HawkesLearningTriggeringKernel implements the multivariate Hawkes process where the triggering kernel can be learned from the data. More...

#include "include/HawkesLearningTriggeringKernel.h"

Inheritance diagram for HawkesLearningTriggeringKernel:

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

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

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

Public Member Functions
	HawkesLearningTriggeringKernel (const unsigned &n, const unsigned &num_dims, const Eigen::VectorXd &tau, const Eigen::VectorXd &sigma)
	The constructor. More...

	HawkesLearningTriggeringKernel (const unsigned &n, const unsigned &num_dims, const Graph *graph, const Eigen::VectorXd &tau, const Eigen::VectorXd &sigma)
	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...

void	PlotTriggeringKernel (const unsigned &dim_m, const unsigned &dim_n, const double &T, const double &delta)
	Visualize the learned triggering kernel between dimension dim_m and dimension dim_n. 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	InitializeWithGraph (const std::vector< Sequence > &data)
	Initialize the temporal features arrayK and arrayG from the input sequences where the dependency structure among the dimensions is given. More...

void	InitializeConstants ()
	Initialize all helper variables. More...

void	PostProcessing ()
	Post process the learned dependency structure. More...

void	GetNegLoglikelihood (double &objvalue, Eigen::VectorXd &gradient)
	Compute the negative loglikelihood. More...

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

Protected Attributes
const double	PI = 3.14159265358979323846

Eigen::VectorXd	tau_
	A column vector of size num_rbfs_. Each component of tau_ is the location of the respective RBF basis function. More...

unsigned	num_rbfs_
	Total number of RBF basis functions. More...

Eigen::VectorXd	sigma_
	A column vector of size num_rbfs_. Each component of sigma_ is the bandwidth of the respective RBF basis function. More...

Eigen::VectorXd	sqrt2sigma_
	Helper variable defined as \(\sqrt{2}\)sigma_. More...

Eigen::VectorXd	sqrt2PIsigma_
	Helper variable defined as \(0.5\sqrt{2\pi}\)sigma_. More...

Eigen::VectorXd	erfctau_sigma_
	Helper variable defined as \(\text{erfc}\bigg(\frac{tau\_}{\sqrt{2}\text{sigma_}}\bigg)\). More...

std::vector< std::vector< std::vector< Eigen::MatrixXd > > >	arrayK
	Temporal features associated with the intensity function. More...

std::vector< std::vector< Eigen::MatrixXd > >	arrayG
	Temporal features derived from the integral of the intensity. 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...

unsigned	num_sequences_
	Total number of observed sequences. More...

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

const Graph *	graph_
	A graph object represents the dependency structure among the dimensions. 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

HawkesLearningTriggeringKernel implements the multivariate Hawkes process where the triggering kernel can be learned from the data.

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\sum_{t^m_j < t}\gamma_{mn}(t, t^m_j), \end{align}

where \(\lambda_0^n\geq 0\) is the base intensity, \(D\) is the number of dimensions, and \(\gamma_{mn}(t, t^m_j)\geq 0\) captures the extent to which an event on dimension \(m\) at the time \(t^m_j\) can trigger an event on dimension \(n\) at the time \(t\). We parametrize \(\gamma_{mn}(t, t^m_j) = \boldsymbol{k}_{mn}(t - t^m_j)^\top\boldsymbol{\alpha}_{mn}\). \(\boldsymbol{k}_{mn}(t)\) is a column vector where the \(l\)-th component is defined as \(\boldsymbol{k}_{mnl}(t)=\exp(-(\frac{\tau_l - t}{\sqrt{2}\sigma_l})^2)\), and \(\tau_l\) and \(\sigma_l\) are the parameters of the basis function.

Definition at line 28 of file HawkesLearningTriggeringKernel.h.

Member Enumeration Documentation

enum HawkesLearningTriggeringKernel::RegCoef

Regularization coefficients.

Enumerator
LAMBDA0	Regularization coefficient for \(\\|\boldsymbol{\lambda}_0\\|\)
LAMBDA	Regularization coefficient for GROUP LASSO

Definition at line 55 of file HawkesLearningTriggeringKernel.h.

enum HawkesLearningTriggeringKernel::Regularizer

Supported regularizations used to fit the standard Hawkes Process.

Enumerator
L22	L22 norm \(\\|\cdot\\|_2^2\)
GROUP	GROUP LASSO \(\sum_{m}\\|\boldsymbol{\alpha}_{mn}\\|_2\)
NONE	No regularization

Definition at line 36 of file HawkesLearningTriggeringKernel.h.

Constructor & Destructor Documentation

HawkesLearningTriggeringKernel::HawkesLearningTriggeringKernel	(	const unsigned &	n,
		const unsigned &	num_dims,
		const Eigen::VectorXd &	tau,
		const Eigen::VectorXd &	sigma
	)

inline

The constructor.

Parameters

[in]	n	the number of parameters in total.
[in]	num_dims	the number of dimensions in the process.
[in]	tau	the location of each basis function.
[in]	sigma	the bandwidth of each basis function.

Definition at line 188 of file HawkesLearningTriggeringKernel.h.

HawkesLearningTriggeringKernel::HawkesLearningTriggeringKernel	(	const unsigned &	n,
		const unsigned &	num_dims,
		const Graph *	graph,
		const Eigen::VectorXd &	tau,
		const Eigen::VectorXd &	sigma
	)

inline

The constructor.

Parameters

[in]	n	the number of parameters in total.
[in]	num_dims	the number of dimensions in the process.
[in]	graph	the graph object representing the dependency structure among the dimensions.
[in]	tau	the location of each basis function.
[in]	sigma	the bandwidth of each basis function.

Definition at line 201 of file HawkesLearningTriggeringKernel.h.

Member Function Documentation

void HawkesLearningTriggeringKernel::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 230 of file HawkesLearningTriggeringKernel.cc.

void HawkesLearningTriggeringKernel::GetNegLoglikelihood	(	double &	objvalue,
		Eigen::VectorXd &	gradient
	)

protected

Compute the negative loglikelihood.

Parameters

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

Definition at line 268 of file HawkesLearningTriggeringKernel.cc.

void HawkesLearningTriggeringKernel::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 392 of file HawkesLearningTriggeringKernel.cc.

void HawkesLearningTriggeringKernel::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 26 of file HawkesLearningTriggeringKernel.cc.

void HawkesLearningTriggeringKernel::InitializeConstants ( )

protected

Initialize all helper variables.

Definition at line 13 of file HawkesLearningTriggeringKernel.cc.

void HawkesLearningTriggeringKernel::InitializeWithGraph ( const std::vector< Sequence > & data )

protected

Initialize the temporal features arrayK and arrayG from the input sequences where the dependency structure among the dimensions is given.

Parameters

[in] data input collection of sequences

Definition at line 97 of file HawkesLearningTriggeringKernel.cc.

double HawkesLearningTriggeringKernel::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 399 of file HawkesLearningTriggeringKernel.cc.

double HawkesLearningTriggeringKernel::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 413 of file HawkesLearningTriggeringKernel.cc.

double HawkesLearningTriggeringKernel::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 406 of file HawkesLearningTriggeringKernel.cc.

void HawkesLearningTriggeringKernel::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\sum_{t^m_{j,c}<t^n_{i,c}}\underbrace{\boldsymbol{k}_{mn}(t^n_{i,c} - t^m_{j,c})^\top}_{\text{arrayK[n][c][i](m,:)}}\boldsymbol{\alpha}_{mn})\bigg) - T_c\lambda_0^n - \sum_{m=1}^D\sum_{t^m_{j,c} < T_c}\underbrace{\int_{t^m_j}^{T_c}\boldsymbol{k}_{mn}(t - t^m_{j,c})^\top dt}_{\text{arrayG[n][c](m,:)}}\cdot\boldsymbol{\alpha}_{mn}\bigg)\bigg\}. \end{align}

Parameters

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

Implements IProcess.

Definition at line 333 of file HawkesLearningTriggeringKernel.cc.

void HawkesLearningTriggeringKernel::PlotTriggeringKernel	(	const unsigned &	dim_m,
		const unsigned &	dim_n,
		const double &	T,
		const double &	delta
	)

Visualize the learned triggering kernel between dimension dim_m and dimension dim_n.

Parameters

dim_m	the influential dimension
dim_n	the influenced dimension
T	observation window.
delta	unit step size to partition the time line.

Definition at line 424 of file HawkesLearningTriggeringKernel.cc.

void HawkesLearningTriggeringKernel::PostProcessing ( )

protected

Post process the learned dependency structure.

Definition at line 171 of file HawkesLearningTriggeringKernel.cc.

double HawkesLearningTriggeringKernel::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 419 of file HawkesLearningTriggeringKernel.cc.

Member Data Documentation

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

protected

Temporal features derived from the integral of 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\sum_{t^m_{j,c}<t^n_{i,c}}\underbrace{\boldsymbol{k}_{mn}(t^n_{i,c} - t^m_{j,c})^\top}_{\text{arrayK[n][c][i](m,:)}}\boldsymbol{\alpha}_{mn})\bigg) - T_c\lambda_0^n - \sum_{m=1}^D\sum_{t^m_{j,c} < T_c}\underbrace{\int_{t^m_j}^{T_c}\boldsymbol{k}_{mn}(t - t^m_{j,c})^\top dt}_{\text{arrayG[n][c](m,:)}}\cdot\boldsymbol{\alpha}_{mn}\bigg)\bigg\}. \end{align}

arrayG[n][c] is a \(D\) by num_rbfs_ matrix which stores the integral of the influence from dimension \(m\) to dimension \(n\) in the sequence c.

Definition at line 137 of file HawkesLearningTriggeringKernel.h.

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

protected

Temporal features associated with the intensity function.

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\sum_{t^m_{j,c}<t^n_{i,c}}\underbrace{\boldsymbol{k}_{mn}(t^n_{i,c} - t^m_{j,c})^\top}_{\text{arrayK[n][c][i](m,:)}}\boldsymbol{\alpha}_{mn})\bigg) - T_c\lambda_0^n - \sum_{m=1}^D\sum_{t^m_{j,c} < T_c}\underbrace{\int_{t^m_j}^{T_c}\boldsymbol{k}_{mn}(t - t^m_{j,c})^\top dt}_{\text{arrayG[n][c](m,:)}}\cdot\boldsymbol{\alpha}_{mn}\bigg)\bigg\}. \end{align}

arrayK[n][c][i] is a \(D\) by num_rbfs_ matrix which stores the influence from the event at the time \(t^m_{j,c}\) on dimension \(m\) to the event at the time \(t^n_{i,c}\) on dimension \(n\) in the sequence c.

Definition at line 126 of file HawkesLearningTriggeringKernel.h.

Eigen::VectorXd HawkesLearningTriggeringKernel::erfctau_sigma_

protected

Helper variable defined as \(\text{erfc}\bigg(\frac{tau\_}{\sqrt{2}\text{sigma_}}\bigg)\).

Definition at line 115 of file HawkesLearningTriggeringKernel.h.

const Graph* HawkesLearningTriggeringKernel::graph_

protected

A graph object represents the dependency structure among the dimensions.

Definition at line 177 of file HawkesLearningTriggeringKernel.h.

unsigned HawkesLearningTriggeringKernel::num_rbfs_

protected

Total number of RBF basis functions.

Definition at line 99 of file HawkesLearningTriggeringKernel.h.

unsigned HawkesLearningTriggeringKernel::num_sequences_

protected

Total number of observed sequences.

Definition at line 145 of file HawkesLearningTriggeringKernel.h.

Eigen::VectorXd HawkesLearningTriggeringKernel::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 141 of file HawkesLearningTriggeringKernel.h.

OPTION HawkesLearningTriggeringKernel::options_

protected

A configuration object which saves the optimization options.

Definition at line 173 of file HawkesLearningTriggeringKernel.h.

const double HawkesLearningTriggeringKernel::PI = 3.14159265358979323846

protected

Definition at line 89 of file HawkesLearningTriggeringKernel.h.

Eigen::VectorXd HawkesLearningTriggeringKernel::sigma_

protected

A column vector of size num_rbfs_. Each component of sigma_ is the bandwidth of the respective RBF basis function.

Definition at line 103 of file HawkesLearningTriggeringKernel.h.

Eigen::VectorXd HawkesLearningTriggeringKernel::sqrt2PIsigma_

protected

Helper variable defined as \(0.5\sqrt{2\pi}\)sigma_.

Definition at line 111 of file HawkesLearningTriggeringKernel.h.

Eigen::VectorXd HawkesLearningTriggeringKernel::sqrt2sigma_

protected

Helper variable defined as \(\sqrt{2}\)sigma_.

Definition at line 107 of file HawkesLearningTriggeringKernel.h.

Eigen::VectorXd HawkesLearningTriggeringKernel::tau_

protected

A column vector of size num_rbfs_. Each component of tau_ is the location of the respective RBF basis function.

Definition at line 94 of file HawkesLearningTriggeringKernel.h.

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

/Users/nandu/Development/MultiVariatePointProcess/include/HawkesLearningTriggeringKernel.h
/Users/nandu/Development/MultiVariatePointProcess/src/HawkesLearningTriggeringKernel.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