tpfit_ils: 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

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

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.

a numerical value greater than one for a constant which controls the growth of the penalization term in the loss function. It is equal to 10 by default.

echo

a logical value; if TRUE, the function prints some information about the optimization. It is FALSE by default.

...

other arguments to pass to the function nlminb.

tpfit

an object tpfit to optimize. If missing, the algorithm starts with a null transition rates matrix.

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).

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.