TerminatingProcessLearningTriggeringKernel implements the multivariate terminating process where the pairwise infection risk function can be learned from the data. More...

#include "include/TerminatingProcessLearningTriggeringKernel.h"

Inheritance diagram for TerminatingProcessLearningTriggeringKernel:

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 { LAMBDA }
	Regularization coefficients. More...

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

	TerminatingProcessLearningTriggeringKernel (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 multivariate terminating point process learning triggering kernels. 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)
	Intensity function of each dimension(or node) 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	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< 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

TerminatingProcessLearningTriggeringKernel implements the multivariate terminating process where the pairwise infection risk function can be learned from the data.

The Multivariate Terminating Point Process is an \(D\)-dimensional temporal point process with the conditional intensity function of each dimension \(d\) is given by \(\lambda_d^*(t) = \mathbb{I}\{N_d(t)\leq 1\}\cdot g(t)\) where \(N_d(t)\) is the number of events on the dimension \(d\), \(g(t)\) is a non-negative function, and \(\mathbb{I}\{{\cdot}\}\) is the indicator function. The Multivariate Terminating Process instantiates the continuous-time information diffusion model with general pairwise infection risk functions \(\gamma_{ji}(t, t^c_j)\), so we have \(g(t) = \sum_{j\neq i}\mathbb{I}(t^c_j < t)\gamma_{ji}(t, t^c_j)\) in a sequence \(c\).

Definition at line 25 of file TerminatingProcessLearningTriggeringKernel.h.

Member Enumeration Documentation

enum TerminatingProcessLearningTriggeringKernel::RegCoef

Regularization coefficients.

Enumerator
LAMBDA	Regularization coefficient for GROUP LASSO

Definition at line 52 of file TerminatingProcessLearningTriggeringKernel.h.

enum TerminatingProcessLearningTriggeringKernel::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 33 of file TerminatingProcessLearningTriggeringKernel.h.

Constructor & Destructor Documentation

TerminatingProcessLearningTriggeringKernel::TerminatingProcessLearningTriggeringKernel	(	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 180 of file TerminatingProcessLearningTriggeringKernel.h.

TerminatingProcessLearningTriggeringKernel::TerminatingProcessLearningTriggeringKernel	(	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 200 of file TerminatingProcessLearningTriggeringKernel.h.

Member Function Documentation

void TerminatingProcessLearningTriggeringKernel::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 193 of file TerminatingProcessLearningTriggeringKernel.cc.

void TerminatingProcessLearningTriggeringKernel::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 230 of file TerminatingProcessLearningTriggeringKernel.cc.

void TerminatingProcessLearningTriggeringKernel::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 327 of file TerminatingProcessLearningTriggeringKernel.cc.

void TerminatingProcessLearningTriggeringKernel::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 13 of file TerminatingProcessLearningTriggeringKernel.cc.

void TerminatingProcessLearningTriggeringKernel::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 74 of file TerminatingProcessLearningTriggeringKernel.cc.

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

virtual

Intensity function of each dimension(or node)

\begin{align} \lambda_i^*(t) = \mathbb{I}\{N_i(t)\leq 1\}\sum_{j\neq i}\mathbb{I}(t^c_j < t)\gamma_{ji}(t, t^c_j) \end{align}

in a given sequence \(c\) where \(\gamma_{ji}(t, t^c_j) = \boldsymbol{k}_{ji}(t - t^c_j)^\top\boldsymbol{\alpha}_{ji}\). \(\boldsymbol{k}_{ji}(t)\) is a column vector where the \(l\)-th component is defined as \(\boldsymbol{k}_{jil}(t)=\exp(-(\frac{\tau_l - t}{\sqrt{2}\sigma_l})^2)\), and \(\tau_l\) and \(\sigma_l\) are the parameters of the basis function.

Parameters

t	current given time.
data	the given sequence of the past events until time t.
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 334 of file TerminatingProcessLearningTriggeringKernel.cc.

double TerminatingProcessLearningTriggeringKernel::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 348 of file TerminatingProcessLearningTriggeringKernel.cc.

double TerminatingProcessLearningTriggeringKernel::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 341 of file TerminatingProcessLearningTriggeringKernel.cc.

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

virtual

Negative loglikelihood of multivariate terminating point process learning triggering kernels.

\begin{align} \sum_{i=1}^D\bigg\{\frac{1}{C}\sum_{c = 1}^C\bigg(\log(\sum_{j\neq i}\underbrace{\mathbb{I}(t^c_j < t^c_i)\boldsymbol{k}_{ji}(t^c_i - t^c_j)^\top}_{\text{arrayK}[i][c](j,:)}\cdot\boldsymbol{\alpha}_{ji}) - \sum_{j\neq i}\underbrace{\mathbb{I}(t^c_j < t^c_i)\int_{t^c_j}^{t^c_i}\boldsymbol{k}_{ji}(t - t^c_j)^\top}_{\text{arrayG[i]}[c](j,:)} dt\cdot\boldsymbol{\alpha}_{ji}\bigg)\bigg\} \end{align}

Parameters

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

Implements IProcess.

Definition at line 295 of file TerminatingProcessLearningTriggeringKernel.cc.

void TerminatingProcessLearningTriggeringKernel::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 359 of file TerminatingProcessLearningTriggeringKernel.cc.

void TerminatingProcessLearningTriggeringKernel::PostProcessing ( )

protected

Post process the learned dependency structure.

Definition at line 134 of file TerminatingProcessLearningTriggeringKernel.cc.

double TerminatingProcessLearningTriggeringKernel::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 354 of file TerminatingProcessLearningTriggeringKernel.cc.

Member Data Documentation

std::vector<std::vector<Eigen::MatrixXd> > TerminatingProcessLearningTriggeringKernel::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_{i=1}^D\bigg\{\frac{1}{C}\sum_{c = 1}^C\bigg(\log(\sum_{j\neq i}\underbrace{\mathbb{I}(t^c_j < t^c_i)\boldsymbol{k}_{ji}(t^c_i - t^c_j)^\top}_{\text{arrayK}[i][c](j,:)}\cdot\boldsymbol{\alpha}_{ji}) - \sum_{j\neq i}\underbrace{\mathbb{I}(t^c_j < t^c_i)\int_{t^c_j}^{t^c_i}\boldsymbol{k}_{ji}(t - t^c_j)^\top}_{\text{arrayG[i]}[c](j,:)} dt\cdot\boldsymbol{\alpha}_{ji}\bigg)\bigg\} \end{align}

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

Definition at line 128 of file TerminatingProcessLearningTriggeringKernel.h.

std::vector<std::vector<Eigen::MatrixXd> > TerminatingProcessLearningTriggeringKernel::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_{i=1}^D\bigg\{\frac{1}{C}\sum_{c = 1}^C\bigg(\log(\sum_{j\neq i}\underbrace{\mathbb{I}(t^c_j < t^c_i)\boldsymbol{k}_{ji}(t^c_i - t^c_j)^\top}_{\text{arrayK}[i][c](j,:)}\cdot\boldsymbol{\alpha}_{ji}) - \sum_{j\neq i}\underbrace{\mathbb{I}(t^c_j < t^c_i)\int_{t^c_j}^{t^c_i}\boldsymbol{k}_{ji}(t - t^c_j)^\top}_{\text{arrayG[i]}[c](j,:)} dt\cdot\boldsymbol{\alpha}_{ji}\bigg)\bigg\} \end{align}

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

Definition at line 117 of file TerminatingProcessLearningTriggeringKernel.h.

Eigen::VectorXd TerminatingProcessLearningTriggeringKernel::erfctau_sigma_

protected

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

Definition at line 105 of file TerminatingProcessLearningTriggeringKernel.h.

const Graph* TerminatingProcessLearningTriggeringKernel::graph_

protected

A graph object represents the dependency structure among the dimensions.

Definition at line 169 of file TerminatingProcessLearningTriggeringKernel.h.

unsigned TerminatingProcessLearningTriggeringKernel::num_rbfs_

protected

Total number of RBF basis functions.