spMC (version 0.3.15)

transiogram: Empirical Transition Probabilities Estimation for 1-D MC

Description

The function estimates transition probabilities matrices for a \(1\)-D continuous lag spatial Markov chain.

Usage

transiogram(data, coords, direction, max.dist = Inf, 
            mpoints = 20, tolerance = pi / 8, reverse = FALSE)

Value

An object of the class transiogram is returned. The function print.transiogram is used to print computed probabilities. The object is a list with the following components:

Tmat

a 3-D array containing the probabilities.

LOSE

a 3-D array containing the standard error calculated for the log odds of the transition probabilities.

lags

a vector containing one-dimensional lags.

type

a character string which specifies that computed probabilities are empirical.

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.

direction

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

max.dist

a numerical value which defines the maximum lag value. It's Inf by default.

mpoints

a numerical value which defines the number of lag intervals.

tolerance

a numerical value for the tolerance angle (in radians). It's pi/8 by default.

reverse

a logical value. If TRUE the transition probabilities of the reversible chain are also computed. It's FALSE by default.

Author

Luca Sartore

Details

Empirical probabilities are estimated by counting such pairs of observations which satisfy some properties, and by normalizing the result.

A generic pair of sample points \(s_i\) and \(s_j\), where \(i \neq j\), must satisfy the following properties:

• \(\Vert s_i - s_j \Vert \in [a, a+\frac{m}{n}],\) where \(a\) is a non negative real value, while \(m\) denotes the maximum lag value (max.dist) and \(n\) is the number of lag intervals (mpoints).

• the lag vector \(h = s_i - s_j\) must have the same direction of the vector \(\phi\) (direction) with a certain angular tolerance.

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.

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

See Also

predict.tpfit, predict.tpfit, plot.transiogram

Examples

# \donttest{
data(ACM)

# Estimate empirical transition 
# probabilities by points
transiogram(ACM$MAT3, ACM[, 1:3], c(0, 0, 1), 200, 5)
# }