def example_random_graph_2(N=100):
print('\n***mds.example_random_graph()***\n')
print('Here we explore the MDS embedding for a random binomial graph with'+\
'different edge probabilities.')
probs = [0.04, 0.05, 0.1, 0.2, 0.5, 1.0]
nums = [4, 5, 10, 20, 50, 100]
dims = [2, 3, 4, 5, 10, 20]
error = np.empty((len(dims), len(probs)))
fig = plt.figure()
for i in range(len(probs)):
p = probs[i]
D = multigraph.binomial(N, p)
for j in range(len(dims)):
dim = dims[j]
mds = MDS(D, dim=dim)
mds.initialize()
mds.stochastic(max_iters=100, approx=.3, lr=5)
mds.stochastic(max_iters=100, approx=.6, lr=10)
mds.stochastic(max_iters=100, approx=.9, lr=15)
mds.agd(min_step=1e-8)
error[j, i] = max(mds.cost, 1e-6)
for i in range(len(dims)):
plt.semilogy(error[i], label=f'dim {dims[i]}')
plt.ylabel('MDS stress')
plt.xlabel('average neighbors')
plt.xticks(range(len(nums)), nums)
plt.legend()
plt.tight_layout
plt.show()
def test_gd_lr(N=100, dim=2):
print('\n***mds.gd_lr()***')
Y = misc.disk(N, dim)
colors = misc.labels(Y)
D = multigraph.from_coordinates(Y, colors=colors)
title = 'recovering random coordinates for different learning rates'
mds = MDS(D, dim=dim, verbose=1, title=title)
mds.initialize()
for lr in [100, 10, 1, .1]:
mds.gd(lr=lr)
mds.figure(title=f'lr = {lr}')
mds.forget()
plt.show()
def example_disk_noisy(N=100, dim=2):
print('\n***mds.example_disk_noisy()***\n')
noise_levels = [0.001, 0.005, 0.01, 0.03, 0.07, 0.1, 0.15, 0.2, 0.7, 1.0]
stress = []
Y = misc.disk(N, dim)
D = distances.compute(Y)
for noise in noise_levels:
D_noisy = distances.add_noise(D, noise)
mds = MDS(D_noisy, dim, verbose=1, title=f'noise : {noise:0.2f}')
mds.initialize()
mds.optimize(algorithm='agd', max_iters=300, verbose=1)
stress.append(mds.ncost)
fig = plt.figure()
plt.loglog(noise_levels, stress, '.-')
plt.xlabel('noise level')
plt.ylabel('stress')
plt.title('Normalized MDS stress for various noise levels')
plt.show()
def example_fewer_edges(N=100, dim=2):
print('\n***mds.example_fewer_edges()***\n')
print(
'Here we explore the MDS embedding for a full graph as far way edges' +
'are removed')
title = 'MDS embedding for multiple proportion of edges'
X = misc.disk(N, dim)
colors = misc.labels(X)
D = multigraph.from_coordinates(X, colors=colors)
X0 = misc.disk(N, dim) * .5
for prop in [.99, .8, .6, .4, .2]:
DD = multigraph.remove_edges(D, proportion=prop)
mds = MDS(DD, dim=dim, verbose=1, title=title)
mds.initialize(X0=X0)
mds.stochastic(verbose=1, max_iters=300, approx=.99, lr=.5)
mds.adaptive(verbose=1, min_step=1e-6, max_iters=300)
mds.figure(title=f'proportion = {prop:0.1f}')
plt.show()
def example_random_graph(N=100, dim=2):
print('\n***mds.example_random_graph()***\n')
print('Here we explore the MDS embedding for a random binomial graph with'+\
'different edge probabilities.')
fig, axes = plt.subplots(2, 3)
#[ax.set_axis_off() for ax in axes.ravel()]
plt.tight_layout()
for p, ax in zip([0.01, 0.02, 0.03, 0.05, 0.1, 1.0], axes.ravel()):
D = multigraph.binomial(N, p)
mds = MDS(D, dim=dim, verbose=1)
mds.initialize()
mds.stochastic(max_iters=100, approx=.6, lr=.5)
mds.agd(min_step=1e-6)
mds.figureX(ax=ax, edges=True)
ax.set_xlabel(f'ave. neighs. : {int(100*p)}')
ax.set_title(f'stress = {mds.cost:0.2e}')
ax.set_yticks([])
ax.set_xticks([])
plt.show()
def embeddability_noise(ax=None):
print('\n**mds.embeddability_noise()')
N = 50
ncost = []
noise_list = [0] + 10**np.arange(-2, 1, 0.5)
X = misc.disk(N, 4)
DD = distances.compute(X)
for noise in noise_list:
D = DD * (1 + np.random.randn(N, N) * noise)
mds = MDS(D, dim=4, verbose=1)
mds.initialize()
mds.optimize()
ncost.append(mds.ncost)
if ax is None:
fig, ax = plt.subplots(1)
plot = True
else:
plot = False
ax.semilogx(noise_list, ncost)
if plot is True:
plt.show()
def example_stochastic(N=100, dim=2):
print('\n***mds.example_stochastic()***\n')
Y = misc.disk(N, dim)
colors = misc.labels(Y)
D = multigraph.from_coordinates(Y, colors=colors)
title = 'recovering random coordinates from full dissimilarity matrix ' +\
'using SGD, same learning rate, and different approx'
mds = MDS(D, dim=dim, verbose=1, title=title)
mds.initialize()
for approx in [1., .8, .6, .4, .2, .1]:
mds.stochastic(verbose=1,
lr=10.0,
min_step=1e-6,
approx=approx,
title=f'SGD using {approx} of edges')
mds.figure(title=f'approx = {approx}, time = {mds.H["time"]:0.2f}')
mds.forget()
plt.show()
def embeddability_dims(ax=None):
print('\n**mds.embeddability_dims()')
N = 50
ncost = []
dims = list(range(2, 50, 5))
#XX = misc.disk(N,20)
XX = misc.box(N, 20)
for dim in dims:
X = XX[:, 0:dim]
D = multigraph.coord2dict(X, weights='relative')
mds = MDS(D, dim=2, verbose=1)
mds.initialize()
mds.optimize()
ncost.append(mds.ncost)
if ax is None:
fig, ax = plt.subplots(1)
plot = True
else:
plot = False
ax.plot(dims, ncost)
if plot is True:
plt.show()
def disk_compare(N=100, dim=2): ###
print('\n***mds.disk_compare()***')
X = misc.disk(N, 2)
labels = misc.labels(X)
plt.figure()
plt.scatter(X[:, 0], X[:, 1], c=labels)
plt.title('original data')
plt.draw()
plt.pause(0.1)
D = distances.compute(X)
mds = MDS(D, dim=dim, verbose=1, title='disk experiments', labels=labels)
mds.initialize()
mds.figureX(title='initial embedding')
title = 'full gradient & agd'
mds.optimize(algorithm='agd', verbose=2, label=title)
mds.figureX(title=title)
mds.figureH(title=title)
mds.forget()
title = 'approx gradient & gd'
mds.approximate(algorithm='gd', verbose=2, label=title)
mds.figureX(title=title)
mds.figureH(title=title)
mds.forget()
title = 'combine'
mds.approximate(algorithm='gd', verbose=2, label=title)
mds.optimize(verbose=2, label=title, max_iters=10)
mds.figureX(title=title)
mds.figureH(title=title)
plt.show()
def example_weights(N=100, dim=2):
print('\n***mds.example_weights()***\n')
print('Here we explore the MDS embedding for a full graph for different' +
'weights')
title = 'MDS embedding for multiple weights'
X = misc.disk(N, dim)
colors = misc.labels(X)
X0 = misc.disk(N, dim)
D = multigraph.from_coordinates(X, colors=colors)
mds = MDS(D, dim=dim, verbose=1, title=title)
mds.initialize(X0=X0)
mds.stochastic(verbose=1, max_iters=50, approx=.6, lr=50)
mds.adaptive(verbose=1, min_step=1e-6, max_iters=300)
mds.figure(title=f'absolute weights')
multigraph.set_weights(D, scaling=.5)
mds = MDS(D, dim=dim, verbose=1, title=title)
mds.initialize(X0=X0)
mds.stochastic(verbose=1, max_iters=50, approx=.6, lr=50)
mds.adaptive(verbose=1, min_step=1e-6, max_iters=300)
mds.figure(title=f'1/sqrt(Dij) weights')
multigraph.set_weights(D, scaling=1)
mds = MDS(D, dim=dim, verbose=1, title=title)
mds.initialize(X0=X0)
mds.stochastic(verbose=1, max_iters=50, approx=.6, lr=50)
mds.adaptive(verbose=1, min_step=1e-6, max_iters=300)
mds.figure(title=f'1/Dij weights')
multigraph.set_weights(D, scaling=2)
mds = MDS(D, dim=dim, verbose=1, title=title)
mds.initialize(X0=X0)
mds.stochastic(verbose=1, max_iters=50, approx=.6, lr=50)
mds.adaptive(verbose=1, min_step=1e-6, max_iters=300)
mds.figure(title=f'relative weights')
plt.show()