def process():
"""
@return: a multi-line string that summarizes the results
"""
np.set_printoptions(linewidth=200)
# define the adjacency matrix
A = g_A
n = 6
# define some mass distributions
m_uniform = np.ones(n) / float(n)
m_weighted = np.array([102, 102, 102, 102, 1, 1], dtype=float) / 410
# make the response
out = StringIO()
# look at the eigendecomposition of -(1/2)HDH where D is the leaf distance matrix
HSH = Euclid.edm_to_dccov(Euclid.g_D_b)
W_HSH, VT_HSH = np.linalg.eigh(HSH)
print >> out, 'W for -(1/2)HDH of the leaf distance matrix:'
print >> out, W_HSH
print >> out, 'VT for -(1/2)HDH of the leaf distance matrix:'
print >> out, VT_HSH
# look at the eigendecomposition of S given a degenerate mass distribution on the full tree
m_degenerate = np.array([.25, .25, .25, .25, 0, 0])
S = Euclid.edm_to_weighted_cross_product(Euclid.g_D_c, m_degenerate)
W_S, VT_S = np.linalg.eigh(S)
print >> out, 'W for -(1/2)(Xi)D(Xi)^T of the full distance matrix with degenerate masses:'
print >> out, W_S
print >> out, 'VT for -(1/2)(Xi)D(Xi)^T of the full distance matrix with degenerate masses:'
print >> out, VT_S
# look at the effects of various mass distributions on the MDS of the full tree
for m in (m_uniform, m_weighted):
# the mass distribution should sum to 1
if not np.allclose(np.sum(m), 1):
raise ValueError('masses should sum to 1')
# to compute the perturbed laplacian matrix first get weighted sums
v = np.dot(m, A)
# now divide elementwise by the masses
v /= m
# subtract the adjacency matrix from the diagonal formed by elements of this vector
Lp = np.diag(v) - A
# now get the eigendecomposition of the pseudoinverse of the perturbed laplacian
W_Lp_pinv, VT_Lp_pinv = np.linalg.eigh(np.linalg.pinv(Lp))
# look at the eigendecomposition of the S matrix associated with the distance matrix of this tree
D = Euclid.g_D_c
S = Euclid.edm_to_weighted_cross_product(D, m)
W_S, VT_S = np.linalg.eigh(S)
print >> out, 'perturbed laplacian:'
print >> out, Lp
print >> out, 'm:', m
print >> out, 'W for the pseudoinverse of the perturbed laplacian:'
print >> out, W_Lp_pinv
print >> out, 'VT for the pseudoinverse of the perturbed laplacian:'
print >> out, VT_Lp_pinv
print >> out, 'W for the cross product matrix:'
print >> out, W_S
print >> out, 'VT for the cross product matrix:'
print >> out, VT_S
return out.getvalue().strip()
def process():
"""
@return: a multi-line string that summarizes the results
"""
np.set_printoptions(linewidth=200)
# define the adjacency matrix
A = g_A
n = 6
# define some mass distributions
m_uniform = np.ones(n) / float(n)
m_weighted = np.array([102, 102, 102, 102, 1, 1], dtype=float) / 410
# make the response
out = StringIO()
# look at the eigendecomposition of -(1/2)HDH where D is the leaf distance matrix
HSH = Euclid.edm_to_dccov(Euclid.g_D_b)
W_HSH, VT_HSH = np.linalg.eigh(HSH)
print >> out, 'W for -(1/2)HDH of the leaf distance matrix:'
print >> out, W_HSH
print >> out, 'VT for -(1/2)HDH of the leaf distance matrix:'
print >> out, VT_HSH
# look at the eigendecomposition of S given a degenerate mass distribution on the full tree
m_degenerate = np.array([.25, .25, .25, .25, 0, 0])
S = Euclid.edm_to_weighted_cross_product(Euclid.g_D_c, m_degenerate)
W_S, VT_S = np.linalg.eigh(S)
print >> out, 'W for -(1/2)(Xi)D(Xi)^T of the full distance matrix with degenerate masses:'
print >> out, W_S
print >> out, 'VT for -(1/2)(Xi)D(Xi)^T of the full distance matrix with degenerate masses:'
print >> out, VT_S
# look at the effects of various mass distributions on the MDS of the full tree
for m in (m_uniform, m_weighted):
# the mass distribution should sum to 1
if not np.allclose(np.sum(m), 1):
raise ValueError('masses should sum to 1')
# to compute the perturbed laplacian matrix first get weighted sums
v = np.dot(m, A)
# now divide elementwise by the masses
v /= m
# subtract the adjacency matrix from the diagonal formed by elements of this vector
Lp = np.diag(v) - A
# now get the eigendecomposition of the pseudoinverse of the perturbed laplacian
W_Lp_pinv, VT_Lp_pinv = np.linalg.eigh(np.linalg.pinv(Lp))
# look at the eigendecomposition of the S matrix associated with the distance matrix of this tree
D = Euclid.g_D_c
S = Euclid.edm_to_weighted_cross_product(D, m)
W_S, VT_S = np.linalg.eigh(S)
print >> out, 'perturbed laplacian:'
print >> out, Lp
print >> out, 'm:', m
print >> out, 'W for the pseudoinverse of the perturbed laplacian:'
print >> out, W_Lp_pinv
print >> out, 'VT for the pseudoinverse of the perturbed laplacian:'
print >> out, VT_Lp_pinv
print >> out, 'W for the cross product matrix:'
print >> out, W_S
print >> out, 'VT for the cross product matrix:'
print >> out, VT_S
return out.getvalue().strip()
def get_weighted_embedding_b(D, m):
"""
This method was suggested by Eric but it has some limitations.
In particular, it fails when some elements of the mass vector are zero.
@param D: a distance matrix
@param m: a mass vector
@return: an embedding
"""
M = np.diag(np.sqrt(m))
cross_product_matrix = Euclid.edm_to_weighted_cross_product(D, m)
Q = np.dot(M, np.dot(cross_product_matrix, M.T))
U, S, VT = np.linalg.svd(Q, full_matrices=False)
Z = np.dot(np.linalg.pinv(M), np.dot(U, np.sqrt(np.diag(S))))
return Z
def process():
"""
@return: a multi-line string that summarizes the results
"""
np.set_printoptions(linewidth=200)
out = StringIO()
# define some distance matrices
D_leaves = Euclid.g_D_b
D_all = Euclid.g_D_c
nvertices = 6
nleaves = 4
# define mass vectors
m_degenerate = np.array([0.25, 0.25, 0.25, 0.25, 0, 0])
m_interesting = np.array([.2, .2, .2, .2, .1, .1])
m_uniform = np.ones(nvertices) / float(nvertices)
# augment a distance matrix by adding leaflets
D_augmented = add_leaflets(D_all, nleaves)
# create the projection of points
X_projected = do_projection(D_all, nleaves)
# show some of the distance matrices
print >> out, 'pairwise distances among vertices in the original tree:'
print >> out, D_all
print >> out, 'pairwise distance matrix augmented with one leaflet per leaf:'
print >> out, D_augmented
# get the distance matrices corresponding to the cases in the docstring
print >> out, 'case 1: embedding of all vertices:'
print >> out, Euclid.edm_to_points(D_all)
print >> out, 'case 2: embedding of leaves and leaflets from the leaflet-augmented distance matrix:'
print >> out, Euclid.edm_to_points(D_augmented)
print >> out, 'case 3: projection of all vertices onto the MDS space of the leaves:'
print >> out, X_projected
# another embedding
print >> out, 'embedding of leaves from the leaf distance matrix:'
print >> out, Euclid.edm_to_points(D_leaves)
# show embeddings of a tree augmented with leaflets
print >> out, 'first few coordinates of the original vertices of the embedded tree with lots of leaflets per leaf:'
D_super_augmented = D_all.copy()
for i in range(20):
D_super_augmented = add_leaflets(D_super_augmented, nleaves)
X_super = Euclid.edm_to_points(D_super_augmented)
X_super_block_small = X_super[:6].T[:3].T
print >> out, X_super_block_small
print >> out, 'ratio of coordinates of projected points to coordinates of this block of the embedding of the augmented tree:'
print >> out, X_projected / X_super_block_small
# test
Z = Euclid.edm_to_weighted_points(D_all, m_uniform)
print >> out, 'generalized case 1:'
print >> out, Z
# test
Z = Euclid.edm_to_weighted_points(D_all, m_interesting)
print >> out, 'generalized case 2:'
print >> out, Z
# test
Z = Euclid.edm_to_weighted_points(D_all, m_degenerate)
print >> out, 'generalized case 3:'
print >> out, Z
# test
Z = get_weighted_embedding_b(D_all, m_uniform)
print >> out, 'eric formula case 1:'
print >> out, Z
# test
Z = get_weighted_embedding_b(D_all, m_interesting)
print >> out, 'eric formula case 2:'
print >> out, Z
# test
Z = get_weighted_embedding_b(D_all, m_degenerate)
print >> out, 'eric formula case 3:'
print >> out, Z
# test stuff
print >> out, 'testing random stuff:'
D = D_all
m = m_degenerate
nvertices = len(m)
sqrtm = np.sqrt(m)
M = np.diag(sqrtm)
cross_product_matrix = Euclid.edm_to_weighted_cross_product(D, m)
U_cross, S_cross, VT_cross = np.linalg.svd(cross_product_matrix, full_matrices=False)
Q = np.dot(M, np.dot(cross_product_matrix, M.T))
U, B, VT = np.linalg.svd(Q, full_matrices=False)
S = np.sqrt(np.diag(B))
US = np.dot(U, S)
M_pinv = np.linalg.pinv(M)
M_pinv_narrow = M_pinv.T[:-2].T
US_short = US[:-2]
print >> out, 'eigenvalues of the abdi cross product:', S_cross
print >> out, 'eigenvalues of the eric cross product:', B
print >> out, M_pinv
print >> out, US
print >> out, M_pinv_narrow
print >> out, US_short
Z = np.dot(M_pinv_narrow, US_short)
print >> out, Z
# return the response
return out.getvalue().strip()
