PlainTerminating implements the multivariate terminating process. More...

#include "include/PlainTerminating.h"

Inheritance diagram for PlainTerminating:

Classes
struct	OPTION
	Options used to configure the fitting of the terminating point process. More...

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

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

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

Public Member Functions
	PlainTerminating (const unsigned &n, const unsigned &num_dims)
	The constructor. More...

	PlainTerminating (const unsigned &n, const unsigned &num_dims, const Graph *graph)
	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. 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 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...

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 network structure among the nodes is given. More...

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

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

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

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

PlainTerminating implements the multivariate terminating process.

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. In this class, we assume the pairwise diffusion time conforms to an exponential distribution, that is, \(f_{ji}(t) = \alpha_{ji}\).

Definition at line 21 of file PlainTerminating.h.

Member Enumeration Documentation

enum PlainTerminating::OptMethod

Optimization algorithms used to fit the standard Hawkes Process.

Enumerator
SGD	stochastic gradient descend.
PLBFGS	projected LBFGS.

Definition at line 29 of file PlainTerminating.h.

enum PlainTerminating::RegCoef

Regularization coefficients.

Enumerator
LAMBDA	Regularization coefficient for \(\\|\mathbf{A}\\|\)

Definition at line 61 of file PlainTerminating.h.

enum PlainTerminating::Regularizer

Supported regularizations used to fit the standard Hawkes Process.

Enumerator
L1	Sparse L1 norm \(\\|\cdot\\|_1\)
L22	L22 norm \(\\|\cdot\\|_2^2\)
NONE	No regularization

Definition at line 43 of file PlainTerminating.h.

Constructor & Destructor Documentation

PlainTerminating::PlainTerminating	(	const unsigned &	n,
		const unsigned &	num_dims
	)

inline

The constructor.

Parameters

[in]	n	the number of parameters in total.
[in]	num_dims	the number of dimensions in the process.

Definition at line 159 of file PlainTerminating.h.

PlainTerminating::PlainTerminating	(	const unsigned &	n,
		const unsigned &	num_dims,
		const Graph *	graph
	)

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.

Definition at line 172 of file PlainTerminating.h.

Member Function Documentation

void PlainTerminating::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 140 of file PlainTerminating.cc.

void PlainTerminating::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 251 of file PlainTerminating.cc.

void PlainTerminating::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 PlainTerminating.cc.

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

protected

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

Parameters

[in] data input collection of sequences

Definition at line 67 of file PlainTerminating.cc.

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

virtual

Intensity function of each node.

The intensity function of each node is defined as \(\lambda^*_{c,i}(t) = \sum_{j\neq i}\mathbb{I}(t^c_j < t)\alpha_{ji}\).

Parameters

[in]	t	the current given time.
[in]	data	sequence of past events.
[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 258 of file PlainTerminating.cc.

double PlainTerminating::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 272 of file PlainTerminating.cc.

double PlainTerminating::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 265 of file PlainTerminating.cc.

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

virtual

Negative loglikelihood of multivariate terminating point process.

\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)}_{\text{arrayK[i]}(c, j)}\alpha_{ji}) - \sum_{j\neq i}\underbrace{\mathbb{I}(t^c_j < t^c_i)(t^c_i - t^c_j)}_{\text{arrayG}[i](c,j)}\alpha_{ji}\bigg)\bigg\}, \end{align}

where \(\alpha_{ji}\) is the pairwise infection risk from node \(j\) to node \(i\).

Parameters

objvalue	negative loglikelihood.
gradient	gradient of the parameters.

Implements IProcess.

Definition at line 165 of file PlainTerminating.cc.

void PlainTerminating::PostProcessing ( )

protected

Post process the learned dependency structure.

Definition at line 114 of file PlainTerminating.cc.

double PlainTerminating::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 278 of file PlainTerminating.cc.

Member Data Documentation

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

protected

Intergral 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)}_{\text{arrayK[i]}(c, j)}\alpha_{ji}) - \sum_{j\neq i}\underbrace{\mathbb{I}(t^c_j < t^c_i)(t^c_i - t^c_j)}_{\text{arrayG}[i](c,j)}\alpha_{ji}\bigg)\bigg\}, \end{align}

where \(\alpha_{ji}\) is the pairwise infection risk from node \(j\) to node \(i\). \(\text{arrayG[i]}(c, j)\) is the time duration between the infection time \(t^c_j\) and \(t^c_i\) in the sequence \(c\).

Definition at line 111 of file PlainTerminating.h.

std::vector<Eigen::MatrixXd> PlainTerminating::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_{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)}_{\text{arrayK[i]}(c, j)}\alpha_{ji}) - \sum_{j\neq i}\underbrace{\mathbb{I}(t^c_j < t^c_i)(t^c_i - t^c_j)}_{\text{arrayG}[i](c,j)}\alpha_{ji}\bigg)\bigg\}, \end{align}

where \(\alpha_{ji}\) is the pairwise infection risk from node \(j\) to node \(i\). \(\text{arrayK[i]}(c, j)\) indicates whether node \(j\) is the infecting parent of node \(i\) in the sequence \(c\).

Definition at line 100 of file PlainTerminating.h.

const Graph* PlainTerminating::graph_

protected

A graph object represents the dependency structure among the dimensions.

Definition at line 149 of file PlainTerminating.h.

unsigned PlainTerminating::num_sequences_

protected

total number of observed sequences

Definition at line 121 of file PlainTerminating.h.

Eigen::VectorXd PlainTerminating::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 116 of file PlainTerminating.h.

OPTION PlainTerminating::options_

protected

A configuration object which saves the optimization options.

Definition at line 144 of file PlainTerminating.h.

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

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