tick.hawkes.SimuHawkesSumExpKernels¶

class tick.hawkes.SimuHawkesSumExpKernels(adjacency, decays, baseline=None, end_time=None, period_length=None, max_jumps=None, seed=None, verbose=True, force_simulation=False)[source]¶

Hawkes process with sum-exponential kernels simulation

They are defined by the intensity:

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

where

• \(D\) is the number of nodes

• \(\mu_i(t)\) are the baseline intensities

• \(\phi_{ij}\) are the kernels

• \(dN_j\) are the processes differentiates

and with an exponential parametrisation of the kernels

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

where \(\alpha^u_{ij}\) are the intensities of the kernel and \(\beta^u\) its decays. The matrix of all \(\alpha\) is called adjacency matrix. Note that all nodes kernels share the same decays list.

Parameters:

baseline : np.ndarray or list

The baseline of all intensities, also noted \(\mu(t)\). It might be three different types:

• np.ndarray, shape=(n_nodes,) : One baseline per node is given. Hence baseline is assumed to be constant, ie. \(\mu_i(t) = \mu_i\)

• np.ndarray, shape=(n_nodes, n_intervals) : n_intervals baselines are given per node. This assumes parameter period_length is also given. In this case baseline is piecewise constant on intervals of size period_length / n_intervals and periodic.

• list of tick.base.TimeFunction, shape=(n_nodes,) : One function is given per node, ie. \(\mu_i(t)\) is explicitely given.

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

Intensities of exponential kernels, also named \(\alpha^u_{ij}\)

decays : np.ndarray, shape=(n_decays, )

Decays of exponential kernels, also named \(\beta^u\) If a float is given, all decays are equal to this float

end_time : float, default=None

Time until which this point process will be simulated

max_jumps : int, default=None

Simulation will stop if this number of jumps in reached

seed : int, default = None

The seed of the random sampling. If it is None then a random seed (different at each run) will be chosen.

force_simulation : bool, default = False

If force is not set to True, simulation won’t be run if the matrix of the L1 norm of kernels has a spectral radius greater or equal to 1 as it would be unstable

Attributes:

timestamps : list of np.ndarray, size=n_nodes

A list of n_nodes timestamps arrays, each array containing the timestamps of all the jumps for this node

n_decays : int

Number of decays of the HawkesSumExpKernel , also noted \(U\)

simulation_time : float

Time until which this point process has been simulated

n_total_jumps : int

Total number of jumps simulated

tracked_intensity : list[np.ndarray], size=n_nodes

A record of the intensity with which this point process has been simulated. Note: you must call track_intensity before simulation to record it

intensity_tracked_times : np.ndarray

The times at which intensity has been recorded. Note: you must call track_intensity before simulation to record it

intensity_track_step : float

Step with which the intensity has been recorded

Examples using tick.hawkes.SimuHawkesSumExpKernels¶