Function to fit a discrete Markov chain

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()
)

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

Arguments

stringchar: It can be a $$n x n$$ matrix or a character vector or a list
toRowProbs: converts a sequence matrix into a probability matrix
sanitize: put 1 in all rows having rowSum equal to zero
possibleStates: Possible states which are not present in the given sequence
data: It can be a character vector or a $$n x n$$ matrix or a $$n x n$$ data frame or a list
method: Method used to estimate the Markov chain. Either "mle", "map", "bootstrap" or "laplace"
byrow: it tells whether the output Markov chain should show the transition probabilities by row.
nboot: Number of bootstrap replicates in case "bootstrap" is used.
laplacian: Laplacian smoothing parameter, default zero. It is only used when "laplace" method is chosen.
name: Optional character for name slot.
parallel: Use parallel processing when performing Boostrap estimates.
confidencelevel: $$\alpha$$ level for conficence intervals width. Used only when method equal to "mle".
confint: a boolean to decide whether to compute Confidence Interval or not.
hyperparam: Hyperparameter matrix for the a priori distribution. If none is provided, default value of 1 is assigned to each parameter. This must be of size $$k x k$$ where k is the number of states in the chain and the values should typically be non-negative integers.
absorbingStates: Character vector of states that are known a priori to be absorbing. The corresponding rows are set to the identity row after MLE fitting when byrow = TRUE; the corresponding columns are set to the identity column when byrow = FALSE. The argument is currently supported only for method = "mle".

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.

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.

When absorbingStates is supplied, the declared states must have no observed outgoing transitions. This allows terminal states in censored customer journeys to be represented as absorbing states without adding artificial observations.

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.

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

Author

Giorgio Spedicato, Tae Seung Kang, Sai Bhargav Yalamanchi

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")