npts = 100
XX, YY = np.meshgrid(np.linspace(xmins[0], xmaxs[0], npts),
np.linspace(xmins[1], xmaxs[1], npts))
data = np.column_stack((XX.ravel(), YY.ravel(), np.zeros((npts**2, D - 2))))
input = np.zeros((data.shape[0], 0))
mask = np.ones_like(data, dtype=bool)
tag = None
lls = hmm.observations.log_likelihoods(data, input, mask, tag)
plt.figure(figsize=(6, 6))
for k in range(K):
plt.contour(XX,
YY,
np.exp(lls[:, k]).reshape(XX.shape),
cmap=white_to_color_cmap(colors[k % len(colors)]))
plt.plot(x[z == k, 0], x[z == k, 1], 'o', mfc=colors[k], mec='none', ms=4)
plt.plot(x[:, 0], x[:, 1], '-k', lw=2, alpha=.5)
plt.xlabel("$x_1$")
plt.ylabel("$x_2$")
plt.title("Observation Distributions")
plt.tight_layout()
if save_figures:
plt.savefig("lds_3.pdf")
# In[16]:
# Simulate from the HMM fit
def plot_place_fields(results, pos, center, radius, data, figdir='.'):
"""
Plot the observation vector associated with a latent state
"""
model = results.samples
model.relabel_by_usage()
N_used = results.N_used[-1]
lmbdas = model.rates[:N_used, :]
stateseq = model.stateseqs[0]
occupancy = model.state_usages
# Plot a figure for each latent state
N_colors = 9
colors = brewer2mpl.get_map('Set1', 'qualitative', N_colors).mpl_colors
# State distributions
dists = []
for s in xrange(N_used):
cd = CircularDistribution(center, radius)
cd.fit_xy(pos[stateseq == s, 0], pos[stateseq == s, 1])
dists.append(cd)
# Plot the log likelihood as a function of iteration
fig = create_figure((5, 4))
plt.figtext(0.05 / 5.0, 3.8 / 4.0, "A")
toplot = [0, 13, 28, 38]
for i, c in enumerate([0, 13, 28, 38]):
left = 1.25 * i + 0.05
print "Plotting cell ", c
color = colors[np.mod(c, N_colors)]
cmap = white_to_color_cmap(color)
# Compute the inferred place field
inf_place_field = dists[0] * lmbdas[0, c] * occupancy[0]
for s in range(1, N_used):
inf_place_field += dists[s] * lmbdas[s, c] * occupancy[s]
# inf_place_field = sum([d*(l*o) for d,l,o in zip(dists, lmbdas[c,:], occupancy)])
spks = np.array(data[:, c] > 0).ravel()
true_place_field = CircularDistribution(center, radius)
true_place_field.fit_xy(pos[spks, 0], pos[spks, 1])
# Plot the locations of this state
ax = create_axis_at_location(fig,
left,
2.65,
1.15,
1.15,
transparent=True)
remove_plot_labels(ax)
# Plot the empirical location distribution
inf_place_field.plot(ax=ax,
cmap=cmap,
plot_data=True,
plot_colorbar=False)
ax.set_title('Inf. Place Field %d' % (c + 1), fontdict={'fontsize': 9})
# Now plot the true place field
ax = create_axis_at_location(fig,
left,
1.25,
1.15,
1.15,
transparent=True)
remove_plot_labels(ax)
true_place_field.plot(ax=ax,
cmap=cmap,
plot_data=True,
plot_colorbar=False)
ax.set_title('True Place Field %d' % (c + 1), fontdict={'fontsize': 9})
# Plot the KL divergence histogram
kls = np.zeros(model.N)
tvs = np.zeros(model.N)
for c in xrange(model.N):
# Compute the inferred place field
inf_place_field = dists[0] * lmbdas[0, c] * occupancy[0]
for s in range(1, N_used):
inf_place_field += dists[s] * lmbdas[s, c] * occupancy[s]
# inf_place_field = sum([d*(l*o) for d,l,o in zip(dists, lmbdas[c,:], occupancy)])
spks = np.array(data[:, c] > 0).ravel()
true_place_field = CircularDistribution(center, radius)
true_place_field.fit_xy(pos[spks, 0], pos[spks, 1])
kls[c] = compute_place_field_KL(inf_place_field, true_place_field)
tvs[c] = compute_place_field_TV(inf_place_field, true_place_field)
bin_centers = np.arange(0.006, 0.0141, 0.001)
bin_width = 0.001
bin_edges = np.concatenate(
(bin_centers - bin_width / 2.0, [bin_centers[-1] + bin_width / 2.0]))
ax = create_axis_at_location(fig, 0.5, 0.5, 4., .5, transparent=True)
ax.hist(tvs, bins=bin_edges, facecolor=allcolors[1])
ax.set_xlim(0.005, 0.015)
ax.set_xticks(bin_centers)
ax.set_xticklabels([
"{0:.3f}".format(bc) if i % 2 == 0 else ""
for i, bc in enumerate(bin_centers)
])
ax.set_xlabel("$TV(p_{inf}, p_{true})$")
ax.set_yticks(np.arange(17, step=4))
ax.set_ylabel("Count")
plt.figtext(0.05 / 5.0, 1.1 / 4.0, "B")
print "TVs of plotted cells: "
print tvs[toplot]
# fig.savefig(os.path.join(figdir,'figure8.pdf'))
fig.savefig(os.path.join(figdir, 'figure8.png'))
plt.show()
def make_figure(results, S_train, pos_train, S_test, pos_test, center, radius, figdir="."):
model = results.samples
model.relabel_by_usage()
N_used = results.N_used[-1]
stateseq = model.stateseqs[0]
occupancy = model.state_usages
T_test = S_test.shape[0]
t_test = np.arange(T_test) * 0.25
fig = create_figure(figsize=(5,3))
# Plot the centers of the latent states
ax = create_axis_at_location(fig, .05, 1.55, 1.15, 1.15, transparent=True)
plt.figtext(0.05/5, 2.8/3, "A")
remove_plot_labels(ax)
circle = matplotlib.patches.Circle(xy=[0,0],
radius= radius,
linewidth=1,
edgecolor="k",
facecolor="white")
ax.add_patch(circle)
plt.figtext(1.2/5, 2.8/3, "B")
for k in xrange(N_used):
relocc = occupancy[k] / np.float(np.amax(occupancy))
cd = CircularDistribution(center, radius)
cd.fit_xy(pos_train[stateseq==k,0], pos_train[stateseq==k,1])
# import pdb; pdb.set_trace()
rm, thm = cd.mean
xm,ym = convert_polar_to_xy(np.array([[rm, thm]]), [0,0])
ax.plot(xm,ym,'o',
markersize=relocc*6,
markerfacecolor='k',
markeredgecolor='k',
markeredgewidth=1)
ax.set_xlim(-radius, radius)
ax.set_ylim(-radius, radius)
ax.set_title('All states', fontdict={'fontsize' : 9})
# Plot a figure for each latent state
for k in xrange(3):
left = 1.25 * (k+1) + 0.05
color = allcolors[k]
cmap = white_to_color_cmap(color)
# Plot the locations of this state
ax = create_axis_at_location(fig, left, 1.55, 1.15, 1.15, transparent=True)
remove_plot_labels(ax)
# Plot the empirical location distribution
cd = CircularDistribution(center, radius)
cd.fit_xy(pos_train[stateseq==k,0], pos_train[stateseq==k,1])
cd.plot(ax=ax, cmap=cmap, plot_data=True, plot_colorbar=False)
ax.set_title('State %d (%.1f%%)' % (k+1, 100.*occupancy[k]),
fontdict={'fontsize' : 9})
# Bottom: Plot the true and predicted locations for heldout data
plt.figtext(0.05/5, 1.55/3, "C")
epdf = estimate_pos(model,
S_train, pos_train,
S_test, pos_test,
center, radius)
# Compute the mean trajectory
mean_location = np.zeros_like(pos_test)
for t in range(T_test):
cd = CircularDistribution(center, radius, pdf=epdf[t,:])
mean_location[t,:] = convert_polar_to_xy(np.atleast_2d(cd.mean), center)
# Convert estimates to x,y and compute mean squared error
sqerr = np.sqrt((mean_location - pos_test)**2).mean(axis=1)
mse = sqerr.mean(axis=0)
stdse = sqerr.std(axis=0)
print "MSE: %f \pm %f" % (mse, stdse)
ax_y = create_axis_at_location(fig, 0.6, 0.4, 3.8, 0.5, box=True, ticks=True)
ax_y.plot(t_test, pos_test[:,1] - center[1], '-k', lw=1)
ax_y.plot(t_test, mean_location[:,1] - center[1], '-', color=allcolors[1])
ax_y.set_ylabel('$y(t)$ [cm]', fontsize=9)
ax_y.set_ylim([-radius,radius])
ax_y.set_xlabel('$t$ [s]', fontsize=9)
ax_y.set_xlim(0,T_test*0.25)
ax_y.tick_params(axis='both', which='major', labelsize=9)
ax_x = create_axis_at_location(fig, 0.6, 1., 3.8, 0.5, box=True, ticks=True)
ax_x.plot(t_test, pos_test[:,0] - center[0], '-k', lw=1)
ax_x.plot(t_test, mean_location[:,0] - center[0], '-', color=allcolors[1])
ax_x.set_ylabel('$x(t)$ [cm]', fontsize=9)
ax_x.set_ylim([-radius,radius])
ax_x.set_xticks(ax_y.get_xticks())
ax_x.set_xticklabels([])
ax_x.tick_params(axis='both', which='major', labelsize=9)
ax_x.set_xlim(0,T_test*0.25)
fig.savefig(os.path.join(figdir, 'figure5.pdf'))
fig.savefig(os.path.join(figdir, 'figure5.png'))
plt.show()
lim = .85 * abs(y).max()
XX, YY = np.meshgrid(np.linspace(-lim, lim, 100), np.linspace(-lim, lim, 100))
data = np.column_stack((XX.ravel(), YY.ravel()))
input = np.zeros((data.shape[0], 0))
mask = np.ones_like(data, dtype=bool)
tag = None
lls = true_hmm.observations.log_likelihoods(data, input, mask, tag)
# In[36]:
plt.figure(figsize=(6, 6))
for k in range(K):
plt.contour(XX,
YY,
np.exp(lls[:, k]).reshape(XX.shape),
cmap=white_to_color_cmap(colors[k]))
plt.plot(y[z == k, 0], y[z == k, 1], 'o', mfc=colors[k], mec='none', ms=4)
plt.plot(y[:, 0], y[:, 1], '-k', lw=1, alpha=.25)
plt.xlabel("$y_1$")
plt.ylabel("$y_2$")
plt.title("Observation Distributions")
if save_figures:
plt.savefig("hmm_1.pdf")
# In[35]:
# Plot the data and the smoothed data
lim = 1.05 * abs(y).max()
plt.figure(figsize=(8, 6))
