sim_ik {spMC}

Conditional Simulation Based on Indicator Kriging

Description

The function simulates a random field through the Indicator Kriging technique.

Usage

sim_ik(x, data, coords, grid, knn = 12, ordinary = TRUE, entropy = FALSE)

Arguments

x

an object of the class multi_tpfit, typically with the output of the function multi_tpfit.

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.

grid

an m \times d matrix where each row denotes the d-D coordinates in the simulation grid.

knn

an integer value which specifies the number of k-nearest neighbours for each simulation point. An optimal number is between 4 and 12. If NULL all observations are considered (just for very small dataset!!). It is 12 by default.

ordinary

a logical value; if FALSE, the probabilities are computed through the Simple Kriging technique, otherwise the Ordinary Kriging method is used.

entropy

a logical value. If TRUE, the prediction uncertainties are computed through the entropy (and standardized entropy). The default value is FALSE.

Details

This method computes an approximation of posterior probabilities

\Pr\left(Z(\mathbf{s}_0) = z_k \left\vert \bigcap_{i = 1}^n Z(\mathbf{s}_i) = z(\mathbf{s}_i)\right.\right).

\hspace{0cm}

The probability is calculated as the sum of the observed proportion and the weighted sum of indicator variables which denote the presence of the k-th category in observed points \mathbf{s}_i. Weights involved in the sum are the solution of a system of equations.

Probabilities approximated are usually truncated and normalized with respect to the probability constraints, because such probabilities might lie outside the interval [0, 1]. The normalization procedure is designed such that it is not possible to obtain vectors such that the sum of their probabilities is always equal to one.

When an initial configuration is simulated, it should be modified to reach a pattern similar to the sample by the use of the quench function.

Value

A data frame containing the simulation grid, the simulated random field, predicted values and the approximated probabilities is returned. Two extra columns respectively denoting the entropy and standardized entorpy are bindend to the data frame when argument entropy = TRUE.

Author(s)

Luca Sartore [e-mail]

References

Carle, S. F., Fogg, G. E. (1996) Transition Probability-Based Indicator Geostatistics. Mathematical Geosciences, 28(4), 453-476.

Carle, S. F. (1999) T-PROGS: Transition Probability Geostatistical Software. University of California, Davis.

Sartore, L. (2010) Geostatistical models for 3-D data. M.Phil. thesis, Ca' Foscari University of Venice.

Weise, T. (2009) Global Optimization Algorithms - Theory and Application. https://archive.org/details/Thomas_Weise__Global_Optimization_Algorithms_Theory_and_Application.

See Also

sim_ck, sim_mcs, sim_path

Examples

data(ACM)

# Model parameters estimation for the
# multinomial categorical simulation
x <- multi_tpfit(ACM$MAT5, ACM[, 1:3])

# Generate the simulation grid
mygrid <- list()
mygrid$X <- seq(min(ACM$X), max(ACM$X), length = 20)
mygrid$Y <- seq(min(ACM$Y), max(ACM$Y), length = 20)
mygrid$Z <- -40 * 0:9 - 1
mygrid <- as.matrix(expand.grid(mygrid$X, mygrid$Y, mygrid$Z))

# Simulate the random field through
# Simple Indicator Kriging algorithm
mySIKSim <- sim_ik(x, ACM$MAT5, ACM[, 1:3], mygrid, ordinary = FALSE)

# Simulate the random field through
# Ordinary Indicator Kriging algorithm
myOIKSim <- sim_ik(x, ACM$MAT5, ACM[, 1:3], mygrid)