multi_tpfit_me {spMC}

Maximum Entropy 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_me(data, coords, tolerance = pi/8, max.it = 9000,
               rotation = NULL, mle = "avg")

Arguments

data

a categorical data vector of length n.

coords

an n \times d matrix where each row denotes the d-D coordinates of data locations.

tolerance

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

max.it

a numerical value which denotes the maximum number of iterations to perform during the optimization phase. It is 9000 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.

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 by the use of the function tpfit_me. 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 "mlk".

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:

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).

Author(s)

Luca Sartore [e-mail]

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.

See Also

predict.multi_tpfit, print.multi_tpfit, image.multi_tpfit, tpfit_me

Examples

data(ACM)

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

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

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