Intensity Raster into Kappa: Simulating Small Trees in Forest Stands

Introduction

As a researcher or developer working with spatial point patterns, you often encounter complex problems that require simulating real-world scenarios. One such challenge is simulating the positions of small trees in forest stands. In this article, we’ll explore how to achieve this using the spatstat package in R and address the limitations of the Thomas clumping model.

Background

The Thomas clumping model is a widely used method for simulating spatial point patterns, including those representing tree locations. It assumes that the number of points at a given location follows a Poisson distribution with an intensity function. However, this approach can lead to unrealistic results when modeling real-world scenarios, such as forest stands, where the density of small trees is often limited.

The Problem

The question from the Stack Overflow post highlights a common issue with simulating small trees in forest stands. Currently, it’s challenging to generate these points in areas with fewer adult and medium-sized trees, resulting in unrealistic patterns. We’ll discuss how to overcome this limitation using an alternative approach.

Inhomogeneous Cluster Models

One possible solution is to use inhomogeneous cluster models, which allow for spatially varying intensity functions. This approach enables the simulation of small trees in areas with fewer adult and medium-sized trees.

Mathematical Background

Mathematically, an inhomogeneous cluster model can be represented as follows:

mu(x) = lambda(x) * h(x)

where mu(x) is the expected number of points at location x, lambda(x) is the intensity function, and h(x) is a kernel function that controls the spread of points.

Controlling Intensity Function

To control the intensity function, you can use various techniques, such as:

Maximum Likelihood Estimation (MLE): Estimate the parameters of the intensity function using MLE methods.
Bayesian Methods: Use Bayesian inference to estimate the parameters of the intensity function.

Implementing Inhomogeneous Cluster Models

To implement inhomogeneous cluster models, you can use the rThomas function from the spatstat package with an additional argument for the intensity function. Here’s an example:

# Define the intensity function
lambda <- function(x) {
  # Example: log-linear model with a kernel function
  exp(10 * (x[1] - 5)^2 + x[2])
}

# Simulate small trees using Thomas clumping model with inhomogeneous cluster model
kappaS <- 0.01 / (mean(lambda(muS) * winarea))
Small.Trees1 <- rThomas(kappa = kappaS,
                      scale = sigmaM,
                      mu = lambda,
                      win = Window222)

Conclusion

Simulating small trees in forest stands is a challenging task that requires careful consideration of spatial patterns and intensity functions. By using inhomogeneous cluster models, you can control the intensity function and generate more realistic simulations.

In this article, we’ve discussed the limitations of the Thomas clumping model and introduced an alternative approach using inhomogeneous cluster models. We provided examples and mathematical background to help you understand how to implement this approach.

Code for Replication

Here is some sample code that can be used to replicate the example:

# Load necessary libraries
library(spatstat)
library(here)

# Define the dataset
here::tempdir()
Window222 <- Matrix(0:1, nrow = 100, ncol = 100, dimnames = list(NULL, NULL))

# Set seed for reproducibility
set.seed(123)

# Define intensity function
lambda <- function(x) {
  # Example: log-linear model with a kernel function
  exp(10 * (x[1] - 5)^2 + x[2])
}

# Simulate small trees using Thomas clumping model with inhomogeneous cluster model
kappaS <- 0.01 / (mean(lambda(muS) * winarea))
Small.Trees1 <- rThomas(kappa = kappaS,
                      scale = sigmaM,
                      mu = lambda,
                      win = Window222)

# Print results
print(Small.Trees1)

Last modified on 2024-09-06