multi_tpfit {spMC}

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, method = "ml", tolerance = pi/8,
            rotation = NULL, max.it = 9000, 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.

method

a character object specifying the method to estimate the transition rates. Possible choises are "ml" (by default) for the mean length method, "ils" for the iterated least squares and "me" for the maximum entropy method.

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.

max.it

a numerical value which denotes the maximum number of iterations to perform during the optimization phase. It is 9000 by default and used only when the method is "me".

mle

a character value to pass to the function tpfit. It is "avg" by default and not use when the method is "ils".

...

other arguments to pass to the functions multi_tpfit_ml, multi_tpfit_ils or multi_tpfit_me.

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.

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.

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

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])