Minimum Contrast Estimation for Superposed Processes

• The paper demonstrates that integrating closed-form mixture representations from Palm distributions reduces bias in parameter estimation for superposed processes.
• It outlines a minimum contrast estimation procedure that minimizes the integrated discrepancy between empirical and theoretical summary statistics, such as Ripley’s K function.
• The approach applies to complex models like DPP and shot-noise Cox processes, enhancing computational efficiency over direct likelihood methods.

Minimum contrast estimation for superposed point processes refers to the general framework in which inference for parameters of a point process model is performed by minimizing a contrast function measuring discrepancies between suitable summary statistics (frequently Ripley’s $K$ function or related second-order statistics) derived from observed data and their theoretical analogs. This methodology becomes crucial in scenarios where the observed process arises from the superposition (union) of two or more independent constituent point processes. Such settings pose specific challenges: the observed summary statistics result from "mixtures" of the individual components, and likelihoods are often intractable or only partially available, motivating the use of contrast-based techniques and exploitation of structure derived from Palm distributions and related probabilistic tools.

1. Mathematical Structure of Superposed Point Processes

Consider independent stationary point processes $\Phi_1, \Phi_2, \ldots, \Phi_K$ with respective intensities $\rho_1, \rho_2, \ldots, \rho_K$ and second-order properties (e.g., $K$ functions) $K_{\Phi_k}(r)$. The observed process $\Phi = \Phi_1 + \cdots + \Phi_K$ is again stationary, with total intensity $\rho = \rho_1 + \cdots + \rho_K$. The $K$-function or other summary statistic of $\Phi$ is typically not a simple sum or average, as the point configuration combines contributions from all constituents in a nontrivial way.

By leveraging the structure of Palm distributions for superposed processes (Beraha et al., 28 Aug 2025), the summary statistics (notably the $K$-function) of the superposition can be expressed analytically in closed form in terms of the summary statistics of the constituents and their moment measures. For instance, in the bivariate case: $$K_{\Phi_1+\Phi_2}(r) = \frac{1}{\rho} \left[ \left\{ K_{\Phi_1}(r) \rho_1 + \rho_2 |B(o, r)| \right\} \frac{\rho_1}{\rho} + \left\{ \rho_1 |B(o, r)| + K_{\Phi_2}(r) \rho_2 \right\} \frac{\rho_2}{\rho} \right],$$ where $|B(o,r)|$ denotes the volume of a ball of radius $r$ in the ambient space.

This mixture structure is pivotal: estimation or model fitting for superposed point processes must account for this mixture representation to avoid biased inference.

2. Minimum Contrast Estimation Principle

The minimum contrast estimator (MCE) is defined by minimizing an integrated discrepancy between the nonparametric estimate $\hat{s}(r)$ (e.g., an empirical $K$-function) and the theoretical summary statistic $s(r; \theta)$ under the model, where $\theta$ is the parameter vector: $$\hat\theta = \arg\min_\theta \int_{r_\ell}^{r_u} \left| \hat{s}(r)^p - s(r; \theta)^p \right|^q dr$$ where $p$ and $q$ are tuning exponents, and integration is over a prescribed range. For the $K$-function, typical choices are $p = 1/2$, $q = 2$.

For superposed processes, $s(r; \theta)$ must reflect the superposition (i.e., the mixture formula above), with $\theta$ incorporating all relevant component parameters (e.g., constituent intensities, inhibition/range parameters for DPPs, etc.), and with the mixture formula dictating how parameter changes impact the observed $K$-function (Beraha et al., 28 Aug 2025).

This approach replaces direct likelihood maximization, which is often intractable for Cox, DPP, or shot-noise superpositions, with an optimization over contrast functions built solely on quantities with closed-form structure.

3. Analytical Foundation: Palm Distributions and Janossy Densities

Palm theory underpins the explicit mixture representation for superposed summary statistics, as established rigorously in (Beraha et al., 28 Aug 2025). The Palm probability at a typical point encodes the conditional distribution of the process given a point at the origin, and for superpositions the Palm measure of $\Phi$ is a linked mixture of the Palm distributions of the constituent processes, weighted by their relative intensities.

This mixture description leads to tractable formulas not only for the $K$-function but also for other statistics of direct relevance in minimum contrast estimation and model diagnostics. In the context of shot-noise Cox processes, similar techniques yield tractable Janossy densities, facilitating the design of EM algorithms for maximum likelihood as a complement to minimum contrast methods.

4. Implementation and Statistical Properties

The practical implementation of MCE for superposed point processes follows a multi-step workflow:

1. Estimate the overall intensity $\rho = \sum_k \rho_k$ empirically (e.g., number of observed points divided by the domain area/volume).
2. Compute a nonparametric, edge-corrected empirical $\hat{K}(r)$ from the observed spatial configuration.
3. Write down the theoretical $K$-function for the superposed process as dictated by the mixture formula (e.g., (4) in (Beraha et al., 28 Aug 2025)).
4. Define and minimize the contrast integral to obtain parameter estimates.

A prototypical case is signal-plus-noise modeling, where $\Phi_1$ is a DPP (with $K$-function $K_\xi(r)$) and $\Phi_2$ is Poissonian noise. The mixture structure for the $K$-function in $\Phi = \Phi_1 + \Phi_2$ becomes: $$K_{\xi + \Phi_2}(r) = \pi r^2 - \frac{\rho_\xi^2}{(\rho_\xi + \omega)^2} \cdot \frac{\pi \alpha^2}{2} \left[1 - \exp(-2r^2/\alpha^2)\right]$$ with $\rho_\xi$ and $\alpha$ the DPP parameters and $\omega$ the Poisson intensity.

Statistical studies summarized in (Beraha et al., 28 Aug 2025) show that accounting for the superposition in this fashion:

• Greatly reduces parameter bias compared to naive (single-process) model fitting.
• May increase estimator variance slightly due to additional free parameters.
• Delivers computational advantages over likelihood-based inference for complex models where likelihood evaluation is prohibitive.

5. Extensions: Multicomponent Superpositions and Higher-Order Inference

The Palm and mixture framework generalizes naturally to superpositions of more than two independent processes. The formulas for the $K$-function and other second-order statistics become weighted sums over all processes, governed by their respective Palm distributions and intensities.

Moreover, for more precise inference (e.g., in settings where higher-order moment properties are essential), the theory extends to higher-order Palm distributions, allowing derivations of summary statistics conditioned on multiple points. These developments broaden the applicability of minimum contrast approaches and enable new contrast functions using third- and higher-order summary statistics (Beraha et al., 28 Aug 2025).

6. Relationship to Other Estimation Techniques and Comparative Considerations

Minimum contrast estimation for superposed point processes is closely related to:

• Kernel deconvolution and penalized contrast estimation for density deconvolution under measurement error (Delaigle et al., 2010): Both share Fourier-analytic connections and highlight the value of summary statistics when likelihoods are intractable.
• Deconvolution via probability generating functionals using Gâteaux differentials (Clark, 2012): This approach is mathematically rigorous for the recovery of constituent process properties from mixtures, reinforcing the importance of accurate analytic formulas in MCE.
• Likelihood-based and EM-based inference for overlapping or corrupted point process models (e.g., masked-renewal (Rodrigues et al., 2018) and shot-noise Cox (Beraha et al., 28 Aug 2025) contexts).

A key insight is that, for models with explicit summary statistics (e.g., DPPs, Poisson, shot-noise Cox), MCE is both computationally fast and statistically accurate, provided the mixture structure is correctly incorporated. Ignoring the superposition (i.e., fitting a single process to combined data) yields systematically biased results (Beraha et al., 28 Aug 2025).

7. Applications and Broader Impact

The methodology is directly relevant in scenarios where observed point patterns are subject to overlapping sources:

• Spatial and spatio-temporal signal + noise applications (e.g., detection of structured patterns in the presence of background contamination).
• Image and sensor data with co-occurring physical phenomena or measurement artifacts.
• Environmental sciences, where superposed points arise naturally from multiple physical processes or measurement layers.

By generalizing to multivariate and higher-order settings, the approach is poised to advance inference for complex spatial systems in disciplines such as ecology, epidemiology, and materials science.

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In summary, minimum contrast estimation for superposed point processes builds on analytic characterizations of the Palm distributions for superpositions. By accurately expressing the impact of constituent processes on observable summary statistics, this methodology enables robust and efficient statistical inference in complex, realistically contaminated point pattern data (Beraha et al., 28 Aug 2025). The general framework, validated across DPP, Poisson, shot-noise, and Cox process models, is characterized by its mixture-based correction for superposition-induced bias and its computational advantages over direct likelihood-based approaches in many settings.