embed_MC: Transition Probabilities Estimation for Embedded Markov Chain

Description

The function estimates the embedded transition probabilities matrix for a $1$-D spatial embedded Markov chain.

Usage

embed_MC(data, coords, loc.id, direction)

Arguments

data

a categorical data vector of length $n$.

coords

an $n x d$ matrix where each row denotes the $d$-D coordinates of data locations.

loc.id

a vector of $n$ values which indicats the directional line of each location. It is usually the output of the function which_lines.

direction

a $d$-D numerical vector (or versor) which represents the chosen direction.

Value

$K x K$ matrix, where $K$ denotes the number of observed categories.

Details

An embedded Markov chain is probabilistic model which defines the transition probabilities between embedded occurrences.

The resulting matrix is given by normalizing a transition count matrix, which doesn't depend on the length of embedded occurrences. Self-transitions of embedded occurrences are not observable, so diagonal entries are set to be NA.

It's also possible to calculate the transition probabilities matrix for several directions in a $d$-D space through arguments direction and loc.id. If the user has no previous knowledge about loc.id, the function which_lines provides a method to compute the right values.

References

Carle, S. F., Fogg, G. E. (1997) Modelling Spatial Variability with One and Multidimensional Continuous-Lag Markov Chains. Mathematical Geology, 29(7), 891-918.

Dynkin, E. B. (1961) Theory of Markov Processes. Englewood Cliffs, N.J.: Prentice-Hall, Inc.

Sartore, L. (2010) Geostatistical models for 3-D data. M.Phil. thesis, Ca' Foscari University of Venice.