Note
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FaDIn on simulated 2-d Hawkes processesΒΆ
This example demonstrates inference performed by FaDIn on multivariate Hawkes processes simulated with specific kernels.
# Authors: Guillaume Staerman <guillaume.staerman@inria.fr>
# Alexandre Gramfort <alexandre.gramfort@inria.fr>
# Thomas Moreau <thomas.moreau@inria.fr>
#
# License: MIT
Let us first define the parameters of our model.
import numpy as np
import torch
import matplotlib.pyplot as plt
import matplotlib.patches as mpatches
from fadin.utils.utils_simu import simu_hawkes_cluster
from fadin.solver import FaDIn
from fadin.kernels import DiscreteKernelFiniteSupport
n_dim = 2
dt = 0.01
T = 1_000_000
kernel_length = 5
L = int(1 / dt)
size_grid = int(T / dt) + 1
discretization = torch.linspace(0, kernel_length, L)
Here, we set the parameters of a Hawkes process with a Exponential(1) distribution.
Here, we simulate the data
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Here, we apply FaDIn.
solver = FaDIn(
n_dim=n_dim,
kernel="truncated_exponential",
kernel_length=kernel_length,
delta=dt, optim="RMSprop",
params_optim={'lr': 1e-3},
max_iter=10000
)
solver.fit(events, T)
# We average on the 10 last values of the optimization.
estimated_baseline = solver.param_baseline[-10:].mean(0)
estimated_alpha = solver.param_alpha[-10:].mean(0)
param_kernel = [solver.param_kernel[0][-10:].mean(0)]
print('Estimated baseline is:', estimated_baseline)
print('Estimated alpha is:', estimated_alpha)
print('Estimated parameters of the truncated Exponential kernel is:',
param_kernel[0])
number of events is: [4025462, 5032651]
precomput: 120.09188032150269
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iterations in 33.389782428741455
Estimated baseline is: tensor([0.1356, 0.5181])
Estimated alpha is: tensor([[0.6359, 0.2643],
[0.2960, 0.6603]])
Estimated parameters of the truncated Exponential kernel is: tensor([[1.0056, 0.8916],
[1.1432, 0.9880]])
Here, we plot the values of the estimated kernels with FaDIn.
kernel = DiscreteKernelFiniteSupport(dt, n_dim, kernel='truncated_exponential',
kernel_length=kernel_length)
kernel_values = kernel.kernel_eval(param_kernel,
discretization)
plt.subplots(figsize=(12, 8))
for i in range(n_dim):
for j in range(n_dim):
plt.subplot(n_dim, n_dim, i*n_dim + j + 1)
plt.plot(discretization[1:], kernel_values[i, j, 1:]/kernel_length)
plt.plot(discretization[1:], torch.exp(-discretization[1:]), c='k')
plt.ylabel(rf'$\phi_{{{i}{j}}}$', size=20)
black_patch = mpatches.Patch(facecolor='k')
blue_patch = mpatches.Patch(facecolor='C0')
labels = ['FaDIn\' estimated kernel', 'True kernel']
colors = [blue_patch, black_patch]
plt.legend(handles=colors, labels=labels, loc="center right",
borderaxespad=0.1, fontsize='x-large')
plt.suptitle('Hawkes influence kernels', size=20)
plt.show()
/github/workspace/examples/plot_multivariate_fadin.py:86: MatplotlibDeprecationWarning: Auto-removal of overlapping axes is deprecated since 3.6 and will be removed two minor releases later; explicitly call ax.remove() as needed.
plt.subplot(n_dim, n_dim, i*n_dim + j + 1)
Total running time of the script: (2 minutes 36.751 seconds)