HawkesGeneralKernel implements the multivariate Hawkes process with customized triggering kernels. More...

#include "include/HawkesGeneralKernel.h"

Inheritance diagram for HawkesGeneralKernel:

Classes
struct	OPTION
	Options used to configure the fitting of the general Hawkes Process with customized triggering kernels. More...

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

enum	RegCoef { LAMBDA0, LAMBDA }
	Regularization coefficients. More...

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

Public Member Functions
	HawkesGeneralKernel (const unsigned &n, const unsigned &num_dims, const std::vector< std::vector< TriggeringKernel * > > &triggeringkernels)
	The constructor. More...

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

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

virtual void	Gradient (const unsigned &k, Eigen::VectorXd &gradient)
	Returns the gradient w.r.t. the model parameters on the k-th sequence. More...

virtual double	Intensity (const double &t, const Sequence &data, Eigen::VectorXd &intensity_dim)
	Returns the summation \(\sum_{d=1}^D\lambda^_d(t)\) of the intensity value \(\lambda^_d(t)\) of each dimension in a given sequence data at the time t. More...

virtual double	IntensityUpperBound (const double &t, const double &L, const Sequence &data, Eigen::VectorXd &intensity_upper_dim)
	Returns the upper bound of the summation of the intensity value on each dimension from time t to t + L given the history of past events in sequence data. Let \({\lambda_d^(t)}\) be the conditional intensity function on the d-th dimension where \(d=1\dotso D\), and num_dims_ = D. This function returns \begin{align} \lambda_0^D(t) \geq \sum_{d=1}^D\sup_{\tau\in[t, t + \tau(t)]}\lambda^_d(\tau), \end{align} where the returned value \(\lambda_0^D(t)\) will be used for Ogata's Thinning algorithm. More...

virtual double	IntensityIntegral (const double &lower, const double &upper, const Sequence &data)
	Returns the integral of the intensity function \(\int_{a}^b\lambda^*(\tau)d\tau\) where \(a = lower\) and \(b = upper\). 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...

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 temporal features arrayK and arrayG from the input sequences More...

void	RestoreOptionToDefault ()
	restore to the default optimization configuration More...

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

Protected Attributes
std::vector< std::vector< Eigen::MatrixXd > >	arrayK
	the temporal features associated with the intensity More...

std::vector< Eigen::MatrixXd >	arrayG
	summation of the integral of the triggering kernels More...

std::vector< std::vector< TriggeringKernel * > >	triggeringkernels_
	a \(D\) by \(D\) grid of triggering-kernels More...

Eigen::VectorXd	observation_window_T_
	a column vector of length \(C\) which is the total number of sequences. Each component records the observation window in the respective sequence. More...

SimpleRNG	RNG_
	internal implementation for random number generator More...

unsigned	num_sequences_
	total number of observed sequences 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

HawkesGeneralKernel implements the multivariate Hawkes process with customized triggering kernels.

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 generally defined as the following:

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

where \(\lambda_0^n\geq 0\) is the base intensity, \(D\) is the number of dimensions, \(\alpha_{mn}\geq 0\), and the triggering kernel \(\gamma(t - t^m_j)\) captures the extent to which an event on dimension m at the time \(t^m_j\) can trigger an event on dimension n in the near future. Normally, in the standard Hawkes process, we have \(\gamma(t - t^m_j) = \exp(-\beta_{mn}(t - t^m_j))\). However, in more general cases, the form of the triggering kernel can be formulated to catpure the phenomena of interest. 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 \(\boldsymbol{\lambda}_0\).

Definition at line 25 of file HawkesGeneralKernel.h.

Member Enumeration Documentation

enum HawkesGeneralKernel::OptMethod

Optimization algorithms used to fit the standard Hawkes Process.

Enumerator
SGD	stochastic gradient descend.
PLBFGS	projected LBFGS.

Definition at line 125 of file HawkesGeneralKernel.h.

enum HawkesGeneralKernel::RegCoef

Regularization coefficients.

Enumerator
LAMBDA0	Regularization coefficient for \(\\|\boldsymbol{\lambda}_0\\|\)
LAMBDA	Regularization coefficient for \(\\|\mathbf{A}\\|\)

Definition at line 111 of file HawkesGeneralKernel.h.

enum HawkesGeneralKernel::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 90 of file HawkesGeneralKernel.h.

Constructor & Destructor Documentation

HawkesGeneralKernel::HawkesGeneralKernel	(	const unsigned &	n,
		const unsigned &	num_dims,
		const std::vector< std::vector< TriggeringKernel * > > &	triggeringkernels
	)

inline

The constructor.

Parameters

[in]	n	the number of parameters in total.
[in]	num_dims	the number of dimensions in the process.
[in]	triggeringkernels	a D-by-D grid of triggering-kernels.

Definition at line 173 of file HawkesGeneralKernel.h.

Member Function Documentation

void HawkesGeneralKernel::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 70 of file HawkesGeneralKernel.cc.

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

virtual

Returns the gradient w.r.t. the model parameters on the k-th sequence.

Parameters

[in]	k	sequence index.
[out]	gradient	the gradient vector w.r.t. the model parameters.

Implements IProcess.

Definition at line 189 of file HawkesGeneralKernel.cc.

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

protected

initialize the temporal features arrayK and arrayG from the input sequences

Parameters

[in] data input collection of sequences

Definition at line 17 of file HawkesGeneralKernel.cc.

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

virtual

Returns the summation \(\sum_{d=1}^D\lambda^*_d(t)\) of the intensity value \(\lambda^*_d(t)\) of each dimension in a given sequence data at the time t.

Parameters

[in]	t	the given time.
[in]	data	the given sequence of the past events until time t.
[out]	intensity_dim	a column vector of size num_dims_ where each component stores the intensity value of the respetive dimension at time t given the past sequence in data.

Returns: the summation of the intensity value from each dimension.

Implements IProcess.

Definition at line 196 of file HawkesGeneralKernel.cc.

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

virtual

Returns the integral of the intensity function \(\int_{a}^b\lambda^*(\tau)d\tau\) where \(a = lower\) and \(b = upper\).

Parameters

[in]	lower	starting point of the integral.
[in]	upper	ending point of the integral.
[in]	data	sequence of past events.

Returns: \(\int_{a}^b\lambda^*(\tau)d\tau\) where \(a = lower\) and \(b = upper\).

Implements IProcess.

Definition at line 266 of file HawkesGeneralKernel.cc.

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

virtual

Returns the upper bound of the summation of the intensity value on each dimension from time t to t + L given the history of past events in sequence data. Let \({\lambda_d^*(t)}\) be the conditional intensity function on the d-th dimension where \(d=1\dotso D\), and num_dims_ = D. This function returns

\begin{align} \lambda_0^D(t) \geq \sum_{d=1}^D\sup_{\tau\in[t, t + \tau(t)]}\lambda^*_d(\tau), \end{align}

where the returned value \(\lambda_0^D(t)\) will be used for Ogata's Thinning algorithm.

Parameters

t	the starting time.
L	the duration.
data	the given sequence of the past events until time t.
intensity_upper_dim	a column vector of size num_dims_ storing the upper bound of the intensity function on each dimension from time t to t + L.

Returns: the summation of the upper-bound of each intensity function from the respetive dimension within the interval [t, t + L].

Implements IProcess.

Definition at line 231 of file HawkesGeneralKernel.cc.

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

virtual

Negative loglikelihood of general Hawkes process.

\begin{align} -\sum_{n=1}^D\bigg\{\frac{1}{C}\sum_{c=1}^C\bigg(\sum_{i = 1}^{n_c}\bigg(\log(\lambda^n_0 + \sum_{m=1}^D\alpha_{mn}\underbrace{\sum_{t^m_{i,c}<t^n_{i,c}}\gamma_{mn}(t^n_{i,c} - t^m_{i,c})}_{\text{arrayK[n][c](i,m)}})\bigg) - T_c\lambda_0^n - \sum_{m=1}^D\alpha_{mn}\underbrace{\sum_{t^m_{j,c} < T_c}\int_{t^m_j}^{T_c}\gamma_{mn}(t - t^m_{j,c})dt)}_{\text{arrayG[n](c,m)}}\bigg)\bigg\}. \end{align}

Parameters

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

Implements IProcess.

Definition at line 83 of file HawkesGeneralKernel.cc.

double HawkesGeneralKernel::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 309 of file HawkesGeneralKernel.cc.

void HawkesGeneralKernel::RestoreOptionToDefault ( )

protected

restore to the default optimization configuration

Definition at line 321 of file HawkesGeneralKernel.cc.

Member Data Documentation

std::vector<Eigen::MatrixXd> HawkesGeneralKernel::arrayG

protected

summation of the integral of the triggering kernels

The log-likelihood of observing a collection of C sequences can be derived as the following:

\begin{align} \sum_{n=1}^D\bigg\{\frac{1}{C}\sum_{c=1}^C\bigg(\sum_{i = 1}^{n_c}\bigg(\log(\lambda^n_0 + \sum_{m=1}^D\alpha_{mn}\underbrace{\sum_{t^m_{i,c}<t^n_{i,c}}\gamma_{mn}(t^n_{i,c} - t^m_{i,c})}_{\text{arrayK[n][c](i,m)}})\bigg) - T_c\lambda_0^n - \sum_{m=1}^D\alpha_{mn}\underbrace{\sum_{t^m_{j,c} < T_c}\int_{t^m_j}^{T_c}\gamma_{mn}(t - t^m_{j,c})dt)}_{\text{arrayG[n](c,m)}}\bigg)\bigg\}. \end{align}

arrayG[n] is an \(C\) by \(D\) matrix where \(n_c\) is the number of events on the nth dimension in the sequence c. arrayK[n][c](i,m) stores the cumulative influence of the past events on dimension \(m\) in the sequence \(c\) to the occurence of the \(i\)th event on dimension \(n\).

Definition at line 52 of file HawkesGeneralKernel.h.

std::vector<std::vector<Eigen::MatrixXd> > HawkesGeneralKernel::arrayK

protected

the temporal features associated with the intensity

The log-likelihood of observing a collection of C sequences can be derived as the following:

\begin{align} \sum_{n=1}^D\bigg\{\frac{1}{C}\sum_{c=1}^C\bigg(\sum_{i = 1}^{n_c}\bigg(\log(\lambda^n_0 + \sum_{m=1}^D\alpha_{mn}\underbrace{\sum_{t^m_{j,c}<t^n_{i,c}}\gamma_{mn}(t^n_{i,c} - t^m_{j,c})}_{\text{arrayK[n][c](i,m)}})\bigg) - T_c\lambda_0^n - \sum_{m=1}^D\alpha_{mn}\underbrace{\sum_{t^m_{j,c} < T_c}\int_{t^m_j}^{T_c}\gamma_{mn}(t - t^m_{j,c})dt)}_{\text{arrayG[n](c,m)}}\bigg)\bigg\}. \end{align}

arrayK[n][c] is an \(n_c\) by \(D\) matrix where \(n_c\) is the number of events on the nth dimension in the sequence c. arrayK[n][c](i,m) stores the cumulative influence of the past events on dimension \(m\) in the sequence \(c\) to the occurence of the \(i\)th event on dimension \(n\).

Definition at line 40 of file HawkesGeneralKernel.h.

unsigned HawkesGeneralKernel::num_sequences_

protected

total number of observed sequences

Definition at line 73 of file HawkesGeneralKernel.h.

Eigen::VectorXd HawkesGeneralKernel::observation_window_T_

protected

a column vector of length \(C\) which is the total number of sequences. Each component records the observation window in the respective sequence.

Definition at line 63 of file HawkesGeneralKernel.h.

OPTION HawkesGeneralKernel::options_

protected

A configuration object which saves the optimization options.

Definition at line 163 of file HawkesGeneralKernel.h.

SimpleRNG HawkesGeneralKernel::RNG_

protected

internal implementation for random number generator

Definition at line 68 of file HawkesGeneralKernel.h.

std::vector<std::vector<TriggeringKernel*> > HawkesGeneralKernel::triggeringkernels_

protected

a \(D\) by \(D\) grid of triggering-kernels

Definition at line 58 of file HawkesGeneralKernel.h.

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

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

Classes

Public Types

Public Member Functions

Protected Member Functions

Protected Attributes

Detailed Description

Member Enumeration Documentation

Constructor & Destructor Documentation

Member Function Documentation

Member Data Documentation