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)

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.

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.

Value

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.

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 $i != j$, must satisfy the following properties:

$a < ||s_i - s_j|| < a + m / n,$ 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.