# Approximating a Markov chain

### Description

Given a vegetation data frame considerd a time series with releves as rows and species as columns transition matrices are derived vor each time step based on some simple assumptions. These are averaged and a model series is derived trough scalar products. Time steps are given in a separate vector t. Missing steps are properly processed.

### Usage

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### Arguments

`veg` |
This is a vegetation data frame, releves are rows, species columns |

`t` |
The time step scale of length according with rows in x |

`x` |
An object of class "fitmarkov" |

`adjust` |
A logical vector adjusting the sum of species scores to 1.0. Default is adjust=FALSE |

`...` |
Vector colors of any length for line colors, vector widths for line widths. See example below. |

### Details

This method yields a possible solution for fitting a Markov series. The true process may be very different.

### Value

An output list of class "fitmarkov" with at least the following intems:

`fitted.data ` |
The fitted time series' |

`raw.data ` |
The input time series' |

`transition.matrix` |
The mean transition matrix' |

`t.measured` |
The time steps upon input where time steps may be missing' |

`t.modeled` |
The time steps upon output, no missing steps' |

### Note

The aim of this method is to provide a smooth curve based on input data. Because this relies on incomplete information, it is just one out of many solutions.

### Author(s)

Otto Wildi

### References

Orloci, L., Anand, M. & He, X. 1993. Markov chain: a realistic model for temporal coenosere? Biom. Praxim 33: 7-26.

Lippe, E., De Smitt, J.T. & Glenn-Lewin, D.C. 1985. Markov models and succession: a test from a heathland in the Netherlands. Journal of Ecology 73: 775-791.

Wildi, O. 2013. Data Analysis in Vegetation Ecology. 2nd ed. Wiley-Blackwell, Chichester.

### Examples

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