def plot_gaussian_mixture(Mu, Sigma, weights=None, x=None, y=None):
if x == None:
x = np.arange(0, 1, 0.01)
if y == None:
y = np.arange(-0.5, 1.2, 0.01)
if len(Mu) == len(Sigma) == len(weights):
pass
else:
print("Error: Mu, Sigma and weights must have the same dimension")
return
X, Y = np.meshgrid(x, y)
Pos = np.dstack((X, Y))
Z = 0
for i in range(len(Mu)):
Z = Z + weights[i] * multivariate_normal(Mu[i].ravel(),
Sigma[i]).pdf(Pos)
fig = plt.figure(figsize=(12, 7))
ax = fig.gca(projection='3d')
ax.plot_surface(X, Y, Z, cmap="copper", lw=0.5, rstride=1, cstride=1)
ax.set_xlabel('X axis')
ax.set_ylabel('Y axis')
ax.set_zlabel('Z axis')
plt.tight_layout()
pml.savefig('mixgaussSurface.pdf')
plt.show()
def plot(self, n_row, n_col, file_name):
'''
Plots the mean of each Bernoulli distribution as an image.
Parameters
----------
n_row : int
The number of rows of the figure
n_col : int
The number of columns of the figure
file_name : str
The path where the figure will be stored
'''
if n_row * n_col != len(self.mixing_coeffs):
raise TypeError('The number of rows and columns does not match with the number of component distribution.')
fig, axes = plt.subplots(n_row, n_col)
for (coeff, mean), ax in zip(zip(self.mixing_coeffs, self.probs), axes.flatten()):
ax.imshow(mean.reshape(28, 28), cmap=plt.cm.gray)
ax.set_title("%1.2f" % coeff)
ax.axis("off")
fig.tight_layout(pad=1.0)
pml.savefig(f"{file_name}.pdf")
plt.show()
def make_convergence_plots():
X = np.random.randn(2, 1000)
X = X - np.mean(X, axis=1).reshape(2, 1)
theta_init = np.array([[1], [-1]])
sf = 3
#theta_trajectory_steepest = theta_init.dot(np.ones((1, 10000)))
theta_trajectory_steepest = theta_init.dot(np.ones((1, 1000)))
theta_trajectory_natural = theta_trajectory_steepest.copy()
L_trajectory_steepest = np.zeros(
(1, theta_trajectory_steepest.shape[1] - 1))
L_trajectory_natural = np.zeros_like(L_trajectory_steepest)
eps_steep = 1 / (sf**2) / 5
eps_nat = eps_steep * sf**2
for i in range(1, theta_trajectory_steepest.shape[1]):
L, dL, G = L_dL_G(theta_trajectory_steepest[:, i - 1], X, sf)
L_trajectory_steepest[:, i - 1] = L
theta_trajectory_steepest[:,
i] = theta_trajectory_steepest[:, i -
1] - eps_steep * dL
L, dL, G = L_dL_G(theta_trajectory_natural[:, i - 1], X, sf)
L_trajectory_natural[:, i - 1] = L
theta_trajectory_natural[:,
i] = theta_trajectory_natural[:, i -
1] - eps_nat * (
np.linalg.
lstsq(
G,
dL)[0])
plt.plot(theta_trajectory_steepest[0, :].T,
theta_trajectory_steepest[1, :].T,
'+r',
label="Steepest descent")
plt.plot(theta_trajectory_natural[0, :].T,
theta_trajectory_natural[1, :].T,
'xb',
label="Natural gradient descent")
plt.xlabel(r"$\theta_1$")
plt.ylabel(r"$\theta_2$")
plt.title("Parameter trajectories")
plt.legend()
#pml.savefig("DescentPathsSteepestNGDescent.pdf")
pml.savefig("natgrad_descent_params.pdf")
plt.show()
plt.loglog(L_trajectory_steepest.flatten(), '+r', label="Steepest descent")
plt.loglog(L_trajectory_natural.flatten(),
'xb',
label="Natural gradient descent")
plt.xlabel("Number of update steps")
plt.ylabel("KL divergence")
plt.title("KL divergence vs. update step")
plt.legend()
#pml.savefig("KLDivergenceSteepestNGDescent.pdf")
pml.savefig("natgrad_descent_kl.pdf")
plt.show()
def plot_pca_vectors(a, b, vectors, w):
mu_a, mu_b = a.mean(axis=0).reshape(-1, 1), b.mean(axis=0).reshape(-1, 1)
mid_point = (mu_a + mu_b) / 2
vector = vectors[:, 0]
slope_pca = vector[1] / vector[0]
c_pca = mid_point[1] - slope_pca * mid_point[0]
slope = w[1] / w[0]
c = mid_point[1] - slope * mid_point[0]
x = np.linspace(xmin + 1, xmax + 1, 100)
z = np.linspace(xmin + 1, xmax + 1, 100)
plt.figure()
plt.xlim(xmin, xmax)
plt.ylim(ymin, ymax)
plt.plot(a[:, 0], a[:, 1], 'b.', b[:, 0], b[:, 1], 'r+')
#plt.plot(x, slope*x + c)
plt.plot(z, slope_pca * z + c_pca)
#plt.plot(mu_a, mu_b, 'black')
plt.legend(['Male', 'Female', 'PCA vector'])
#plt.legend(['Male', 'Female', 'FisherLDA vector', 'PCA vector', 'Means'])
pml.savefig("fisher_lda_lines_pca.pdf")
plt.show()
def callback(X_next, Y_next, i):
global X_sample, Y_sample
# Plot samples, surrogate function, noise-free objective and next sampling location
#plt.subplot(n_iter, 2, 2 * i + 1)
plt.figure()
plot_approximation(gpr,
X,
Y,
X_sample,
Y_sample,
X_next,
show_legend=i == 0)
plt.title(f'Iteration {i+1}')
if save_figures: pml.savefig('bayes-opt-surrogate-{}.pdf'.format(i + 1))
plt.show()
plt.figure()
#plt.subplot(n_iter, 2, 2 * i + 2)
plot_acquisition(X,
expected_improvement(X, X_sample, Y_sample, gpr),
X_next,
show_legend=i == 0)
if save_figures: pml.savefig('bayes-opt-acquisition-{}.pdf'.format(i + 1))
plt.show()
# Add sample to previous samples
X_sample = np.append(X_sample, np.atleast_2d(X_next), axis=0)
Y_sample = np.append(Y_sample, np.atleast_2d(Y_next), axis=0)
def MakePlot(ypreds, SaveN, Title, lowerb=None, upperb=None):
#Function for creating and saving plots
fig, ax = plt.subplots()
ax.scatter(xtrain,
ytrain,
s=140,
facecolors='none',
edgecolors='r',
label='training data')
#plt.ylim([-10,80])
#plt.xlim([-8,8])
Errlogi = lowerb is not None or upperb is not None #Determines where we will be plotting error bars as well
if Errlogi:
errspacing = [
int(round(s)) for s in np.linspace(0, xtest.shape[0] - 1, 30)
]
ax.errorbar(xtest[errspacing],
ypreds[errspacing, 0],
yerr=[lowerb[errspacing], upperb[errspacing]])
for j in range(ypreds.shape[1]):
ax.plot(xtest,
ypreds[:, j],
color='k',
linewidth=2.0,
label='prediction',
alpha=vis)
if Errlogi:
plt.legend(loc=2)
plt.title(Title)
pml.savefig(SaveN + '.pdf')
def MakeG(Probs,SaveN):
fig, ax = plt.subplots()
ax.bar(X, Probs, align='center')
plt.xlim([min(X) - .5, max(X) + .5])
plt.xticks(X)
plt.yticks(np.linspace(0, 1, 5))
pml.savefig(SaveN)
def main():
cmap = create_colormap()
X = np.loadtxt(rawdata)
# Normalise data
X = (X - X.mean(axis=0)) / (X.std(axis=0))
mu1 = np.array([-1.5, 1.5])
mu2 = np.array([1.5, -1.5])
# Initial configuration
Sigma1 = np.identity(2) * 0.1
Sigma2 = np.identity(2) * 0.1
pi = [0.5, 0.5]
mu = [mu1, mu2]
Sigma = [Sigma1, Sigma2]
res = gmm_lib.apply_em(X, pi, mu, Sigma)
# Create grid-plot
hist_index = [0, 10, 25, 30, 35, 40]
fig, ax = plt.subplots(2, 3)
ax = ax.ravel()
for ix, axi in zip(hist_index, ax):
pi, mu, Sigma = res["coeffs"][ix]
r = res["rvals"][ix]
if ix == 0:
r = np.ones_like(r)
colors = cmap if ix > 0 else "Dark2"
gmm_lib.plot_mixtures(X, mu, pi, Sigma, r, cmap=colors, ax=axi)
axi.set_title("Iteration {ix}".format(ix=ix))
plt.tight_layout()
pml.savefig('gmm_faithful.pdf', dpi=300)
plt.show()
def plot_data(a, b):
plt.figure()
plt.plot(a[:, 0], a[:, 1], 'b.', b[:, 0], b[:, 1], 'r+')
#plt.plot(mu_a, mu_b, 'black')
plt.legend(['Male', 'Female', 'Means'])
pml.savefig("fisher_lda_data.pdf")
plt.show()
def plot_sigma_vector(Mu, Sigma):
n = len(Mu)
plt.figure(figsize=(12, 7))
for i in range(n):
plot_sigma_levels(Mu[i], Sigma[i])
plt.tight_layout()
pml.savefig('mixgaussSurface.pdf')
plt.show()
def plot_signals(signals, suptitle, file_name):
plt.figure(figsize=(8, 4))
for i, signal in enumerate(signals, 1):
plt.subplot(n_signals, 1, i)
plt.plot(signal)
plt.xlim([0, N])
plt.tight_layout()
plt.suptitle(suptitle)
plt.subplots_adjust(top=0.85)
pml.savefig(f'{file_name}.pdf')
plt.show()
def plot_samples(S, title, file_name):
min_x, max_x = -4, 4
min_y, max_y = -3, 3
plt.scatter(S[:, 0], S[:, 1], marker='o', s=16)
plt.hlines(0, min_x, max_x, linewidth=2)
plt.vlines(0, min_y, max_y, linewidth=2)
plt.xlim(min_x, max_x)
plt.ylim(min_y, max_y)
plt.title(title)
pml.savefig(f'{file_name}.pdf')
plt.show()
def plot_intermediate_steps_single(method, fwd_func, intermediate_steps, xtest, mu_hist, Sigma_hist):
"""
Plot the intermediate steps of the training process, each one in a different plot.
"""
for step in intermediate_steps:
W_step, SW_step = mu_hist[step], Sigma_hist[step]
x_step, y_step = x[:step], y[:step]
_, axi = plt.subplots()
plot_mlp_prediction(key, x_step, y_step, xtest, fwd_func, W_step, SW_step, axi)
axi.set_title(f"step={step}")
plt.tight_layout()
pml.savefig(f'{method}-mlp-step-{step}.pdf')
def MakeGraph(Data,SaveName):
prior = MakeBeta(Data['prior'])(x)
likelihood = MakeBeta(Data['lik'])(x)
posterior = MakeBeta(Data['post'])(x)
fig, ax = plt.subplots()
ax.plot(x, prior, 'r', label=MakeLabel(Data, "prior"), linewidth=2.0)
ax.plot(x, likelihood, 'k--', label=MakeLabel(Data, "lik"), linewidth=2.0)
ax.plot(x, posterior, 'b--', label=MakeLabel(Data, "post"), linewidth=2.0)
ax.legend(loc='upper left', shadow=True)
pml.savefig(SaveName)
plt.show()
def demo(priorVar, plot_num):
np.random.seed(1)
colors = dict(mcolors.BASE_COLORS, **mcolors.CSS4_COLORS)
N = 10 # number of interior observed points
D = 150 # number of points we evaluate function at
xs = np.linspace(0, 1, D)
allNdx = np.arange(0, D, 1)
perm = np.random.permutation(D)
obsNdx = perm[:N]
obsNdx = np.concatenate((np.array([0]), obsNdx, np.array([D - 1])))
Nobs = len(obsNdx)
hidNdx = np.setdiff1d(allNdx, obsNdx)
Nhid = len(hidNdx)
xobs = np.random.randn(Nobs)
obsNoiseVar = 1
y = xobs + np.sqrt(obsNoiseVar) * np.random.randn(Nobs)
L = (0.5 * scipy.sparse.diags([-1, 2, -1], [0, 1, 2],
(D - 2, D))).toarray()
Lambda = 1 / priorVar
L = L * Lambda
L1 = L[:, hidNdx]
L2 = L[:, obsNdx]
B11 = np.dot(np.transpose(L1), L1)
B12 = np.dot(np.transpose(L1), L2)
B21 = np.transpose(B12)
mu = np.zeros(D)
mu[hidNdx] = -np.dot(np.dot(np.linalg.inv(B11), B12), xobs)
mu[obsNdx] = xobs
inverseB11 = np.linalg.inv(B11)
Sigma = np.zeros((D, D))
# https://stackoverflow.com/questions/22927181/selecting-specific-rows-and-columns-from-numpy-array/22927889#22927889
Sigma[hidNdx[:, None], hidNdx] = inverseB11
plt.figure()
plt.plot(obsNdx, xobs, 'bo', markersize=10)
plt.plot(allNdx, mu, 'r-')
S2 = np.diag(Sigma)
upper = (mu + 2 * np.sqrt(S2))
lower = (mu - 2 * np.sqrt(S2))
plt.fill_between(allNdx, lower, upper, alpha=0.2)
for i in range(0, 3):
fs = np.random.multivariate_normal(mu, Sigma)
plt.plot(allNdx, fs, 'k-', alpha=0.7)
plt.title(f'prior variance {priorVar:0.2f}')
pml.savefig(f'gaussian_interpolation_1d_{plot_num}.pdf')
def sample_plot_mh(x0, τ, π, μ, σ, n_iterations, xmin, xmax):
x_hist = metropolis_sample(x0, τ, π, μ, σ, n_iterations)
fig = plt.figure()
axs = plt.axes(projection="3d")
plot_gmm_3d_trace(x_hist, π, μ, σ, f"MH with $N(0,{τ}^2)$ proposal", xmin, xmax, axs)
style3d(axs, 1.5, 1, 0.8)
plt.subplots_adjust(left=0.001, bottom=0.208)
pml.savefig(f"mh_trace_{τ}tau.pdf", pad_inches=0, bbox_inches="tight")
fig, axs = plt.subplots()
sm.graphics.tsa.plot_acf(x_hist, lags=45, alpha=None, title=f"MH with $N(0,{τ}^2)$ proposal", ax=axs)
pml.savefig(f"mh_autocorrelation_{τ}tau.pdf")
def main():
cmap = create_colormap()
colors = ["tab:red", "tab:blue"]
observations = np.loadtxt("../data/faithful.txt")
# Normalize data
observations = (observations -
observations.mean(axis=0)) / (observations.std(axis=0))
# Initial configuration
mixing_coeffs = jnp.array([0.5, 0.5])
means = jnp.vstack([jnp.array([-1.5, 1.5]), jnp.array([1.5, -1.5])])
covariances = jnp.array([jnp.eye(2) * 0.1, jnp.eye(2) * 0.1])
gmm = GMM(mixing_coeffs, means, covariances)
num_epochs = 2000
history = gmm.fit_sgd(jnp.array(observations),
batch_size=observations.shape[0],
num_epochs=num_epochs)
ll_hist, mix_dist_probs_hist, comp_dist_loc_hist, comp_dist_cov_hist, responsibility_hist = history
# Create grid-plot
hist_index = [0, 10, 125, 320, 1450, 1999]
fig, ax = plt.subplots(2, 3)
ax = ax.ravel()
for idx, axi in zip(hist_index, ax):
means = comp_dist_loc_hist[idx]
covariances = comp_dist_cov_hist[idx]
responsibility = responsibility_hist[idx]
if idx == 0:
responsibility = jnp.ones_like(responsibility)
color_map = cmap if idx > 0 else "Dark2"
gmm.plot(observations,
means,
covariances,
responsibility[:, 0],
cmap=color_map,
colors=colors,
ax=axi)
axi.set_title(f"Iteration {idx}")
plt.tight_layout()
pml.savefig('../figures/gmm_faithful.pdf')
plt.show()
def plt_patterns(patterns, ndisplay=None, figsize=30, name=None):
assert patterns.nimages >= ndisplay, "number of images in the datset cannot \
be less than number of images to be displayed"
if not ndisplay:
ndisplay = self.nimages
fig, axs = plt.subplots(1, ndisplay, figsize=(figsize, figsize * ndisplay))
fig.suptitle(f'{name}', fontsize=16, y=0.55)
for i in range(ndisplay):
axs[i].imshow(patterns[:, i * patterns.width:(i + 1) * patterns.width],
cmap="Greys")
pml.savefig(f'{name}.pdf')
plt.show()
def plot_var_em_bound(x, true_log_ll, lower_bound, title):
plt.plot(x, true_log_ll, '-b', linewidth=3)
plt.text(2.5, y1[-1] + 0.02, 'true log-likelihood', fontweight='bold')
plt.plot(x, lower_bound, ':r', linewidth=3)
plt.text(2.8, 0.9, 'lower bound', fontweight='bold')
plt.xlim([0, 4])
plt.ylim([0.5, np.max(y1) + 0.05])
plt.xlabel('training time', fontweight='bold')
plt.xticks([])
plt.yticks([])
pml.savefig(f'{title}.pdf', dpi=300)
plt.show()
def MakeDirSampleFig(alpha):
AlphaVec = np.repeat(alpha, NSamples)
samps = np.random.dirichlet(AlphaVec, NSamples)
fig, ax = plt.subplots(NSamples)
fig.suptitle('Samples from Dir (alpha=' + str(alpha) + ')', y=1)
fig.tight_layout()
for i in range(NSamples):
ax[i].bar(X, samps[i, :], align='center')
ax[i].set_ylim([0, 1])
ax[i].yaxis.set_ticks([0, .5, 1])
ax[i].set_xlim([min(X) - .5, max(X) + .5])
plt.draw()
SaveN = "dirSample" + str(int(np.round(10 * alpha))) + ".pdf"
pml.savefig(SaveN)
def main():
cmap = create_colormap()
colors = ["tab:red", "tab:blue"]
url = 'https://raw.githubusercontent.com/probml/probml-data/main/data/faithful.txt'
response = requests.get(url)
rawdata = BytesIO(response.content)
observations = np.loadtxt(rawdata)
# Normalize data
observations = (observations - observations.mean(axis=0)) / (observations.std(axis=0))
# Initial configuration
mixing_coeffs = jnp.array([0.5, 0.5])
means = jnp.vstack([jnp.array([-1.5, 1.5]),
jnp.array([1.5, -1.5])])
covariances = jnp.array([jnp.eye(2) * 0.1,
jnp.eye(2) * 0.1])
gmm = GMM(mixing_coeffs, means, covariances)
num_of_iters = 50
history = gmm.fit_em(observations, num_of_iters=num_of_iters)
ll_hist, mix_dist_probs_hist, comp_dist_loc_hist, comp_dist_cov_hist, responsibility_hist = history
# Create grid-plot
hist_index = [0, 10, 25, 30, 35, 40]
fig, ax = plt.subplots(2, 3)
ax = ax.ravel()
for idx, axi in zip(hist_index, ax):
means = comp_dist_loc_hist[idx]
covariances = comp_dist_cov_hist[idx]
responsibility = responsibility_hist[idx]
if idx == 0:
responsibility = np.ones_like(responsibility)
color_map = cmap if idx > 0 else "Dark2"
gmm.plot(observations, means, covariances, responsibility, cmap=color_map, colors=colors, ax=axi)
axi.set_title(f"Iteration {idx}")
plt.tight_layout()
pml.savefig('gmm_faithful.pdf')
plt.show()
def plot_proj(name, argument):
plt.figure()
if name == 'pca':
Xproj_pca_male = argument[:nMale]
Xproj_pca_female = argument[nMale:nFemale]
plt.hist(Xproj_pca_male, color='red', ec='black')
plt.hist(Xproj_pca_female, color='blue', ec='black')
plt.title('Projection of points onto PCA vector')
pml.savefig("fisher_lda_pca.pdf")
plt.show()
else:
Xproj_fish_male = argument[:nMale]
Xproj_fish_female = argument[nMale:nFemale]
plt.hist(Xproj_fish_male, color='red', ec='black')
plt.hist(Xproj_fish_female, color='blue', ec='black')
plt.title('Projection of points onto Fisher vector')
pml.savefig("fisher_lda_flda.pdf")
plt.show()
def plot_data(data, assignments, title, data_type):
fig = plt.figure()
ax = fig.add_subplot(1, 1, 1)
ax.plot(data[assignments == 0, 0],
data[assignments == 0, 1],
'o',
color='r')
ax.plot(data[assignments == 1, 0],
data[assignments == 1, 1],
'o',
color='b')
ax.set_xlabel('x')
ax.set_ylabel('y')
ax.axis('square')
ax.grid(True)
ax.set_title(title)
plt.tight_layout()
pml.savefig(f"{data_type}_{title.replace(' ', '_')}.pdf")
def plot_surface(clf, X, y, filename, xnames, ynames):
n_classes = 3
plot_step = 0.02
markers = ['o', 's', '^']
plt.figure()
x_min, x_max = X[:, 0].min() - 1, X[:, 0].max() + 1
y_min, y_max = X[:, 1].min() - 1, X[:, 1].max() + 1
xx, yy = np.meshgrid(np.arange(x_min, x_max, plot_step),
np.arange(y_min, y_max, plot_step))
plt.tight_layout(h_pad=0.5, w_pad=0.5, pad=2.5)
Z = clf.predict(np.c_[xx.ravel(), yy.ravel()])
Z = Z.reshape(xx.shape)
#cmap=plt.cm.jet
#cmap=plt.cm.RdYlBu
#cmap = ListedColormap(['orange', 'green', 'purple'])
cmap = ListedColormap(['blue', 'orange', 'green'])
cs = plt.contourf(xx, yy, Z, cmap=cmap, alpha=0.5)
#plot_colors = "ryb"
#plot_colors = "byg"
plot_colors = [cmap(i) for i in range(3)]
plt.xlabel(xnames[0])
plt.ylabel(xnames[1])
# Plot the training points
for i, color, marker in zip(range(n_classes), plot_colors, markers):
idx = np.where(y == i)
plt.scatter(X[idx, 0],
X[idx, 1],
label=ynames[i],
edgecolor='black',
color=color,
s=50,
cmap=cmap,
marker=marker)
plt.legend()
pml.savefig(filename)
plt.show()
def main():
np.random.seed(12)
data_dim = 8
n_data = 10
threshold_missing = 0.5
mu = np.random.randn(data_dim, 1)
sigma = make_spd_matrix(
n_dim=data_dim) # Generate a random positive semi-definite matrix
# test if the matrix is positive definite
# print(is_pos_def(sigma))
x_full = gauss.gauss_sample(mu, sigma, n_data)
missing = np.random.rand(n_data, data_dim) < threshold_missing
x_miss = np.copy(x_full)
x_miss[missing] = np.nan
x_imputed = gauss.gauss_impute(mu, sigma, x_miss)
#Create a matrix from x_miss by replacing the NaNs with 0s to display the hinton_diagram
xmiss0 = np.copy(x_miss)
for g in np.argwhere(np.isnan(x_miss)):
xmiss0[g[0], g[1]] = 0
plot_1 = plt.figure(1)
pml.hinton_diagram(xmiss0, ax=plot_1.gca())
plot_1.suptitle('Observed')
pml.savefig("gauss_impute_observed.pdf", dpi=300)
plot_2 = plt.figure(2)
pml.hinton_diagram(x_full, ax=plot_2.gca())
plot_2.suptitle('Hidden truth')
pml.savefig("gauss_impute_truth.pdf", dpi=300)
plot_3 = plt.figure(3)
pml.hinton_diagram(x_imputed, ax=plot_3.gca())
plot_3.suptitle('Imputation with true params')
pml.savefig("gauss_impute_pred.pdf", dpi=300)
plt.show()
def sample_plot_gibbs(x0, z0, kv, π, μ, σ, n_iterations, xmin, xmax):
x_hist, z_hist = gibbs_sample(x0, z0, kv, π, μ, σ, n_iterations)
colors = ["tab:blue" if z else "tab:red" for z in z_hist]
fig, axs = plt.subplots()
axs.scatter(np.arange(n_iterations),
x_hist,
s=20,
facecolors="none",
edgecolors=colors)
pml.savefig("gibbs_scatter.pdf")
fig = plt.figure()
axs = plt.axes(projection="3d")
plot_gmm_3d_trace(x_hist, π, μ, σ, "Gibbs sampling", xmin, xmax, axs)
pml.style3d(axs, 1.5, 1, 0.8)
plt.subplots_adjust(left=0.001, bottom=0.208, right=0.7)
pml.savefig("gibbs_trace.pdf", pad_inches=0, bbox_inches="tight")
fig, axs = plt.subplots()
sm.graphics.tsa.plot_acf(x_hist,
lags=45,
alpha=None,
title="Gibbs",
ax=axs)
pml.savefig("gibbs_autocorrelation.pdf")
def make_graph(data, save_name):
prior = beta.pdf(x, a=data["prior"]["a"], b=data["prior"]["b"])
n_0 = data["likelihood"]["n_0"]
n_1 = data["likelihood"]["n_1"]
samples = jnp.concatenate([jnp.zeros(n_0), jnp.ones(n_1)])
likelihood_function = jnp.vectorize(
lambda p: jnp.exp(bernoulli.logpmf(samples, p).sum()))
likelihood = likelihood_function(x)
posterior = beta.pdf(x, a=data["posterior"]["a"], b=data["posterior"]["b"])
fig, ax = plt.subplots()
axt = ax.twinx()
fig1 = ax.plot(
x,
prior,
"k",
label=f"prior Beta({data['prior']['a']}, {data['prior']['b']})",
linewidth=2.0,
)
fig2 = axt.plot(x,
likelihood,
"r:",
label=f"likelihood Bernoulli",
linewidth=2.0)
fig3 = ax.plot(
x,
posterior,
"b-.",
label=
f"posterior Beta({data['posterior']['a']}, {data['posterior']['b']})",
linewidth=2.0,
)
fig_list = fig1 + fig2 + fig3
labels = [fig.get_label() for fig in fig_list]
ax.legend(fig_list, labels, loc="upper left", shadow=True)
axt.set_ylabel("Likelihood")
ax.set_ylabel("Prior/Posterior")
ax.set_title(f"$N_0$:{n_0}, $N_1$:{n_1}")
pml.savefig(save_name)
def make_vector_field_plots():
# initialize the theta domain
theta1, theta2 = np.meshgrid(np.linspace(-1, 1, 9), np.linspace(-1, 1, 9))
theta = np.array([theta1.T.flatten(), theta2.T.flatten()])
sf = 3
# get random values and subtract their mean
X = np.random.randn(2, 10000)
X = X - np.mean(X, axis=1).reshape(2, 1)
dL = np.zeros_like(theta)
for i in range(0, theta.shape[1]):
_, dL[:, i], G = L_dL_G(theta[:, i], X, sf)
# change derivative to get steepest descent
dL = -dL
plt.quiver(theta[0, :], theta[1, :], dL[0, :], dL[1, :])
plt.xlabel(r"$\theta_1$")
plt.ylabel(r"$\theta_2$")
plt.title("Steepest descent vectors in original parameter space")
#pml.savefig("SDOriginalParam.pdf")
pml.savefig("natgrad_descent_vectors_orig.pdf")
plt.show()
phi = theta.copy()
theta = np.linalg.inv(sqrtm(G)).dot(phi)
dL = np.zeros_like(theta)
for i in range(0, theta.shape[1]):
_, dL[:, i], G = L_dL_G(theta[:, i], X, sf)
dL = -dL
dLphi = sqrtm(np.linalg.inv(G)).dot(dL)
plt.quiver(phi[0, :], phi[1, :], dLphi[0, :], dLphi[1, :])
plt.xlabel(r"$\phi_1$")
plt.ylabel(r"$\phi_2$")
plt.title("Steepest descent vectors in natural parameter space")
#pml.savefig("SDNaturalParam.pdf")
pml.savefig("natgrad_descent_vectors_natural.pdf")
plt.show()
def main():
base = plt.gca().transData
rot = tr.Affine2D().rotate_deg(30)
plot_ellipses(0, 0, 9, 81, 3, c="tab:red", linewidth=3.0, transform=rot + base)
plt.arrow(-1.2, 0, 2.4, 0, length_includes_head=True, width=0.015, head_width=0.05, head_length=0.05, color='blue',
transform=rot + base)
plt.arrow(0, -0.5, 0, 1, length_includes_head=True, width=0.015, head_width=0.05, head_length=0.05, color='blue',
transform=rot + base)
plt.arrow(-1.2, 0, 0, 0.5, length_includes_head=True, width=0.015, head_width=0.05, head_length=0.05,
color='black', transform=rot + base)
plt.arrow(0, -0.5, 1.2, 0, length_includes_head=True, width=0.015, head_width=0.05, head_length=0.05,
color='black', transform=rot + base)
plt.arrow(-1.2, 0.5, 0, -0.5, length_includes_head=True, width=0.015, head_width=0.05, head_length=0.05,
color='black', transform=rot + base)
plt.arrow(1.2, -0.5, -1.2, 0, length_includes_head=True, width=0.015, head_width=0.05, head_length=0.05,
color='black', transform=rot + base)
plt.arrow(1, 0.3, 0, 0.3, length_includes_head=True, width=0.015, head_width=0.05, head_length=0.05, color='black',
transform=rot + base)
plt.text(0.658, 1, r"$u_2$", fontsize=14)
plt.text(1, 0.94, r"$u_1$", fontsize=14)
plt.text(-1.4, -0.48, r"$λ_2^{1/2}$", fontsize=14)
plt.text(0.85, -0.3, r"$λ_1^{1/2}$", fontsize=14)
plt.arrow(1, 0.3, 0.3, 0, length_includes_head=True, width=0.015, head_width=0.05, head_length=0.05, color='black',
transform=rot + base)
plt.xticks([])
plt.yticks([])
plt.xlabel("$x_1$", fontsize=14)
plt.ylabel("$x_2$", fontsize=14)
xmin, xmax = plt.xlim()
ymin, ymax = plt.ylim()
scale_factor = 1.2
plt.xlim(xmin * scale_factor, xmax * scale_factor)
plt.ylim(ymin * scale_factor, ymax * scale_factor)
plt.tight_layout()
pml.savefig("gaussEvec.pdf", dpi=300)
plt.show()
def sensor_fusion():
sigmas = [0.01 * np.eye(2), 0.01 * np.eye(2)]
helper(sigmas)
pml.savefig("demoGaussBayes2dEqualSpherical.pdf")
plt.show()
sigmas = [0.05 * np.eye(2), 0.01 * np.eye(2)]
helper(sigmas)
pml.savefig("demoGaussBayes2dUnequalSpherical.pdf")
plt.show()
sigmas = [
0.01 * np.array([[10, 1], [1, 1]]), 0.01 * np.array([[1, 1], [1, 10]])
]
helper(sigmas)
pml.savefig("demoGaussBayes2dUnequal.pdf")
plt.show()
