# 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

1 2 3 4 5 6 | ```
createSequenceMatrix(stringchar, toRowProbs = FALSE, sanitize = FALSE,
possibleStates = character())
markovchainFit(data, method = "mle", byrow = TRUE, nboot = 10L,
laplacian = 0, name = "", parallel = FALSE, confidencelevel = 0.95,
hyperparam = matrix(), sanitize = FALSE, possibleStates = character())
``` |

### Arguments

`stringchar` |
Equivalent to data. Either a nx2 matrix or a character vector. |

`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` |
A character 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` |
level for conficence intervals width.
Used only when |

`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 kxk where k is the number of states in the chain and the values should typically be non-negative integers. |

### 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 "euristically".
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

1 2 3 4 5 | ```
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")
``` |