def process(ntaxa):
np.set_printoptions(linewidth=200)
out = StringIO()
# sample an xtree topology
xtree = TreeSampler.sample_agglomerated_tree(ntaxa)
# sample an xtree with exponentially distributed branch lengths
mu = 2.0
for branch in xtree.get_branches():
branch.length = random.expovariate(1 / mu)
# convert the xtree to a FelTree so we can use the internal vertices
tree_string = xtree.get_newick_string()
tree = NewickIO.parse(tree_string, FelTree.NewickTree)
# get ordered ids and the number of leaves and some auxiliary variables
ordered_ids = get_ordered_ids(tree)
nleaves = len(list(tree.gen_tips()))
id_to_index = dict((myid, i) for i, myid in enumerate(ordered_ids))
# get the distance matrix relating all of the points
D_full = np.array(tree.get_full_distance_matrix(ordered_ids))
# Now do the projection so that
# the resulting points are in the subspace whose basis vectors are the axes of the leaf ellipsoid.
# First get the points such that the n rows in X are points in n-1 dimensional space.
X = Euclid.edm_to_points(D_full)
print >> out, 'points with centroid at origin:'
print >> out, X
print >> out
# Translate all of the points so that the origin is at the centroid of the leaves.
X -= np.mean(X[:nleaves], 0)
print >> out, 'points with centroid of leaves at origin:'
print >> out, X
print >> out
# Extract the subset of points that define the leaves.
L = X[:nleaves]
# Find the orthogonal transformation of the leaves onto their MDS axes.
# According to the python svd documentation, singular values are sorted most important to least important.
U, s, Vt = np.linalg.svd(L)
# Transform all of the points (including the internal vertices) according to this orthogonal transformation.
# The axes are now the axes of the Steiner circumscribed ellipsoid of the leaf vertices.
# I am using M.T[:k].T to get the first k columns of M.
Z = np.dot(X, Vt.T)
print >> out, 'orthogonally transformed points (call this Z):'
print >> out, Z
print >> out
Y = Z.T[:(nleaves - 1)].T
print >> out, 'projection of the points onto the axes of the leaf ellipsoid,'
print >> out, '(these are the first columns of Z; call this projected matrix Y):'
print >> out, Y
print >> out
# Show the inner products.
inner_products_of_columns = np.dot(Y.T, Y)
print >> out, "pairwise inner products of the columns of Y (that is, Y'Y)"
print >> out, inner_products_of_columns
print >> out
# Show other inner products.
inner_products_of_columns = np.dot(Y[:5].T, Y[:5])
print >> out, "pairwise inner products of the first few columns of Y"
print >> out, inner_products_of_columns
print >> out
# Extract the subset of points that define the points of articulation.
# Note that the origin is the centroid of the leaves.
R = X[nleaves:]
Y_leaves = Y[:nleaves]
W = np.dot(np.linalg.pinv(L), Y_leaves)
print >> out, 'leaf projection using pseudoinverse (first few rows of Y):'
print >> out, np.dot(L, W)
print >> out
print >> out, 'projection of points of articulation using pseudoinverse (remaining rows of Y):'
print >> out, np.dot(R, W)
print >> out
# Get all of the points in high dimensional space.
X = Euclid.edm_to_points(D_full)
# Get the MDS onto the lower dimensional space.
X = X.T[:(nleaves - 1)].T
assert np.allclose(sum(X, 0), 0)
print >> out, 'all points projected onto the first principal axes of the full ellipsoid:'
print >> out, X
print >> out
# Look at only the leaves in this space.
L = X[:nleaves]
L -= np.mean(L, 0)
print >> out, 'leaves projected onto the first principal axes of the full ellipsoid and then centered:'
print >> out, L
print >> out
# Re-project the leaves onto the axes of leaf ellipsoid.
D_leaves = Euclid.dccov_to_edm(np.dot(L, L.T))
Y = Euclid.edm_to_points(D_leaves)
print >> out, 'leaves further projected onto principal axes of their own ellipsoid:'
print >> out, Y
print >> out
# Try something else
D_all = Euclid.dccov_to_edm(np.dot(X, X.T))
Y = Euclid.edm_to_points(D_all).T[:(nleaves - 1)].T
print >> out, 'all points further projected onto their own principal axes of inertia:'
print >> out, Y
print >> out
# Try the same thing some more
D_again = Euclid.dccov_to_edm(np.dot(Y, Y.T))
Z = Euclid.edm_to_points(D_again).T[:(nleaves - 1)].T
print >> out, 'all points further projected onto their own principal axes of inertia (second iteration):'
print >> out, Z
print >> out
return out.getvalue().strip()
def process(ntaxa):
np.set_printoptions(linewidth=200)
out = StringIO()
# sample an xtree topology
xtree = TreeSampler.sample_agglomerated_tree(ntaxa)
# sample an xtree with exponentially distributed branch lengths
mu = 2.0
for branch in xtree.get_branches():
branch.length = random.expovariate(1/mu)
# convert the xtree to a FelTree so we can use the internal vertices
tree_string = xtree.get_newick_string()
tree = NewickIO.parse(tree_string, FelTree.NewickTree)
# get ordered ids and the number of leaves and some auxiliary variables
ordered_ids = get_ordered_ids(tree)
nleaves = len(list(tree.gen_tips()))
id_to_index = dict((myid, i) for i, myid in enumerate(ordered_ids))
# get the distance matrix relating all of the points
D_full = np.array(tree.get_full_distance_matrix(ordered_ids))
# Now do the projection so that
# the resulting points are in the subspace whose basis vectors are the axes of the leaf ellipsoid.
# First get the points such that the n rows in X are points in n-1 dimensional space.
X = Euclid.edm_to_points(D_full)
print >> out, 'points with centroid at origin:'
print >> out, X
print >> out
# Translate all of the points so that the origin is at the centroid of the leaves.
X -= np.mean(X[:nleaves], 0)
print >> out, 'points with centroid of leaves at origin:'
print >> out, X
print >> out
# Extract the subset of points that define the leaves.
L = X[:nleaves]
# Find the orthogonal transformation of the leaves onto their MDS axes.
# According to the python svd documentation, singular values are sorted most important to least important.
U, s, Vt = np.linalg.svd(L)
# Transform all of the points (including the internal vertices) according to this orthogonal transformation.
# The axes are now the axes of the Steiner circumscribed ellipsoid of the leaf vertices.
# I am using M.T[:k].T to get the first k columns of M.
Z = np.dot(X, Vt.T)
print >> out, 'orthogonally transformed points (call this Z):'
print >> out, Z
print >> out
Y = Z.T[:(nleaves-1)].T
print >> out, 'projection of the points onto the axes of the leaf ellipsoid,'
print >> out, '(these are the first columns of Z; call this projected matrix Y):'
print >> out, Y
print >> out
# Show the inner products.
inner_products_of_columns = np.dot(Y.T, Y)
print >> out, "pairwise inner products of the columns of Y (that is, Y'Y)"
print >> out, inner_products_of_columns
print >> out
# Show other inner products.
inner_products_of_columns = np.dot(Y[:5].T, Y[:5])
print >> out, "pairwise inner products of the first few columns of Y"
print >> out, inner_products_of_columns
print >> out
# Extract the subset of points that define the points of articulation.
# Note that the origin is the centroid of the leaves.
R = X[nleaves:]
Y_leaves = Y[:nleaves]
W = np.dot(np.linalg.pinv(L), Y_leaves)
print >> out, 'leaf projection using pseudoinverse (first few rows of Y):'
print >> out, np.dot(L, W)
print >> out
print >> out, 'projection of points of articulation using pseudoinverse (remaining rows of Y):'
print >> out, np.dot(R, W)
print >> out
# Get all of the points in high dimensional space.
X = Euclid.edm_to_points(D_full)
# Get the MDS onto the lower dimensional space.
X = X.T[:(nleaves-1)].T
assert np.allclose(sum(X, 0), 0)
print >> out, 'all points projected onto the first principal axes of the full ellipsoid:'
print >> out, X
print >> out
# Look at only the leaves in this space.
L = X[:nleaves]
L -= np.mean(L, 0)
print >> out, 'leaves projected onto the first principal axes of the full ellipsoid and then centered:'
print >> out, L
print >> out
# Re-project the leaves onto the axes of leaf ellipsoid.
D_leaves = Euclid.dccov_to_edm(np.dot(L, L.T))
Y = Euclid.edm_to_points(D_leaves)
print >> out, 'leaves further projected onto principal axes of their own ellipsoid:'
print >> out, Y
print >> out
# Try something else
D_all = Euclid.dccov_to_edm(np.dot(X, X.T))
Y = Euclid.edm_to_points(D_all).T[:(nleaves-1)].T
print >> out, 'all points further projected onto their own principal axes of inertia:'
print >> out, Y
print >> out
# Try the same thing some more
D_again = Euclid.dccov_to_edm(np.dot(Y, Y.T))
Z = Euclid.edm_to_points(D_again).T[:(nleaves-1)].T
print >> out, 'all points further projected onto their own principal axes of inertia (second iteration):'
print >> out, Z
print >> out
return out.getvalue().strip()
