tpfit_me: Maximum Entropy Method for One-dimensional Model Parameters Estimation

Description

The function estimates the model parameters of a 1-D continuous lag spatial Markov chain by the use of the maximum entropy method. Transition rates matrix along a user defined direction and proportions of categories are computed.

Usage

tpfit_me(data, coords, direction, tolerance = pi/8, max.it = 9000, mle = "avg")

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.

tolerance

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

max.it

a numerical value which denotes the maximum number of iterations to perform during the optimization phase. It is 9000 by default.

mle

a character value to pass to the function mlen. It is "avg" by default.

Value

coefficients: the transition rates matrix computed for the user defined direction.
prop: a vector containing the proportions of each observed category.
tolerance: a numerical value which denotes the tolerance angle (in radians).

Details

A 1-D continuous-lag spatial Markov chain is probabilistic model which involves a transition rate matrix $R$ computed for the direction $phi$. It defines the transition probability $Pr(Z(s + h) = z_k | Z(s) = z_j)$ through the entry $t_(jk)$ of the following matrix $$T = \mbox{expm} (h R),$$ where $h$ is a positive lag value.

To calculate entries of the transition rate matrix, we need to maximize the entropy of the transition probabilities of embedded occurrences along a given direction $phi$. The entropy is defined as $$e = - \sum_{k}^K \sum_{j \neq k}^K \tau_{jk, \phi} \log \tau_{jk, \phi},$$ where $tau_(jk, phi)$ are transition probabilities of embedded occurrences. It is maximized by the use of the iterative proportion fitting method.

When some entries of the matrix $R$ are not identifiable, it is suggested to vary the tolerance coefficient or to set the input argument mle to "mlk".

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.