multi_tpfit_ml: Mean Length Method for Multidimensional Model Parameters Estimation

Description

The function estimates the model parameters of a $d$-D continuous lag spatial Markov chain. Transition rates matrices along axial directions and proportions of categories are computed.

Usage

multi_tpfit_ml(data, coords, tolerance = pi/8, rotation = NULL, 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.

tolerance

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

rotation

a numerical vector of length $d - 1$ with rotation angles (in radians), in order to perform the main axes rotation. No rotation is performed by default.

mle

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

Value

coordsnames: a character vector containing the name of each axis.
coefficients: a list containing the transition rates matrices computed for each axial direction.
prop: a vector containing the proportions of each observed category.
tolerance: a numerical value which denotes the tolerance angle (in radians).

Details

A $d$-D continuous-lag spatial Markov chain is probabilistic model which is developed by interpolation of the transition rate matrices computed for the main directions. It defines transition probabilities $Pr(Z(s + h) = z_k | Z(s) = z_j)$ through $$\mbox{expm} (\Vert h \Vert R),$$ where $h$ is the lag vector and the entries of $R$ are ellipsoidally interpolated.

The ellipsoidal interpolation is given by $$\vert r_{jk} \vert = \sqrt{\sum_{i = 1}^d \left( \frac{h_i}{\Vert h \Vert} r_{jk, \mathbf{e}_i} \right)^2},$$ where $e_i$ is a standard basis for a $d$-D space.

If $h_i < 0$ the respective entries $r_(jk, e_i)$ are replaced by $r_(jk, -e_i)$, which is computed as $$r_{jk, -\mathbf{e}_i} = \frac{p_k}{p_j} \, r_{kj, \mathbf{e}_i},$$ where $p_k$ and $p_j$ respectively denote the proportions for the $k$-th and $j$-th categories. In so doing, the model may describe the anisotropy of the process.

When some entries of the rates matrices are not identifiable, it is suggested to vary the tolerance coefficient and the rotation angles. This problem may be also avoided if the input argument mle is to set to be "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.

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