prob_zk: Posterior Density Estimation at a Single Location

In BMEmapping: Spatial Interpolation using Bayesian Maximum Entropy (BME)

Posterior Density Estimation at a Single Location

Description

Computes the posterior and plots probability density function (PDF) at a single unobserved spatial location using the Bayesian Maximum Entropy (BME) framework. This function integrates both hard data (precise measurements) and soft data (interval or uncertain observations), together with a specified variogram model, to numerically estimate the posterior density across a range of possible values. Optionally displays a plot of the posterior density function for the specified location.

Usage

prob_zk(x, ch, cs, zh, a, b,
        model, nugget, sill, range, nsmax = 5,
        nhmax = 5, n = 50, zk_range = extended_range(zh, a, b),
        plot = FALSE)

Arguments

x	A two-column matrix of spatial coordinates for a single estimation location.
ch	A two-column matrix of spatial coordinates for hard data locations.
cs	A two-column matrix of spatial coordinates for soft (interval) data locations.
zh	A numeric vector of observed values at the hard data locations.
a	A numeric vector of lower bounds for the soft interval data.
b	A numeric vector of upper bounds for the soft interval data.
model	A string specifying the variogram or covariance model to use (e.g., "exp", "sph", etc.).
nugget	A non-negative numeric value for the nugget effect in the variogram model.
sill	A numeric value representing the sill (total variance) in the variogram model.
range	A positive numeric value for the range (or effective range) parameter of the variogram model.
nsmax	An integer specifying the maximum number of nearby soft data points to include for estimation (default is 5).
nhmax	An integer specifying the maximum number of nearby hard data points to include for estimation (default is 5).
n	An integer indicating the number of points at which to evaluate the posterior density over zk_range.
zk_range	A numeric vector specifying the range over which to evaluate the unobserved value at the estimation location (zk). Although zk is unknown, it is assumed to lie within a range similar to the observed data (zh, a, and b). It is advisable to explore the posterior distribution at a few locations using prob_zk() before finalizing this range. The default is extended_range(zh, a, b).
plot	Logical; if TRUE, plots the posterior density curve.

Value

Two elements:

data frame

A data frame with two columns: zk_i (assumed zk values) and prob_zk_i (corresponding posterior densities).

plot

An optional plot of posterior density of the estimation location.

Examples

data("utsnowload")
x <- utsnowload[1, c("latitude", "longitude")]
ch <- utsnowload[2:67, c("latitude", "longitude")]
cs <- utsnowload[68:232, c("latitude", "longitude")]
zh <- utsnowload[2:67, "hard"]
a <- utsnowload[68:232, "lower"]
b <- utsnowload[68:232, "upper"]
prob_zk(x, ch, cs, zh, a, b, model = "exp", nugget = 0.0953, sill = 0.3639,
        range = 1.0787, plot = TRUE)

BMEmapping documentation built on July 2, 2025, 9:07 a.m.