multi_tpfit_ils {spMC} | R Documentation |
Iterated Least Squares Method for Multidimensional Model Parameters Estimation
Description
The function estimates the model parameters of a d
-D continuous lag spatial Markov chain by the use of the iterated least squares and the bound-constrained Lagrangian methods. Transition rates matrices along axial directions and proportions of categories are computed.
Usage
multi_tpfit_ils(data, coords, max.dist = Inf, mpoints = 20,
tolerance = pi/8, rotation = NULL, q = 10,
echo = FALSE, ..., mtpfit)
Arguments
data |
a categorical data vector of length |
coords |
an |
max.dist |
a numerical value which defines the maximum lag value. It is |
mpoints |
a numerical value which defines the number of lag intervals. |
tolerance |
a numerical value for the tolerance angle (in radians). It is |
rotation |
a numerical vector of length |
q |
a numerical value greater than one for a constant which controls the growth of the penalization term in the loss function. It is equal to |
echo |
a logical value; if |
... |
other arguments to pass to the function |
mtpfit |
an object |
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.
In particular, to estimate entries of transition rate matrices computed for the main axial directions, we need to minimize the discrepancies between the empirical transiograms (see transiogram
) and the predicted transition probabilities.
By the use of the iterated least squares, the diagonal entries of R
are constrained to be negative, while the off-diagonal transition rates are constrained to be positive. Further constraints are considered in order to obtain a proper transition rates matrix.
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). |
Warning
If the process is not stationary, the optimization algorithm does not converge.
Author(s)
Luca Sartore drwolf85@gmail.com
References
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_ils
, transiogram
Examples
data(ACM)
# Estimate the parameters of a
# multidimensional MC model
multi_tpfit_ils(ACM$MAT3, ACM[, 1:3], 100)