tpfit: One-dimensional Model Parameters Estimation

Description

The function estimates the model parameters of a 1-D continuous lag spatial Markov chain. Transition rates matrix along a user defined direction and proportions of categories are computed.

Usage

tpfit(data, coords, direction, method = "ml", tolerance = pi/8, max.it = 9000, 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.

direction

a $d$-D numerical vector (or versor) which represents the chosen direction.

method

a character object specifying the method to estimata 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.

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 mlen. It is "avg" by default and not use when the method is "ils".

...

other arguments to pass to the functions tpfit_ml, tpfit_ils or tpfit_me.

Value

coefficients: the transition rates matrix computed for the user defined direction.
prop: a vector containing the proportions of each observed category.
tolerance: a numerical value which denotes the tolerance angle (in radians).

Details

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.

Three methods are available to calculate entries of the transition rate matrix. The mean length method is performed by the use of the function tpfit_ml, the iterated least squares are applied through the function tpfit_ils, while the function tpfit_me implements the maximum entropy method.

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.