def integral_mean_uniform(num_samples: int, model: VanillaBayesianQuadrature):
bounds = model.integral_bounds.bounds
samples = _sample_uniform(num_samples, bounds)
gp_mean_at_samples, _ = model.predict(samples)
differences = np.array([x[1] - x[0] for x in bounds])
volume = np.prod(differences)
return np.mean(gp_mean_at_samples) * volume
def bq_loop(config, ai_model=None):
user_function, integral_bounds = eval(config.name)()
lb = integral_bounds[0][0] # lower bound
ub = integral_bounds[0][1] # upper bound
data_dim = config.data_dim
init_num_data = config.init_num_data
interval_std = config.interval_std
interval = np.zeros((1, data_dim))
std = np.zeros((1, data_dim))
mean = np.zeros((1, data_dim))
integral_bounds_scaled = integral_bounds.copy()
for ii in range(data_dim):
interval[0, ii] = integral_bounds[ii][1] - integral_bounds[ii][0]
std[0, ii] = interval[0, ii] / interval_std
mean[0, ii] = (integral_bounds[ii][1] + integral_bounds[ii][0]) / 2
integral_bounds_scaled[ii] = ((integral_bounds[ii] - mean[0, ii]) /
std[0, ii]).tolist()
lb_scaled = integral_bounds_scaled[0][0] # lower bound
ub_scaled = integral_bounds_scaled[0][1] # upper bound
lb = integral_bounds[0][0] # lower bound
ub = integral_bounds[0][1] # upper bound
results_list = [None] * config.repeated_runs
npr = np.random.RandomState(config.seed)
for kk in tqdm(range(config.repeated_runs)):
integral_mean_list = np.zeros(config.bq_iter + 1)
integral_std_list = np.zeros(config.bq_iter + 1)
#initialize data points
X_init = (npr.rand(init_num_data, data_dim) - 0.5) * interval + mean
Y_init = user_function.f(X_init)
X_init_norm = (X_init - mean) / std
Y_init_norm, mean_Y, std_Y = standardize(Y_init)
X = X_init_norm
Y = Y_init_norm
X[np.abs(X) < 1.0e-5] = 1.0e-5
#normalized function
function_norm = lambda x: (user_function.f(x * std + mean) - mean_Y
) / std_Y
if data_dim == 1:
ground_truth = univariate_approximate_ground_truth_integral(
function_norm, (lb_scaled, ub_scaled))[0]
elif data_dim == 2:
ground_truth = bivariate_approximate_ground_truth_integral(
function_norm, integral_bounds_scaled)[0]
#Set up Emukit_BO_BQ_GP_Model
emukit_gp_model = Emukit_BO_BQ_GP_Model(X_init_norm, Y_init_norm,
config, ai_model)
emukit_gp_model.optimize()
emukit_gp_model.set_kernel()
emukit_quad_kern = QuadratureKernelCustom(emukit_gp_model,
integral_bounds_scaled)
emukit_model = BaseGaussianProcessCustomModel(kern=emukit_quad_kern,
gp_model=emukit_gp_model)
emukit_method = VanillaBayesianQuadrature(base_gp=emukit_model,
X=X,
Y=Y)
#set up Bayesian quadrature
if config.plot:
x_plot = np.linspace(integral_bounds_scaled[0][0],
integral_bounds_scaled[0][1], 300)[:, None]
y_plot = function_norm(x_plot)
mu_plot, var_plot = emukit_method.predict(x_plot)
plt.figure(figsize=FIGURE_SIZE)
plt.plot(X_init_norm,
Y_init_norm,
"ro",
markersize=10,
label="Observations")
plt.plot(x_plot, y_plot, "k", label="The Integrand")
plt.plot(x_plot, mu_plot, "C0", label="Model")
plt.fill_between(x_plot[:, 0],
mu_plot[:, 0] + np.sqrt(var_plot)[:, 0],
mu_plot[:, 0] - np.sqrt(var_plot)[:, 0],
color="C0",
alpha=0.6)
plt.fill_between(x_plot[:, 0],
mu_plot[:, 0] + 2 * np.sqrt(var_plot)[:, 0],
mu_plot[:, 0] - 2 * np.sqrt(var_plot)[:, 0],
color="C0",
alpha=0.4)
plt.fill_between(x_plot[:, 0],
mu_plot[:, 0] + 3 * np.sqrt(var_plot)[:, 0],
mu_plot[:, 0] - 3 * np.sqrt(var_plot)[:, 0],
color="C0",
alpha=0.2)
plt.legend(loc=2, prop={'size': LEGEND_SIZE})
plt.xlabel(r"$x$")
plt.ylabel(r"$f(x)$")
plt.grid(True)
plt.xlim(lb_scaled, ub_scaled)
plt.show()
initial_integral_mean, initial_integral_variance = emukit_method.integrate(
)
integral_mean_list[0] = initial_integral_mean
integral_std_list[0] = np.sqrt(initial_integral_variance)
if config.plot:
x_plot_integral = np.linspace(
initial_integral_mean - 3 * np.sqrt(initial_integral_variance),
initial_integral_mean + 3 * np.sqrt(initial_integral_variance),
200)
y_plot_integral_initial = 1/np.sqrt(initial_integral_variance * 2 * np.pi) * \
np.exp( - (x_plot_integral - initial_integral_mean)**2 / (2 * initial_integral_variance) )
plt.figure(figsize=FIGURE_SIZE)
plt.plot(x_plot_integral,
y_plot_integral_initial,
"k",
label="initial integral density")
plt.axvline(initial_integral_mean, color="red", label="initial integral estimate", \
linestyle="--")
plt.axvline(ground_truth, color="blue", label="ground truth integral", \
linestyle="--")
plt.legend(loc=2, prop={'size': LEGEND_SIZE})
plt.xlabel(r"$F$")
plt.ylabel(r"$p(F)$")
plt.grid(True)
plt.xlim(np.min(x_plot_integral), np.max(x_plot_integral))
plt.show()
print('The initial estimated integral is: ',
round(initial_integral_mean, 4))
print('with a credible interval: ',
round(2 * np.sqrt(initial_integral_variance), 4), '.')
print('The ground truth rounded to 2 digits for comparison is: ',
round(ground_truth, 4), '.')
for ii in range(config.bq_iter):
time_count = 0
result = {}
ivr_acquisition = IntegralVarianceReduction(emukit_method)
space = ParameterSpace(emukit_method.reasonable_box_bounds.
convert_to_list_of_continuous_parameters())
num_steps = 200
num_init_points = 5
optimizer = LocalSearchAcquisitionOptimizer(
space, num_steps, num_init_points)
x_new, _ = optimizer.optimize(ivr_acquisition)
y_new = function_norm(x_new)
X = np.append(X, x_new, axis=0)
Y = np.append(Y, y_new, axis=0)
X[np.abs(X) < 1.0e-5] = 1.0e-5
emukit_method.set_data(X, Y)
start_time = time.time()
emukit_model.optimize()
time_count = time_count + time.time() - start_time
integral_mean, integral_variance = emukit_method.integrate()
integral_mean_list[ii + 1] = integral_mean
integral_std_list[ii + 1] = np.sqrt(integral_variance)
if config.plot:
mu_plot_final, var_plot_final = emukit_method.predict(x_plot)
y_plot_integral = 1/np.sqrt(integral_variance * 2 * np.pi) * \
np.exp( - (x_plot_integral - integral_mean)**2 / (2 * integral_variance) )
plt.figure(figsize=FIGURE_SIZE)
plt.plot(x_plot_integral,
y_plot_integral_initial,
"gray",
label="initial integral density")
plt.plot(x_plot_integral,
y_plot_integral,
"k",
label="new integral density")
plt.axvline(initial_integral_mean,
color="gray",
label="initial integral estimate",
linestyle="--")
plt.axvline(integral_mean,
color="red",
label="new integral estimate",
linestyle="--")
plt.axvline(ground_truth, color="blue", label="ground truth integral", \
linestyle="--")
plt.legend(loc=2, prop={'size': LEGEND_SIZE})
plt.xlabel(r"$F$")
plt.ylabel(r"$p(F)$")
plt.grid(True)
plt.xlim(np.min(x_plot_integral), np.max(x_plot_integral))
plt.show()
plt.figure(figsize=FIGURE_SIZE)
plt.plot(emukit_model.X,
emukit_model.Y,
"ro",
markersize=10,
label="Observations")
plt.plot(x_plot, y_plot, "k", label="The Integrand")
plt.plot(x_plot, mu_plot_final, "C0", label="Model")
plt.fill_between(
x_plot[:, 0],
mu_plot_final[:, 0] + np.sqrt(var_plot_final)[:, 0],
mu_plot_final[:, 0] - np.sqrt(var_plot_final)[:, 0],
color="C0",
alpha=0.6)
plt.fill_between(
x_plot[:, 0],
mu_plot_final[:, 0] + 2 * np.sqrt(var_plot_final)[:, 0],
mu_plot_final[:, 0] - 2 * np.sqrt(var_plot_final)[:, 0],
color="C0",
alpha=0.4)
plt.fill_between(
x_plot[:, 0],
mu_plot_final[:, 0] + 3 * np.sqrt(var_plot_final)[:, 0],
mu_plot_final[:, 0] - 3 * np.sqrt(var_plot_final)[:, 0],
color="C0",
alpha=0.2)
plt.legend(loc=2, prop={'size': LEGEND_SIZE})
plt.xlabel(r"$x$")
plt.ylabel(r"$f(x)$")
plt.grid(True)
plt.xlim(lb_scaled, ub_scaled)
plt.show()
print('The estimated integral is: ', round(integral_mean, 4))
print('with a credible interval: ',
round(2 * np.sqrt(integral_variance), 4), '.')
print('The ground truth rounded to 2 digits for comparison is: ',
round(ground_truth, 4), '.')
integral_error_list = np.abs(integral_mean_list - ground_truth)
result['integral_error_list'] = integral_error_list
integral_error_list_scaledback = integral_error_list * std_Y.item()
for jj in range(data_dim):
integral_error_list_scaledback = integral_error_list_scaledback * std[
0, jj]
result[
'integral_error_list_scaledback'] = integral_error_list_scaledback
result['integral_std_list'] = integral_std_list
result['time_elapsed'] = time_count
results_list[kk] = result
print(time_count)
if config.plot:
plt.figure(figsize=(12, 8))
plt.fill_between(np.arange(config.bq_iter + 1) + 1,
integral_error_list - 0.2 * integral_std_list,
integral_error_list + 0.2 * integral_std_list,
color='red',
alpha=0.15)
plt.plot(np.arange(config.bq_iter + 1) + 1,
integral_error_list,
'or-',
lw=2,
label='Estimated integral')
plt.legend(loc=2, prop={'size': LEGEND_SIZE})
plt.xlabel(r"iteration")
plt.ylabel(r"$f(x)$")
plt.grid(True)
plt.show()
#end of one run
return results_list