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

#include "include/LowRankHawkesProcess.h"

Inheritance diagram for LowRankHawkesProcess:

Classes
struct	OPTION
	Options used to configure the fitting of the Low-rank Hawkes process. More...

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

Public Member Functions
	LowRankHawkesProcess (const unsigned &num_rows, const unsigned &num_cols, const Eigen::VectorXd &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 LowRankHawkesProcess::OPTION &options, const Eigen::MatrixXd &TrueLambda0, const Eigen::MatrixXd &TrueAlpha)
	Maximum likelihood estimation for the model parameters. More...

virtual void	NegLoglikelihood (double &objvalue, Eigen::VectorXd &gradient)
	Negative log-likelihood of the sequences from the observed user-item pairs. 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 event_intensity_features_ and integral_intensity_features_. More...

unsigned	Vec2Ind (const unsigned &i, const unsigned &j)
	Helper function returns the respective index in a column vector of the given (row, column) pair. More...

void	Ind2Vec (const unsigned &ind, unsigned &i, unsigned &j)
	Helper function returns the respective (row, column) pair given the index in the corresponding column vector. More...

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

Protected Attributes
Eigen::VectorXd	beta_
	A column vector of size $\|\Omega\|$ where each component is the decaying rate of the respective exponential triggering kernel for each observed user-item pair $(u,i)\in\Omega$. More...

Eigen::VectorXd	event_intensity_features_
	Temporal features associated with each event. More...

Eigen::VectorXd	integral_intensity_features_
	Temporal features associated with each user-item pair. More...

Eigen::SparseMatrix< double >	pair_event_map_
	A sparse bitmap mapping matrix. More...

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

Eigen::VectorXi	observed_idx_
	Stores the index of each observed user-item pair in the total $m$ by $n$ grid. More...

unsigned	num_rows_
	Total number of rows(or users). More...

unsigned	num_cols_
	Total number of columns(or items). More...

LowRankHawkesProcess::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

LowRankHawkesProcess implements the standard multivariate Hawkes process.

The Low-rank Hawkes Process is an $mn$-dimensional Hawkes process arranged in an $m$-by- $n$ grid. For instance, we have $m$ users and $n$ items. The conditional intensity function of the entry $(u,i)$ between user $u$ and item $i$ is $\lambda^{u,i}(t) = \boldsymbol{\Lambda}_0(u,i) + \boldsymbol{A}(u,i)\sum_{t^{u,i}_k < t}\exp(-\beta_{u,i}(t - t^{u,i}_k))$ where $\boldsymbol{\Lambda}_0$ and $\boldsymbol{A}$ are the low-rank base intensity matrix and the self-exciting matrix, respectively.

Definition at line 20 of file LowRankHawkesProcess.h.

Member Enumeration Documentation

enum LowRankHawkesProcess::RegCoef

Regularization coefficients.

Enumerator
LAMBDA0	Regularization coefficient for $\\|\boldsymbol{\Lambda}_0\\|$
LAMBDA	Regularization coefficient for $\\|\mathbf{A}\\|$

Definition at line 101 of file LowRankHawkesProcess.h.

Constructor & Destructor Documentation

LowRankHawkesProcess::LowRankHawkesProcess	(	const unsigned &	num_rows,
		const unsigned &	num_cols,
		const Eigen::VectorXd &	beta
	)

inline

The constructor.

Parameters

[in]	num_rows	the total number of rows(or users).
[in]	num_dims	the total number of columns(or items).
[in]	beta	a column vector of size $\|\Omega\|$ where each component is the decaying rate of the respective exponential triggering kernel for each observed user-item pair $(u,i)\in\Omega$.

Definition at line 159 of file LowRankHawkesProcess.h.

Member Function Documentation

void LowRankHawkesProcess::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 containing different configurations for the optimization algorithm and the respective regularizations.

Definition at line 135 of file LowRankHawkesProcess.cc.

void LowRankHawkesProcess::fit	(	const std::vector< Sequence > &	data,
		const LowRankHawkesProcess::OPTION &	options,
		const Eigen::MatrixXd &	TrueLambda0,
		const Eigen::MatrixXd &	TrueAlpha
	)

Maximum likelihood estimation for the model parameters.

Parameters

[in]	data	vectors of observed sequences.
[in]	options	data structure containing different configurations for the optimization algorithm and the respective regularizations.
[in]	TrueLambda0	ground truth $\boldsymbol{\Lambda}_0$.
[in]	TrueAlpha	ground truth $\boldsymbol{A}$.

Definition at line 147 of file LowRankHawkesProcess.cc.

void LowRankHawkesProcess::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 130 of file LowRankHawkesProcess.cc.

void LowRankHawkesProcess::Ind2Vec	(	const unsigned &	ind,
		unsigned &	i,
		unsigned &	j
	)

protected

Helper function returns the respective (row, column) pair given the index in the corresponding column vector.

Parameters

[in]	ind	index in the long column vector.
[out]	i	returned row index.
[out]	j	returned column index.

Definition at line 21 of file LowRankHawkesProcess.cc.

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

protected

Initialize the temporal features event_intensity_features_ and integral_intensity_features_.

Parameters

[in] data collection of input sequences induced from each observed user-item pair.

Definition at line 28 of file LowRankHawkesProcess.cc.

double LowRankHawkesProcess::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 89 of file LowRankHawkesProcess.cc.

double LowRankHawkesProcess::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 164 of file LowRankHawkesProcess.cc.

double LowRankHawkesProcess::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 94 of file LowRankHawkesProcess.cc.

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

virtual

Negative log-likelihood of the sequences from the observed user-item pairs.

\begin{align} - \frac{1}{|\Omega|}\sum_{\mathcal{T}^{u,i}}\bigg\{\sum_{t^{u,i}_j\in\mathcal{T}^{u,i}}\log\bigg(\boldsymbol{\Lambda}_0(u,i) + \boldsymbol{A}(u,i)\sum_{t^{u,i}_k < t^{u,i}_j}\exp(-\beta_{u,i}(t^{u,i}_j - t^{u,i}_k))\bigg) - T^{u,i}\boldsymbol{\Lambda}_0(u,i) - \boldsymbol{A}(u,i)\sum_{t^{u,i}_k < T^{u,i}}\int_{t^{u,i}_k}^{T^{u,i}}\lambda^{u,i}(t)dt\bigg\}, \end{align}

where $\mathcal{T}^{u,i}$ is the sequence of events induced from the pair $(u,i)$.

Parameters

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

Implements IProcess.

Definition at line 99 of file LowRankHawkesProcess.cc.

double LowRankHawkesProcess::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 159 of file LowRankHawkesProcess.cc.

unsigned LowRankHawkesProcess::Vec2Ind	(	const unsigned &	i,
		const unsigned &	j
	)

protected

Helper function returns the respective index in a column vector of the given (row, column) pair.

Parameters

[in]	i	row index.
[in]	j	column index.

Returns: the index of the pair in a column vector.

Definition at line 16 of file LowRankHawkesProcess.cc.

Member Data Documentation

Eigen::VectorXd LowRankHawkesProcess::beta_

protected

A column vector of size $|\Omega|$ where each component is the decaying rate of the respective exponential triggering kernel for each observed user-item pair $(u,i)\in\Omega$.

Definition at line 28 of file LowRankHawkesProcess.h.

Eigen::VectorXd LowRankHawkesProcess::event_intensity_features_

protected

Temporal features associated with each event.

Suppose we have observed a collection $\Omega = \{(u,i)\}$ of user-item pairs. Each user-item pair $(u,i)$ induces a sequence of $n_{u,i}$ events. The total number of events is thus $N = \sum_{(u,i)\in\Omega}n_{u,i}$. event_intensity_features_ is a column vector of size $N$, which can be regarded as the sequential concatenation of each sequence from the respective user-item pair. Each component of event_intensity_features_ is the summation of the influence from the past events to the current event in the respective sequence, that is, $\sum_{t^{u,i}_k < t^{u,i}_j}\exp(-\beta_{u,i}(t^{u,i}_j - t^{u,i}_k))$.

Definition at line 35 of file LowRankHawkesProcess.h.

Eigen::VectorXd LowRankHawkesProcess::integral_intensity_features_

protected

Temporal features associated with each user-item pair.

Suppose we have observed a collection $\Omega = \{(u,i)\}$ of user-item pairs. integral_intensity_features_ is a column vector of size $|\Omega|$ where each component stores $\sum_{t^{u,i}_k < T^{u,i}}\int_{t^{u,i}_k}^{T^{u,i}}\lambda^{u,i}(t)dt$ for the respective user-item pair $(u,i)$ where $T^{u,i}$ is the corresponding observation window.

Definition at line 42 of file LowRankHawkesProcess.h.

unsigned LowRankHawkesProcess::num_cols_

protected

Total number of columns(or items).

Definition at line 71 of file LowRankHawkesProcess.h.

unsigned LowRankHawkesProcess::num_rows_

protected

Total number of rows(or users).

Definition at line 66 of file LowRankHawkesProcess.h.

Eigen::VectorXd LowRankHawkesProcess::observation_window_T_

protected

observation_window_T_ stores the observation window for each sequence.

For each sequence induced from the pair $(u,i)$, $\text{observation_window_T_}(c) = T^{u,i}$.

Definition at line 56 of file LowRankHawkesProcess.h.

Eigen::VectorXi LowRankHawkesProcess::observed_idx_

protected

Stores the index of each observed user-item pair in the total $m$ by $n$ grid.

Definition at line 61 of file LowRankHawkesProcess.h.

LowRankHawkesProcess::OPTION LowRankHawkesProcess::options_

protected

A configuration object which saves the optimization options.

Definition at line 148 of file LowRankHawkesProcess.h.

Eigen::SparseMatrix<double> LowRankHawkesProcess::pair_event_map_

protected

A sparse bitmap mapping matrix.

pair_event_map_ is a sparse binary matrix of size $|\Omega|$ by $\sum_{(u,i)\in\Omega}n_{u,i}$. For each pair $(u,i)$, the $n_{u,i}$ entries (out of the total $\sum_{(u,i)\in\Omega}n_{u,i}$ columns) in the respective row are set to be one. Because each row is of size $\sum_{(u,i)\in\Omega}n_{u,i}$ which is the total number of events induced from all observed user-item pairs, the non-zero entries in each row marks the location of the events belonging to the respective $(u,i)$ pair.

Definition at line 49 of file LowRankHawkesProcess.h.

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

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

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

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