spMC (version 0.2.2)

ilstpfit: Iterated Least Squares 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 iterated least squares and the bound-constrained Lagrangian methods. Transition rates matrix along a user defined direction and proportions of categories are computed.

Usage

ilstpfit(data, coords, direction, max.dist = Inf, mpoints = 20, 
         tolerance = pi/8, q = 10, echo = FALSE, ..., tpfit)

Value

An object of the class tpfit is returned. The function print.tpfit is used to print the fitted model. The object is a list with the following components:coefficientsthe transition rates matrix computed for the user defined direction.propa vector containing the proportions of each observed category.tolerancea numerical value which denotes the tolerance angle (in radians).

Rdversion

1.1

Warning

If the process is not stationary, the optimization algorithm does not converge.

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 minimize the discrepancies between the empirical transiogram (see transiogram) and the predicted transition probabilities.

By the use of the iterated least squares, the diagonal entries of $R$ are constrained to be negative, while the off-diagonal transition rates are constrained to be positive. Further constraints are considered in order to obtain a proper transition rates matrix.

References

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

See Also

predict.tpfit, print.tpfit, multi.tpfit, transiogram

Examples

data(ACM)

# Estimate the parameters of a 
# one-dimensional MC model
ilstpfit(ACM$MAT3, ACM[, 1:3], c(0,0,1), 100)