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.
tpfit_ils(data, coords, direction, max.dist = Inf, mpoints = 20,
tolerance = pi/8, q = 10, echo = FALSE, ..., tpfit)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:
the transition rates matrix computed for the user defined direction.
a vector containing the proportions of each observed category.
a numerical value which denotes the tolerance angle (in radians).
a categorical data vector of length \(n\).
an \(n \times d\) matrix where each row denotes the \(d\)-D coordinates of data locations.
a \(d\)-D numerical vector (or versor) which represents the chosen direction.
a numerical value which defines the maximum lag value. It's Inf by default.
a numerical value which defines the number of lag intervals.
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.
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.
an object tpfit to optimize. If missing, the algorithm starts with a null transition rates matrix.
Luca Sartore drwolf85@gmail.com
If the process is not stationary, the optimization algorithm does not converge.
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.
Sartore, L. (2010) Geostatistical models for 3-D data. M.Phil. thesis, Ca' Foscari University of Venice.
predict.tpfit, print.tpfit, multi_tpfit_ils, transiogram
# \donttest{
data(ACM)
# Estimate the parameters of a
# one-dimensional MC model
tpfit_ils(ACM$MAT3, ACM[, 1:3], c(0,0,1), 100)
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