PlainHawkes implements the standard multivariate Hawkes process. More...

#include "include/PlainHawkes.h"

Inheritance diagram for PlainHawkes:

Classes
struct	OPTION
	Options used to configure the fitting of the standard Hawkes Process. More...

Public Types
enum	OptMethod { SGD, PLBFGS }
	Optimization algorithms used to fit the standard Hawkes Process. More...

enum	Regularizer { L1, L22, NUCLEAR, NONE }
	Supported regularizations used to fit the standard Hawkes Process. More...

enum	RegCoef { LAMBDA, BETA }
	Regularization coefficients. More...

Public Member Functions
	PlainHawkes (const unsigned &n, const unsigned &num_dims, const Eigen::MatrixXd &Beta)
	The constructor. More...

void	fit (const std::vector< Sequence > &data, const OPTION &options)
	Maximum likelihood estimation for the model parameters. More...

void	fit (const std::vector< Sequence > &data, const OPTION &options, const Eigen::VectorXd &trueparameters)
	Maximum likelihood estimation for the model parameters. More...

virtual void	NegLoglikelihood (double &objvalue, Eigen::VectorXd &gradient)
	Negative loglikelihood of Hawkes process. More...

virtual void	Gradient (const unsigned &k, Eigen::VectorXd &gradient)
	Gradient on the \(k\)-th data sample. More...

virtual double	Intensity (const double &t, const Sequence &data, Eigen::VectorXd &intensity_dim)
	Intensity function of the standard multi-dimensional Hawkes process. More...

virtual double	IntensityUpperBound (const double &t, const double &L, const Sequence &data, Eigen::VectorXd &intensity_upper_dim)
	The upper bound of the intensity function between \([t, t + L]\). More...

virtual double	IntensityIntegral (const double &lower, const double &upper, const Sequence &data)
	The integral of the intensity function within the range \([\text{lower}, \text{upper}]\). More...

virtual double	PredictNextEventTime (const Sequence &data, const unsigned &num_simulations)
	Predict the next event timing by the expectation \(\int_{t_n}^\infty tf^(t)dt\). Currently, we use the sample average by simulations to approximate the expectation since the conditional density \(f^(t)\) normally does not have an analytic form. More...

void	Simulate (const std::vector< double > &vec_T, std::vector< Sequence > &sequences)
	Fast simulation method based on the property of the exponential triggering kernel using dynamic programming. More...

void	Simulate (const unsigned &n, const unsigned &num_sequences, std::vector< Sequence > &sequences)
	Fast simulation method based on the property of the exponential triggering kernel using dynamic programming. More...

Public Member Functions inherited from IProcess
	IProcess (const unsigned &n, const unsigned &num_dims)
	The constructor. More...

const Eigen::VectorXd &	GetParameters ()
	Return the column vector of model parameters. More...

unsigned	GetNumDims ()
	Return the number of dimensions in the process. More...

void	SetParameters (const Eigen::VectorXd &v)
	Set the model parameters. More...

void	PlotIntensityFunction (const Sequence &data)
	Plots the intensity functions based on the given sequence. It plots the intensity function and the associated event points up of each dimension in the same figure. More...

void	PlotIntensityFunction (const Sequence &data, const unsigned &dim_id)
	Plots the intensity function and the associated event points of the dimension dim_id. More...

Protected Member Functions
void	Initialize (const std::vector< Sequence > &data)
	Initialize the internal feature varables all_exp_kernel_recursive_sum_ and intensity_itegral_features_. More...

void	RestoreOptionToDefault ()
	Restores the optimization methods and regualrization into their default values. More...

unsigned	AssignDim (const Eigen::VectorXd &lambda)
	Auxiliary function of sampling dimension for fast simulation of Hawkes process. More...

void	UpdateExpSum (double t, const Eigen::VectorXd &last_event_per_dim, Eigen::MatrixXd &expsum)
	Auxiliary function of updating the intensity function of each dimension for fast simulation of Hawkes process. More...

Protected Member Functions inherited from IProcess
void	InitializeDimension (const std::vector< Sequence > &data)

Protected Attributes
Eigen::MatrixXd	Beta_
	Beta_ is a D-by-D matrix, which stores the \(\{\beta_{mn}\geq 0\}\) decaying rates of the exponential kernels. More...

std::vector< std::vector< std::vector< Eigen::VectorXd > > >	all_exp_kernel_recursive_sum_
	all_exp_kernel_recursive_sum_ stores the cumulative influence between dimensions. More...

std::vector< Eigen::MatrixXd >	intensity_itegral_features_
	intensity_itegral_features_ stores the integral of each exponential kernel. More...

Eigen::VectorXd	observation_window_T_
	observation_window_T_ stores the observation window for each sequence. More...

SimpleRNG	RNG_
	RNG_ implements a simple random number generator. More...

unsigned	num_sequences_
	num_sequences_ is the total number of sequences in the given data. More...

OPTION	options_
	A configuration object which saves the optimization options. More...

Protected Attributes inherited from IProcess
Eigen::VectorXd	parameters_
	A column vector represents all model parameters of the process. More...

unsigned	num_dims_
	The total number of dimensions of the process. More...

std::vector< std::vector< std::vector< double > > >	all_timestamp_per_dimension_
	all_timestamp_per_dimension_ is a 3-d array where all_timestamp_per_dimension_[c][n][i] records the i-th event on the n-th dimension in the c-th sequence. More...

Detailed Description

PlainHawkes implements the standard multivariate Hawkes process.

A multivariate Hawkes Process is a process where the occurrence of an event to a dimension will trigger more events on this dimension and other related dimensions in the near future. The intensity function of each dimension of the Hawkes process is defined as the following :

\begin{align} \lambda^n(t) = \lambda_0^n + \sum_{m=1}^D\alpha_{mn}\sum_{t^m_j < t}\exp(-\beta_{mn}(t - t^m_j)), \end{align}

where \(\lambda_0^n\geq 0\) is the base intensity, \(D\) is the number of dimensions, and \(\alpha_{mn}\geq 0\) captures the extent to which an event on dimension m can trigger an event on dimension n in the near future. The collection of \(\{\alpha_{mn}\}\) can be represented as a matrix \(\mathbf{A}(m,n) = \alpha_{mn}\), and the collection of \(\{\lambda_0^n\}\) can be represented as a column vector \(\boldsymbol{\lambda}_0\).

\(\boldsymbol{\lambda}_0\) and \(\mathbf{A}\) are the model parameters of a multivariate Hawkes process. The collection of \(\{\beta_{mn}\geq 0\}\) is often given in advance.

Definition at line 25 of file PlainHawkes.h.

Member Enumeration Documentation

enum PlainHawkes::OptMethod

Optimization algorithms used to fit the standard Hawkes Process.

Enumerator
SGD	stochastic gradient descend.
PLBFGS	projected LBFGS.

Definition at line 105 of file PlainHawkes.h.

enum PlainHawkes::RegCoef

Regularization coefficients.

Enumerator
LAMBDA	Regularization coefficient for \(\\|\boldsymbol{\lambda}_0\\|\)
BETA	Regularization coefficient for \(\\|\mathbf{A}\\|\)

Definition at line 141 of file PlainHawkes.h.

enum PlainHawkes::Regularizer

Supported regularizations used to fit the standard Hawkes Process.

Enumerator
L1	Sparse L1 norm \(\\|\cdot\\|_1\)
L22	L22 norm \(\\|\cdot\\|_2^2\)
NUCLEAR	Nuclear norm \(\\|\mathbf{A}\\|_* = \sum_{i=1}^{\min(m,n)}\sigma_i\) where \(sigma_i\) is the singular value of matrix \(\mathbf{A}\)
NONE	No regularization

Definition at line 119 of file PlainHawkes.h.

Constructor & Destructor Documentation

PlainHawkes::PlainHawkes	(	const unsigned &	n,
		const unsigned &	num_dims,
		const Eigen::MatrixXd &	Beta
	)

inline

The constructor.

Parameters

[in]	n	the number of parameters in total.
[in]	num_dims	the number of dimensions in the process.
[in]	Beta	a D-by-D matrix, which stores the \(\{\beta_{mn}\geq 0\}\) decaying rates of the exponential kernels.

Definition at line 204 of file PlainHawkes.h.

Member Function Documentation

unsigned PlainHawkes::AssignDim ( const Eigen::VectorXd & lambda )

protected

Auxiliary function of sampling dimension for fast simulation of Hawkes process.

Parameters

lambda A column vector where each component is the intensity of the respective dimension.

Returns: the dimension which can most likely explain the generation of the current event point.

Definition at line 494 of file PlainHawkes.cc.

void PlainHawkes::fit	(	const std::vector< Sequence > &	data,
		const OPTION &	options
	)

Maximum likelihood estimation for the model parameters.

Parameters

[in]	data	vectors of observed sequences.
[in]	options	data structure sotring different configuration for the optimization algorithm and the respective regularizations.

Definition at line 355 of file PlainHawkes.cc.

void PlainHawkes::fit	(	const std::vector< Sequence > &	data,
		const OPTION &	options,
		const Eigen::VectorXd &	trueparameters
	)

Maximum likelihood estimation for the model parameters.

Parameters

[in]	data	vectors of observed sequences.
[in]	options	data structure sotring different configuration for the optimization algorithm and the respective regularizations.
[in]	trueparameters	a column vector storing the true parameters of the Hawkes process used to generate the set of observed sequences in data to compare.

Definition at line 395 of file PlainHawkes.cc.

void PlainHawkes::Gradient	(	const unsigned &	k,
		Eigen::VectorXd &	gradient
	)

virtual

Gradient on the \(k\)-th data sample.

Parameters

[in]	data	sample index.
[out]	gradient	of the objective function w.r.t. the current data sample

Implements IProcess.

Definition at line 302 of file PlainHawkes.cc.

void PlainHawkes::Initialize ( const std::vector< Sequence > & data )

protected

Initialize the internal feature varables all_exp_kernel_recursive_sum_ and intensity_itegral_features_.

Parameters

[in] data A collection of input sequences.

Definition at line 19 of file PlainHawkes.cc.

double PlainHawkes::Intensity	(	const double &	t,
		const Sequence &	data,
		Eigen::VectorXd &	intensity_dim
	)

virtual

Intensity function of the standard multi-dimensional Hawkes process.

Parameters

[in]	t	the given time t.
[in]	data	the sequence of past events.
[out]	intensity_dim	a column vector where the n-th component is the intensity function of dimension n at the current given time t, that is, \begin{align} \text{intensity_dim[n]} = \lambda^n(t) = \lambda^n_0 + \sum_{m=1}^D\alpha_{mn}\sum_{t^m_j < t}\exp(-\beta_{mn}(t - t^m_j)). \end{align}

Returns: the total intensity value from all dimensions at the time t, that is, \(\sum_{n=1}^D\lambda^n(t)\).

Implements IProcess.

Definition at line 111 of file PlainHawkes.cc.

double PlainHawkes::IntensityIntegral	(	const double &	lower,
		const double &	upper,
		const Sequence &	data
	)

virtual

The integral of the intensity function within the range \([\text{lower}, \text{upper}]\).

Parameters

[in]	lower	the inegeral lower bound.
[in]	upper	the integral upper bound.
[in]	data	the sequence of past events.

Returns: the intensity integral \(\int_{\text{lower}}^{\text{upper}}\lambda(t)dt\).

Implements IProcess.

Definition at line 447 of file PlainHawkes.cc.

double PlainHawkes::IntensityUpperBound	(	const double &	t,
		const double &	L,
		const Sequence &	data,
		Eigen::VectorXd &	intensity_upper_dim
	)

virtual

The upper bound of the intensity function between \([t, t + L]\).

Parameters

[in]	t	the given time t.
[in]	L	the length we look into the future starting from the time t.
[in]	data	the sequence of past events.
[out]	intensity_upper_dim	a column vector where the n-th component is the upper bound for the n-th dimension between \([t, t + L]\).

Returns: the total intensity upper bound from all dimensions, that is, \(\sum_{n=1}^D\text{intensity_dim[n]}\).

Implements IProcess.

Definition at line 146 of file PlainHawkes.cc.

void PlainHawkes::NegLoglikelihood	(	double &	objvalue,
		Eigen::VectorXd &	gradient
	)

virtual

Negative loglikelihood of Hawkes process.

Parameters

[out]	objvalue	negative loglikelihood.
[out]	gradient	gradient of the parameters.

Implements IProcess.

Definition at line 180 of file PlainHawkes.cc.

double PlainHawkes::PredictNextEventTime	(	const Sequence &	data,
		const unsigned &	num_simulations
	)

virtual

Predict the next event timing by the expectation \(\int_{t_n}^\infty tf^*(t)dt\). Currently, we use the sample average by simulations to approximate the expectation since the conditional density \(f^*(t)\) normally does not have an analytic form.

Parameters

[in]	data	the sequence of past events.
[in]	num_simulations	number of simulations we use to calculate the sample average.

Returns: the prediction of the next event timing.

Implements IProcess.

Definition at line 435 of file PlainHawkes.cc.

void PlainHawkes::RestoreOptionToDefault ( )

protected

Restores the optimization methods and regualrization into their default values.

Definition at line 485 of file PlainHawkes.cc.

void PlainHawkes::Simulate	(	const std::vector< double > &	vec_T,
		std::vector< Sequence > &	sequences
	)

Fast simulation method based on the property of the exponential triggering kernel using dynamic programming.

The intensity function on the n-th dimension is defined as \(\lambda^n(t) = \lambda^n_0 + \sum_{m=1}^D\alpha_{mn}\sum_{t^m_j < t}\exp(-\beta_{mn}(t - t^m_j))\). Let \(A_{mn}(t) = \sum_{t^m_j < t}exp(-\beta_{mn}(t - t^m_j))\). Then, \(A_{mn}(t) = \exp(-\beta_{mn}(t - t^n_i))A_{mn}(t^n_i) + \sum_{t^n_i\leq t^m_j < t}\exp(-\beta_{mn}(t - t^m_j))\).

Parameters

[in]	vec_T	a columne vector where each component is the observation window for each simulated sequence. We can simulate multiple sequences. Each component of the column vector vec_T is the given observation window of the respective sequence, so the dimension of vec_T is equal to the number of simulated sequences in total.
[out]	sequences	a vector of simulated sequences.

Definition at line 531 of file PlainHawkes.cc.

void PlainHawkes::Simulate	(	const unsigned &	n,
		const unsigned &	num_sequences,
		std::vector< Sequence > &	sequences
	)

Fast simulation method based on the property of the exponential triggering kernel using dynamic programming.

Parameters

[in]	n	number of events per simulated sequence.
[in]	num_sequences	number of simulated sequences.
[out]	sequences	a vector of simulated sequences.

Definition at line 634 of file PlainHawkes.cc.

void PlainHawkes::UpdateExpSum	(	double	t,
		const Eigen::VectorXd &	last_event_per_dim,
		Eigen::MatrixXd &	expsum
	)

protected

Auxiliary function of updating the intensity function of each dimension for fast simulation of Hawkes process.

Parameters

[in]	t	current given time.
[in]	last_event_per_dim	a column vector where each component is the last event of the respective dimension.
[out]	expsum	cumulative influence between dimensions at time t. \(\text{expsum(m,n)(t)} = \sum_{t^m_j < t}\exp(-\beta_{mn}(t - t^m_j)) = \exp(-\beta_{mn}(t - t^m_{n_m}))(1 + \text{expsum(m,n)}(t^m_{n_m}))\) where \(n_m\) is the index of the last event on dimension m.

Definition at line 519 of file PlainHawkes.cc.

Member Data Documentation

std::vector<std::vector<std::vector<Eigen::VectorXd> > > PlainHawkes::all_exp_kernel_recursive_sum_

protected

all_exp_kernel_recursive_sum_ stores the cumulative influence between dimensions.

For each sequence c, given a pair of dimension m and n, and a point \(t^{c,n}_{i}\) on the dimension n,

\begin{align} \text{all_exp_kernel_recursive_sum_[c][m][n][i]} & = \sum_{t^{c,m}_{j} < t^{c,n}_{i}}\exp(-\beta_{mn}(t^{c,n}_i - t^{c,m}_j))\\ & = \exp(-\beta_{mn}(t^{c,n}_i - t^{c,n}_{i-1}))\cdot\text{all_exp_kernel_recursive_sum_[c][m][n][i - 1]}\quad + \sum_{t^{c,n}_{i-1}\leq t^{c,m}_j < t^{c,n}_i} \exp(-\beta_{mn}(t^{c,n}_i - t^{c,m}_j)) \end{align}

which is the sum of the influence from all past events \(t^{c,m}_{j} < t^{c,n}_{i}\) on the dimension m in the sequence c.

Definition at line 45 of file PlainHawkes.h.

Eigen::MatrixXd PlainHawkes::Beta_

protected

Beta_ is a D-by-D matrix, which stores the \(\{\beta_{mn}\geq 0\}\) decaying rates of the exponential kernels.

Definition at line 33 of file PlainHawkes.h.

std::vector<Eigen::MatrixXd> PlainHawkes::intensity_itegral_features_

protected

intensity_itegral_features_ stores the integral of each exponential kernel.

For each sequence c, given a pair of dimension m and n,

\begin{align} \text{intensity_itegral_features_[c](m,n)} = \sum_{t^{c,m}_j < T^c}(1 - \exp(-\beta_{mn}(T_c - t^{c,m}_j))) \end{align}

Definition at line 55 of file PlainHawkes.h.

unsigned PlainHawkes::num_sequences_

protected

num_sequences_ is the total number of sequences in the given data.

Definition at line 72 of file PlainHawkes.h.

Eigen::VectorXd PlainHawkes::observation_window_T_

protected

observation_window_T_ stores the observation window for each sequence.

For each sequence c, \(\text{observation_window_T_}(c) = T_c\).

Definition at line 62 of file PlainHawkes.h.

OPTION PlainHawkes::options_

protected

A configuration object which saves the optimization options.

Definition at line 194 of file PlainHawkes.h.

SimpleRNG PlainHawkes::RNG_

protected

RNG_ implements a simple random number generator.

Definition at line 67 of file PlainHawkes.h.

The documentation for this class was generated from the following files:

/Users/nandu/Development/MultiVariatePointProcess/include/PlainHawkes.h
/Users/nandu/Development/MultiVariatePointProcess/src/PlainHawkes.cc

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