tick.hawkes.HawkesSumGaussians¶

class tick.hawkes.HawkesSumGaussians(max_mean_gaussian, n_gaussians=5, step_size=1e-07, C=1000.0, lasso_grouplasso_ratio=0.5, max_iter=50, tol=1e-05, n_threads=1, verbose=False, print_every=10, record_every=10, approx=0, em_max_iter=30, em_tol=None)[source]¶

A class that implements parametric inference for Hawkes processes with parametrisation of the kernels as sum of Gaussian basis functions and a mix of Lasso and group-lasso 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 parametrisation of the kernels as sum of Gaussian basis functions

\[\phi_{ij}(t) = \sum_{m=1}^M \alpha^{ij}_m f (t - t_m), \quad f(t) = (2 \pi \sigma^2)^{-1} \exp(- t^2 / (2 \sigma^2))\]

In our implementation we denote:

• Integer \(D\) by the attribute n_nodes

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

• Vector \((t_m) \in \mathbb{R}^{M}\) by the variable means_gaussians

• Number \(\sigma\) by the variable std_gaussian

• Tensor \(A = (\alpha^{ij}_m)_{ijm} \in \mathbb{R}^{D \times D \times M}\) by the attribute amplitudes

Parameters:

max_mean_gaussian : float

The mean of the last Gaussian basis function. This can be considered a proxy of the kernel support.

n_gaussians : int

The number of Gaussian basis functions used to approximate each kernel.

step_size : float

The step-size used in the optimization for the EM algorithm.

C : float, default=1e3

Level of penalization

lasso_grouplasso_ratio : float, default=0.5

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

• For ratio = 0 this is Group-Lasso regularization

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

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

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

n_threads : int, default=1

Number of threads used for parallel computation.

verbose : bool, default=False

If True, we verbose things

• 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

amplitudes : np.ndarray, shape=(n_nodes, n_nodes, n_gaussians)

Inferred adjacency matrix

means_gaussians : np.array, shape=(n_gaussians,)

The means of the Gaussian basis functions.

std_gaussian : float

The standard deviation of each Gaussian basis function.

References

Xu, Farajtabar, and Zha (2016, June) in ICML, Learning Granger Causality for Hawkes Processes.

Examples using tick.hawkes.HawkesSumGaussians¶