Version:	1.0.0
Title:	PH/MAP Parameter Estimation
Type:	Package
Maintainer:	Hiroyuki Okamura <okamu@hiroshima-u.ac.jp>
Description:	Estimation methods for phase-type distribution (PH) and Markovian arrival process (MAP) from empirical data (point and grouped data) and density function. The tool is based on the following researches: Okamura et al. (2009) <doi:10.1109/TNET.2008.2008750>, Okamura and Dohi (2009) <doi:10.1109/QEST.2009.28>, Okamura et al. (2011) <doi:10.1016/j.peva.2011.04.001>, Okamura et al. (2013) <doi:10.1002/asmb.1919>, Horvath and Okamura (2013) <doi:10.1007/978-3-642-40725-3_10>, Okamura and Dohi (2016) <doi:10.15807/jorsj.59.72>.
Encoding:	UTF-8
License:	MIT + file LICENSE
RoxygenNote:	7.2.2
Imports:	R6, deformula, Matrix, methods, Rcpp
URL:	https://github.com/okamumu/mapfit
BugReports:	https://github.com/okamumu/mapfit/issues
Suggests:	covr, ggplot2, testthat (≥ 3.0.0)
LinkingTo:	Rcpp
Config/testthat/edition:	3
NeedsCompilation:	yes
Packaged:	2022-11-22 08:02:30 UTC; r346502
Author:	Hiroyuki Okamura [aut, cre]
Repository:	CRAN
Date/Publication:	2022-11-22 14:50:02 UTC

mapfit: PH/MAP Parameter Estimation

Description

Estimation methods for phase-type distribution (PH) and Markovian arrival process (MAP) from empirical data (point and grouped data) and density function. The tool is based on the following researches: Okamura et al. (2009) doi:10.1109/TNET.2008.2008750, Okamura and Dohi (2009) doi:10.1109/QEST.2009.28, Okamura et al. (2011) doi:10.1016/j.peva.2011.04.001, Okamura et al. (2013) doi:10.1002/asmb.1919, Horvath and Okamura (2013) doi:10.1007/978-3-642-40725-3_10, Okamura and Dohi (2016) doi:10.15807/jorsj.59.72.

Author(s)

Maintainer: Hiroyuki Okamura okamu@hiroshima-u.ac.jp (ORCID)

See Also

Useful links:

• https://github.com/okamumu/mapfit

• Report bugs at https://github.com/okamumu/mapfit/issues

ErlangHMM for MAP with fixed phases

Description

ErlangHMM for MAP with fixed phases

ErlangHMM for MAP with fixed phases

Details

A special case of MAP.

Methods

Public methods

• AERHMMClass$alpha()

• AERHMMClass$shape()

• AERHMMClass$rate()

• AERHMMClass$P()

• AERHMMClass$xi()

• AERHMMClass$new()

• AERHMMClass$copy()

• AERHMMClass$size()

• AERHMMClass$df()

• AERHMMClass$print()

• AERHMMClass$mmoment()

• AERHMMClass$jmoment()

• AERHMMClass$acf()

• AERHMMClass$emfit()

• AERHMMClass$init()

• AERHMMClass$clone()

Method alpha()

Get alpha

Usage

AERHMMClass$alpha()

Returns

A vector of alpha

Method shape()

Get shape

Usage

AERHMMClass$shape()

Returns

A vector of shapes

Method rate()

Get rate

Usage

AERHMMClass$rate()

Returns

A vector of rates

Method P()

Get P

Usage

AERHMMClass$P()

Returns

A matrix of P

Method xi()

Get exit rates

Usage

AERHMMClass$xi()

Returns

A vector of exit rates

Method new()

Create an AERHMM

Usage

AERHMMClass$new(size, erhmm)

Arguments

Returns

An instance of AERHMM

Method copy()

copy

Usage

AERHMMClass$copy()

Returns

A new instance

Method size()

The number of components

Usage

AERHMMClass$size()

Returns

The number of components

Method df()

Degrees of freedom

Usage

AERHMMClass$df()

Returns

The degrees of freedom

Method print()

Print

Usage

AERHMMClass$print(...)

Arguments

Method mmoment()

Marginal moments

Usage

AERHMMClass$mmoment(k, ...)

Arguments

Returns

A vector of moments

Method jmoment()

Joint moments

Usage

AERHMMClass$jmoment(lag, ...)

Arguments

Returns

A matrix of moments

Method acf()

k-lag correlation

Usage

AERHMMClass$acf(...)

Arguments

Returns

A vector for k-lag correlation

Method emfit()

Run EM

Usage

AERHMMClass$emfit(data, options, ...)

Arguments

Method init()

Initialize with data

Usage

AERHMMClass$init(data, ...)

Arguments

Method clone()

The objects of this class are cloneable with this method.

Usage

AERHMMClass$clone(deep = FALSE)

Arguments

Hyper-Erlang distribution with a fixed phase

Description

Hyper-Erlang distribution with a fixed phase

Hyper-Erlang distribution with a fixed phase

Details

A mixture of Erlang distributions. A subclass of PH distributions.

Methods

Public methods

• AHerlangClass$mixrate()

• AHerlangClass$shape()

• AHerlangClass$rate()

• AHerlangClass$new()

• AHerlangClass$copy()

• AHerlangClass$size()

• AHerlangClass$df()

• AHerlangClass$moment()

• AHerlangClass$print()

• AHerlangClass$pdf()

• AHerlangClass$cdf()

• AHerlangClass$ccdf()

• AHerlangClass$sample()

• AHerlangClass$emfit()

• AHerlangClass$init()

• AHerlangClass$clone()

Method mixrate()

Get mixrate

Usage

AHerlangClass$mixrate()

Returns

A vector of mixrate

Method shape()

Get shape

Usage

AHerlangClass$shape()

Returns

A vector of shapes

Method rate()

Get rate

Usage

AHerlangClass$rate()

Returns

A vector of rates

Method new()

Create a hyper-Erlang distribution with fixed phases

Usage

AHerlangClass$new(size, herlang)

Arguments

Returns

An instance of AHerlang

Method copy()

copy

Usage

AHerlangClass$copy()

Returns

A new instance

Method size()

The number of components

Usage

AHerlangClass$size()

Returns

The number of components

Method df()

Degrees of freedom

Usage

AHerlangClass$df()

Returns

The degrees of freedom

Method moment()

Moments of HErlang

Usage

AHerlangClass$moment(k, ...)

Arguments

Returns

A vector of moments from 1st to k-th moments

Method print()

Print

Usage

AHerlangClass$print(...)

Arguments

Method pdf()

PDF

Usage

AHerlangClass$pdf(x, ...)

Arguments

Returns

A vector of densities.

Method cdf()

CDF

Usage

AHerlangClass$cdf(q, ...)

Arguments

Returns

A vector of probabilities

Method ccdf()

Complementary CDF

Usage

AHerlangClass$ccdf(q, ...)

Arguments

Returns

A vector of probabilities

Method sample()

Make a sample

Usage

AHerlangClass$sample(...)

Arguments

Returns

A sample of HErlang

Method emfit()

Run EM

Usage

AHerlangClass$emfit(data, options, ...)

Arguments

Method init()

Initialize with data

Usage

AHerlangClass$init(data, ...)

Arguments

Method clone()

The objects of this class are cloneable with this method.

Usage

AHerlangClass$clone(deep = FALSE)

Arguments

Packet Trace Data

Description

The data contains packet arrivals seen on an Ethernet at the Bellcore Morristown Research and Engineering facility. Two of the traces are LAN traffic (with a small portion of transit WAN traffic), and two are WAN traffic. The original trace BC-pAug89 began at 11:25 on August 29, 1989, and ran for about 3142.82 seconds (until 1,000,000 packets had been captured). The trace BC-pOct89 began at 11:00 on October 5, 1989, and ran for about 1759.62 seconds. These two traces captured all Ethernet packets. The number of arrivals in the original trace is one million.

Format

BCpAug89 is a vector for the inter-arrival time in seconds for 1000 arrivals.

Source

The original trace data are published in http://ita.ee.lbl.gov/html/contrib/BC.html.

Canonical phase-type distribution

Description

Canonical phase-type distribution

Canonical phase-type distribution

Details

A continuous distribution dominated by a continuous-time Markov chain. A random time is given by an absorbing time. In the CF1 (canonical form 1), the infinitesimal generator is given by a bi-diagonal matrix, and whose order is determined by the ascending order.

Super class

mapfit::GPHClass -> CF1Class

Methods

Public methods

• CF1Class$rate()

• CF1Class$new()

• CF1Class$copy()

• CF1Class$print()

• CF1Class$sample()

• CF1Class$emfit()

• CF1Class$init()

• CF1Class$clone()

Method rate()

Get rate

Usage

CF1Class$rate()

Returns

An instance of rate

Method new()

Create a CF1

Usage

CF1Class$new(alpha, rate)

Arguments

Returns

An instance of CF1

Method copy()

copy

Usage

CF1Class$copy()

Returns

A new instance

Method print()

Print

Usage

CF1Class$print(...)

Arguments

Method sample()

Generate a sample of CF1

Usage

CF1Class$sample(...)

Arguments

Returns

A sample of CF1

Method emfit()

Run EM

Usage

CF1Class$emfit(data, options, ...)

Arguments

Method init()

Initialize with data

Usage

CF1Class$init(data, options, ...)

Arguments

Method clone()

The objects of this class are cloneable with this method.

Usage

CF1Class$clone(deep = FALSE)

Arguments

ErlangHMM for MAP

Description

ErlangHMM for MAP

ErlangHMM for MAP

Details

A special case of MAP.

Methods

Public methods

• ERHMMClass$alpha()

• ERHMMClass$shape()

• ERHMMClass$rate()

• ERHMMClass$P()

• ERHMMClass$xi()

• ERHMMClass$new()

• ERHMMClass$copy()

• ERHMMClass$size()

• ERHMMClass$df()

• ERHMMClass$print()

• ERHMMClass$mmoment()

• ERHMMClass$jmoment()

• ERHMMClass$acf()

• ERHMMClass$emfit()

• ERHMMClass$init()

• ERHMMClass$clone()

Method alpha()

Get alpha

Usage

ERHMMClass$alpha()

Returns

A vector of alpha

Method shape()

Get shape

Usage

ERHMMClass$shape()

Returns

A vector of shapes

Method rate()

Get rate

Usage

ERHMMClass$rate()

Returns

A vector of rates

Method P()

Get P

Usage

ERHMMClass$P()

Returns

A matrix of P

Method xi()

Get exit rates

Usage

ERHMMClass$xi()

Returns

A vector of exit rates

Method new()

Create an ERHMM

Usage

ERHMMClass$new(alpha, shape, rate, P, xi)

Arguments

Returns

An instance of ERHMM

Method copy()

copy

Usage

ERHMMClass$copy()

Returns

A new instance

Method size()

The number of components

Usage

ERHMMClass$size()

Returns

The number of components

Method df()

Degrees of freedom

Usage

ERHMMClass$df()

Returns

The degrees of freedom

Method print()

Print

Usage

ERHMMClass$print(...)

Arguments

Method mmoment()

Marginal moments

Usage

ERHMMClass$mmoment(k, ...)

Arguments

Returns

A vector of moments

Method jmoment()

Joint moments

Usage

ERHMMClass$jmoment(lag, ...)

Arguments

Returns

A matrix of moments

Method acf()

k-lag correlation

Usage

ERHMMClass$acf(...)

Arguments

Returns

A vector for k-lag correlation

Method emfit()

Run EM

Usage

ERHMMClass$emfit(data, options, ...)

Arguments

Method init()

Initialize with data

Usage

ERHMMClass$init(data, ...)

Arguments

Method clone()

The objects of this class are cloneable with this method.

Usage

ERHMMClass$clone(deep = FALSE)

Arguments

GMMPP: Approximation for MAP

Description

GMMPP: Approximation for MAP

GMMPP: Approximation for MAP

Details

A point process dominated by a continuous-time Markov chain.

Super class

mapfit::MAPClass -> GMMPPClass

Methods

Public methods

• GMMPPClass$new()

• GMMPPClass$copy()

• GMMPPClass$emfit()

• GMMPPClass$clone()

Method new()

Create a MAP

Usage

GMMPPClass$new(alpha, D0, D1, xi)

Arguments

Returns

An instance of MAP

Method copy()

copy

Usage

GMMPPClass$copy()

Returns

A new instance

Method emfit()

Run EM

Usage

GMMPPClass$emfit(data, options, ...)

Arguments

Method clone()

The objects of this class are cloneable with this method.

Usage

GMMPPClass$clone(deep = FALSE)

Arguments

General phase-type distribution

Description

General phase-type distribution

General phase-type distribution

Details

A continuous distribution dominated by a continuous-time Markov chain. A random time is given by an absorbing time.

Methods

Public methods

• GPHClass$alpha()

• GPHClass$Q()

• GPHClass$xi()

• GPHClass$new()

• GPHClass$copy()

• GPHClass$size()

• GPHClass$df()

• GPHClass$moment()

• GPHClass$print()

• GPHClass$pdf()

• GPHClass$cdf()

• GPHClass$ccdf()

• GPHClass$sample()

• GPHClass$emfit()

• GPHClass$init()

• GPHClass$clone()

Method alpha()

Get alpha

Usage

GPHClass$alpha()

Returns

A vector of alpha

Method Q()

Get Q

Usage

GPHClass$Q()

Returns

A matrix of Q

Method xi()

Get xi

Usage

GPHClass$xi()

Returns

A vector of xi

Method new()

Create a GPH

Usage

GPHClass$new(alpha, Q, xi)

Arguments

Returns

An instance of GPH

Method copy()

copy

Usage

GPHClass$copy()

Returns

A new instance

Method size()

The number of phases

Usage

GPHClass$size()

Returns

The number of phases

Method df()

Degrees of freedom

Usage

GPHClass$df()

Returns

The degrees of freedom

Method moment()

Moments of GPH

Usage

GPHClass$moment(k, ...)

Arguments

Returns

A vector of moments from 1st to k-th moments

Method print()

Print

Usage

GPHClass$print(...)

Arguments

Method pdf()

PDF

Usage

GPHClass$pdf(x, poisson.eps = 1e-08, ufactor = 1.01, ...)

Arguments

Returns

A vector of densities.

Method cdf()

CDF

Usage

GPHClass$cdf(x, poisson.eps = 1e-08, ufactor = 1.01, ...)

Arguments

Returns

A vector of probabilities

Method ccdf()

Complementary CDF

Usage

GPHClass$ccdf(x, poisson.eps = 1e-08, ufactor = 1.01, ...)

Arguments

Returns

A vector of probabilities

Method sample()

Make a sample

Usage

GPHClass$sample(...)

Arguments

Returns

A sample of GPH

Method emfit()

Run EM

Usage

GPHClass$emfit(data, options, ...)

Arguments

Method init()

Initialize with data

Usage

GPHClass$init(data, ...)

Arguments

Method clone()

The objects of this class are cloneable with this method.

Usage

GPHClass$clone(deep = FALSE)

Arguments

Note

This function provides the values of p.d.f. for PH distribution with the uniformization technique.

This function provides the values of c.d.f. for PH distribution with the uniformization technique.

This function provides the values of complementary c.d.f. for PH distribution with the uniformization technique.

Hyper-Erlang distribution

Description

Hyper-Erlang distribution

Hyper-Erlang distribution

Details

A mixture of Erlang distributions. A subclass of PH distributions.

Methods

Public methods

• HErlangClass$mixrate()

• HErlangClass$shape()

• HErlangClass$rate()

• HErlangClass$new()

• HErlangClass$copy()

• HErlangClass$size()

• HErlangClass$df()

• HErlangClass$moment()

• HErlangClass$print()

• HErlangClass$pdf()

• HErlangClass$cdf()

• HErlangClass$ccdf()

• HErlangClass$sample()

• HErlangClass$emfit()

• HErlangClass$init()

• HErlangClass$clone()

Method mixrate()

Get mixrate

Usage

HErlangClass$mixrate()

Returns

A vector of mixrate

Method shape()

Get shape

Usage

HErlangClass$shape()

Returns

A vector of shapes

Method rate()

Get rate

Usage

HErlangClass$rate()

Returns

A vector of rates

Method new()

Create a hyper-Erlang distribution

Usage

HErlangClass$new(mixrate, shape, rate)

Arguments

Returns

An instance of HErlang

Method copy()

copy

Usage

HErlangClass$copy()

Returns

A new instance

Method size()

The number of components

Usage

HErlangClass$size()

Returns

The number of components

Method df()

Degrees of freedom

Usage

HErlangClass$df()

Returns

The degrees of freedom

Method moment()

Moments of HErlang

Usage

HErlangClass$moment(k, ...)

Arguments

Returns

A vector of moments from 1st to k-th moments

Method print()

Print

Usage

HErlangClass$print(...)

Arguments

Method pdf()

PDF

Usage

HErlangClass$pdf(x, ...)

Arguments

Returns

A vector of densities.

Method cdf()

CDF

Usage

HErlangClass$cdf(q, ...)

Arguments

Returns

A vector of probabilities

Method ccdf()

Complementary CDF

Usage

HErlangClass$ccdf(q, ...)

Arguments

Returns

A vector of probabilities

Method sample()

Make a sample

Usage

HErlangClass$sample(...)

Arguments

Returns

A sample of HErlang

Method emfit()

Run EM

Usage

HErlangClass$emfit(data, options, ...)

Arguments

Method init()

Initialize with data

Usage

HErlangClass$init(data, ...)

Arguments

Method clone()

The objects of this class are cloneable with this method.

Usage

HErlangClass$clone(deep = FALSE)

Arguments

General Markovian arrival process

Description

General Markovian arrival process

General Markovian arrival process

Details

A point process dominated by a continuous-time Markov chain.

Methods

Public methods

• MAPClass$alpha()

• MAPClass$D0()

• MAPClass$D1()

• MAPClass$xi()

• MAPClass$new()

• MAPClass$copy()

• MAPClass$size()

• MAPClass$df()

• MAPClass$print()

• MAPClass$mmoment()

• MAPClass$jmoment()

• MAPClass$acf()

• MAPClass$emfit()

• MAPClass$init()

• MAPClass$clone()

Method alpha()

Get alpha

Usage

MAPClass$alpha()

Returns

A vector of alpha

Method D0()

Get D0

Usage

MAPClass$D0()

Returns

A matrix of D0

Method D1()

Get D1

Usage

MAPClass$D1()

Returns

A matrix of D1

Method xi()

Get xi

Usage

MAPClass$xi()

Returns

A vector of xi

Method new()

Create a MAP

Usage

MAPClass$new(alpha, D0, D1, xi)

Arguments

Returns

An instance of MAP

Method copy()

copy

Usage

MAPClass$copy()

Returns

A new instance

Method size()

The number of phases

Usage

MAPClass$size()

Returns

The number of phases

Method df()

Degrees of freedom

Usage

MAPClass$df()

Returns

The degrees of freedom

Method print()

Print

Usage

MAPClass$print(...)

Arguments

Method mmoment()

Marginal moments

Usage

MAPClass$mmoment(k, ...)

Arguments

Returns

A vector of moments

Method jmoment()

Joint moments

Usage

MAPClass$jmoment(lag, ...)

Arguments

Returns

A matrix of moments

Method acf()

k-lag correlation

Usage

MAPClass$acf(...)

Arguments

Returns

A vector for k-lag correlation

Method emfit()

Run EM

Usage

MAPClass$emfit(data, options, ...)

Arguments

Method init()

Initialize with data

Usage

MAPClass$init(data, ...)

Arguments

Method clone()

The objects of this class are cloneable with this method.

Usage

MAPClass$clone(deep = FALSE)

Arguments

Convert from HErlang to GPH

Description

Convert from hyper-Erlang distribution to the general PH distribution

Usage

as.gph(h)

Arguments

Value

An instance of GPH

Examples

#' ## create a hyper Erlang with specific parameters
(param <- herlang(shape=c(2,3), mixrate=c(0.3,0.7), rate=c(1.0,10.0)))

## convert to a general PH
as.gph(param)

Convert from ERHMM to MAP

Description

Convert from ERHMM to the general MAP

Usage

as.map(x)

Arguments

Value

An instance of MAP

Examples

## create a hyper Erlang with specific parameters
(param <- erhmm(shape=c(2,3), alpha=c(0.3,0.7), rate=c(1.0,10.0)))

## convert to a general PH
as.map(param)

Create CF1

Description

Create an instance of CF1.

Usage

cf1(size, alpha, rate)

Arguments

Value

An instance of CF1.

Examples

## create a CF1 with 5 phases
(param1 <- cf1(5))

## create a CF1 with 5 phases
(param1 <- cf1(size=5))

## create a CF1 with specific parameters
(param2 <- cf1(alpha=c(1,0,0), rate=c(1.0,2.0,3.0)))

Create CF1 with data information

Description

Crate CF1 with the first moment of a given data. This function calls cf1.param.linear and cf1.param.power to determine CF1. After execute 5 EM steps, the model with the smallest LLF is selected.

Usage

cf1.param(data, size, options, ...)

Arguments

Examples

## Generate group data
dat <- data.frame.phase.group(c(1,2,0,4), seq(0,10,length.out=5))

## Create an instance of CF1
p <- cf1.param(data=dat, size=5)

Determine CF1 parameters

Description

Determine CF1 parameters based on the linear rule.

Usage

cf1.param.linear(size, mean, s)

Arguments

Value

A list of alpha and rate

Determine CF1 parameters

Description

Determine CF1 parameters based on the power rule.

Usage

cf1.param.power(size, mean, s)

Arguments

Value

A list of alpha and rate

Markov stationary

Description

Compute the stationary vector with GTH

Usage

ctmc.st(Q)

Arguments

Value

The stationary vector of DTMC/CTMC

Create group data for map

Description

Provide the data.frame for group data.

Usage

data.frame.map.group(counts, breaks, intervals, instants)

Arguments

Value

A dataframe

Examples

t <- c(1,1,1,1,1)
n <- c(1,3,0,0,1)

dat <- data.frame.map.group(counts=n, intervals=t)
mean(dat)
print(dat)

Create data for map

Description

Provide a data.frame with samples.

Usage

data.frame.map.time(time, intervals)

Arguments

Value

A dataframe

Note

• If both time and intervals are used, time is used.

• map.time is given by a special case of map.group.

Examples

x <- runif(10)

dat <- data.frame.map.time(time=x)
mean(dat)
print(dat)

Create group data for phase

Description

Provide the data.frame for group data.

Usage

data.frame.phase.group(counts, breaks, intervals, instants)

Arguments

Value

A dataframe

Examples

dat <- data.frame.phase.group(counts=c(1,2,1,1,0,0,1,4))
print(dat)
mean(dat)

Create data for phase with weighted sample

Description

Provide a data.frame with weighted samples.

Usage

data.frame.phase.time(x, weights)

Arguments

Value

A dataframe

Note

The point time is sorted and their differences are stored as the column of time

Examples

x <- runif(10)
w <- runif(10)

dat <- data.frame.phase.time(x=x, weights=w)
print(dat)
mean(dat)

Probability density function of PH distribution

Description

Compute the probability density function (p.d.f.) for a given PH distribution

Usage

dphase(x, ph, log = FALSE, ...)

Arguments

Value

A vector of densities.

Examples

## create a PH with specific parameters
(phdist <- ph(alpha=c(1,0,0),
              Q=rbind(c(-4,2,0),c(2,-5,1),c(1,0,-1)),
              xi=c(2,2,0)))

## p.d.f. for 0, 0.1, ..., 1
dphase(x=seq(0, 1, 0.1), ph=phdist)

EM Options

Description

A list of options for EM

Usage

emoptions()

Value

A list of options with default values

Create ERHMM

Description

Create an instance of ERHMM

Usage

erhmm(
  size,
  shape,
  alpha = rep(1/length(shape), length(shape)),
  rate = rep(1, length(shape)),
  P = matrix(1/length(shape), length(shape), length(shape))
)

Arguments

Value

An instance of ERHMM

Note

If shape is given, shape is used even though size is set.

Determine ERHMM parameters

Description

Determine ERHMM parameters with k-means.

Usage

erhmm.param(data, skel, ...)

Arguments

Value

An instance of ERHMM

Create GMMPP

Description

Create an instance of GMMPP

Usage

gmmpp(size, alpha, D0, D1)

Arguments

Value

An instance of GMMPP

Note

This function can omit several patterns of arguments. For example, map(5) omit the arguments alpha, Q and xi. In this case, the default values are assigned to them.

Examples

## create a map (full matrix) with 5 phases
(param1 <- gmmpp(5))

## create a map with specific parameters
(param2 <- gmmpp(alpha=c(1,0,0),
              D0=rbind(c(-4,2,0),c(2,-5,1),c(1,0,-1)),
              D1=rbind(c(2,0,0),c(0,2,0),c(0,0,0))))

Generate GPH using the information on data

Description

Generate GPH randomly and adjust parameters to fit its first moment to the first moment of data.

Usage

gph.param(data, skel, ...)

Arguments

Value

An instance of GPH

Examples

## Create data
wsample <- rweibull(10, shape=2)
(dat <- data.frame.phase.time(x=wsample))

## Generate PH that is fitted to dat
(model <- gph.param(data=dat, skel=ph(5)))

Create HErlang distribution

Description

Create an instance of Hyper-Erlang distribution

Usage

herlang(
  size,
  shape,
  mixrate = rep(1/length(shape), length(shape)),
  rate = rep(1, length(shape))
)

Arguments

Value

An instance of HErlang

Note

If shape is given, shape is used even though size is set.

Examples

## create a hyper Erlang consisting of two Erlang
## with shape parameters 2 and 3.
(param1 <- herlang(shape=c(2,3)))

## create a hyper Erlang with specific parameters
(param2 <- herlang(shape=c(2,3), mixrate=c(0.3,0.7), rate=c(1.0,10.0)))

## convert to a general PH
as.gph(param2)

## p.d.f. for 0, 0.1, ..., 1
(dphase(x=seq(0, 1, 0.1), ph=param2))

## c.d.f. for 0, 0.1, ..., 1
(pphase(q=seq(0, 1, 0.1), ph=param2))

## generate 10 samples
(rphase(n=10, ph=param2))

Determine hyper-Erlang parameters

Description

Determine the hyper-Erlang parameters with k-means.

Usage

herlang.param(data, shape, ...)

Arguments

Value

An instance of HErlang

Examples

## Create data
wsample <- rweibull(10, shape=2)
(dat <- data.frame.phase.time(x=wsample))

## Generate PH that is fitted to dat
(model <- herlang.param(data=dat, shape=c(1,2,3)))

Create MAP

Description

Create an instance of MAP

Usage

map(size, alpha, D0, D1)

Arguments

Value

An instance of MAP

Note

This function can omit several patterns of arguments. For example, map(5) omit the arguments alpha, D0 D1 and xi. In this case, the default values are assigned to them.

Examples

## create a map (full matrix) with 5 phases
(param1 <- map(5))

## create a map with specific parameters
(param2 <- map(alpha=c(1,0,0),
              D0=rbind(c(-4,2,0),c(2,-5,1),c(1,0,-1)),
              D1=rbind(c(2,0,0),c(0,2,0),c(0,0,0))))

k-lag correlation of MAP

Description

Compute k-lag correlation

Usage

map.acf(map, ...)

Arguments

Value

A vector of k-lag correlation

Examples

## create an MAP with specific parameters
(param1 <- map(alpha=c(1,0,0),
               D0=rbind(c(-4,2,0),c(2,-5,1),c(1,0,-4)),
               D1=rbind(c(1,1,0),c(1,0,1),c(2,0,1))))

## create an ER-HMM with specific parameters
(param2 <- erhmm(shape=c(2,3), alpha=c(0.3,0.7),
                 rate=c(1.0,10.0),
                 P=rbind(c(0.3, 0.7), c(0.1, 0.9))))

map.acf(map=param1)
map.acf(map=param2)

Joint moments of MAP

Description

Compute joint moments for a given MAP

Usage

map.jmoment(lag, map, ...)

Arguments

Value

A vector of moments

Examples

## create an MAP with specific parameters
(param1 <- map(alpha=c(1,0,0),
               D0=rbind(c(-4,2,0),c(2,-5,1),c(1,0,-4)),
               D1=rbind(c(1,1,0),c(1,0,1),c(2,0,1))))

## create an ER-HMM with specific parameters
(param2 <- erhmm(shape=c(2,3), alpha=c(0.3,0.7),
                 rate=c(1.0,10.0),
                 P=rbind(c(0.3, 0.7), c(0.1, 0.9))))

map.jmoment(lag=1, map=param1)
map.jmoment(lag=1, map=param2)

Marginal moments of MAP

Description

Compute up to k-th marginal moments for a given MAP

Usage

map.mmoment(k, map, ...)

Arguments

Value

A vector of moments

Examples

## create an MAP with specific parameters
(param1 <- map(alpha=c(1,0,0),
               D0=rbind(c(-4,2,0),c(2,-5,1),c(1,0,-4)),
               D1=rbind(c(1,1,0),c(1,0,1),c(2,0,1))))

## create an ER-HMM with specific parameters
(param2 <- erhmm(shape=c(2,3), alpha=c(0.3,0.7),
                 rate=c(1.0,10.0),
                 P=rbind(c(0.3, 0.7), c(0.1, 0.9))))

map.mmoment(k=3, map=param1)
map.mmoment(k=3, map=param2)

Generate MAP using the information on data

Description

Generate MAP randomly and adjust parameters to fit its first moment to the first moment of data.

Usage

map.param(data, skel, ...)

Arguments

Value

An instance of MAP

MAP fitting with grouped data

Description

Estimates MAP parameters from grouped data.

Usage

mapfit.group(map, counts, breaks, intervals, instants, ...)

Arguments

Value

Returns a list with components, which is an object of S3 class mapfit.result;

Examples

## load trace data
data(BCpAug89)
BCpAug89s <- head(BCpAug89, 50)

## make grouped data
BCpAug89.group <- hist(cumsum(BCpAug89s),
                         breaks=seq(0, 0.15, 0.005),
                         plot=FALSE)
                         
## MAP fitting for general MAP
(result1 <- mapfit.group(map=map(2),
                        counts=BCpAug89.group$counts,
                        breaks=BCpAug89.group$breaks))
## MAP fitting for MMPP
(result2 <- mapfit.group(map=mmpp(2),
                         counts=BCpAug89.group$counts,
                         breaks=BCpAug89.group$breaks))
                         
## MAP fitting with approximate MMPP
(result3 <- mapfit.group(map=gmmpp(2),
                         counts=BCpAug89.group$counts,
                         breaks=BCpAug89.group$breaks))

## marginal moments for estimated MAP
map.mmoment(k=3, map=result1$model)
map.mmoment(k=3, map=result2$model)
map.mmoment(k=3, map=result3$model)

## joint moments for estimated MAP
map.jmoment(lag=1, map=result1$model)
map.jmoment(lag=1, map=result2$model)
map.jmoment(lag=1, map=result3$model)

## lag-k correlation
map.acf(map=result1$model)
map.acf(map=result2$model)
map.acf(map=result3$model)

MAP fitting with point data

Description

Estimates MAP parameters from point data.

Usage

mapfit.point(map, x, intervals, ...)

Arguments

Value

Returns a list with components, which is an object of S3 class mapfit.result;

Examples

## load trace data
data(BCpAug89)
BCpAug89s <- head(BCpAug89, 50)

## MAP fitting for general MAP
(result1 <- mapfit.point(map=map(2), x=cumsum(BCpAug89s)))

## MAP fitting for MMPP
(result2 <- mapfit.point(map=mmpp(2), x=cumsum(BCpAug89s)))

## MAP fitting for ER-HMM
(result3 <- mapfit.point(map=erhmm(3), x=cumsum(BCpAug89s)))

## marginal moments for estimated MAP
map.mmoment(k=3, map=result1$model)
map.mmoment(k=3, map=result2$model)
map.mmoment(k=3, map=result3$model)

## joint moments for estimated MAP
map.jmoment(lag=1, map=result1$model)
map.jmoment(lag=1, map=result2$model)
map.jmoment(lag=1, map=result3$model)

## lag-k correlation
map.acf(map=result1$model)
map.acf(map=result2$model)
map.acf(map=result3$model)

Create an MMPP

Description

Create an instance of MMPP (Markov-Modulated Poisson Process)

Usage

mmpp(size)

Arguments

Value

An instance of MMPP

Note

MMPP is a MAP whose D1 is given by a diagonal matrix.

Examples

## create an MMPP with 5 phases
(param1 <- mmpp(5))

Create GPH distribution

Description

Create an instance of GPH

Usage

ph(size, alpha, Q, xi)

Arguments

Value

An instance of GPH

Note

This function can omit several patterns of arguments. For example, ph(5) omit the arguments alpha, Q and xi. In this case, the default values are assigned to them.

Examples

## create a PH (full matrix) with 5 phases
(param1 <- ph(5))

## create a PH (full matrix) with 5 phases
(param1 <- ph(size=5))

## create a PH with specific parameters
(param2 <- ph(alpha=c(1,0,0),
              Q=rbind(c(-4,2,0),c(2,-5,1),c(1,0,-1)),
              xi=c(2,2,0)))

Create a bi-diagonal PH distribution

Description

Create an instance of bi-diagonal PH distribution.

Usage

ph.bidiag(size)

Arguments

Value

An instance of bi-diagonal PH distribution

Note

Bi-diagonal PH distribution is the PH distribution whose infinitesimal generator is given by a upper bi-diagonal matrix. This is similar to canonical form 1. But there is no restriction on the order for diagonal elements.

Examples

## create a bidiagonal PH with 5 phases
(param1 <- ph.bidiag(5))

Create a Coxian PH distribution

Description

Create an instance of coxian PH distribution.

Usage

ph.coxian(size)

Arguments

Value

An instance of coxian PH distribution

Note

Coxian PH distribution is the PH distribution whose infinitesimal generator is given by a upper bi-diagonal matrix. This is also called canonical form 3.

Examples

## create a Coxian PH with 5 phases
(param1 <- ph.coxian(5))

Mean of PH distribution

Description

Compute the mean of a given PH distribution.

Usage

ph.mean(ph, ...)

Arguments

Value

A value of mean

Examples

## create a PH with specific parameters
(param1 <- ph(alpha=c(1,0,0), 
              Q=rbind(c(-4,2,0),c(2,-5,1),c(1,0,-1)), 
              xi=c(2,2,0)))

## create a CF1 with specific parameters
(param2 <- cf1(alpha=c(1,0,0), rate=c(1.0,2.0,3.0)))

## create a hyper Erlang with specific parameters
(param3 <- herlang(shape=c(2,3), mixrate=c(0.3,0.7), rate=c(1.0,10.0)))

## mean
ph.mean(param1)
ph.mean(param2)
ph.mean(param3)

Moments of PH distribution

Description

Generate moments up to k-th moments for a given PH distribution.

Usage

ph.moment(k, ph, ...)

Arguments

Value

A vector of moments

Examples

## create a PH with specific parameters
(param1 <- ph(alpha=c(1,0,0), 
              Q=rbind(c(-4,2,0),c(2,-5,1),c(1,0,-1)), 
              xi=c(2,2,0)))

## create a CF1 with specific parameters
(param2 <- cf1(alpha=c(1,0,0), rate=c(1.0,2.0,3.0)))

## create a hyper Erlang with specific parameters
(param3 <- herlang(shape=c(2,3), mixrate=c(0.3,0.7), rate=c(1.0,10.0)))

## up to 5 moments 
ph.moment(5, param1)
ph.moment(5, param2)
ph.moment(5, param3)

Create a tri-diagonal PH distribution

Description

Create an instance of tri-diagonal PH distribution.

Usage

ph.tridiag(size)

Arguments

Value

An instance of tri-diagonal PH distribution

Note

Tri-diagonal PH distribution is the PH distribution whose infinitesimal generator is given by a tri-diagonal matrix (band matrix).

Examples

## create a tridiagonal PH with 5 phases
(param1 <- ph.tridiag(5))

Variance of PH distribution

Description

Compute the variance of a given PH distribution.

Usage

ph.var(ph, ...)

Arguments

Value

A value of variance

Examples

## create a PH with specific parameters
(param1 <- ph(alpha=c(1,0,0), 
              Q=rbind(c(-4,2,0),c(2,-5,1),c(1,0,-1)), 
              xi=c(2,2,0)))

## create a CF1 with specific parameters
(param2 <- cf1(alpha=c(1,0,0), rate=c(1.0,2.0,3.0)))

## create a hyper Erlang with specific parameters
(param3 <- herlang(shape=c(2,3), mixrate=c(0.3,0.7), rate=c(1.0,10.0)))

## variance
ph.var(param1)
ph.var(param2)
ph.var(param3)

PH fitting with three moments

Description

Estimates PH parameters from three moments.

Usage

phfit.3mom(
  m1,
  m2,
  m3,
  method = c("Osogami06", "Bobbio05"),
  max.phase = 50,
  epsilon = sqrt(.Machine$double.eps)
)

Arguments

Value

An object of GPH.

Note

The method "Osogami06" checks the first three moments on whether there exists a PH whose three moments match to them. In such case, the method "Bobbio05" often returns an error.

References

Osogami, T. and Harchol-Balter, M. (2006) Closed Form Solutions for Mapping General Distributions to Minimal PH Distributions. Performance Evaluation, 63(6), 524–552.

Bobbio, A., Horvath, A. and Telek, M. (2005) Matching Three Moments with Minimal Acyclic Phase Type Distributions. Stochastic Models, 21(2-3), 303–326.

Examples

## Three moment matching
## Moments of Weibull(shape=2, scale=1); (0.886227, 1.0, 1.32934)
(result1 <- phfit.3mom(0.886227, 1.0, 1.32934))

## Three moment matching
## Moments of Weibull(shape=2, scale=1); (0.886227, 1.0, 1.32934)
(result2 <- phfit.3mom(0.886227, 1.0, 1.32934, method="Bobbio05"))

## mean
ph.mean(result1)
ph.mean(result2)

## variance
ph.var(result1)
ph.var(result2)

## up to 5 moments 
ph.moment(5, result1)
ph.moment(5, result2)

PH fitting with density function

Description

Estimates PH parameters from density function.

Usage

phfit.density(
  ph,
  f,
  deformula = deformula.zeroinf,
  weight.zero = 1e-12,
  weight.reltol = 1e-08,
  start.divisions = 8,
  max.iter = 12,
  ...
)

Arguments

Value

Returns a list with components, which is an object of S3 class phfit.result;

Note

Any of density function can be applied to the argument f, where f should be defined f <- function(x, ...). The first argument of f should be an integral parameter. The other parameters are set in the argument ... of phfit.density. The truncated density function can also be used directly.

Examples

####################
##### truncated density
####################

## PH fitting for general PH
(result1 <- phfit.density(ph=ph(2), f=dnorm, mean=3, sd=1))

## PH fitting for CF1
(result2 <- phfit.density(ph=cf1(2), f=dnorm, mean=3, sd=1))

## PH fitting for hyper Erlang
(result3 <- phfit.density(ph=herlang(3), f=dnorm, mean=3, sd=1))

## mean
ph.mean(result1$model)
ph.mean(result2$model)
ph.mean(result3$model)

## variance
ph.var(result1$model)
ph.var(result2$model)
ph.var(result3$model)

## up to 5 moments 
ph.moment(5, result1$model)
ph.moment(5, result2$model)
ph.moment(5, result3$model)

PH fitting with grouped data

Description

Estimates PH parameters from grouped data.

Usage

phfit.group(ph, counts, breaks, intervals, instants, ...)

Arguments

Value

Returns a list with components, which is an object of S3 class phfit.result;

Note

In this method, we can handle truncated data using NA and Inf; phfit.group(ph=cf1(5), counts=c(countsdata, NA), breaks=c(breakdata, +Inf)) NA means missing of count data at the corresponding interval, and Inf is allowed to put the last of breaks or intervals which represents a special interval [the last break point,infinity).

Examples

## make sample
wsample <- rweibull(n=100, shape=2, scale=1)
wgroup <- hist(x=wsample, breaks="fd", plot=FALSE)

## PH fitting for general PH
(result1 <- phfit.group(ph=ph(2), counts=wgroup$counts, breaks=wgroup$breaks))

## PH fitting for CF1
(result2 <- phfit.group(ph=cf1(2), counts=wgroup$counts, breaks=wgroup$breaks))

## PH fitting for hyper Erlang
(result3 <- phfit.group(ph=herlang(3), counts=wgroup$counts, breaks=wgroup$breaks))

## mean
ph.mean(result1$model)
ph.mean(result2$model)
ph.mean(result3$model)

## variance
ph.var(result1$model)
ph.var(result2$model)
ph.var(result3$model)

## up to 5 moments 
ph.moment(5, result1$model)
ph.moment(5, result2$model)
ph.moment(5, result3$model)

PH fitting with point data

Description

Estimates PH parameters from point data.

Usage

phfit.point(ph, x, weights, ...)

Arguments

Value

Returns a list with components, which is an object of S3 class phfit.result;

Examples

## make sample
wsample <- rweibull(n=100, shape=2, scale=1)

## PH fitting for general PH
(result1 <- phfit.point(ph=ph(2), x=wsample))

## PH fitting for CF1
(result2 <- phfit.point(ph=cf1(2), x=wsample))

## PH fitting for hyper Erlang
(result3 <- phfit.point(ph=herlang(3), x=wsample))

## mean
ph.mean(result1$model)
ph.mean(result2$model)
ph.mean(result3$model)

## variance
ph.var(result1$model)
ph.var(result2$model)
ph.var(result3$model)

## up to 5 moments 
ph.moment(5, result1$model)
ph.moment(5, result2$model)
ph.moment(5, result3$model)

Distribution function of PH distribution

Description

Compute the cumulative distribution function (c.d.f.) for a given PH distribution

Usage

pphase(q, ph, lower.tail = TRUE, log.p = FALSE, ...)

Arguments

Value

A vector of densities

Examples

## create a PH with specific parameters
(phdist <- ph(alpha=c(1,0,0),
              Q=rbind(c(-4,2,0),c(2,-5,1),c(1,0,-1)),
              xi=c(2,2,0)))

## c.d.f. for 0, 0.1, ..., 1
pphase(q=seq(0, 1, 0.1), ph=phdist)

Sampling of PH distributions

Description

Generate a sample from a given PH distribution.

Usage

rphase(n, ph, ...)

Arguments

Value

A vector of samples.

Examples

## create a PH with specific parameters
(phdist <- ph(alpha=c(1,0,0),
              Q=rbind(c(-4,2,0),c(2,-5,1),c(1,0,-1)),
              xi=c(2,2,0)))

## generate 10 samples
rphase(n=10, ph=phdist)