Package 'ppdiag'

Draw the intensity of Hawkes Process

Description

Draw the intensity of a Hawkes Process

Usage

drawHPIntensity(
  hp = NULL,
  events,
  int_title = "Hawkes Intensity",
  start = 0,
  end = max(events),
  history = NULL,
  color = 1,
  i = 1,
  add = FALSE,
  fit = FALSE,
  plot_events = TRUE,
  verbose = FALSE
)

Arguments

hp	object parameters for Hawkes process.
events	the event times happened in this state
int_title	title of the intensity plot
start	the start time of current state
end	the end time of current state
history	the past event times
color	specify the default plotting color.
i	state number, used only for drawUniMMHPIntensity
add	whether to add the hawkes intensity to an existing plot, used for drawUniMMHPIntensity
fit	a boolean indicating whether to fit a new HP to events
plot_events	indicate whether events will be plotted
verbose	whether to output informative messages as running

Value

no return value, intensity plot of Hawkes process

Examples

set.seed(100)
hp_obj <- pp_hp(lambda0 = 0.5, alpha = 0.45, beta = 0.5)
events <- pp_simulate(hp_obj, start = 0, end = 20)
drawHPIntensity(hp_obj, events)

---

Draw intensity of homogeneous Poisson process

Description

Draw the intensity for a homogeneous Poisson process

Usage

drawHPPIntensity(
  hpp = NULL,
  events,
  int_title = "Homogeneous Poisson Process",
  start = 0,
  end = max(events),
  color = "red",
  plot_events = TRUE,
  fit = FALSE,
  add = FALSE,
  verbose = FALSE
)

Arguments

hpp	object for homogeneous Poisson process
events	event times input
int_title	the plot title
start	start of events
end	end of events
color	a specification for the default plotting color.
plot_events	a boolean indicating whether input events will be plotted
fit	a boolean indicating whether to fit a hpp or use the passed object
add	whether to add the hpp intensity to an existing plot
verbose	whether to output informative messages as running

Value

no return value, intensity plot of homogeneous Poisson process

Examples

pois_y <- pp_hpp(lambda = 1)
drawHPPIntensity(pois_y, events = pp_simulate(pois_y, end = 10))

---

Draw the intensity of the Markov-modulated Hawkes Process(MMHP)

Description

Take a mmhp object and draw its intensity accordingly

Usage

drawUniMMHPIntensity(
  mmhp,
  simulation,
  int_title = "Intensity of MMHP",
  leg_location = "topright",
  color = 1,
  add = FALSE
)

Arguments

mmhp	a mmhp object including its state, state_time, events, lambda0, lambda1, beta and alpha.
simulation	the simulated Markov-modulated Hawkes Process(MMHP)
int_title	title of the plot.
leg_location	location of legend, if moving needed
color	A specification for the default plotting color.
add	logical; if TRUE add to an already existing plot; if NA start a new plot taking the defaults for the limits and log-scaling of the x-axis from the previous plot. Taken as FALSE (with a warning if a different value is supplied) if no graphics device is open.

Value

no return value, intensity plot of Markov-modulated Hawkes process

Examples

Q <- matrix(c(-0.4, 0.4, 0.2, -0.2), ncol = 2, byrow = TRUE)
x <- pp_mmhp(Q,
  delta = c(1 / 3, 2 / 3), lambda0 = 0.9, lambda1 = 1.1,
  alpha = 0.8, beta = 1.2
)
y <- pp_simulate(x, n = 25)
drawUniMMHPIntensity(x, y)

---

Draw the intensity of the Markov-modulated Poisson Process(MMPP)

Description

Take a mmpp object and draw its intensity accordingly

Usage

drawUniMMPPIntensity(
  mmpp,
  simulation,
  add = FALSE,
  color = 1,
  fit = FALSE,
  int_title = "Intensity Plot of MMPP"
)

Arguments

mmpp	a mmpp object including its transition probability matrix, lambda0, delta, and c.
simulation	the simulated Markov-modulated Poisson Process(MMPP)
add	logical; if TRUE add to an already existing plot; if NA start a new plot taking the defaults for the limits and log-scaling of the x-axis from the previous plot. Taken as FALSE (with a warning if a different value is supplied) if no graphics device is open.
color	A specification for the default plotting color.
fit	a boolean indicating whether to fit the events provided
int_title	title of the plot.

Value

no return value, intensity plot of Markov-modulated Poisson process

Examples

Q <- matrix(c(-0.4, 0.4, 0.2, -0.2), ncol = 2, byrow = TRUE)
x <- pp_mmpp(Q, delta = c(1 / 3, 2 / 3), lambda0 = 0.9, c = 1.2)
y <- pp_simulate(x, n = 10)
drawUniMMPPIntensity(x, y)

---

Determine the MLE of Hawkes process numerically

Description

Determine the MLE of Hawkes process numerically

Usage

fithp(events, end = max(events), vec = c(0.1, 0.2, 0.3))

Arguments

events	event times
end	end of observation period starting from 0 (default last event)
vec	vector of initial parameter values

Value

a hp object indicating the maximum likelihood parameter values (lambda0,alpha,beta) for Hawkes process. This is a non-convex problem and a (unique) solution is not guaranteed.

Examples

hp_obj <- pp_hp(lambda0 = 0.1, alpha = 0.45, beta = 0.5)
sims <- pp_simulate(hp_obj, start = 0, n = 10)
fithp(sims)

---

Fit a homogeneous poisson process to event data

Description

Compute maximum likelihood estimator of the rate of a homogeneous Poisson process for the given events.

Usage

fithpp(events, end = max(events))

Arguments

events	vector containing the event times.
end	end of observation period, starting from 0 (default is last event)

Value

a hpp object containing the events and the estimated parameter

Examples

pois_y <- pp_hpp(lambda = 1)
events <- pp_simulate(pois_y, end = 10)
fithpp(events)

---

Draw intensity of fitted point process and QQ-Plot of rescaled events

Description

Draw the intensity and q-q plot for models

Usage

intensityqqplot(object, events, markov_states)

## Default S3 method:
intensityqqplot(object, events, markov_states)

## S3 method for class 'hp'
intensityqqplot(object, events, markov_states = NULL)

## S3 method for class 'hpp'
intensityqqplot(object, events, markov_states = NULL)

## S3 method for class 'mmpp'
intensityqqplot(object, events = markov_states$events, markov_states)

## S3 method for class 'mmhp'
intensityqqplot(object, events = markov_states$events, markov_states)

Arguments

object	parameters for the models: hp, hpp, and mmhp
events	event times
markov_states	only for mmhp and mmpp, markov states simulation output

Value

no return value, intensity and qq-plot in a single plot

---

Compensators for point processes

Description

Computes the compensator for included point processes

Usage

pp_compensator(object, events)

## Default S3 method:
pp_compensator(object, events)

## S3 method for class 'mmpp'
pp_compensator(object, events)

## S3 method for class 'hp'
pp_compensator(object, events)

## S3 method for class 'mmhp'
pp_compensator(object, events)

## S3 method for class 'hpp'
pp_compensator(object, events)

Arguments

object	a point process model
events	event times, which can have first value as 0

Value

compensator vector of rescaled interevent times

Examples

hpp_obj <- pp_hpp(lambda = 1)
events <- pp_simulate(hpp_obj, end = 10)
comp <- pp_compensator(hpp_obj, events)

---

Summarise diagnostics for point process models

Description

Generate diagnostic tools for different point process models, including quantile-quantile plot, ks plot, raw residual and pearson residual.

Usage

pp_diag(object, events)

## Default S3 method:
pp_diag(object, events)

## S3 method for class 'hp'
pp_diag(object, events)

## S3 method for class 'mmhp'
pp_diag(object, events)

## S3 method for class 'mmpp'
pp_diag(object, events)

## S3 method for class 'hpp'
pp_diag(object, events)

Arguments

object	a point process model
events	event times

Value

Invisibly returns NULL. Outputs plots and summary of diagnostics to console

Examples

hpp_obj <- pp_hpp(lambda = 1)
events <- pp_simulate(hpp_obj, end = 50)
pp_diag(hpp_obj, events)

---

Create a Hawkes process object

Description

Create a Hawkes Process with an exponential kernel according to the given parameters: lambda0, alpha, beta and events. If events are missing, then it means that data will be added later(simulated from this process)

Usage

pp_hp(lambda0, alpha, beta, events = NULL)

Arguments

lambda0	initial intensity at the start time
alpha	jump size in increase of intensity
beta	exponential decay of intensity
events	vector containing the event times. Note that the first event is at time zero. Alternatively, events could be specified as NULL, meaning that the data will be added later (e.g. simulated).

Value

hp object

Examples

pp_hp(lambda0 = 0.1, alpha = 0.45, beta = 0.5)

---

Create a homogeneous Poisson process object

Description

Create a homogeneous Poisson object according to given parameters: lambda, and events. If events are missing, then it means that data will be added later(simulated from this process).

Usage

pp_hpp(lambda, events = NULL)

Arguments

lambda	rate of the Poisson process
events	event times, optional

Value

hpp object

Examples

pp_hpp(lambda = 1)

---

KS plot of empirical and theoretical cdf curve of fitted point process

Description

Plot empirical cdf plot for rescaled-inter-event-times and exponential cdf as a reference curve

Usage

pp_ksplot(r, ...)

Arguments

r	rescaled-inter-event-times
...	other arguments for plots

Value

no return value, KS plot for rescaled-inter-event-times and exponential cdf curve

---

Create a Markov-modulated Hawkes Process(MMHP) object

Description

Create a Markov-modulated Hawkes Process(MMHP) model according to the given parameters: lambda0, lambda1, alpha, beta, event times and transition probability matrix. If event time events is missing, then it means that data will be added later(e.g. simulated)

Usage

pp_mmhp(lambda0, lambda1, alpha, beta, Q = NULL, delta = NULL, events = NULL)

Arguments

lambda0	intensity for homogeneous Poisson process.
lambda1	base intensity for Hawkes process.
alpha	jump size of the increase in intensity in the hawkes process
beta	exponential decrease of intensity in the hawkes process
Q	transition probability matrix.
delta	initial state probability.
events	vector containing the event times. Note that the first event is at time zero. Alternatively, events could be specified as NULL, meaning that the data will be added later (e.g. simulated).

Value

mmhp object

Examples

Q <- matrix(c(-0.4, 0.4, 0.2, -0.2), ncol = 2, byrow = TRUE)
pp_mmhp(Q,
  delta = c(1 / 3, 2 / 3), lambda0 = 0.9, lambda1 = 1.1,
  alpha = 0.8, beta = 1.2
)

---

Create a Markov-modulated Poisson Process(MMPP) object

Description

Create a Markov-modulated Poisson Process(MMPP) model according to the given parameters: lambda0, c, q1, q2 and event times. If event time tau is missing, then it means that data will be added later(e.g. simulated)

Usage

pp_mmpp(lambda0, c, Q, events = NULL, delta = NULL)

Arguments

lambda0	parameters for Poisson process.
c	the proportion of intensity 1 over intensity 2
Q	transition probability matrix
events	vector containing the event times. Note that the first event is often specified as zero. Alternatively, events could be specified as NULL, meaning that the data will be added later (e.g. simulated).
delta	initial state probability.

Value

mmpp object

Examples

Q <- matrix(c(-0.4, 0.4, 0.2, -0.2), ncol = 2, byrow = TRUE)
pp_mmpp(Q = Q, lambda0 = 1, c = 1.5, delta = c(1 / 3, 2 / 3))

---

Plot QQ-plot for rescaled-inter-event-times of fitted point process

Description

Generate Quantile-quantile plot for rescaled-inter-event-times, which are independently and identically distributed as exponential random variables with rate 1 under the true point process.

Usage

pp_qqexp(r, ...)

Arguments

r	rescaled-inter-event-times
...	other arguments for plots

Value

no return value, quantile-quantile plot for rescaled-inter-event-times

---

Compute raw and pearson residuals for point process models

Description

Compute raw and pearson residuals for point process models

Usage

pp_residual(object, events, start = 0, end = max(events), steps = 1000)

Arguments

object	point process model containing the parameters
events	vector of event times
start	start of observation period (default 0)
end	end of observation period (default final event)
steps	number of steps for numeric integration (if needed)

Value

the raw and pearson residuals

Examples

Q <- matrix(c(-0.4, 0.4, 0.2, -0.2), ncol = 2, byrow = TRUE)
x <- pp_mmhp(Q,
  delta = c(1 / 3, 2 / 3), lambda0 = 0.9,
  lambda1 = 1.1, alpha = 0.8, beta = 1.2
)
y <- pp_simulate(x, n = 10)
pp_residual(x, events = y$events)

---

Simulate events from a temporal point process

Description

Currently available point processes are homogeneous Poisson, Hawkes with exponential kernel, MMHP and MMPP

Usage

pp_simulate(object, start = 0, end = 1, n = NULL, verbose = FALSE)

## Default S3 method:
pp_simulate(object, start = 0, end = 1, n = NULL, verbose = FALSE)

## S3 method for class 'hpp'
pp_simulate(object, start = 0, end = 1, n = NULL, verbose = FALSE)

## S3 method for class 'hp'
pp_simulate(object, start = 0, end = 1, n = NULL, verbose = FALSE)

## S3 method for class 'mmpp'
pp_simulate(object, start = 0, end = 1, n = NULL, verbose = FALSE)

## S3 method for class 'mmhp'
pp_simulate(object, start = 0, end = 1, n = NULL, verbose = FALSE)

Arguments

object	point process model object of type hpp, hp, mmhp, or mmpp
start	start time of events simulated. Not used for Markov modulated models
end	end time of events simulated. Not used for Markov modulated models
n	number of events simulated. Required for Markov modulated models, optional otherwise
verbose	whether to output informative messages as running

Value

a vector of event times for all models. For Markov modulated models, also returns details on the underlying latent process

Examples

hpp_obj <- pp_hpp(lambda = 1)
s <- pp_simulate(hpp_obj, n = 50)

---