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:
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
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
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 CPUotherwise the desired number of threads
print_every : int
, default=10
Print history information when
n_iter
(iteration number) is a multiple ofprint_every
record_every : int
, default=10
Record history information when
n_iter
(iteration number) is a multiple ofrecord_every
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.
__init__
(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]¶Initialize self. See help(type(self)) for accurate signature.
fit
(events, end_times=None, baseline_start=None, amplitudes_start=None)[source]¶Fit the model according to the given training data.
events : list
of list
of np.ndarray
List of Hawkes processes realizations. Each realization of the Hawkes process is a list of n_node for each component of the Hawkes. Namely
events[i][j]
contains a one-dimensionalnumpy.array
of the events’ timestamps of component j of realization i.If only one realization is given, it will be wrapped into a list
end_times : np.ndarray
or float
, default = None
List of end time of all hawkes processes that will be given to the model. If None, it will be set to each realization’s latest time. If only one realization is provided, then a float can be given.
baseline_start : None
or np.ndarray
, shape=(n_nodes)
Set initial value of baseline parameter If
None
starts with uniform 1 values
amplitudes_start : None
or np.ndarray
, shape=(n_nodes, n_nodes, n_gaussians)
Set initial value of amplitudes parameter If
None
starts with random values uniformly sampled between 0.5 and 0.9`
get_history
(key=None)¶Returns history of the solver
key : str
, default=None
If
None
all history is returned as adict
If
str
, name of the history element to retrieve
output : list
or dict
If
key
is None orkey
is not in history then output is a dict containing history of all keysIf
key
is the name of an element in the history, output is alist
containing the history of this element
get_kernel_norms
()[source]¶Computes kernel norms. This makes our learner compliant with
tick.plot.plot_hawkes_kernel_norms
API
norms : np.ndarray
, shape=(n_nodes, n_nodes)
2d array in which each entry i, j corresponds to the norm of kernel i, j
get_kernel_supports
()[source]¶Computes kernel support. This makes our learner compliant with
tick.plot.plot_hawkes_kernels
API
output : np.ndarray
, shape=(n_nodes, n_nodes)
2d array in which each entry i, j corresponds to the support of kernel i, j
get_kernel_values
(i, j, abscissa_array)[source]¶Computes value of the specified kernel on given time values. This
makes our learner compliant with tick.plot.plot_hawkes_kernels
API
i : int
First index of the kernel
j : int
Second index of the kernel
abscissa_array : np.ndarray
, shape=(n_points, )
1d array containing all the times at which this kernel will computes it value
output : np.ndarray
, shape=(n_points, )
1d array containing the values of the specified kernels at the given times.
objective
(coeffs, loss: float = None)[source]¶Compute the objective minimized by the solver at coeffs
coeffs : numpy.ndarray
, shape=(n_coeffs,)
The objective is computed at this point
loss : float
, default=`None`
Gives the value of the loss if already known (allows to avoid its computation in some cases)
output : float
Value of the objective at given
coeffs
tick.hawkes.HawkesSumGaussians
¶