Type:	Package
Title:	Easy Handling Discrete Time Markov Chains
Version:	0.10.0
Maintainer:	Giorgio Alfredo Spedicato <spedicato_giorgio@yahoo.it>
Description:	Functions and S4 methods to create and manage discrete time Markov chains more easily. In addition functions to perform statistical (fitting and drawing random variates) and probabilistic (analysis of their structural proprieties) analysis are provided. See Spedicato (2017) <doi:10.32614/RJ-2017-036>. Some functions for continuous times Markov chains depend on the suggested ctmcd package.
License:	MIT + file LICENSE
Depends:	R (≥ 4.2.0), Matrix (≥ 1.5-0), methods
Imports:	igraph, expm, stats4, parallel, Rcpp (≥ 1.0.2), RcppParallel, utils, stats, grDevices
Suggests:	knitr, testthat, diagram, DiagrammeR, msm, Rsolnp, rmarkdown, ctmcd, bookdown, rticles
Enhances:	etm
VignetteBuilder:	rmarkdown, knitr, bookdown, rticles
LinkingTo:	Rcpp, RcppParallel, RcppArmadillo (≥ 0.9.600.4.0)
SystemRequirements:	GNU make
LazyLoad:	yes
ByteCompile:	yes
Encoding:	UTF-8
BugReports:	https://github.com/spedygiorgio/markovchain/issues
URL:	https://github.com/spedygiorgio/markovchain/
RoxygenNote:	7.3.2
NeedsCompilation:	yes
Packaged:	2024-11-13 23:26:14 UTC; Utente
Author:	Giorgio Alfredo Spedicato [aut, cre], Tae Seung Kang [aut], Sai Bhargav Yalamanchi [aut], Mildenberger Thoralf [ctb], Deepak Yadav [aut], Ignacio Cordón [aut], Vandit Jain [ctb], Toni Giorgino [ctb], Richèl J.C. Bilderbeek [ctb], Daniel Ebbert [ctb], Shreyash Maheshwari [ctb], Reinhold Koch [ctb]
Repository:	CRAN
Date/Publication:	2024-11-14 00:00:02 UTC

Easy Handling Discrete Time Markov Chains

Description

The package contains classes and method to create and manage (plot, print, export for example) discrete time Markov chains (DTMC). In addition it provide functions to perform statistical (fitting and drawing random variates) and probabilistic (analysis of DTMC proprieties) analysis

Author(s)

Giorgio Alfredo Spedicato Maintainer: Giorgio Alfredo Spedicato <spedicato_giorgio@yahoo.it>

References

Discrete-Time Markov Models, Bremaud, Springer 1999

See Also

Useful links:

• https://github.com/spedygiorgio/markovchain/

• Report bugs at https://github.com/spedygiorgio/markovchain/issues

Examples

# create some markov chains
statesNames=c("a","b")
mcA<-new("markovchain", transitionMatrix=matrix(c(0.7,0.3,0.1,0.9),byrow=TRUE,
         nrow=2, dimnames=list(statesNames,statesNames)))
         
statesNames=c("a","b","c")
mcB<-new("markovchain", states=statesNames, transitionMatrix=
         matrix(c(0.2,0.5,0.3,0,1,0,0.1,0.8,0.1), nrow=3, 
         byrow=TRUE, dimnames=list(statesNames, statesNames)))

statesNames=c("a","b","c","d")
matrice<-matrix(c(0.25,0.75,0,0,0.4,0.6,0,0,0,0,0.1,0.9,0,0,0.7,0.3), nrow=4, byrow=TRUE)
mcC<-new("markovchain", states=statesNames, transitionMatrix=matrice)
mcD<-new("markovchain", transitionMatrix=matrix(c(0,1,0,1), nrow=2,byrow=TRUE))


#operations with S4 methods
mcA^2
steadyStates(mcB)
absorbingStates(mcB)
markovchainSequence(n=20, markovchain=mcC, include=TRUE)

Returns expected hitting time from state i to state j

Description

Returns expected hitting time from state i to state j

Usage

ExpectedTime(C,i,j,useRCpp)

Arguments

Details

According to the theorem, holding times for all states except j should be greater than 0.

Value

A numerical value that returns expected hitting times from i to j

Author(s)

Vandit Jain

References

Markovchains, J. R. Norris, Cambridge University Press

Examples

states <- c("a","b","c","d")
byRow <- TRUE
gen <- matrix(data = c(-1, 1/2, 1/2, 0, 1/4, -1/2, 0, 1/4, 1/6, 0, -1/3, 1/6, 0, 0, 0, 0),
nrow = 4,byrow = byRow, dimnames = list(states,states))
ctmc <- new("ctmc",states = states, byrow = byRow, generator = gen, name = "testctmc")
ExpectedTime(ctmc,1,4,TRUE)

Higher order Markov Chains class

Description

The S4 class that describes HigherOrderMarkovChain objects.

Absorption probabilities

Description

Computes the absorption probability from each transient state to each recurrent one (i.e. the (i, j) entry or (j, i), in a stochastic matrix by columns, represents the probability that the first not transient state we can go from the transient state i is j (and therefore we are going to be absorbed in the communicating recurrent class of j)

Usage

absorptionProbabilities(object)

Arguments

Value

A named vector with the expected number of steps to go from a transient state to any of the recurrent ones

Author(s)

Ignacio Cordón

References

C. M. Grinstead and J. L. Snell. Introduction to Probability. American Mathematical Soc., 2012.

Examples

m <- matrix(c(1/2, 1/2, 0,
              1/2, 1/2, 0,
                0, 1/2, 1/2), ncol = 3, byrow = TRUE)
mc <- new("markovchain", states = letters[1:3], transitionMatrix = m)
absorptionProbabilities(mc)

Mobility between income quartiles

Description

This table show mobility between income quartiles for father and sons for the 1970 cohort born

Usage

data(blanden)

Format

An object of class table with 4 rows and 4 columns.

Details

The rows represent fathers' income quartile when the son is aged 16, whilst the columns represent sons' income quartiles when he is aged 30 (in 2000).

Source

Personal reworking

References

Jo Blanden, Paul Gregg and Stephen Machin, Intergenerational Mobility in Europe and North America, Center for Economic Performances (2005)

Examples

data(blanden)
mobilityMc<-as(blanden, "markovchain")

Calculates committor of a markovchain object with respect to set A, B

Description

Returns the probability of hitting states rom set A before set B with different initial states

Usage

committorAB(object,A,B,p)

Arguments

Details

The function solves a system of linear equations to calculate probaility that the process hits a state from set A before any state from set B

Value

Return a vector of probabilities in case initial state is not provided else returns a number

Examples

transMatr <- matrix(c(0,0,0,1,0.5,
                      0.5,0,0,0,0,
                      0.5,0,0,0,0,
                      0,0.2,0.4,0,0,
                      0,0.8,0.6,0,0.5),
                      nrow = 5)
object <- new("markovchain", states=c("a","b","c","d","e"),transitionMatrix=transMatr)
committorAB(object,c(5),c(3))

conditionalDistribution of a Markov Chain

Description

It extracts the conditional distribution of the subsequent state, given current state.

Usage

conditionalDistribution(object, state)

Arguments

Value

A named probability vector

Author(s)

Giorgio Spedicato, Deepak Yadav

References

A First Course in Probability (8th Edition), Sheldon Ross, Prentice Hall 2010

See Also

markovchain

Examples

# define a markov chain
statesNames <- c("a", "b", "c")
markovB <- new("markovchain", states = statesNames, transitionMatrix = 
               matrix(c(0.2, 0.5, 0.3, 0, 1, 0, 0.1, 0.8, 0.1),nrow = 3, 
                      byrow = TRUE, dimnames = list(statesNames, statesNames)))
                      
conditionalDistribution(markovB, "b")                       

CD4 cells counts on HIV Infects between zero and six month

Description

This is the table shown in Craig and Sendi paper showing zero and six month CD4 cells count in six brakets

Usage

data(craigsendi)

Format

The format is: table [1:3, 1:3] 682 154 19 33 64 19 25 47 43 - attr(*, "dimnames")=List of 2 ..$ : chr [1:3] "0-49" "50-74" "75-UP" ..$ : chr [1:3] "0-49" "50-74" "75-UP"

Details

Rows represent counts at the beginning, cols represent counts after six months.

Source

Estimation of the transition matrix of a discrete time Markov chain, Bruce A. Craig and Peter P. Sendi, Health Economics 11, 2002.

References

see source

Examples

data(craigsendi)
csMc<-as(craigsendi, "markovchain")
steadyStates(csMc)

Function to fit a discrete Markov chain

Description

Given a sequence of states arising from a stationary state, it fits the underlying Markov chain distribution using either MLE (also using a Laplacian smoother), bootstrap or by MAP (Bayesian) inference.

Usage

createSequenceMatrix(
  stringchar,
  toRowProbs = FALSE,
  sanitize = FALSE,
  possibleStates = character()
)

markovchainFit(
  data,
  method = "mle",
  byrow = TRUE,
  nboot = 10L,
  laplacian = 0,
  name = "",
  parallel = FALSE,
  confidencelevel = 0.95,
  confint = TRUE,
  hyperparam = matrix(),
  sanitize = FALSE,
  possibleStates = character()
)

Arguments

Details

Disabling confint would lower the computation time on large datasets. If data or stringchar contain NAs, the related NA containing transitions will be ignored.

Value

A list containing an estimate, log-likelihood, and, when "bootstrap" method is used, a matrix of standards deviations and the bootstrap samples. When the "mle", "bootstrap" or "map" method is used, the lower and upper confidence bounds are returned along with the standard error. The "map" method also returns the expected value of the parameters with respect to the posterior distribution.

Note

This function has been rewritten in Rcpp. Bootstrap algorithm has been defined "heuristically". In addition, parallel facility is not complete, involving only a part of the bootstrap process. When data is either a data.frame or a matrix object, only MLE fit is currently available.

Author(s)

Giorgio Spedicato, Tae Seung Kang, Sai Bhargav Yalamanchi

References

A First Course in Probability (8th Edition), Sheldon Ross, Prentice Hall 2010

Inferring Markov Chains: Bayesian Estimation, Model Comparison, Entropy Rate, and Out-of-Class Modeling, Christopher C. Strelioff, James P. Crutchfield, Alfred Hubler, Santa Fe Institute

Yalamanchi SB, Spedicato GA (2015). Bayesian Inference of First Order Markov Chains. R package version 0.2.5

See Also

markovchainSequence, markovchainListFit

Examples

sequence <- c("a", "b", "a", "a", "a", "a", "b", "a", "b", "a", "b", "a", "a", 
              "b", "b", "b", "a")        
sequenceMatr <- createSequenceMatrix(sequence, sanitize = FALSE)
mcFitMLE <- markovchainFit(data = sequence)
mcFitBSP <- markovchainFit(data = sequence, method = "bootstrap", nboot = 5, name = "Bootstrap Mc")

na.sequence <- c("a", NA, "a", "b")
# There will be only a (a,b) transition        
na.sequenceMatr <- createSequenceMatrix(na.sequence, sanitize = FALSE)
mcFitMLE <- markovchainFit(data = na.sequence)

# data can be a list of character vectors
sequences <- list(x = c("a", "b", "a"), y = c("b", "a", "b", "a", "c"))
mcFitMap <- markovchainFit(sequences, method = "map")
mcFitMle <- markovchainFit(sequences, method = "mle")

Continuous time Markov Chains class

Description

The S4 class that describes ctmc (continuous time Markov chain) objects.

Arguments

Methods

dim

signature(x = "ctmc"): method to get the size

initialize

signature(.Object = "ctmc"): initialize method

states

signature(object = "ctmc"): states method.

steadyStates

signature(object = "ctmc"): method to get the steady state vector.

plot

signature(x = "ctmc", y = "missing"): plot method for ctmc objects

Note

1. ctmc classes are written using S4 classes

2. Validation method is used to assess whether either columns or rows totals to zero. Rounding is used up to 5th decimal. If state names are not properly defined for a generator matrix, coercing to ctmc object leads to overriding states name with artificial "s1", "s2", ... sequence

References

Introduction to Stochastic Processes with Applications in the Biosciences (2013), David F. Anderson, University of Wisconsin at Madison. Sai Bhargav Yalamanchi, Giorgio Spedicato

See Also

generatorToTransitionMatrix,rctmc

Examples

energyStates <- c("sigma", "sigma_star")
byRow <- TRUE
gen <- matrix(data = c(-3, 3,
                       1, -1), nrow = 2,
              byrow = byRow, dimnames = list(energyStates, energyStates))
molecularCTMC <- new("ctmc", states = energyStates, 
                     byrow = byRow, generator = gen, 
                     name = "Molecular Transition Model")
                     steadyStates(molecularCTMC)
## Not run: plot(molecularCTMC)

Function to fit a CTMC

Description

This function fits the underlying CTMC give the state transition data and the transition times using the maximum likelihood method (MLE)

Usage

ctmcFit(data, byrow = TRUE, name = "", confidencelevel = 0.95)

Arguments

Details

Note that in data, there must exist an element wise corresponding between the two elements of the list and that data[[2]][1] is always 0.

Value

It returns a list containing the CTMC object and the confidence intervals.

Author(s)

Sai Bhargav Yalamanchi

References

Continuous Time Markov Chains (vignette), Sai Bhargav Yalamanchi, Giorgio Alfredo Spedicato 2015

See Also

rctmc

Examples

data <- list(c("a", "b", "c", "a", "b", "a", "c", "b", "c"), c(0, 0.8, 2.1, 2.4, 4, 5, 5.9, 8.2, 9))
ctmcFit(data)

Expected Rewards for a markovchain

Description

Given a markovchain object and reward values for every state, function calculates expected reward value after n steps.

Usage

expectedRewards(markovchain,n,rewards)

Arguments

Details

the function uses a dynamic programming approach to solve a recursive equation described in reference.

Value

returns a vector of expected rewards for different initial states

Author(s)

Vandit Jain

References

Stochastic Processes: Theory for Applications, Robert G. Gallager, Cambridge University Press

Examples

transMatr<-matrix(c(0.99,0.01,0.01,0.99),nrow=2,byrow=TRUE)
simpleMc<-new("markovchain", states=c("a","b"),
             transitionMatrix=transMatr)
expectedRewards(simpleMc,1,c(0,1))

Expected first passage Rewards for a set of states in a markovchain

Description

Given a markovchain object and reward values for every state, function calculates expected reward value for a set A of states after n steps.

Usage

expectedRewardsBeforeHittingA(markovchain, A, state, rewards, n)

Arguments

Details

The function returns the value of expected first passage rewards given rewards coressponding to every state, an initial state and number of steps.

Value

returns a expected reward (numerical value) as described above

Author(s)

Sai Bhargav Yalamanchi, Vandit Jain

First passage across states

Description

This function compute the first passage probability in states

Usage

firstPassage(object, state, n)

Arguments

Details

Based on Feres' Matlab listings

Value

A matrix of size 1:n x number of states showing the probability of the first time of passage in states to be exactly the number in the row.

Author(s)

Giorgio Spedicato

References

Renaldo Feres, Notes for Math 450 Matlab listings for Markov chains

See Also

conditionalDistribution

Examples

simpleMc <- new("markovchain", states = c("a", "b"),
                 transitionMatrix = matrix(c(0.4, 0.6, .3, .7), 
                                    nrow = 2, byrow = TRUE))
firstPassage(simpleMc, "b", 20)

function to calculate first passage probabilities

Description

The function calculates first passage probability for a subset of states given an initial state.

Usage

firstPassageMultiple(object, state, set, n)

Arguments

Value

A vector of size n showing the first time proabilities

Author(s)

Vandit Jain

References

Renaldo Feres, Notes for Math 450 Matlab listings for Markov chains; MIT OCW, course - 6.262, Discrete Stochastic Processes, course-notes, chap -05

See Also

firstPassage

Examples

statesNames <- c("a", "b", "c")
markovB <- new("markovchain", states = statesNames, transitionMatrix =
matrix(c(0.2, 0.5, 0.3,
         0, 1, 0,
         0.1, 0.8, 0.1), nrow = 3, byrow = TRUE,
       dimnames = list(statesNames, statesNames)
))
firstPassageMultiple(markovB,"a",c("b","c"),4)  

Function to fit Higher Order Multivariate Markov chain

Description

Given a matrix of categorical sequences it fits Higher Order Multivariate Markov chain.

Usage

fitHighOrderMultivarMC(seqMat, order = 2, Norm = 2)

Arguments

Value

an hommc object

Author(s)

Giorgio Spedicato, Deepak Yadav

References

W.-K. Ching et al. / Linear Algebra and its Applications

Examples

data <- matrix(c('2', '1', '3', '3', '4', '3', '2', '1', '3', '3', '2', '1', 
               c('2', '4', '4', '4', '4', '2', '3', '3', '1', '4', '3', '3')), 
               ncol = 2, byrow = FALSE)
               
fitHighOrderMultivarMC(data, order = 2, Norm = 2)                

Functions to fit a higher order Markov chain

Description

Given a sequence of states arising from a stationary state, it fits the underlying Markov chain distribution with higher order.

Usage

fitHigherOrder(sequence, order = 2)
seq2freqProb(sequence)
seq2matHigh(sequence, order)

Arguments

Value

A list containing lambda, Q, and X.

Author(s)

Giorgio Spedicato, Tae Seung Kang

References

Ching, W. K., Huang, X., Ng, M. K., & Siu, T. K. (2013). Higher-order markov chains. In Markov Chains (pp. 141-176). Springer US.

Ching, W. K., Ng, M. K., & Fung, E. S. (2008). Higher-order multivariate Markov chains and their applications. Linear Algebra and its Applications, 428(2), 492-507.

Examples

sequence<-c("a", "a", "b", "b", "a", "c", "b", "a", "b", "c", "a", "b",
            "c", "a", "b", "c", "a", "b", "a", "b")
fitHigherOrder(sequence)

Returns a generator matrix corresponding to frequency matrix

Description

The function provides interface to calculate generator matrix corresponding to a frequency matrix and time taken

Usage

freq2Generator(P, t = 1, method = "QO", logmethod = "Eigen")

Arguments

Value

returns a generator matix with same dimnames

References

E. Kreinin and M. Sidelnikova: Regularization Algorithms for Transition Matrices. Algo Research Quarterly 4(1):23-40, 2001

Examples

sample <- matrix(c(150,2,1,1,1,200,2,1,2,1,175,1,1,1,1,150),nrow = 4,byrow = TRUE)
sample_rel = rbind((sample/rowSums(sample))[1:dim(sample)[1]-1,],c(rep(0,dim(sample)[1]-1),1)) 
freq2Generator(sample_rel,1)

data(tm_abs)
tm_rel=rbind((tm_abs/rowSums(tm_abs))[1:7,],c(rep(0,7),1))
## Derive quasi optimization generator matrix estimate
freq2Generator(tm_rel,1)

Function to obtain the transition matrix from the generator

Description

The transition matrix of the embedded DTMC is inferred from the CTMC's generator

Usage

generatorToTransitionMatrix(gen, byrow = TRUE)

Arguments

Value

Returns the transition matrix.

Author(s)

Sai Bhargav Yalamanchi

References

Introduction to Stochastic Processes with Applications in the Biosciences (2013), David F. Anderson, University of Wisconsin at Madison

See Also

rctmc,ctmc-class

Examples

energyStates <- c("sigma", "sigma_star")
byRow <- TRUE
gen <- matrix(data = c(-3, 3, 1, -1), nrow = 2,
              byrow = byRow, dimnames = list(energyStates, energyStates))
generatorToTransitionMatrix(gen)

Hitting probabilities for markovchain

Description

Given a markovchain object, this function calculates the probability of ever arriving from state i to j

Usage

hittingProbabilities(object)

Arguments

Value

a matrix of hitting probabilities

Author(s)

Ignacio Cordón

References

R. Vélez, T. Prieto, Procesos Estocásticos, Librería UNED, 2013

Examples

M <- markovchain:::zeros(5)
M[1,1] <- M[5,5] <- 1
M[2,1] <- M[2,3] <- 1/2
M[3,2] <- M[3,4] <- 1/2
M[4,2] <- M[4,5] <- 1/2

mc <- new("markovchain", transitionMatrix = M)
hittingProbabilities(mc)

Holson data set

Description

A data set containing 1000 life histories trajectories and a categorical status (1,2,3) observed on eleven evenly spaced steps.

Usage

data(holson)

Format

A data frame with 1000 observations on the following 12 variables.

id

unique id

time1

observed status at i-th time

time2

observed status at i-th time

time3

observed status at i-th time

time4

observed status at i-th time

time5

observed status at i-th time

time6

observed status at i-th time

time7

observed status at i-th time

time8

observed status at i-th time

time9

observed status at i-th time

time10

observed status at i-th time

time11

observed status at i-th time

Details

The example can be used to fit a markovchain or a markovchainList object.

Source

Private communications

References

Private communications

Examples

data(holson)
head(holson)

An S4 class for representing High Order Multivariate Markovchain (HOMMC)

Description

An S4 class for representing High Order Multivariate Markovchain (HOMMC)

Slots

order

an integer equal to order of Multivariate Markovchain

states

a vector of states present in the HOMMC model

P

array of transition matrices

Lambda

a vector which stores the weightage of each transition matrices in P

byrow

if FALSE each column sum of transition matrix is 1 else row sum = 1

name

a name given to hommc

Author(s)

Giorgio Spedicato, Deepak Yadav

Examples

statesName <- c("a", "b")

P <- array(0, dim = c(2, 2, 4), dimnames = list(statesName, statesName))
P[,,1] <- matrix(c(0, 1, 1/3, 2/3), byrow = FALSE, nrow = 2)
P[,,2] <- matrix(c(1/4, 3/4, 0, 1), byrow = FALSE, nrow = 2)
P[,,3] <- matrix(c(1, 0, 1/3, 2/3), byrow = FALSE, nrow = 2)
P[,,4] <- matrix(c(3/4, 1/4, 0, 1), byrow = FALSE, nrow = 2)

Lambda <- c(0.8, 0.2, 0.3, 0.7)

ob <- new("hommc", order = 1, states = statesName, P = P, 
          Lambda = Lambda, byrow = FALSE, name = "FOMMC")
          

An S4 class for representing Imprecise Continuous Time Markovchains

Description

An S4 class for representing Imprecise Continuous Time Markovchains

Slots

states

a vector of states present in the ICTMC model

Q

matrix representing the generator demonstrated in the form of variables

range

a matrix that stores values of range of variables

name

name given to ICTMC

Calculating full conditional probability using lower rate transition matrix

Description

This function calculates full conditional probability at given time s using lower rate transition matrix

Usage

impreciseProbabilityatT(C,i,t,s,error,useRCpp)

Arguments

Author(s)

Vandit Jain

References

Imprecise Continuous-Time Markov Chains, Thomas Krak et al., 2016

Examples

states <- c("n","y")
Q <- matrix(c(-1,1,1,-1),nrow = 2,byrow = TRUE,dimnames = list(states,states))
range <- matrix(c(1/52,3/52,1/2,2),nrow = 2,byrow = 2)
name <- "testictmc"
ictmc <- new("ictmc",states = states,Q = Q,range = range,name = name)
impreciseProbabilityatT(ictmc,2,0,1,10^-3,TRUE)

Function to infer the hyperparameters for Bayesian inference from an a priori matrix or a data set

Description

Since the Bayesian inference approach implemented in the package is based on conjugate priors, hyperparameters must be provided to model the prior probability distribution of the chain parameters. The hyperparameters are inferred from a given a priori matrix under the assumption that the matrix provided corresponds to the mean (expected) values of the chain parameters. A scaling factor vector must be provided too. Alternatively, the hyperparameters can be inferred from a data set.

Usage

inferHyperparam(transMatr = matrix(), scale = numeric(), data = character())

Arguments

Details

transMatr and scale need not be provided if data is provided.

Value

Returns the hyperparameter matrix in a list.

Note

The hyperparameter matrix returned is such that the row and column names are sorted alphanumerically, and the elements in the matrix are correspondingly permuted.

Author(s)

Sai Bhargav Yalamanchi, Giorgio Spedicato

References

Yalamanchi SB, Spedicato GA (2015). Bayesian Inference of First Order Markov Chains. R package version 0.2.5

See Also

markovchainFit, predictiveDistribution

Examples

data(rain, package = "markovchain")
inferHyperparam(data = rain$rain)
 
weatherStates <- c("sunny", "cloudy", "rain")
weatherMatrix <- matrix(data = c(0.7, 0.2, 0.1, 
                                 0.3, 0.4, 0.3, 
                                 0.2, 0.4, 0.4), 
                        byrow = TRUE, nrow = 3, 
                        dimnames = list(weatherStates, weatherStates))
inferHyperparam(transMatr = weatherMatrix, scale = c(10, 10, 10))
 

Check if CTMC is irreducible

Description

This function verifies whether a CTMC object is irreducible

Usage

is.CTMCirreducible(ctmc)

Arguments

Value

a boolean value as described above.

Author(s)

Vandit Jain

References

Continuous-Time Markov Chains, Karl Sigman, Columbia University

Examples

energyStates <- c("sigma", "sigma_star")
byRow <- TRUE
gen <- matrix(data = c(-3, 3,
                       1, -1), nrow = 2,
              byrow = byRow, dimnames = list(energyStates, energyStates))
molecularCTMC <- new("ctmc", states = energyStates, 
                     byrow = byRow, generator = gen, 
                     name = "Molecular Transition Model")
is.CTMCirreducible(molecularCTMC)

checks if ctmc object is time reversible

Description

The function returns checks if provided function is time reversible

Usage

is.TimeReversible(ctmc)

Arguments

Value

Returns a boolean value stating whether ctmc object is time reversible

a boolean value as described above

Author(s)

Vandit Jain

References

INTRODUCTION TO STOCHASTIC PROCESSES WITH R, ROBERT P. DOBROW, Wiley

Examples

energyStates <- c("sigma", "sigma_star")
byRow <- TRUE
gen <- matrix(data = c(-3, 3,
                       1, -1), nrow = 2,
              byrow = byRow, dimnames = list(energyStates, energyStates))
molecularCTMC <- new("ctmc", states = energyStates, 
                     byrow = byRow, generator = gen, 
                     name = "Molecular Transition Model")
is.TimeReversible(molecularCTMC)

Verify if a state j is reachable from state i.

Description

This function verifies if a state is reachable from another, i.e., if there exists a path that leads to state j leaving from state i with positive probability

Usage

is.accessible(object, from, to)

Arguments

Details

It wraps an internal function named reachabilityMatrix.

Value

A boolean value.

Author(s)

Giorgio Spedicato, Ignacio Cordón

References

James Montgomery, University of Madison

See Also

is.irreducible

Examples

statesNames <- c("a", "b", "c")
markovB <- new("markovchain", states = statesNames, 
               transitionMatrix = matrix(c(0.2, 0.5, 0.3,
                                             0,   1,   0,
                                           0.1, 0.8, 0.1), nrow = 3, byrow = TRUE, 
                                         dimnames = list(statesNames, statesNames)
                                        )
               )
is.accessible(markovB, "a", "c")

Function to check if a Markov chain is irreducible (i.e. ergodic)

Description

This function verifies whether a markovchain object transition matrix is composed by only one communicating class.

Usage

is.irreducible(object)

Arguments

Details

It is based on .communicatingClasses internal function.

Value

A boolean values.

Author(s)

Giorgio Spedicato

References

Feres, Matlab listings for Markov Chains.

See Also

summary

Examples

statesNames <- c("a", "b")
mcA <- new("markovchain", transitionMatrix = matrix(c(0.7,0.3,0.1,0.9),
                                             byrow = TRUE, nrow = 2, 
                                             dimnames = list(statesNames, statesNames)
           ))
is.irreducible(mcA)

Check if a DTMC is regular

Description

Function to check wether a DTCM is regular

Usage

is.regular(object)

Arguments

Details

A Markov chain is regular if some of the powers of its matrix has all elements strictly positive

Value

A boolean value

Author(s)

Ignacio Cordón

References

Matrix Analysis. Roger A.Horn, Charles R.Johnson. 2nd edition. Corollary 8.5.8, Theorem 8.5.9

See Also

is.irreducible

Examples

P <- matrix(c(0.5,  0.25, 0.25,
              0.5,     0, 0.5,
              0.25, 0.25, 0.5), nrow = 3)
colnames(P) <- rownames(P) <- c("R","N","S")
ciao <- as(P, "markovchain")
is.regular(ciao)

Example from Kullback and Kupperman Tests for Contingency Tables

Description

A list of two matrices representing raw transitions between two states

Usage

data(kullback)

Format

A list containing two 6x6 non - negative integer matrices

Markov Chain class

Description

The S4 class that describes markovchain objects.

Arguments

Creation of objects

Objects can be created by calls of the form new("markovchain", states, byrow, transitionMatrix, ...).

Methods

*

signature(e1 = "markovchain", e2 = "markovchain"): multiply two markovchain objects

*

signature(e1 = "markovchain", e2 = "matrix"): markovchain by matrix multiplication

*

signature(e1 = "markovchain", e2 = "numeric"): markovchain by numeric vector multiplication

*

signature(e1 = "matrix", e2 = "markovchain"): matrix by markov chain

*

signature(e1 = "numeric", e2 = "markovchain"): numeric vector by markovchain multiplication

[

signature(x = "markovchain", i = "ANY", j = "ANY", drop = "ANY"): ...

^

signature(e1 = "markovchain", e2 = "numeric"): power of a markovchain object

==

signature(e1 = "markovchain", e2 = "markovchain"): equality of two markovchain object

!=

signature(e1 = "markovchain", e2 = "markovchain"): non-equality of two markovchain object

absorbingStates

signature(object = "markovchain"): method to get absorbing states

canonicForm

signature(object = "markovchain"): return a markovchain object into canonic form

coerce

signature(from = "markovchain", to = "data.frame"): coerce method from markovchain to data.frame

conditionalDistribution

signature(object = "markovchain"): returns the conditional probability of subsequent states given a state

coerce

signature(from = "data.frame", to = "markovchain"): coerce method from data.frame to markovchain

coerce

signature(from = "table", to = "markovchain"): coerce method from table to markovchain

coerce

signature(from = "msm", to = "markovchain"): coerce method from msm to markovchain

coerce

signature(from = "msm.est", to = "markovchain"): coerce method from msm.est (but only from a Probability Matrix) to markovchain

coerce

signature(from = "etm", to = "markovchain"): coerce method from etm to markovchain

coerce

signature(from = "sparseMatrix", to = "markovchain"): coerce method from sparseMatrix to markovchain

coerce

signature(from = "markovchain", to = "igraph"): coercing to igraph objects

coerce

signature(from = "markovchain", to = "matrix"): coercing to matrix objects

coerce

signature(from = "markovchain", to = "sparseMatrix"): coercing to sparseMatrix objects

coerce

signature(from = "matrix", to = "markovchain"): coercing to markovchain objects from matrix one

dim

signature(x = "markovchain"): method to get the size

names

signature(x = "markovchain"): method to get the names of states

names<-

signature(x = "markovchain", value = "character"): method to set the names of states

initialize

signature(.Object = "markovchain"): initialize method

plot

signature(x = "markovchain", y = "missing"): plot method for markovchain objects

predict

signature(object = "markovchain"): predict method

print

signature(x = "markovchain"): print method.

show

signature(object = "markovchain"): show method.

sort

signature(x = "markovchain", decreasing=FALSE): sorting the transition matrix.

states

signature(object = "markovchain"): returns the names of states (as names.

steadyStates

signature(object = "markovchain"): method to get the steady vector.

summary

signature(object = "markovchain"): method to summarize structure of the markov chain

transientStates

signature(object = "markovchain"): method to get the transient states.

t

signature(x = "markovchain"): transpose matrix

transitionProbability

signature(object = "markovchain"): transition probability

Note

1. markovchain object are backed by S4 Classes.

2. Validation method is used to assess whether either columns or rows totals to one. Rounding is used up to .Machine$double.eps * 100. If state names are not properly defined for a probability matrix, coercing to markovchain object leads to overriding states name with artificial "s1", "s2", ... sequence. In addition, operator overloading has been applied for +,*,^,==,!= operators.

Author(s)

Giorgio Spedicato

References

A First Course in Probability (8th Edition), Sheldon Ross, Prentice Hall 2010

See Also

markovchainSequence,markovchainFit

Examples

#show markovchain definition
showClass("markovchain")
#create a simple Markov chain
transMatr<-matrix(c(0.4,0.6,.3,.7),nrow=2,byrow=TRUE)
simpleMc<-new("markovchain", states=c("a","b"),
              transitionMatrix=transMatr, 
              name="simpleMc")
#power
simpleMc^4
#some methods
steadyStates(simpleMc)
absorbingStates(simpleMc)
simpleMc[2,1]
t(simpleMc)
is.irreducible(simpleMc)
#conditional distributions
conditionalDistribution(simpleMc, "b")
#example for predict method
sequence<-c("a", "b", "a", "a", "a", "a", "b", "a", "b", "a", "b", "a", "a", "b", "b", "b", "a")
mcFit<-markovchainFit(data=sequence)
predict(mcFit$estimate, newdata="b",n.ahead=3)
#direct conversion
myMc<-as(transMatr, "markovchain")

#example of summary
summary(simpleMc)
## Not run: plot(simpleMc)

Non homogeneus discrete time Markov Chains class

Description

A class to handle non homogeneous discrete Markov chains

Arguments

Objects from the Class

A markovchainlist is a list of markovchain objects. They can be used to model non homogeneous discrete time Markov Chains, when transition probabilities (and possible states) change by time.

Methods

[[

signature(x = "markovchainList"): extract the i-th markovchain

dim

signature(x = "markovchainList"): number of markovchain underlying the matrix

predict

signature(object = "markovchainList"): predict from a markovchainList

print

signature(x = "markovchainList"): prints the list of markovchains

show

signature(object = "markovchainList"): same as print

Note

The class consists in a list of markovchain objects. It is aimed at working with non homogeneous Markov chains.

Author(s)

Giorgio Spedicato

References

A First Course in Probability (8th Edition), Sheldon Ross, Prentice Hall 2010

See Also

markovchain

Examples

showClass("markovchainList")
#define a markovchainList
statesNames=c("a","b")

mcA<-new("markovchain",name="MCA", 
         transitionMatrix=matrix(c(0.7,0.3,0.1,0.9),
                          byrow=TRUE, nrow=2, 
                          dimnames=list(statesNames,statesNames))
        )
                                                           
mcB<-new("markovchain", states=c("a","b","c"), name="MCB",
         transitionMatrix=matrix(c(0.2,0.5,0.3,0,1,0,0.1,0.8,0.1),
         nrow=3, byrow=TRUE))
 
mcC<-new("markovchain", states=c("a","b","c","d"), name="MCC",
         transitionMatrix=matrix(c(0.25,0.75,0,0,0.4,0.6,
                                   0,0,0,0,0.1,0.9,0,0,0.7,0.3), 
                                 nrow=4, byrow=TRUE)
)
mcList<-new("markovchainList",markovchains=list(mcA, mcB, mcC), 
           name="Non - homogeneous Markov Chain")

markovchainListFit

Description

Given a data frame or a matrix (rows are observations, by cols the temporal sequence), it fits a non - homogeneous discrete time markov chain process (storing row). In particular a markovchainList of size = ncol - 1 is obtained estimating transitions from the n samples given by consecutive column pairs.

Usage

markovchainListFit(data, byrow = TRUE, laplacian = 0, name)

Arguments

Details

If data contains NAs then the transitions containing NA will be ignored.

Value

A list containing two slots: estimate (the estimate) name

Examples

# using holson dataset
data(holson)
# fitting a single markovchain
singleMc <- markovchainFit(data = holson[,2:12])
# fitting a markovchainList
mclistFit <- markovchainListFit(data = holson[, 2:12], name = "holsonMcList")

Function to generate a sequence of states from homogeneous Markov chains.

Description

Provided any markovchain object, it returns a sequence of states coming from the underlying stationary distribution.

Usage

markovchainSequence(
  n,
  markovchain,
  t0 = sample(markovchain@states, 1),
  include.t0 = FALSE,
  useRCpp = TRUE
)

Arguments

Details

A sequence of size n is sampled.

Value

A Character Vector

Author(s)

Giorgio Spedicato

References

A First Course in Probability (8th Edition), Sheldon Ross, Prentice Hall 2010

See Also

markovchainFit

Examples

# define the markovchain object
statesNames <- c("a", "b", "c")
mcB <- new("markovchain", states = statesNames, 
   transitionMatrix = matrix(c(0.2, 0.5, 0.3, 0, 0.2, 0.8, 0.1, 0.8, 0.1), 
   nrow = 3, byrow = TRUE, dimnames = list(statesNames, statesNames)))

# show the sequence
outs <- markovchainSequence(n = 100, markovchain = mcB, t0 = "a")

Mean absorption time

Description

Computes the expected number of steps to go from any of the transient states to any of the recurrent states. The Markov chain should have at least one transient state for this method to work

Usage

meanAbsorptionTime(object)

Arguments

Value

A named vector with the expected number of steps to go from a transient state to any of the recurrent ones

Author(s)

Ignacio Cordón

References

C. M. Grinstead and J. L. Snell. Introduction to Probability. American Mathematical Soc., 2012.

Examples

m <- matrix(c(1/2, 1/2, 0,
              1/2, 1/2, 0,
                0, 1/2, 1/2), ncol = 3, byrow = TRUE)
mc <- new("markovchain", states = letters[1:3], transitionMatrix = m)
times <- meanAbsorptionTime(mc)

Mean First Passage Time for irreducible Markov chains

Description

Given an irreducible (ergodic) markovchain object, this function calculates the expected number of steps to reach other states

Usage

meanFirstPassageTime(object, destination)

Arguments

Details

For an ergodic Markov chain it computes:

• If destination is empty, the average first time (in steps) that takes the Markov chain to go from initial state i to j. (i, j) represents that value in case the Markov chain is given row-wise, (j, i) in case it is given col-wise.

• If destination is not empty, the average time it takes us from the remaining states to reach the states in destination

Value

a Matrix of the same size with the average first passage times if destination is empty, a vector if destination is not

Author(s)

Toni Giorgino, Ignacio Cordón

References

C. M. Grinstead and J. L. Snell. Introduction to Probability. American Mathematical Soc., 2012.

Examples

m <- matrix(1 / 10 * c(6,3,1,
                       2,3,5,
                       4,1,5), ncol = 3, byrow = TRUE)
mc <- new("markovchain", states = c("s","c","r"), transitionMatrix = m)
meanFirstPassageTime(mc, "r")


# Grinstead and Snell's "Oz weather" worked out example
mOz <- matrix(c(2,1,1,
                2,0,2,
                1,1,2)/4, ncol = 3, byrow = TRUE)

mcOz <- new("markovchain", states = c("s", "c", "r"), transitionMatrix = mOz)
meanFirstPassageTime(mcOz)

Mean num of visits for markovchain, starting at each state

Description

Given a markovchain object, this function calculates a matrix where the element (i, j) represents the expect number of visits to the state j if the chain starts at i (in a Markov chain by columns it would be the element (j, i) instead)

Usage

meanNumVisits(object)

Arguments

Value

a matrix with the expect number of visits to each state

Author(s)

Ignacio Cordón

References

R. Vélez, T. Prieto, Procesos Estocásticos, Librería UNED, 2013

Examples

M <- markovchain:::zeros(5)
M[1,1] <- M[5,5] <- 1
M[2,1] <- M[2,3] <- 1/2
M[3,2] <- M[3,4] <- 1/2
M[4,2] <- M[4,5] <- 1/2

mc <- new("markovchain", transitionMatrix = M)
meanNumVisits(mc)

Mean recurrence time

Description

Computes the expected time to return to a recurrent state in case the Markov chain starts there

Usage

meanRecurrenceTime(object)

Arguments

Value

For a Markov chain it outputs is a named vector with the expected time to first return to a state when the chain starts there. States present in the vector are only the recurrent ones. If the matrix is ergodic (i.e. irreducible), then all states are present in the output and order is the same as states order for the Markov chain

Author(s)

Ignacio Cordón

References

C. M. Grinstead and J. L. Snell. Introduction to Probability. American Mathematical Soc., 2012.

Examples

m <- matrix(1 / 10 * c(6,3,1,
                       2,3,5,
                       4,1,5), ncol = 3, byrow = TRUE)
mc <- new("markovchain", states = c("s","c","r"), transitionMatrix = m)
meanRecurrenceTime(mc)

A function to compute multinomial confidence intervals of DTMC

Description

Return estimated transition matrix assuming a Multinomial Distribution

Usage

multinomialConfidenceIntervals(
  transitionMatrix,
  countsTransitionMatrix,
  confidencelevel = 0.95
)

Arguments

Value

Two matrices containing the confidence intervals.

References

Constructing two-sided simultaneous confidence intervals for multinomial proportions for small counts in a large number of cells. Journal of Statistical Software 5(6) (2000)

See Also

markovchainFit

Examples

seq<-c("a", "b", "a", "a", "a", "a", "b", "a", "b", "a", "b", "a", "a", "b", "b", "b", "a")
mcfit<-markovchainFit(data=seq,byrow=TRUE)
seqmat<-createSequenceMatrix(seq)
multinomialConfidenceIntervals(mcfit$estimate@transitionMatrix, seqmat, 0.95)

Method to retrieve name of markovchain object

Description

This method returns the name of a markovchain object

Usage

name(object)

## S4 method for signature 'markovchain'
name(object)

Arguments

Author(s)

Giorgio Spedicato, Deepak Yadav

Examples

statesNames <- c("a", "b", "c")
markovB <- new("markovchain", states = statesNames, transitionMatrix =
                matrix(c(0.2, 0.5, 0.3, 0, 1, 0, 0.1, 0.8, 0.1), nrow = 3,
                byrow = TRUE, dimnames=list(statesNames,statesNames)),
                name = "A markovchain Object" 
)
name(markovB)

Method to set name of markovchain object

Description

This method modifies the existing name of markovchain object

Usage

name(object) <- value

## S4 replacement method for signature 'markovchain'
name(object) <- value

Arguments

Author(s)

Giorgio Spedicato, Deepak Yadav

Examples

statesNames <- c("a", "b", "c")
markovB <- new("markovchain", states = statesNames, transitionMatrix =
                matrix(c(0.2, 0.5, 0.3, 0, 1, 0, 0.1, 0.8, 0.1), nrow = 3,
                byrow = TRUE, dimnames=list(statesNames,statesNames)),
                name = "A markovchain Object" 
)
name(markovB) <- "dangerous mc"

Returns the states for a Markov chain object

Description

Returns the states for a Markov chain object

Usage

## S4 method for signature 'markovchain'
names(x)

Arguments

return a joint pdf of the number of visits to the various states of the DTMC

Description

This function would return a joint pdf of the number of visits to the various states of the DTMC during the first N steps.

Usage

noofVisitsDist(markovchain,N,state)

Arguments

Details

This function would return a joint pdf of the number of visits to the various states of the DTMC during the first N steps.

Value

a numeric vector depicting the above described probability density function.

Author(s)

Vandit Jain

Examples

transMatr<-matrix(c(0.4,0.6,.3,.7),nrow=2,byrow=TRUE)
simpleMc<-new("markovchain", states=c("a","b"),
             transitionMatrix=transMatr, 
             name="simpleMc")   
noofVisitsDist(simpleMc,5,"a")

Returns an Identity matrix

Description

Returns an Identity matrix

Usage

ones(n)

Arguments

Value

a identity matrix

Various function to perform structural analysis of DTMC

Description

These functions return absorbing and transient states of the markovchain objects.

Usage

period(object)

communicatingClasses(object)

recurrentClasses(object)

transientClasses(object)

transientStates(object)

recurrentStates(object)

absorbingStates(object)

canonicForm(object)

Arguments

Value

period

returns a integer number corresponding to the periodicity of the Markov chain (if it is irreducible)

absorbingStates

returns a character vector with the names of the absorbing states in the Markov chain

communicatingClasses

returns a list in which each slot contains the names of the states that are in that communicating class

recurrentClasses

analogously to communicatingClasses, but with recurrent classes

transientClasses

analogously to communicatingClasses, but with transient classes

transientStates

returns a character vector with all the transient states for the Markov chain

recurrentStates

returns a character vector with all the recurrent states for the Markov chain

canonicForm

returns the Markov chain reordered by a permutation of states so that we have blocks submatrices for each of the recurrent classes and a collection of rows in the end for the transient states

Author(s)

Giorgio Alfredo Spedicato, Ignacio Cordón

References

Feres, Matlab listing for markov chain.

See Also

markovchain

Examples

statesNames <- c("a", "b", "c")
mc <- new("markovchain", states = statesNames, transitionMatrix =
          matrix(c(0.2, 0.5, 0.3,
                   0,   1,   0,
                   0.1, 0.8, 0.1), nrow = 3, byrow = TRUE,
                 dimnames = list(statesNames, statesNames))
         )

communicatingClasses(mc)
recurrentClasses(mc)
recurrentClasses(mc)
absorbingStates(mc)
transientStates(mc)
recurrentStates(mc)
canonicForm(mc)

# periodicity analysis
A <- matrix(c(0, 1, 0, 0, 0.5, 0, 0.5, 0, 0, 0.5, 0, 0.5, 0, 0, 1, 0), 
            nrow = 4, ncol = 4, byrow = TRUE)
mcA <- new("markovchain", states = c("a", "b", "c", "d"), 
          transitionMatrix = A,
          name = "A")

is.irreducible(mcA) #true
period(mcA) #2

# periodicity analysis
B <- matrix(c(0, 0, 1/2, 1/4, 1/4, 0, 0,
                   0, 0, 1/3, 0, 2/3, 0, 0,
                   0, 0, 0, 0, 0, 1/3, 2/3,
                   0, 0, 0, 0, 0, 1/2, 1/2,
                   0, 0, 0, 0, 0, 3/4, 1/4,
                   1/2, 1/2, 0, 0, 0, 0, 0,
                   1/4, 3/4, 0, 0, 0, 0, 0), byrow = TRUE, ncol = 7)
mcB <- new("markovchain", transitionMatrix = B)
period(mcB)

Simulate a higher order multivariate markovchain

Description

This function provides a prediction of states for a higher order multivariate markovchain object

Usage

predictHommc(hommc,t,init)

Arguments

Details

The user is required to provide a matrix of giving n previous coressponding every categorical sequence. Dimensions of the init are s X n, where s is number of categorical sequences and n is order of the homc.

Value

The function returns a matrix of size s X t displaying t predicted states in each row coressponding to every categorical sequence.

Author(s)

Vandit Jain

predictiveDistribution

Description

The function computes the probability of observing a new data set, given a data set

Usage

predictiveDistribution(stringchar, newData, hyperparam = matrix())

Arguments

Details

The underlying method is Bayesian inference. The probability is computed by averaging the likelihood of the new data with respect to the posterior. Since the method assumes conjugate priors, the result can be represented in a closed form (see the vignette for more details), which is what is returned.

Value

The log of the probability is returned.

Author(s)

Sai Bhargav Yalamanchi

References

Inferring Markov Chains: Bayesian Estimation, Model Comparison, Entropy Rate, and Out-of-Class Modeling, Christopher C. Strelioff, James P. Crutchfield, Alfred Hubler, Santa Fe Institute

Yalamanchi SB, Spedicato GA (2015). Bayesian Inference of First Order Markov Chains. R package version 0.2.5

See Also

markovchainFit

Examples

sequence<- c("a", "b", "a", "a", "a", "a", "b", "a", "b", "a", "b", "a", "a", 
             "b", "b", "b", "a")
hyperMatrix<-matrix(c(1, 2, 1, 4), nrow = 2,dimnames=list(c("a","b"),c("a","b")))
predProb <- predictiveDistribution(sequence[1:10], sequence[11:17], hyperparam =hyperMatrix )
hyperMatrix2<-hyperMatrix[c(2,1),c(2,1)]
predProb2 <- predictiveDistribution(sequence[1:10], sequence[11:17], hyperparam =hyperMatrix2 )
predProb2==predProb

Preprogluccacon DNA protein bases sequences

Description

Sequence of bases for preproglucacon DNA protein

Usage

data(preproglucacon)

Format

A data frame with 1572 observations on the following 2 variables.

V1

a numeric vector, showing original coding

preproglucacon

a character vector, showing initial of DNA bases (Adenine, Cytosine, Guanine, Thymine)

Source

Avery Henderson

References

Averuy Henderson, Fitting markov chain models on discrete time series such as DNA sequences

Examples

data(preproglucacon)
preproglucaconMc<-markovchainFit(data=preproglucacon$preproglucacon)

priorDistribution

Description

Function to evaluate the prior probability of a transition matrix. It is based on conjugate priors and therefore a Dirichlet distribution is used to model the transitions of each state.

Usage

priorDistribution(transMatr, hyperparam = matrix())

Arguments

Details

The states (dimnames) of the transition matrix and the hyperparam may be in any order.

Value

The log of the probabilities for each state is returned in a numeric vector. Each number in the vector represents the probability (log) of having a probability transition vector as specified in corresponding the row of the transition matrix.

Note

This function can be used in conjunction with inferHyperparam. For example, if the user has a prior data set and a prior transition matrix, he can infer the hyperparameters using inferHyperparam and then compute the probability of their prior matrix using the inferred hyperparameters with priorDistribution.

Author(s)

Sai Bhargav Yalamanchi, Giorgio Spedicato

References

Yalamanchi SB, Spedicato GA (2015). Bayesian Inference of First Order Markov Chains. R package version 0.2.5

See Also

predictiveDistribution, inferHyperparam

Examples

priorDistribution(matrix(c(0.5, 0.5, 0.5, 0.5), 
                  nrow = 2, 
                  dimnames = list(c("a", "b"), c("a", "b"))), 
                  matrix(c(2, 2, 2, 2), 
                  nrow = 2, 
                  dimnames = list(c("a", "b"), c("a", "b"))))

Calculating probability from a ctmc object

Description

This function returns the probability of every state at time t under different conditions

Usage

probabilityatT(C,t,x0,useRCpp)

Arguments

Details

The initial state is not mandatory, In case it is not provided, function returns a matrix of transition function at time t else it returns vector of probaabilities of transition to different states if initial state was x0

Value

returns a vector or a matrix in case x0 is provided or not respectively.

Author(s)

Vandit Jain

References

INTRODUCTION TO STOCHASTIC PROCESSES WITH R, ROBERT P. DOBROW, Wiley

Examples

states <- c("a","b","c","d")
byRow <- TRUE
gen <- matrix(data = c(-1, 1/2, 1/2, 0, 1/4, -1/2, 0, 1/4, 1/6, 0, -1/3, 1/6, 0, 0, 0, 0),
nrow = 4,byrow = byRow, dimnames = list(states,states))
ctmc <- new("ctmc",states = states, byrow = byRow, generator = gen, name = "testctmc")
probabilityatT(ctmc,1,useRCpp = TRUE)

Alofi island daily rainfall

Description

Rainfall measured in Alofi Island

Usage

data(rain)

Format

A data frame with 1096 observations on the following 2 variables.

V1

a numeric vector, showing original coding

rain

a character vector, showing daily rainfall millilitres brackets

Source

Avery Henderson

References

Avery Henderson, Fitting markov chain models on discrete time series such as DNA sequences

Examples

data(rain)
rainMc<-markovchainFit(data=rain$rain)

rctmc

Description

The function generates random CTMC transitions as per the provided generator matrix.

Usage

rctmc(n, ctmc, initDist = numeric(), T = 0, include.T0 = TRUE,
  out.type = "list")

Arguments

Details

In order to use the T0 argument, set n to Inf.

Value

Based on out.type, a list or a data frame is returned. The returned list has two elements - a character vector (states) and a numeric vector (indicating time of transitions). The data frame is similarly structured.

Author(s)

Sai Bhargav Yalamanchi

References

Introduction to Stochastic Processes with Applications in the Biosciences (2013), David F. Anderson, University of Wisconsin at Madison

See Also

generatorToTransitionMatrix,ctmc-class

Examples

energyStates <- c("sigma", "sigma_star")
byRow <- TRUE
gen <- matrix(data = c(-3, 3, 1, -1), nrow = 2,
             byrow = byRow, dimnames = list(energyStates, energyStates))
molecularCTMC <- new("ctmc", states = energyStates, 
                     byrow = byRow, generator = gen, 
                     name = "Molecular Transition Model")   

statesDist <- c(0.8, 0.2)
rctmc(n = Inf, ctmc = molecularCTMC, T = 1)
rctmc(n = 5, ctmc = molecularCTMC, initDist = statesDist, include.T0 = FALSE)

Function to generate a sequence of states from homogeneous or non-homogeneous Markov chains.

Description

Provided any markovchain or markovchainList objects, it returns a sequence of states coming from the underlying stationary distribution.

Usage

rmarkovchain(
  n,
  object,
  what = "data.frame",
  useRCpp = TRUE,
  parallel = FALSE,
  num.cores = NULL,
  ...
)

Arguments

Details

When a homogeneous process is assumed (markovchain object) a sequence is sampled of size n. When a non - homogeneous process is assumed, n samples are taken but the process is assumed to last from the begin to the end of the non-homogeneous markov process.

Value

Character Vector, data.frame, list or matrix

Note

Check the type of input

Author(s)

Giorgio Spedicato

References

A First Course in Probability (8th Edition), Sheldon Ross, Prentice Hall 2010

See Also

markovchainFit, markovchainSequence

Examples

# define the markovchain object
statesNames <- c("a", "b", "c")
mcB <- new("markovchain", states = statesNames, 
   transitionMatrix = matrix(c(0.2, 0.5, 0.3, 0, 0.2, 0.8, 0.1, 0.8, 0.1), 
   nrow = 3, byrow = TRUE, dimnames = list(statesNames, statesNames)))

# show the sequence
outs <- rmarkovchain(n = 100, object = mcB, what = "list")


#define markovchainList object
statesNames <- c("a", "b", "c")
mcA <- new("markovchain", states = statesNames, transitionMatrix = 
   matrix(c(0.2, 0.5, 0.3, 0, 0.2, 0.8, 0.1, 0.8, 0.1), nrow = 3, 
   byrow = TRUE, dimnames = list(statesNames, statesNames)))
mcB <- new("markovchain", states = statesNames, transitionMatrix = 
   matrix(c(0.2, 0.5, 0.3, 0, 0.2, 0.8, 0.1, 0.8, 0.1), nrow = 3, 
   byrow = TRUE, dimnames = list(statesNames, statesNames)))
mcC <- new("markovchain", states = statesNames, transitionMatrix = 
   matrix(c(0.2, 0.5, 0.3, 0, 0.2, 0.8, 0.1, 0.8, 0.1), nrow = 3, 
   byrow = TRUE, dimnames = list(statesNames, statesNames)))
mclist <- new("markovchainList", markovchains = list(mcA, mcB, mcC)) 

# show the list of sequence
rmarkovchain(100, mclist, "list")
     

Sales Demand Sequences

Description

Sales demand sequences of five products (A, B, C, D, E). Each row corresponds to a sequence. First row corresponds to Sequence A, Second row to Sequence B and so on.

Usage

data("sales")

Format

An object of class matrix (inherits from array) with 269 rows and 5 columns.

Details

The example can be used to fit High order multivariate markov chain.

Examples

data("sales")
# fitHighOrderMultivarMC(seqMat = sales, order = 2, Norm = 2)

Function to display the details of hommc object

Description

This is a convenience function to display the slots of hommc object in proper format

Usage

## S4 method for signature 'hommc'
show(object)

Arguments

Defined states of a transition matrix

Description

This method returns the states of a transition matrix.

Usage

states(object)

## S4 method for signature 'markovchain'
states(object)

Arguments

Value

The character vector corresponding to states slot.

Author(s)

Giorgio Spedicato

References

A First Course in Probability (8th Edition), Sheldon Ross, Prentice Hall 2010

See Also

markovchain

Examples

statesNames <- c("a", "b", "c")
markovB <- new("markovchain", states = statesNames, transitionMatrix =
                matrix(c(0.2, 0.5, 0.3, 0, 1, 0, 0.1, 0.8, 0.1), nrow = 3,
                byrow = TRUE, dimnames=list(statesNames,statesNames)),
                name = "A markovchain Object" 
)
states(markovB)
names(markovB)

Stationary states of a markovchain object

Description

This method returns the stationary vector in matricial form of a markovchain object.

Usage

steadyStates(object)

Arguments

Value

A matrix corresponding to the stationary states

Note

The steady states are identified starting from which eigenvectors correspond to identity eigenvalues and then normalizing them to sum up to unity. When negative values are found in the matrix, the eigenvalues extraction is performed on the recurrent classes submatrix.

Author(s)

Giorgio Spedicato

References

A First Course in Probability (8th Edition), Sheldon Ross, Prentice Hall 2010

See Also

markovchain

Examples

statesNames <- c("a", "b", "c")
markovB <- new("markovchain", states = statesNames, transitionMatrix =
                matrix(c(0.2, 0.5, 0.3, 0, 1, 0, 0.1, 0.8, 0.1), nrow = 3,
                byrow = TRUE, dimnames=list(statesNames,statesNames)),
               name = "A markovchain Object" 
)       
steadyStates(markovB)

Single Year Corporate Credit Rating Transititions

Description

Matrix of Standard and Poor's Global Corporate Rating Transition Frequencies 2000 (NR Removed)

Usage

data(tm_abs)

Format

The format is: num [1:8, 1:8] 17 2 0 0 0 0 0 0 1 455 ... - attr(*, "dimnames")=List of 2 ..$ : chr [1:8] "AAA" "AA" "A" "BBB" ... ..$ : chr [1:8] "AAA" "AA" "A" "BBB" ...

References

European Securities and Markets Authority, 2016 https://cerep.esma.europa.eu/cerep-web/statistics/transitionMatrice.xhtml

Examples

data(tm_abs)

Return the generator matrix for a corresponding transition matrix

Description

Calculate the generator matrix for a corresponding transition matrix

Usage

transition2Generator(P, t = 1, method = "logarithm")

Arguments

Value

A matrix that represent the generator of P

See Also

rctmc

Examples

mymatr <- matrix(c(.4, .6, .1, .9), nrow = 2, byrow = TRUE)
Q <- transition2Generator(P = mymatr)
expm::expm(Q)
 

Function to get the transition probabilities from initial to subsequent states.

Description

This is a convenience function to get transition probabilities.

Usage

transitionProbability(object, t0, t1)

## S4 method for signature 'markovchain'
transitionProbability(object, t0, t1)

Arguments

Value

Numeric Vector

Author(s)

Giorgio Spedicato

References

A First Course in Probability (8th Edition), Sheldon Ross, Prentice Hall 2010

See Also

markovchain

Examples

statesNames <- c("a", "b", "c")
markovB <- new("markovchain", states = statesNames, transitionMatrix =
                matrix(c(0.2, 0.5, 0.3, 0, 1, 0, 0.1, 0.8, 0.1), nrow = 3,
                byrow = TRUE, dimnames=list(statesNames,statesNames)),
               name = "A markovchain Object" 
)    
transitionProbability(markovB,"b", "c")

Various functions to perform statistical inference of DTMC

Description

These functions verify the Markov property, assess the order and stationarity of the Markov chain.

This function tests whether an empirical transition matrix is statistically compatible with a theoretical one. It is a chi-square based test. In case a cell in the empirical transition matrix is >0 that is 0 in the theoretical transition matrix the null hypothesis is rejected.

Verifies that the s elements in the input list belongs to the same DTMC

Usage

verifyMarkovProperty(sequence, verbose = TRUE)

assessOrder(sequence, verbose = TRUE)

assessStationarity(sequence, nblocks, verbose = TRUE)

verifyEmpiricalToTheoretical(data, object, verbose = TRUE)

verifyHomogeneity(inputList, verbose = TRUE)

Arguments

Value

Verification result

a list with following slots: statistic (the chi - square statistic), dof (degrees of freedom), and corresponding p-value. In case a cell in the empirical transition matrix is >0 that is 0 in the theoretical transition matrix the null hypothesis is rejected. In that case a p-value of 0 and statistic and dof of NA are returned.

a list of transition matrices?

Author(s)

Tae Seung Kang, Giorgio Alfredo Spedicato

References

Anderson and Goodman.

See Also

markovchain

Examples

sequence <- c("a", "b", "a", "a", "a", "a", "b", "a", "b",
              "a", "b", "a", "a", "b", "b", "b", "a")
mcFit <- markovchainFit(data = sequence, byrow = FALSE)
verifyMarkovProperty(sequence)
assessOrder(sequence)
assessStationarity(sequence, 1)



#Example taken from Kullback Kupperman Tests for Contingency Tables and Markov Chains

sequence<-c(0,1,2,2,1,0,0,0,0,0,0,1,2,2,2,1,0,0,1,0,0,0,0,0,0,1,1,
2,0,0,2,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,0,0,0,0,2,1,0,
0,2,1,0,0,0,0,0,0,1,1,1,2,2,0,0,2,1,1,1,1,2,1,1,1,1,1,1,1,1,1,0,2,
0,1,1,0,0,0,1,2,2,0,0,0,0,0,0,2,2,2,1,1,1,1,0,1,1,1,1,0,0,2,1,1,
0,0,0,0,0,2,2,1,1,1,1,1,2,1,2,0,0,0,1,2,2,2,0,0,0,1,1)

mc=matrix(c(5/8,1/4,1/8,1/4,1/2,1/4,1/4,3/8,3/8),byrow=TRUE, nrow=3)
rownames(mc)<-colnames(mc)<-0:2; theoreticalMc<-as(mc, "markovchain")

verifyEmpiricalToTheoretical(data=sequence,object=theoreticalMc)


data(kullback)
verifyHomogeneity(inputList=kullback,verbose=TRUE)

Matrix to create zeros

Description

Matrix to create zeros

Usage

zeros(n)

Arguments

Value

a square matrix of zeros