Binned Poisson Convolution Model

• Binned Poisson Convolution Model is a class of discrete count models that convolves independent Poisson variables and applies factorial tilting to capture aggregation, overdispersion, and underdispersion.
• It employs discrete dilation and binomial thinning to generalize classical Poisson models, enabling precise scaling and limit behavior analysis for binned data.
• Multivariate extensions, including zero-inflation detection, make the model applicable to various fields such as ecology, insurance, and time series analysis.

A Binned Poisson Convolution Model represents a class of discrete count models where the observed data, typically organized into bins or aggregates, is modeled as resulting from the convolution (sum) of independent or conditionally independent Poisson random variables. This framework generalizes the classical Poisson model to accommodate overdispersion, underdispersion, aggregation effects, and convolutional or mixture structure, providing the basis for inference in applications ranging from high-dimensional signal recovery to statistical mechanics, spatial processes, and binned or thinning-transformed time series.

1. Structural Foundations and Factorial Dispersion Models

A foundational formulation for binned Poisson convolution models is provided by the class of factorial dispersion models constructed via convolution (summation of independent integer-valued random variables) and factorial tilting. The key analytic tool is the factorial cumulant generating function (FCGF),

$C(t) = \log \mathbb{E}[(1+t)^X]$

for a non-negative integer random variable $X$. The FCGF is additive under convolution, and the family of models generated by "shifting" the FCGF—factorial tilting $C_\theta(t) = C(\theta + t) - C(\theta)$—admits a two-parameter structure indexed by the mean $\mu = C'(\theta)$ and dispersion $v(\mu) = C''(\theta)$. When $v(\mu)$ is of the power-law form $v(\mu) \propto \mu^p$, the resulting models are termed Poisson-Tweedie factorial dispersion models. These capture familiar overdispersed count distributions as special cases (e.g., negative binomial, Poisson-inverse Gaussian) and feature variance functions

$\operatorname{Var}(Y) = \mu + \gamma\mu^p.$

This generic form, along with closure under convolution, underpins the theoretical foundation for binned Poisson convolution models (Jørgensen et al., 2014).

2. Discrete Dilation, Binomial Thinning, and Power Asymptotics

In binned or aggregated data, counts are often subject to thinning or rescaling. The discrete dilation operator, defined as

$C(t; c \cdot X) = C(ct; X),$

serves as a natural analogue to continuous scaling, coinciding with binomial thinning when $0 < c < 1$. For $X$ supported on $\mathbb{N}$,

$c \cdot X \eqd \sum_{i=1}^X N_i, \quad N_i \sim \mathrm{Bern}(c).$

This operation preserves the structure of the FCGF under convolution and is extended for $c > 1$ via geometric compounding. The dilation operation facilitates the paper of limiting behavior—specifically, Poisson-Tweedie asymptotics—by establishing that, under suitable scaling,

$$c^{-1} \cdot \mathrm{FD}(c\mu, c^{2-p}\gamma) \xrightarrow{d} \mathrm{PT}_p(\mu, \gamma c_0),$$

where $c\to 0$ or $c\to \infty$, as governed by the shape of the dispersion function (Jørgensen et al., 2014). This generalizes results such as the law of thin numbers (the discrete analogue of the law of large numbers), Hermite convergence (central limit-like behavior in discrete models), and the convergence of specific binned sums to Poisson-Tweedie limits under binning or thinning regimes.

3. Duality Transformations and Dispersion Modulation

A duality or M-transformation combines reflection of the random variable with dilation, yielding

$C(t; -X) = C(-t/(1+t); X),\quad t > -1,$

and for $c > 0$,

$$X_a = c^{-1} \cdot [-c \cdot (-X)],\quad a = (1-c)/c > -1,$$

with

$C(t; X_a) = C\left(\frac{t}{1+at}; X\right).$

The transformation leaves the mean invariant, but modifies the dispersion: $S(X_a) = S(X) - 2a \mathbb{E}(X).$ Thus, a model initially exhibiting overdispersion (variance exceeding the mean) can be mapped to one exhibiting underdispersion, and vice versa. This bridges binomial and negative binomial models, providing a powerful tool for systematic control of dispersion properties in binned convolution models (Jørgensen et al., 2014).

4. Multivariate Generalizations and Zero-Inflation

Extending to the multivariate case, the FCGF is defined for $\mathbf{X}\in \mathbb{N}^k$ by

$$C(\mathbf{t}; \mathbf{X}) = \log\mathbb{E}\left[\prod_{i=1}^k (1+t_i)^{X_i}\right],\quad t_i > -1.$$

Multivariate Poisson-Tweedie models arise via Poisson mixtures of multivariate Tweedie random variables, with covariance structure

$$\operatorname{Cov}(\mathbf{Y}) = [\mu]^{p/2} \Sigma [\mu]^{p/2},$$

and latent Poisson observation: $X_i | Y_i \sim \mathrm{Poisson}(Y_i).$ Multivariate dispersion is quantified using the dispersion matrix,

$$S(\mathbf{X}) = \operatorname{Cov}(\mathbf{X}) - \operatorname{diag}(\mathbb{E}[\mathbf{X}]),$$

with equi-, over-, and underdispersion defined via definiteness. The zero-inflation index generalizes to

$$ZI(\mathbf{X}) = 1 + \frac{C(-\mathbf{1}; \mathbf{X})}{\mathbb{E}[X]},$$

with directional extensions for projections, enabling the detection and quantification of excessive zero counts (inflation or deflation) in high-dimensional or aggregated binned data (Jørgensen et al., 2014).

5. Asymptotic Convergence and Limit Theorems

The Poisson–Tweedie framework yields unification of several classical discrete asymptotic results for binned data:

• Law of Thin Numbers: Under binomial thinning or binning, counts in each bin converge weakly to Poisson or Poisson–Tweedie distributions.
• Hermite Convergence: For $p = 0$, appropriately scaled and centralized sums of discrete random variables converge to the Hermite distribution, capturing higher-order concentration properties analogous to the central limit theorem, but in a discrete (binned) setting (Jørgensen et al., 2014).

These results validate the use of Poisson-type or overdispersed models in practical contexts where aggregation, thinning, or convolution by design or process is inherent.

6. Applications, Example Models, and Interpretation

Applications

Binned Poisson convolution models, in their factorial dispersion form, are prevalent in:

• Ecology/Epidemiology: Modeling spatial or temporal aggregates over habitats or individuals.
• Insurance and Claims Modeling: Aggregating claims over policies or time.
• Time Series and Aggregated Panel Data: Natural binning by temporal intervals (e.g., daily, weekly sums) or spatial regions.

Example Models

Table: Representative Subclasses and Correspondence

Model Class	Dispersion Function $v(\mu)$	Example Special Case
Poisson	$v(\mu)=\mu$	Classical equidispersed Poisson
Negative Binomial	$v(\mu)=\mu+\gamma\mu^2$	Overdispersed Poisson–Tweedie
Poisson–Inverse Gaussian	$v(\mu)=\mu+\gamma\mu^{3}$	Higher-level overdispersion
Hermite	$v(\mu)=\mathbf{constant}$	Central discrete limit

The ability to encompass these models ensures that binned Poisson convolution models can be tuned to exhibit the observed dispersion, aggregation, or zero-inflation properties inherent in real-world data.

Modeling Insights

The closure properties (under convolution, thinning, dilation), explicit control over dispersion, and multivariate generalizability render Poisson-Tweedie factorial dispersion models a comprehensive theoretical framework for binned count data. The dilation operator formalizes aggregation and binning; duality transformations enable systematic conversion between overdispersed and underdispersed regimes, facilitating model selection and misspecification correction as dictated by data features.

7. Theoretical and Practical Significance

The rigorous construction, generalizable limiting results, and explicit connection to classical models position the Binned Poisson Convolution Model as a central object in discrete stochastic modeling. Its flexibility in representing various forms of count-data aggregation—subject to thinning, grouping, and convolution—makes it directly applicable to contemporary problems in signal processing, spatial and temporal inference, and high-throughput quantitative disciplines where binned data structures dominate (Jørgensen et al., 2014).

The factorial dispersion model approach, particularly via Poisson–Tweedie mixtures and their multivariate extensions, provides both practitioners and theorists with a unified family of models possessing strong asymptotic guarantees, explicit analytical tools (e.g., FCGF), and provable properties with practical relevance for modeling structured, binned, and aggregated count data.