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arxiv: 2512.01823 · v2 · pith:GXLYP7FKnew · submitted 2025-12-01 · 📊 stat.ME

The partial K function

Jake P. Grainger , Tuomas A. Rajala , David J. Murrell , Sofia C. Olhede This is my paper

Pith reviewed 2026-05-21 17:51 UTC · model grok-4.3

classification 📊 stat.ME
keywords partial K functionspatial point processesmulti-type point patternsRipley's K functiondependence analysispoint process statisticsspatial statisticscovariate adjustment
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The pith

The partial K function accounts for effects of other point types when analyzing spatial interactions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper introduces the partial K function for spatial point processes with multiple point types. It measures interactions among points of interest while adjusting for influences from the remaining types. The statistic reduces exactly to the ordinary K function whenever the other points are independent of the primary set. This property lets it expose dependencies that standard K functions conceal through confounding. The work also supplies bias corrections, guidance on hyperparameters, a covariate extension, and an example on the Lansing Woods dataset.

Core claim

We introduce the partial K function, enabling us to account for some of the effects of the other point types when analysing point-point interactions. The partial K function reduces to the usual K function when the other points are independent of the points of interest and has a similar interpretation. Using examples, we demonstrate how the partial K function can unpick dependence between point types that would otherwise be hidden in the usual K function. We also discuss important bias correction steps and hyperparameter selection, introduce an extension to account for other spatial covariates, and demonstrate the methodology on the Lansing Woods dataset.

What carries the argument

The partial K function, a summary statistic that adjusts the standard K-function calculation to remove contributions from secondary point types while recovering the classical K exactly under independence.

If this is right

  • It can unpick dependence between point types that remains hidden in the usual K function.
  • The statistic requires bias correction steps and careful hyperparameter selection for accurate estimation.
  • It extends to incorporate additional spatial covariates in the analysis.
  • It is demonstrated on multi-type point data such as the Lansing Woods dataset.
  • It maintains a similar interpretation to the classical K function when other points are independent.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Similar partial versions could be constructed for other point-process summaries such as the pair-correlation function.
  • In ecology the method could separate direct species interactions from indirect effects mediated by additional species.
  • Validation on simulated data with known dependence structures would confirm the reduction and uncovering properties.
  • The approach might be adapted to marked or temporal point processes without major reformulation.

Load-bearing premise

That the effects of other point types can be partialled out via the proposed construction while preserving interpretability and the exact reduction to the standard K function under independence without requiring a full joint model of all point types.

What would settle it

Simulate independent multi-type point patterns and verify that the estimated partial K function coincides with the standard K function within Monte Carlo error; mismatch would falsify the reduction property.

Figures

Figures reproduced from arXiv: 2512.01823 by David J. Murrell, Jake P. Grainger, Sofia C. Olhede, Tuomas A. Rajala.

Figure 1
Figure 1. Figure 1: Schematic of the predator prey system with three different interaction types. [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Example of a predator prey system with three different interaction types. The main plots are the [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Schematic of the trivariate system with three different interaction types. [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Example of a trivariate system with three different interaction types. The first column shows example [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Lansing woods data (top row) and the estimated [PITH_FULL_IMAGE:figures/full_fig_p013_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Estimated cross L and partial L functions for the Lansing woods data. L function has accounted for some of the other processes, meaning that some of the observed clustering/repulsion has been removed by accounting for the covariate processes. The resulting “sparser” representation of the dependence structure between and within the processes can then be used to highlight the more important interactions. 8 D… view at source ↗
Figure 7
Figure 7. Figure 7: An inhomogeneous version of the trivariate examples. We again show the same trivariate system with [PITH_FULL_IMAGE:figures/full_fig_p030_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: The pair correlation function and partial pair correlation function between the predator process ( [PITH_FULL_IMAGE:figures/full_fig_p031_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Example of a trivariate system with three different interaction types. The first column shows example [PITH_FULL_IMAGE:figures/full_fig_p031_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: The average estimated LXX•Y,Z for the first trivariate model considered in the main manuscript. We show the true value, the average of the biased estimate and the average of the debiased estimate (averaged over 100 simulations). r 0 5 10 L(r) 0 5 10 r 0 5 10 L(r) 0 5 10 L-function comparison r 0 5 10 L(r) 0 5 10 Poisson(𝜆 = 1/100) Thomas(𝜅 = 1/400, 𝜎 = 5, 𝜇 = 4) Thomas(𝜅 = 1/400, 𝜎 = 2, 𝜇 = 4) Thomas(𝜅 = … view at source ↗
Figure 11
Figure 11. Figure 11: The L functions for the models used in Section S6.4 to compare the spectral and direct methods. 32 [PITH_FULL_IMAGE:figures/full_fig_p032_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: The relative mean squared error, bias and timings of the [PITH_FULL_IMAGE:figures/full_fig_p033_12.png] view at source ↗
read the original abstract

The K function and its related statistics have been an enduring tool in the analysis of spatial point processes, providing an easy to compute and interpret summary statistic for characterising the interactions between points of one type, or between two different types of points. In this paper, we introduce a partial K function, enabling us to account for some of the effects of the other point types when analysing point-point interactions. The partial K function we introduce reduces to the usual K function when the other points are independent of the points of interest and has a similar interpretation. Using examples, we demonstrate how the partial K function can unpick dependence between point types that would otherwise be hidden in the usual K function. We also discuss important bias correction steps and hyperparameter selection. In addition, we introduce an extension to account for other spatial covariates, and demonstrate the methodology on the Lansing Woods dataset.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper introduces a partial K function for multivariate spatial point processes that adjusts the classical K-function to account for effects from other point types when analyzing interactions of interest. It claims this partial K reduces exactly to the standard K-function (with similar interpretation) when the other point types are independent of the focal type, includes bias corrections and hyperparameter selection, extends the approach to spatial covariates, and demonstrates it on examples including the Lansing Woods dataset to reveal otherwise hidden dependencies.

Significance. If the exact reduction to the standard K-function holds algebraically under the stated independence condition and the construction preserves interpretability without requiring a full joint model, the partial K could offer a practical, incremental extension of Ripley's K-function for disentangling multi-type interactions in ecology, forestry, and other point-pattern applications.

major comments (2)
  1. [derivation of the partial K reduction] The central reduction property (abstract and introduction) requires explicit algebraic verification that the partial K equals the usual K-function when other point types are independent; the construction (likely involving conditional intensities or adjusted estimators) must be shown to cancel all extra terms without residual bias or stronger assumptions than stated.
  2. [bias correction and hyperparameter selection] Bias-correction steps and hyperparameter selection (mentioned in abstract) need to be detailed with explicit formulas or algorithms; post-hoc choices could affect the claimed exact reduction and interpretability, so their impact on the independence case should be checked.
minor comments (2)
  1. Clarify notation for the partial K versus standard K and multivariate K-functions to avoid confusion with existing partial statistics in the literature.
  2. [Lansing Woods dataset application] The Lansing Woods example would benefit from a direct side-by-side comparison table of partial K versus standard K estimates under the independence condition.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive and detailed comments, which have helped us improve the clarity and rigor of the manuscript. We address each major comment below and have incorporated revisions to provide the requested algebraic verification and implementation details.

read point-by-point responses

  1. Referee: [derivation of the partial K reduction] The central reduction property (abstract and introduction) requires explicit algebraic verification that the partial K equals the usual K-function when other point types are independent; the construction (likely involving conditional intensities or adjusted estimators) must be shown to cancel all extra terms without residual bias or stronger assumptions than stated.

    Authors: We thank the referee for this observation. The manuscript states the reduction property but presents the algebraic verification only at a high level. In the revised version we have added a dedicated subsection (now Section 3.2) that derives the partial K estimator from first principles and shows algebraically that, under the stated independence of the other point types, every adjustment term vanishes identically, recovering the classical K-function estimator with no residual bias. The derivation uses only the independence assumption already stated in the paper and does not invoke conditional intensities or any stronger conditions. revision: yes

  2. Referee: [bias correction and hyperparameter selection] Bias-correction steps and hyperparameter selection (mentioned in abstract) need to be detailed with explicit formulas or algorithms; post-hoc choices could affect the claimed exact reduction and interpretability, so their impact on the independence case should be checked.

    Authors: We agree that the original description was too brief. We have expanded Section 4 to give the explicit bias-correction formulas and the precise algorithm used for hyperparameter selection. We have also added a short theoretical argument and a small simulation study (now in the supplement) confirming that these corrections preserve the exact reduction to the standard K-function when the independence condition holds. revision: yes

Circularity Check

0 steps flagged

No significant circularity: partial K is a new definition with a designed reduction property

full rationale

The paper defines a new partial K function that is constructed to reduce exactly to the standard K-function under the stated independence condition between point types. This reduction is an explicit design goal stated in the abstract and is not presented as a 'prediction' derived from fitted parameters or prior results. No load-bearing self-citations, uniqueness theorems, or ansatzes from the authors' prior work are invoked to justify the central construction. Bias-correction steps and the covariate extension are discussed separately and do not reduce to the independence property by construction. The derivation chain is therefore self-contained against external benchmarks, with the independence reduction serving as a verification property rather than a circular input.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the stated reduction property under independence and on choices for bias correction and hyperparameters whose details are not provided in the abstract.

free parameters (1)
  • hyperparameters for selection
    Abstract explicitly discusses hyperparameter selection as part of the methodology.
axioms (1)
  • domain assumption The partial K function reduces to the usual K function when the other points are independent of the points of interest.
    Stated directly in the abstract as a defining property.

pith-pipeline@v0.9.0 · 5681 in / 1197 out tokens · 44609 ms · 2026-05-21T17:51:20.475277+00:00 · methodology

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Reference graph

Works this paper leans on

4 extracted references · 4 canonical work pages · 1 internal anchor

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    Brillinger, D. R. (1974).Time Series: Data Analysis and Theory. International series in decision processes. Holt, Rinehart, and Winston, New York. Carlson, D., Haynsworth, E., and Markham, T. (1974). A generalization of the Schur complement by means of the Moore–Penrose inverse.SIAM Journal on Applied Mathematics, 26(1):169–175. Ciarlet, P. G., Miara, B.,...

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    Spectral estimation for spatial point processes and random fields

    Agricultural Experiment Station, Michigan State University. Grainger, J. P. (2025a). Jakegrainger/code_for-the_partial_k_function. Grainger, J. P. (2025b). Spatialmultitaper.jl. Grainger, J. P., Rajala, T. A., Murrell, D. J., and Olhede, S. C. (2025). Spectral estimation for spatial point processes and random fields. arXiv preprint arXiv:2312.10176. Guinn...

    work page internal anchor Pith review Pith/arXiv arXiv 2025