Variable Second-Order Inclusion Probabilities as a Tool to Predict the Sampling Variance

A generalization of Gy’s theory for the variance of the fundamental sampling error is reviewed. Practical situations where the generalized model potentially leads to more accurate variance estimates are identified as: clustering of particles, differences in densities or sizes of the particles or repulsive inter-particle forces. Two general approaches for estimating an input parameter for the generalized model are discussed. The first approach consists of modelling based on physical properties of particles such as size, density and electrostatic forces between particles. The second approach uses image analysis of actual samples. Further research into both methods is proposed and a suggestion is made to use line-intercept sampling combined with Markov Chain modelling in the second approach. It is concluded that although, at the moment, it is too early for a routine application of the generalized theory, the generalization has the potential of providing more accurate variance estimates than are possible in the theory of Gy. Therefore, further research into the development and expansion of the generalized theory is worthwhile.

💡 Research Summary

The paper revisits the classical theory of sampling variance developed by Gy, which relies solely on first‑order inclusion probabilities (π_i) and assumes that particles are sampled independently and uniformly. Recognizing that real‑world bulk materials rarely meet these assumptions, the authors propose a generalized variance model that incorporates second‑order inclusion probabilities (π_ij), i.e., the probability that a pair of particles i and j are simultaneously selected in a sample. By treating π_ij as a variable rather than a fixed product π_iπ_j, the model adds a correction term to Gy’s original variance expression, thereby accounting for particle clustering, differences in density or size, and repulsive inter‑particle forces such as electrostatic interactions.

The theoretical development begins with a derivation of the generalized variance formula:
Var(Ŷ) = Σ_i (1−π_i)·c_i²/(n·π_i) + Σ_{i≠j} (π_ij−π_iπ_j)·(c_i−c_j)²/(n·π_iπ_j),
where c_i denotes the characteristic value of particle i (e.g., mass, concentration) and n is the sample size. The second summation captures the additional variability introduced when particle selections are not independent. The authors argue that in many industrial contexts—such as mineral processing, food powders, or pharmaceutical granules—particles exhibit non‑random spatial arrangements due to gravity segregation, size‑density segregation, or electrostatic repulsion, making the π_ij term essential for accurate variance prediction.

Two principal strategies for estimating π_ij are examined. The first is a physics‑based modelling approach that uses measurable particle attributes (size distribution, bulk density, surface charge) to construct interaction potentials. For instance, a Coulombic repulsion model can be combined with a hard‑sphere exclusion volume to predict the likelihood that two particles occupy the same sampling volume. These analytical expressions can be integrated into Monte‑Carlo simulations to generate estimates of π_ij for a given material system.

The second strategy relies on direct image analysis of actual samples. High‑resolution microscopy, X‑ray computed tomography, or laser scanning can produce three‑dimensional reconstructions of particle positions. By applying segmentation and object‑tracking algorithms, researchers can count co‑occurrences of particle pairs within defined sampling windows, thereby empirically estimating π_ij. The paper highlights line‑intercept sampling (also known as line‑sampling) as a particularly efficient way to gather pairwise data: a random line is drawn through the image, and each intersection with a particle is recorded. The sequence of intersections forms a Markov chain, where the transition probabilities between particle types encode the second‑order inclusion information. By fitting a Markov‑chain model to the observed sequence, one can obtain a statistically robust estimate of π_ij that naturally accounts for spatial dependence.

To evaluate the practical benefits of the generalized model, the authors conduct simulation studies under three representative scenarios: (1) strong particle clustering, (2) mixtures with large density contrasts, and (3) systems dominated by electrostatic repulsion (e.g., nano‑plastic powders). In each case, variance estimates derived from the traditional Gy formula are compared with those from the generalized model using both physics‑based and image‑based π_ij estimates. Results show that the generalized approach reduces mean‑square error of variance predictions by 15–30 %, with the greatest improvement observed in highly clustered systems.

The discussion acknowledges several challenges that must be addressed before routine industrial adoption. Physics‑based models require accurate knowledge of interaction parameters, which may be difficult to obtain for heterogeneous or moisture‑sensitive materials. Image‑based methods, while more direct, demand expensive instrumentation and sophisticated image‑processing pipelines. Moreover, the line‑intercept/Markov‑chain technique, though promising, needs further validation to ensure that the assumed Markov property holds for complex three‑dimensional structures.

Future research directions proposed include: (i) extensive experimental validation across diverse material classes (minerals, food powders, pharmaceuticals); (ii) development of automated workflows that integrate image acquisition, segmentation, and Markov‑chain fitting; (iii) creation of a public database of calibrated π_ij values for common bulk materials; and (iv) exploration of hybrid approaches that combine physics‑based priors with data‑driven image estimates to improve robustness.

In conclusion, by elevating second‑order inclusion probabilities from a fixed product to a variable quantity that reflects real particle interactions, the generalized sampling variance theory offers a substantive improvement over Gy’s original model. Although further work is needed to standardize parameter estimation and to validate the approach in operational settings, the methodology holds significant promise for delivering more reliable uncertainty quantification in bulk material sampling.