mixle.stats.processes.multivariate_hawkes module

Multivariate (mutually-exciting) Hawkes process with an exponential kernel.

A D-dimensional Hawkes process over marked events (t, mark) on a fixed window [0, T]. Each mark has a baseline intensity mu_d and an event of mark j excites the intensity of mark d by alpha_{dj} exp(-beta (t - t_k)), so

lambda_d(t) = mu_d + sum_{t_k < t} alpha_{d, mark_k} exp(-beta (t - t_k)),

coupling the dimensions through the D x D excitation matrix alpha (a shared decay beta). The exact log-likelihood sum_n log lambda_{mark_n}(t_n) - sum_d \int_0^T lambda_d is computed in O(n D) by a per-mark excitation recursion, sampling is by multivariate Ogata thinning, and the parameters are fit by the Veen-Schoenberg branching EM (each event is an immigrant of its mark or the offspring of an earlier event, with the excitation kernel as the parent likelihood). The process is stationary when the spectral radius of alpha / beta is below 1.

Reference: Hawkes, ‘Spectra of some self-exciting and mutually exciting point processes’, Biometrika (1971).

class MultivariateHawkesProcessDistribution(mu, alpha, beta, window, name=None, keys=None)

Bases: SequenceEncodableProbabilityDistribution

Multivariate Hawkes process: baselines mu (D), excitation alpha (D, D), decay beta.

Parameters:

• mu (ndarray)

• alpha (ndarray)

• beta (float)

• window (float)

• name (str | None)

• keys (str | None)

intensity(t, times, marks)

Per-mark conditional rate vector (the vector-valued variant of intensity).

Returns lambda(t) of shape (D,) with lambda_k(t) = mu_k + sum_{(t_i, m_i) < t} alpha[k, m_i] exp(-beta (t - t_i)).

Parameters:

• t (float)

• times (Any)

• marks (Any)

Return type:

ndarray

expected_count(t_start, t_end, times, marks)

Per-mark compensator vector (the vector-valued variant of expected_count).

Returns (D,) with the integral of lambda_k over [t_start, t_end] given the history.

Parameters:

• t_start (float)

• t_end (float)

• times (Any)

• marks (Any)

Return type:

ndarray

density(x)

Probability density of one realization (a sequence of (time, mark) events).

Parameters:

x (Any)

Return type:

float

log_density(x)

Exact log-likelihood of one realization of marked events sorted by time.

Parameters:

x (Any)

Return type:

float

seq_log_density(x)

Log-likelihood for a list of realizations.

Parameters:

x (list[Any])

Return type:

ndarray

sampler(seed=None)

Return a sampler (multivariate Ogata thinning).

Parameters:

seed (int | None)

Return type:

MultivariateHawkesProcessSampler

estimator(pseudo_count=None)

Return a branching-EM estimator over the same window and dimension.

Parameters:

pseudo_count (float | None)

Return type:

MultivariateHawkesProcessEstimator

dist_to_encoder()

Return the data encoder (passes realizations through; the likelihood is per-realization).

Return type:

MultivariateHawkesProcessDataEncoder

class MultivariateHawkesProcessSampler(dist, seed=None)

Bases: DistributionSampler

Draw realizations by multivariate Ogata thinning.

Parameters:

• dist (MultivariateHawkesProcessDistribution)

• seed (int | None)

sample(size=None)

Draw one marked-event realization, or size iid realizations.

Parameters:

size (int | None)

class MultivariateHawkesProcessAccumulator(dim, window, name=None, keys=None)

Bases: SequenceEncodableStatisticAccumulator

Accumulate the multivariate branching sufficient statistics.

Parameters:

• dim (int)

• window (float)

• name (str | None)

• keys (str | None)

update(x, weight, estimate)

Accumulate branching EM statistics for one marked-event realization.

Parameters:

• x (Any)

• weight (float)

• estimate (MultivariateHawkesProcessDistribution | None)

Return type:

None

initialize(x, weight, rng)

Initialize branching statistics for one marked-event realization.

Parameters:

• x (Any)

• weight (float)

• rng (RandomState | None)

Return type:

None

seq_update(x, weights, estimate)

Accumulate branching EM statistics from encoded realizations.

Parameters:

• x (list[Any])

• weights (ndarray)

• estimate (MultivariateHawkesProcessDistribution)

Return type:

None

seq_initialize(x, weights, rng)

Initialize branching statistics from encoded realizations.

Parameters:

• x (list[Any])

• weights (ndarray)

• rng (RandomState | None)

Return type:

None

combine(suff_stat)

Merge another multivariate-Hawkes sufficient-statistic tuple.

Parameters:

suff_stat (tuple)

Return type:

MultivariateHawkesProcessAccumulator

value()

Return immigrant, offspring, delay, mass, and window statistics.

Return type:

tuple

from_value(x)

Replace accumulator contents from branching sufficient statistics.

Parameters:

x (tuple)

Return type:

MultivariateHawkesProcessAccumulator

scale(c)

Scale all weight-linear sufficient statistics by c.

Parameters:

c (float)

Return type:

MultivariateHawkesProcessAccumulator

key_merge(stats_dict)

Merge keyed statistics into stats_dict when keys are configured.

Parameters:

stats_dict (dict[str, Any])

Return type:

None

key_replace(stats_dict)

Replace this accumulator from keyed statistics when available.

Parameters:

stats_dict (dict[str, Any])

Return type:

None

acc_to_encoder()

Return the marked-event encoder used by this accumulator.

Return type:

MultivariateHawkesProcessDataEncoder

class MultivariateHawkesProcessAccumulatorFactory(dim, window, name=None, keys=None)

Bases: StatisticAccumulatorFactory

Factory for MultivariateHawkesProcessAccumulator.

Parameters:

• dim (int)

• window (float)

• name (str | None)

• keys (str | None)

make()

Create a fresh multivariate-Hawkes accumulator.

Return type:

MultivariateHawkesProcessAccumulator

class MultivariateHawkesProcessEstimator(dim, window, name=None, keys=None)

Bases: ParameterEstimator

Veen-Schoenberg branching-EM estimator for the multivariate Hawkes parameters.

Parameters:

• dim (int)

• window (float)

• name (str | None)

• keys (str | None)

accumulator_factory()

Return an accumulator factory for branching EM statistics.

Return type:

MultivariateHawkesProcessAccumulatorFactory

estimate(nobs, suff_stat)

Estimate baseline, excitation, and decay from branching statistics.

Parameters:

• nobs (float | None)

• suff_stat (tuple)

Return type:

MultivariateHawkesProcessDistribution

class MultivariateHawkesProcessDataEncoder(window, dim)

Bases: DataSequenceEncoder

Validate and pass through realizations of marked events.

Parameters:

• window (float)

• dim (int)

seq_encode(x)

Validate and normalize marked-event realizations.

Parameters:

x (Sequence[Any])

Return type:

list[Any]