)
# show the results
out = StringIO()
np.set_printoptions(linewidth=200)
print >> out, 'random symmetric matrix:'
print >> out, M
print >> out, 'after centering:'
print >> out, W
print >> out, 'after adding a constant eigenvector with unit eigenvalue:'
print >> out, WpQ
print >> out
print >> out, 'eigendecomposition of Schur complement of centered matrix:'
print >> out, W_SC_w
print >> out, W_SC_vt
print >> out
print >> out, 'eigendecomposition of Schur complement of decentered matrix:'
print >> out, WpQ_SC_w
print >> out, WpQ_SC_vt
print >> out
return out.getvalue()
def assert_pinvproj(fs, M):
"""
Raise a Counterexample if one is found.
"""
M_sub = M[:fs.nsmall, :fs.nsmall]
pinvproj_of_sub = pinvproj(M_sub)
schur_of_pinvproj = schur(pinvproj(M), fs.nsmall)
bottduff_of_sub = numpyutils.bott_duffin_const(M_sub)
schur_of_bottduff = schur(numpyutils.bott_duffin_const(M), fs.nsmall)
"""
assert_named_equation(
(np.array([[1]]), 'one'),
(np.array([[2]]), 'two'),
MatrixUtil.double_centered(M))
"""
assert_named_equation((pinvproj_of_sub, 'pinvproj of sub'),
(schur_of_pinvproj, 'schur of pinvproj'),
double_centered_slow(M))
assert_named_equation((pinvproj_of_sub, 'pinvproj of sub'),
(bottduff_of_sub, 'bottduff of sub'),
double_centered_slow(M))
assert_named_equation((schur_of_pinvproj, 'schur of pinvproj'),
(schur_of_bottduff, 'schur of bottduff'),
double_centered_slow(M))
