spMC (version 0.3.2)

mlen: Mean Length Estimation for Embedded Markov Chain

Description

The function estimates the mean length for a $d$-D spatial embedded Markov chain for a specified direction $\phi$.

Usage

mlen(data, coords, loc.id, direction, mle = "trm")

Value

A numeric vector containing the mean length for each observed category.

Rdversion

1.1

Details

The mean length is the total length occupied by the $k$-th category divided by the number of its embedded occurrences along lines in the direction $\phi$. More robust methods are implemented, such as the trimmed mean and the trimmed median.

If the stratum lengths are censored, the maximum likelihood approach is more appropriate than the arithmetic mean. In this case, the stratum lengths are assumed to be independent realizations from a log-normal random variable. The quantity to maximize is $$L(\mu_1, \ldots, \mu_K, \sigma_1, \ldots, \sigma_K) = \prod_{i = 1}^m \prod_{k = 1}^K \left[ \int_{l_i}^{l_i+u_i} \frac{1}{x \sigma_k \sqrt{2}} \exp \left\lbrace - \frac{(\log x - \mu_k)^2}{2 \sigma_k^2} \right\rbrace \right]^{z_{k, i}} \mbox{d}x,$$ where $\boldsymbol{\mu} = (\mu_1, \ldots, \mu_K)^\top$ and $\boldsymbol{\sigma} = (\sigma_1, \ldots, \sigma_K)^\top$ are vectors of parameters, $l_i$ is the observed stratum length, $u_i$ denotes the upper bound of the censor and $z_{k, i}$ denotes a dummy variable which assumes value 1 if and only if the $i$-th stratum is referred to the $k$-th category.

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

which_lines

Examples

data(ACM)
direction <- c(0,0,1)

# Compute the appartaining directional line for each location
loc.id <- which_lines(ACM[, 1:3], direction)

# Estimate the mean lengths for each observed category
ml <- mlen(ACM$MAT5, ACM[, 1:3], loc.id, direction, mle = "avg")

# Equivalently
gl <- getlen(ACM$MAT5, ACM[, 1:3], loc.id, direction, zero.allowed = TRUE)
ml1 <- tapply(gl$length, gl$categories, mean)