Note

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FaDIn on simulated univariate Hawkes processesΒΆ

This example demonstrates inference performed by FaDIn on univariate 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

from fadin.utils.utils_simu import simu_hawkes_cluster
from fadin.solver import FaDIn
from fadin.kernels import DiscreteKernelFiniteSupport

n_dim = 1
dt = 0.01
T = 10_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 an Exponential(1) distribution.

baseline = np.array([.4])
alpha = np.array([[0.8]])
beta = 2.

Here, we simulate the data.

# standard parameter is beta, the parameter of the exponential distribution,
# equal to one.
kernel = 'expon'
events = simu_hawkes_cluster(T, baseline, alpha, kernel,
                             params_kernel={'scale': 1 / beta})

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Here, we apply FaDIn.

solver = FaDIn(n_dim=1,
               kernel="truncated_exponential",
               kernel_length=kernel_length,
               delta=dt, optim="RMSprop",
               params_optim={'lr': 1e-3},
               max_iter=2000
               )
solver.fit(events, T)

# We average on the 10 last values of the optimization.

estimated_baseline = solver.param_baseline[-10:].mean().item()
estimated_alpha = solver.param_alpha[-10:].mean().item()
param_kernel = [solver.param_kernel[0][-10:].mean().item()]

print('Estimated baseline is:', estimated_baseline)
print('Estimated alpha is:', estimated_alpha)
print('Estimated beta parameter of the exponential kernel is:', param_kernel[0])

number of events is: [20346]
precomput: 1.5059421062469482
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iterations in  2.5219380855560303
Estimated baseline is: 0.4023369252681732
Estimated alpha is: 0.8023015260696411
Estimated beta parameter of the exponential kernel is: 0.9666296243667603

Here, we plot the values of the estimated kernel with FaDIn.

kernel = DiscreteKernelFiniteSupport(dt, n_dim, kernel='truncated_exponential',
                                     kernel_length=kernel_length)
kernel_values = kernel.kernel_eval([torch.Tensor([param_kernel])],
                                   discretization)

plt.plot(discretization[1:], kernel_values.squeeze()[1:]/kernel_length,
         label='FaDIn\' estimated kernel')
plt.plot(discretization[1:], beta * torch.exp(-discretization[1:]*beta),
         label='True kernel', c='k')
plt.title('Hawkes influence kernel', size=20)
plt.xlabel('Time', size=20)
plt.ylabel(r'$\phi(t)$', size=25)
plt.legend(fontsize='x-large')
plt.show()
Hawkes influence kernel

Total running time of the script: (0 minutes 5.024 seconds)

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