spMC (version 0.3.15)

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)

Value

A \(K \times K\) transition probability matrix, where \(K\) denotes the number of observed categories. Another \(K \times K\) matrix with the counts of transitions is attached as an attribute.

Arguments

data

a categorical data vector of length \(n\).

coords

an \(n \times d\) 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.

Author

Luca Sartore

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.

See Also

which_lines, predict.tpfit, predict.multi_tpfit

Examples

# \donttest{
data(ACM)
direction <- c(0, 0, 1)

# Compute the appertaining directional line for each location
loc.id <- which_lines(ACM[, 1:3], direction, pi/8)

# Estimate the embedded transition probabilities
# matrix for the categorical variable MAT5
embed_MC(ACM$MAT5, ACM[, 1:3], loc.id, direction)

# Estimate the embedded transition probabilities
# matrix for the categorical variable MAT3
embed_MC(ACM$MAT3, ACM[, 1:3], loc.id, direction)

# Estimate the embedded transition probabilities
# matrix for the categorical variable PERM
embed_MC(ACM$PERM, ACM[, 1:3], loc.id, direction)
# }