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ldmppr: Location Dependent Marked Point Processes in R

This paper introduces **ldmppr**, an R package designed to estimate, simulate, evaluate, and visualize location-dependent marked spatial point processes, thereby addressing the limitations of assuming mark-location independence in applications such as forestry.

Original authors: Lane Drew, Andee Kaplan

Published 2026-05-20

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CC BY 4.0

Original authors: Lane Drew, Andee Kaplan

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a forester walking through a forest. You see trees scattered across the land. In the old way of thinking, statisticians assumed two things:

The Trees: They are scattered randomly, like raindrops hitting a roof (or perhaps slightly repelling each other so they don't grow too close).
The Sizes: The size of each tree (its height or volume) is completely random and has nothing to do with where it is standing.

The Problem: The authors of this paper argue that this second assumption is wrong. In the real world, a tree's size is heavily influenced by its location. A tree standing on a sunny, moist hillside will be huge, while a tree in a dry, rocky valley will be small. The "size" (the mark) depends on the "location."

The Solution: The authors created a new tool called ldmppr (pronounced "L-D-M-P-R"). Think of this tool as a sophisticated digital forest simulator that understands this connection between location and size.

Here is how the paper explains the tool, using simple analogies:

1. The "Time-Travel" Trick

The math behind the tool is complex, but the core idea is clever. The authors realized that modeling "trees that repel each other" is very hard to do directly. So, they invented a time-travel trick.

They pretend that the biggest trees in the forest were the first to "arrive" in a timeline, and the smallest trees arrived last. By turning Size into Time, they can use a well-known, easier method called a "self-correcting process."

The Analogy: Imagine a crowded room where people want to keep their personal space. If you try to model this directly, it's messy. But if you imagine people entering the room one by one, and each new person checks the distance to everyone already inside before picking a spot, it becomes much easier to calculate. The ldmppr package does exactly this: it treats the largest trees as the "first arrivals" and the smallest as the "last arrivals" to make the math work.

2. The Two-Part Engine

The package works like a two-part engine that learns from real data:

Part A: The "Space" Engine (The Self-Correcting Model)
This part learns the rules of the forest floor. It figures out how much space trees need between them. Do they hate being close? Do they cluster? It calculates the "repulsion" rules so the simulated trees don't overlap or clump up unrealistically.
Part B: The "Size" Engine (The Machine Learning Model)
This part learns the rules of the environment. It looks at maps (raster images) showing elevation, sunlight, and soil moisture. It uses a "smart guesser" (like a Random Forest or XGBoost algorithm) to predict: "If a tree is at this specific spot with this much sun, how big will it be?"

3. The Workflow: Learn, Check, and Create

The paper outlines a simple three-step recipe for using the tool:

Learn (Estimation): You feed the package a photo of a real forest (data) and a map of the terrain (covariates). The package runs a complex optimization game to find the perfect settings for both the "Space" and "Size" engines. It tries millions of combinations to find the one that matches your real forest best.
Check (Goodness-of-Fit): Before trusting the tool, you ask it to generate 500 fake forests based on what it learned. Then, you compare the fake forests to the real one.
- The "Envelope Test": The paper uses a statistical test called a "Global Envelope." Imagine drawing a safety zone around the average of the fake forests. If your real forest falls outside that zone, the tool says, "Oops, my model is wrong." If the real forest stays safely inside the zone, the model is a good fit.
Create (Simulation): Once the model passes the test, you can use it to generate brand new, realistic forests. You can say, "Show me what a forest would look like if the soil was wetter," or "Generate 100 possible forests for this specific mountain."

4. Why This Matters

The paper highlights that while other tools exist for studying point patterns (like the spatstat package), they often assume tree sizes are random or are too computationally heavy to use.

ldmppr is special because:

It is fast: It uses a smart mathematical shortcut (the time-mapping trick) instead of slow, brute-force methods.
It is flexible: You can swap out the "Size" engine for any machine learning model you like.
It is realistic: It finally allows scientists to say, "The size of the tree depends on where it is," and actually model that relationship.

The Bottom Line

The paper presents ldmppr as a user-friendly R package that helps scientists simulate complex, realistic patterns where location dictates size. It turns a difficult mathematical problem into a step-by-step workflow: Map the terrain, learn the spacing rules, learn the size rules, check if the simulation looks real, and then generate new scenarios.

The authors tested this on a real dataset of conifer trees in the Rocky Mountains. Initially, their simple model failed (the fake trees didn't look like the real ones). But by refining the "Space" engine and training the "Size" engine more carefully, they achieved a perfect match, proving the tool works for real-world forestry data.

Technical Summary of ldmppr: Location Dependent Marked Point Processes in R

Problem Statement

Point process models are widely used to analyze events occurring in space and time, often augmented with "marks" (attributes like size or type). A common simplifying assumption in existing literature is that these marks are independent of the spatial locations of the points. However, in many scientific domains, particularly forestry, this assumption is untenable. For instance, tree sizes (canopy volumes) are often influenced by location-specific covariates such as elevation, soil moisture, and sunlight, as well as by the spatial distribution of neighboring trees.

While R packages like spatstat and ptProcess exist for point process analysis, they often lack the flexibility to model the dependence between marks and locations, especially when the spatial pattern exhibits regularity (inhibition), where points tend to repel one another. Traditional approaches to modeling regularity, such as Gibbs processes, are computationally expensive due to intractable normalizing constants and reliance on MCMC algorithms. There is a need for a computationally scalable framework that can estimate, simulate, and evaluate models where marks depend on both spatial location and the history of the process.

Methodology

The paper introduces ldmppr, an R package that implements a likelihood-based framework for location-dependent marked point processes characterized by regularity. The core methodology involves mapping a spatial marked point process onto a marked spatio-temporal point process, allowing for the use of established likelihood-based estimation and simulation techniques.

1. Spatio-Temporal Mapping

The authors extend the mechanistic approach of Møller et al. (2016) and the self-correcting process of Isham and Westcott (1979).

Time Mapping: Primary marks (e.g., tree size) are mapped to arrival times $t$ in the interval $[0, 1]$ using a power-law transformation:
$\tilde{T}_i = 1 - \left[ \frac{M_{i1} - \min(M_{j1})}{\max(M_{j1}) - \min(M_{j1})} \right]^\delta$
where $\delta$ controls the shape of the mapping. This ordering processes points from largest to smallest mark, conditioning the process on an "anchor point" (the point with the largest mark).
Process Decomposition: The conditional intensity function $\lambda^*(t, y, x)$ $λ^{*} (t, y, x)$ is decomposed into:
- Arrival Time Process ( $\lambda^*_1$ ): A self-correcting process where intensity decreases as the number of events increases, enforcing regularity.
- Spatial Process ( $h^*_2$ ): An interaction function based on Euclidean distance that defines inhibitory circular regions around existing points.
- Conditional Mark Process ( $g^*_3$ ): A flexible, non-linear model (e.g., Random Forest or Gradient Boosted Trees) predicting the secondary mark given time, location, and covariates.
- Spatio-Temporal Interaction ( $f^*_4$ ): A term regulating the joint occurrence of points in space and time.

2. Estimation and Simulation

Parameter Estimation: The framework separates the estimation of the spatio-temporal parameters (via maximum likelihood optimization using the nloptr library) and the mark model parameters (via machine learning). This modularity allows for the use of complex non-linear models for marks while maintaining computational efficiency for the spatial process.
Optimization Strategies: The package supports various optimization strategies, including "local," "global_local," and "multires_global_local," utilizing algorithms like NLOPT_GN_CRS2_LM and NLOPT_LN_BOBYQA.
Simulation: Realizations are generated using a time-integrated intensity function and rejection sampling, allowing for the generation of new point patterns conditional on the estimated parameters.

3. Goodness-of-Fit

The package employs Global Envelope Tests (GET) using a suite of non-parametric summary functions (L, F, G, J, E, and V) to assess model fit. This includes combined tests to control for multiplicity across distances and summary types.

Key Contributions

The primary contribution is the ldmppr R package, which provides a modular workflow for:

Estimating parameters for self-correcting marked point processes with location-dependent marks.
Training flexible machine learning models for the conditional mark distribution using spatial covariates and competition indices.
Evaluating model fit via global envelope tests and visual diagnostics.
Simulating realistic point patterns from the fitted model.

The package is designed to be adaptable, allowing users to substitute specific components (e.g., different mark models or optimization strategies) while maintaining the overall structure.

Results

The paper demonstrates the package's utility using a forestry dataset from the Southern Rocky Mountains, containing conifer tree locations and canopy volumes, along with topographic covariates (Aspect, Wetness Index, Elevation, Slope).

Initial Fit: An initial model estimation with a coarse grid and moderate optimization budget resulted in a poor fit (combined p-value = 0.002). The simulated envelopes failed to capture the nearest-neighbor distance distribution ( $G(r)$ ) and mark variance ( $V(r)$ ) of the reference data.
Refined Fit: By increasing the grid resolution, expanding the optimization budget, utilizing a multi-resolution strategy ("multires_global_local"), increasing the number of optimization starts, and refining the mark model training (increasing cross-validation folds and tuning grid size), the model fit improved significantly.
Outcome: The refined model achieved a combined global envelope test p-value of 0.2819, indicating the reference data is compatible with the fitted model. Visual comparisons and histograms confirmed that the improved model successfully replicated both the spatial inhibition and the distribution of canopy volumes observed in the reference data.

Significance and Claims

The authors claim that ldmppr offers a computationally feasible alternative to Gibbs processes for modeling regular, marked point processes with location dependence. By leveraging a likelihood-based approach and separating the spatial and mark modeling tasks, the package enables:

Scalability: Efficient estimation and simulation for processes that would be computationally prohibitive with MCMC-based Gibbs approaches.
Flexibility: The ability to incorporate complex, non-linear relationships between marks and spatial covariates using modern machine learning techniques.
Practical Utility: A streamlined workflow for researchers in ecology, forestry, and other fields to generate realistic point patterns and rigorously assess model goodness-of-fit.

The paper acknowledges that the current implementation focuses on processes exhibiting regularity (inhibition) and notes that future development aims to extend the framework to handle clustering behaviors while maintaining the focus on location-dependent marks.