def histogram_test(X0=1,
U=lambda x : np.log(2*np.cosh(0.5*np.pi*x)),
gradU=lambda x : 0.5*np.pi*np.tanh(0.5*np.pi*x),
d=1,
n=int(1e5)):
model = PotentialProcess(U, gradU, X0, d)
coeff = PolynomialStep(1/3, 1/3)
simulation = Euler(model, coeff)
simulation.run(n)
nu_x, nu_x2 = simulation.test_functions([0, 1])
plt.figure()
plt.hist(x=simulation.invariant_law,
weights=simulation.weights,
bins=20,
density=True,
range=(-5, 5),
)
x = np.linspace(-10,10,num=1000)
y = np.exp(-U(x))
plt.plot(x,y)
plt.xlabel("x")
plt.ylabel("density")
plt.title("Invariant measure with hyperbolic secante potential. X0={},\nnu(x)={:.2f}, nu(x^2)={:.2f}\nn={} ".format(X0, nu_x[0], nu_x2[0],n))
plt.show()
def heatmap(nb_functions=list(range(l)),
theta=0.5,
mu=0,
sigma=1,
X0=0,
d=1,
n=int(1e4),
alpha_poly_min=0.11,
alpha_poly_sup=0.5,
nb_division_alpha=10,
beta_poly_min=0.11,
beta_poly_sup=0.5,
nb_division_beta=10):
model = OrnsteinUhlenbeck(theta, mu, sigma, X0, d)
grille_puissance_alpha = np.linspace(alpha_poly_min,
alpha_poly_sup,
num=nb_division_alpha)
grille_puissance_beta = np.linspace(beta_poly_min,
beta_poly_sup,
num=nb_division_beta)
def show_heatmap(nth, mean):
plt.imshow(np.abs(values_list[:, :, nth] - mean))
plt.colorbar()
plt.xticks(ticks=[i for i in range(nb_division_beta)],
labels=np.round(grille_puissance_beta, 2))
plt.yticks(ticks=[i for i in range(nb_division_alpha)],
labels=np.round(grille_puissance_alpha, 2))
plt.title(
"Heatmap with different alpha-beta with the function = {}".format(
test_collection[nth]))
plt.show()
values_list = []
for alpha in grille_puissance_alpha:
values_with_same_alpha = []
for beta in grille_puissance_beta:
coeff = PolynomialStep(alpha, beta)
simulation = Euler(model, coeff)
simulation.run(n)
values_with_same_alpha.append(simulation.test_functions())
values_list.append(values_with_same_alpha)
values_list = np.array(values_list)
for nb_function in nb_functions:
show_heatmap(nb_function, 0.0)
def limite_law(X0=1, c=0.5, n=int(1e5), M=int(1e3), shift=0):
model = MultipleInvariantMeasure(X0, c)
coeff = PolynomialShiftedStep(1 / 3, 1 / 3, shift)
simulation = Euler(model, coeff)
means = []
for i in tqdm(range(M)):
simulation.run(n)
means.append(simulation.test_functions([0])[0][0])
plt.hist(
x=means,
bins=20,
density=True,
)
plt.xlabel("x_mean")
plt.ylabel("density")
plt.title(
"Histogram of the mean(x) for n = {}, M={}\n X0={}, c={}, shift={}".
format(n, M, X0, c, shift))
plt.show()
def distrib_result(coeff):
values = []
phi = test_collection[2]
simulation = Euler(model, coeff)
for i in range(M):
simulation.reset()
simulation.run(n)
values.append(simulation.nu_f(phi))
return np.array(values)
def histogram_test(X0=1, c=0.5, n=int(1e5), shift=0):
model = MultipleInvariantMeasure(X0, c)
#coeff = PolynomialStep(1/3, 1/3)
coeff = PolynomialShiftedStep(1 / 3, 1 / 3, shift)
simulation = Euler(model, coeff)
simulation.run(n)
nu_x, nu_x2 = simulation.test_functions([0, 1])
plt.figure()
plt.hist(
x=simulation.invariant_law,
weights=simulation.weights,
bins=20,
density=True,
range=(-5, 5),
)
plt.xlabel("x")
plt.ylabel("density")
plt.title(
"Different invariant measures test. X0={}, c={}\nnu(x)={:.2f}, nu(x^2)={:.2f}\nn={}"
.format(X0, c, nu_x[0], nu_x2[0], n))
plt.show()
def run(coeff):
simulation = Euler(model, coeff)
simulation.run(n)
simulation.test_functions(nb_functions, MODE_DISPLAY=True)