tick.hawkes.HawkesConditionalLaw¶

class tick.hawkes.HawkesConditionalLaw(delta_lag=0.1, min_lag=0.0001, max_lag=40, n_quad=50, max_support=40, min_support=0.0001, quad_method='gauss', marked_components=None, delayed_component=None, delay=1e-05, model=None, n_threads=1, claw_method='lin')[source]¶

This class is used for performing non parametric estimation of multi-dimensional marked Hawkes processes based on conditional laws.

Marked Hawkes processes are point processes defined by the intensity:

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

where

• \(D\) is the number of nodes

• \(\mu_i\) are the baseline intensities

• \(\phi_{ij}\) are the kernels

• \(v_j\) are the marks (considered iid) of the process \(N_j\)

• \(f_{ij}\) the mark functions supposed to be piece-wise constant on intervals \(I^j(l)\)

The estimation is made from empirical computations of

\[\lim_{\epsilon \rightarrow 0} E [ (N_i[t + lag + \delta + \epsilon] - \Lambda[t + lag + \epsilon]) | N_j[t]=1 \quad \& \quad v_j(t) \in I^j(l) ]\]

For all the possible values of \(i\), \(i\) and \(l\). The \(lag\) is sampled on a uniform grid defined by \(\delta\): \(lag = n * \delta\).

Estimation can be performed using several realizations.

Parameters:

claw_method : {‘lin’, ‘log’}, default=’lin’

Specifies the way the conditional laws are sampled. It can be either:

• ‘lin’ : sampling is linear on [0, max_lag] using sampling period delta_lag

• ‘log’ : sampling is semi-log. It uses linear sampling on [0, min_lag] with sampling period delta_lag and log sampling on [min_lag, max_lag] using \(\exp(\delta)\) sampling period.

delta_lag : float, default=0.1

See claw_methods

min_lag : float, default=1e-4

See claw_methods

max_lag : float, default=40

See claw_methods

quad_method : {‘gauss’, ‘lin’, ‘log’}, default=gauss

Sampling used for quadrature

• ‘gauss’ for gaussian quadrature

• ‘lin’ for linear quadrature

• ‘log’ for log quadrature

min_support : float, default=1e-4

Start value of kernel estimation. It is used for ‘log’ quadrature method only, otherwise it is set to 0.

max_support : float, default=40

End value of kernel estimation

n_quad : int

The number of quadrature points between [min_support, max_support] used for solving the system. Be aware that the complexity increase as this number squared.

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

Attributes:

n_nodes : int

Number of nodes of the estimated Hawkes process

n_realizations : int

Number of given realizations

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

Estimation of the baseline

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

L1 norm matrix of the kernel norms

kernels : list of list

Kernel’s estimation on the quadrature points

mean_intensity : list of float

The estimated mean intensity

symmetries1d : list of 2-tuple

List of component index pairs for imposing symmetries on the mean intensity (e.g, [(0,1),(2,3)] means that the mean intensity of the components 0 and 1 must be the same and the mean intensity of the components 2 and 3 also Can be set using can be set using the set_model method.

symmetries2d : list of 2-tuple of 2-tuple

List of kernel coordinates pairs to impose symmetries on the kernel matrix (e.g., [[(0,0),(1,1)],[(1,0),(0,1)]] for a bidiagonal kernel in dimension 2) Can be set using can be set using the set_model method.

mark_functions : list of 2-tuple

The mark functions as a list (lexical order on i,j and l, see below)

References

Bacry, E., & Muzy, J. F. (2014). Second order statistics characterization of Hawkes processes and non-parametric estimation. arXiv preprint arXiv:1401.0903.

Examples using tick.hawkes.HawkesConditionalLaw¶