Work in progress package for providing functions in R for simulations of Markov chains, estimation of probability transition matrices and transition rate matrices, and computation of stationary distributions (when they exist) for both discrete time and continuous time Markov chains.
- Simulating both discrete and continuous time Markov chains
- Estimating the transition matrices of discrete and continuous time chains
- Computing the stationary distribution if it exists of DTMCs
- Sample text generation
- Computing the stationary distribution if it exists of CTMCs
- Common transition matrix functions available for quickly populating matrices
- More to come...
You can install the github version of markovChains with:
devtools::install_github("shill1729/markovChains")The plan is to include functions for populating transition matrices for common models. Here is an example simulating a birth-death process.
library(markovChains)
# Birth rate, death rate, carrying capacity, time-horizon
lambda <- 0.5
mu <- 0.25
m <- 100
tt <- 100
# Initial distribution: P(X_0=1)=1, start with population of 1 at time zero with 100% chance.
im <- c(0, 1, rep(0, m-2))
# Transition-rate matrix
Q <- birth_death_Q(lambda, mu, m) # actually a RW with reflecting barriers transition rate matrix
Q[1,] <- rep(0, m) # Ensures extinction if we reach zero population, a true BD process transition-rate matrix
# Simulating the BD process
x <- rctmc(tt, Q, im, states = c(0:(m-1)))
# Plotting the process
plot(x, type = "s", main = "Birth and Death Process")A hemoglobin molecule can carry one oxygen or one carbon monoxide molecule, exclusively. Suppose that the two types of gas molecules arrive at rates 1 and 2 and attach for an exponentiall amount of time of 3 and 4, respectively. The + denotes attached oxygen, - denotes attached carbon monoxide and 0 denotes free hemoglobin molecule. Find long run fraction of the time the hemoglobin spends in each three states.
library(markovChains)
# Transition rate matrix
Q <- rbind(c(-3, 3, 0),
c(1, -3, 2),
c(0, 4, -4))
# Embedded jump-chain
J <- jump_chain(Q)
# Semigroup P(t)
Pt <- semigroup(100, Q)
# States the hemoglobin molecule is in: + oxygen, 0 neither, - carbon monoxide
states <- c("+", "0", "-")
# Simulate the CTMC
x <- rctmc(10, Q = Q, mu = c(0, 1, 0), states = states)
print("Infinitesimal generator")
print(Q)
print("Jump-chain transition probability matrix")
print(J)
print("Long term probability transition semigroup matrix")
print(Pt)
print("Simulation")
print(head(x))
# Sample output
# [1] "Infinitesimal generator"
# [,1] [,2] [,3]
# [1,] -3 3 0
# [2,] 1 -3 2
# [3,] 0 4 -4
# [1] "Jump-chain transition probability matrix"
# [,1] [,2] [,3]
# [1,] 0.0000000 1 0.0000000
# [2,] 0.3333333 0 0.6666667
# [3,] 0.0000000 1 0.0000000
# [1] "Long term probability transition semigroup matrix"
# [,1] [,2] [,3]
# [1,] 0.1818182 0.5454545 0.2727273
# [2,] 0.1818182 0.5454545 0.2727273
# [3,] 0.1818182 0.5454545 0.2727273
# [1] "Simulation"
# time state
# 1 0.0000000 0
# 2 0.6923294 +
# 3 0.8641708 0
# 4 0.9841894 +
# 5 1.4583014 0
# 6 2.1225525 -We provide some rough but amusing text generators devoid of punctuation and lower/uppercase distinctions. The first example is from using Shakespeare's 154 sonnets the second is Bukowski's Ham on Rye.
Okay this will not actually follow the rhyming scheme of a sonnet, but it will give off a mild Shakespearean effect by using his vocab and first-order word frequencies. Included is a data.frame of all 154 sonnets by Shakespeare.
library(markovChains)
# Load the sonnet data and format as an observable Markov chain
sonnet_data <- observe_sonnet_chain()
# Estimate the first-order transition matrix
P <- estimate_dtmc(chain = sonnet_data$encodedChain, m = length(sonnet_data$stateSpace))
# Generate your new sonnet
sonnet <- markov_text(sonnet_data, P)
print(sonnet) # an example output:
# [1] "in pays bore paws last forebemoaned starved four composition possessing"
# [2] "follow esteeming excusing nine worst word foregone wardrobe poverty famine"
# [3] "huge dost love delights receives offences lovekindling refuse paid lame"
# [4] "minion twixt paper sword us beweep hammered ear honest his"
# [5] "wastes preserve indirectly point crave dearpurchased alien towers oergreen disperse"
# [6] "brood each reeleth being weeds far wanton selfdoing played fashion"
# [7] "servant grow ragged hast womans unto estimate widows snow wights"
# [8] "fee betwixt loathsome hardest repair bath these wardrobe eat slumbers"
# [9] "doing riot greet pent growst unknown once sufferance watching closet"
# [10] "injurious woe fingers action flown shaken whose chronicle prevent dwellers"
# [11] "souls selflove phrase potions lace whether no bearing hooks cunning"
# [12] "reigns anothers view carve workings for comments thriftless awakes sufferance"
# [13] "thing lacking returnd print attend trees impiety temptation jewels slain"
# [14] "loathsome care made roses commits titles rather healthful sire wars"This one is going to be real non-sensical.
library(markovChains)
# Load the text data and format as an observable Markov chain
text_data <- observe_ham_on_rye() # use reducedSize = NULL for full 80,000 word chain
# Estimate the first-order transition matrix
P <- estimate_dtmc(chain = text_data$encodedChain, m = length(text_data$stateSpace))
# Generate your new text: default 14 lines with 10 words per line.
textout <- markov_text(text_data, P)
print(textout) # an example output:
# [1] "it 35 satin ship track worship peek checking hari cs"
# [2] "prevent striped hunk jab dried mrs windmilled instructed depicted ma"
# [3] "nod doorways relentless tuition care stab sessions 523 purple rooting"
# [4] "platform christ fenster valves semi aplomb melodrama apes reach hoho"
# [5] "tits suicided meaningless next starched durham sisters dried 8th batter"
# [6] "by staring 8 mutilated kittens refused disrobe shit let homo"
# [7] "handkerchief fall jungle track going flower webs surrendered glistening items"
# [8] "blades stand jap bill successful handkerchiefs triumph scratched pomp corps"
# [9] "docked pearshaped kicked equipment butts mere switch unmarred everybodys fly"
# [10] "limping befriended toughest sit werent stopped homemade mates adopted energy"
# [11] "indian mound effective crash tanks fantastic dull girlwoman candle fidget"
# [12] "admired passes roast arguing gum timecard appointment worthwhile to rightwing"
# [13] "clumps except fighter biologist greener nomansland meatballs kickoff equipment firm"
# [14] "umpire windmilled deserved stretched mustard stockboy leading blackboards points morris"