tick.hawkes.HawkesADM4¶

class tick.hawkes.HawkesADM4(decay, C=1000.0, lasso_nuclear_ratio=0.5, max_iter=50, tol=1e-05, n_threads=1, verbose=False, print_every=10, record_every=10, rho=0.1, approx=0, em_max_iter=30, em_tol=None)[source]¶

A class that implements parametric inference for Hawkes processes with an exponential parametrisation of the kernels and a mix of Lasso and nuclear regularization

Hawkes processes are point processes defined by the intensity:

\[\forall i \in [1 \dots D], \quad \lambda_i(t) = \mu_i + \sum_{j=1}^D \sum_{t_k^j < t} \phi_{ij}(t - t_k^j)\]

where

• \(D\) is the number of nodes

• \(\mu_i\) are the baseline intensities

• \(\phi_{ij}\) are the kernels

• \(t_k^j\) are the timestamps of all events of node \(j\)

and with an exponential parametrisation of the kernels

\[\phi_{ij}(t) = \alpha^{ij} \beta \exp (- \beta t) 1_{t > 0}\]

In our implementation we denote:

• Integer \(D\) by the attribute n_nodes

• Vector \(\mu \in \mathbb{R}^{D}\) by the attribute baseline

• Matrix \(A = (\alpha^{ij})_{ij} \in \mathbb{R}^{D \times D}\) by the attribute adjacency

• Number \(\beta \in \mathbb{R}\) by the parameter decay. This parameter is given to the model

Parameters:

decay : float

The decay used in the exponential kernel

C : float, default=1e3

Level of penalization

lasso_nuclear_ratio : float, default=0.5

Ratio of Lasso-Nuclear regularization mixing parameter with 0 <= ratio <= 1.

• For ratio = 0 this is nuclear regularization

• For ratio = 1 this is lasso (L1) regularization

• For 0 < ratio < 1, the regularization is a linear combination of Lasso and nuclear.

max_iter : int, default=50

Maximum number of iterations of the solving algorithm

tol : float, default=1e-5

The tolerance of the solving algorithm (iterations stop when the stopping criterion is below it). If not reached it does max_iter iterations

verbose : bool, default=False

If True, we verbose things

n_threads : int, default=1

Number of threads used for parallel computation.

• if int <= 0: the number of physical cores available on the CPU

• otherwise the desired number of threads

print_every : int, default=10

Print history information when n_iter (iteration number) is a multiple of print_every

record_every : int, default=10

Record history information when n_iter (iteration number) is a multiple of record_every

Attributes:

n_nodes : int

Number of nodes / components in the Hawkes model

baseline : np.array, shape=(n_nodes,)

Inferred baseline of each component’s intensity

adjacency : np.ndarray, shape=(n_nodes, n_nodes)

Inferred adjacency matrix

References

Zhou, K., Zha, H., & Song, L. (2013, May). Learning Social Infectivity in Sparse Low-rank Networks Using Multi-dimensional Hawkes Processes. In AISTATS (Vol. 31, pp. 641-649).

Examples using tick.hawkes.HawkesADM4¶