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spMC (version 0.2.2)

multi.tpfit: 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(data, coords, tolerance = pi/8,
            rotation = NULL, mle = FALSE)

Arguments

Value

An object of the class multi.tpfit is returned. The function print.multi.tpfit is used to print the fitted model. The object is a list with the following components:coordsnamesa character vector containing the name of each axis.coefficientsa list containing the transition rates matrices computed for each axial direction.propa vector containing the proportions of each observed category.tolerancea numerical value which denotes the tolerance angle (in radians).

Rdversion

1.1

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 $\mathbf{e}_i$ is a standard basis for a $d$-D space.

If $h_i < 0$ the respective entries $r_{jk, \mathbf{e}_i}$ are replaced by $r_{jk, -\mathbf{e}_i}$, 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 TRUE.

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.

See Also

predict.multi.tpfit, print.multi.tpfit, image.multi.tpfit, tpfit

Examples

Run this code
data(ACM)

# Estimate transition rates matrices and 
# proportions for the categorical variable MAT5
multi.tpfit(ACM$MAT5, ACM[, 1:3])

# Estimate transition rates matrices and
# proportions for the categorical variable MAT3
multi.tpfit(ACM$MAT3, ACM[, 1:3])

# Estimate transition rates matrices and
# proportions for the categorical variable PERM
multi.tpfit(ACM$PERM, ACM[, 1:3])

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