Bounded Nonstationary Gamma Process (BNGP)

• BNGP is a gamma-process-based model with a bounded shape function ensuring that mean and variance saturate over time.
• In infrastructure deterioration, it models bounded stochastic degradation while retaining characteristics like independent increments.
• In latent variable contexts (e.g., SGLDS), BNGP supports sparse state selection and scalable latent dynamics using gamma-process priors.

Searching arXiv for the cited papers and related BNGP context. A Bounded Nonstationary Gamma Process (BNGP) is a gamma-process-based deterioration model that extends the standard gamma process to encode both boundedness and nonstationarity. In the infrastructure deterioration formulation, the standard gamma process $X(t)$ is a continuous-time, monotone-increasing process with independent increments, and the BNGP modifies the shape function so that the mean and variance are bounded as time grows (Chen et al., 18 Aug 2025). In a distinct latent-variable modeling context, the term also appears in connection with the Sparse Graph Linear Dynamical System (SGLDS), where a gamma-process construction is used to control support and scale of latent states and their transition graph through a Bernoulli-Poisson link (Kalantari et al., 2018). These usages share an underlying gamma-process perspective, but they emphasize different modeling objectives: bounded stochastic deterioration in one case and sparse latent-state structure in the other.

1. Mathematical formulation

In the infrastructure deterioration setting, the standard gamma process $X(t)$ is specified so that, at time $t$, $X(t)$ is Gamma-distributed with density

$$f_{X(t)}(x) = \frac{x^{a(t)-1} e^{-x/\beta}}{\beta^{a(t)} \Gamma(a(t))}$$

where $a(t)$ is a shape function and $\beta$ is a scale parameter (Chen et al., 18 Aug 2025). Its mean and variance are

$$\mathbb{E}[X(t)] = \beta a(t), \qquad \mathrm{Var}[X(t)] = \beta^2 a(t).$$

The BNGP introduces boundedness through a bounded shape function,

$$a(t) = x_{\lim}\left[1 - \exp\left(-\theta_2 t^{\theta_3}\right)\right],$$

where $x_{\lim}$ is the upper limit for the process and $X(t)$0 are shape parameters (Chen et al., 18 Aug 2025). As $X(t)$1, $X(t)$2, and therefore

$X(t)$3

This captures bounded mean and variance, unlike the standard unbounded gamma process (Chen et al., 18 Aug 2025).

A representative bounded performance measure is written as

$X(t)$4

where $X(t)$5 evolves as a BNGP (Chen et al., 18 Aug 2025). In this formulation, boundedness is imposed through the time-dependent shape function rather than through a pathwise transformation.

2. Stochastic properties

The BNGP retains several structural properties of the standard gamma process. Its sample paths are monotonically increasing and continuous; its increments over disjoint intervals are independent; and nonstationarity is introduced through the time-dependent nonlinear shape function $X(t)$6 (Chen et al., 18 Aug 2025). The same source states that the process preserves the Markov property in the sense that increments over disjoint intervals are independent.

A central technical point is that the BNGP is not strictly bounded pathwise. Individual realizations can exceed $X(t)$7, so the process is bounded only in the mean square sense, not with probability $X(t)$8 (Chen et al., 18 Aug 2025). This distinction is fundamental in applications where physical or managerial limits are strict. The increment $X(t)$9 remains gamma-distributed, with shape $t$0, because the independent-increments structure is preserved (Chen et al., 18 Aug 2025).

This suggests that the BNGP should be understood as a bounded-moment model rather than a pathwise-bounded stochastic process. A plausible implication is that it is well suited when asymptotic saturation in expectation is the main modeling requirement, but less suitable when every realized trajectory must remain within hard bounds.

3. Relation to the gamma process as a completely random measure

The broader gamma-process literature provides the nonparametric Bayesian background needed to interpret BNGP constructions. The gamma process has been characterized as a completely random measure (CRM) with rate measure

$t$1

and an explicit stick-breaking representation was introduced for the gamma process in “Gamma Processes, Stick-Breaking, and Variational Inference” (Roychowdhury et al., 2014). In that work, the construction

$t$2

is shown to generate exactly the classical gamma process CRM, with correctness established through marked Poisson process theory and superposition arguments (Roychowdhury et al., 2014).

The same paper states that the construction is general in the sense of allowing base measures $t$3 that can be arbitrary finite measures on the atom space $t$4, and it notes that, to incorporate boundedness, one would simply replace the base measure $t$5 with a measure supported on the desired subset (Roychowdhury et al., 2014). For nonstationarity, it gives the generalized rate measure

$t$6

and states that the stick-breaking construction is potentially extendable to the BNGP case by making $t$7 and/or $t$8 dependent on $t$9 (Roychowdhury et al., 2014).

These observations do not constitute a full BNGP theory in the same sense as the infrastructure formulation, but they situate boundedness and nonstationarity within standard CRM machinery. This suggests a conceptual bridge between deterioration models and Bayesian nonparametric random-measure models: in both, boundedness and nonstationarity are introduced by modifying the governing measure or its parameterization rather than abandoning the gamma-process foundation.

4. BNGP in Sparse Graph Linear Dynamical Systems

In SGLDS, a gamma-process-based construction is used to model sequentially observed multivariate data through an infinite-dimensional sparse random graph over latent states (Kalantari et al., 2018). The gamma process is written as

$X(t)$0

and, in the model summary, the relevant weights are also given as

$X(t)$1

The same description states that a BNGP variant is used to control both support and scale, with boundedness interpreted as almost sure finiteness of the total measure $X(t)$2, and nonstationarity interpreted as variation of the base measure across the domain (Kalantari et al., 2018).

Sparsity is induced through the Bernoulli-Poisson link:

$X(t)$3

The state-transition matrix is then

$X(t)$4

where $X(t)$5 is the binary mask generated from the Bernoulli-Poisson link (Kalantari et al., 2018). The gamma process governs which states exist, their strengths, and which transitions are allowed.

Within this framework, a latent state is categorized as dynamic if at least one row or column in $X(t)$6 is nonzero, and non-dynamic if its row and column are all zero (Kalantari et al., 2018). The paper further distinguishes dynamic states into live, absorbing, or noise-injection states, while a normal-gamma construction shrinks the energy captured by non-dynamic states (Kalantari et al., 2018). In this usage, “BNGP” functions less as a deterioration model and more as a sparse structural prior over an unbounded latent-state space.

5. Infrastructure deterioration modeling

The principal applied role of the BNGP in the supplied material is infrastructure asset deterioration modeling. The model is motivated by the fact that many infrastructure performance deterioration processes are constrained by physical or managerial limits, while empirical degradation rates can be time-varying (Chen et al., 18 Aug 2025). Gamma processes are attractive in this setting because of their monotonic sample paths, independent increments, and mathematical tractability (Chen et al., 18 Aug 2025).

In this application, the BNGP is used to model slow, stochastic, and eventually limiting degradation for physical infrastructure such as bridge health and wall thinning in pipes (Chen et al., 18 Aug 2025). It is particularly relevant when the performance metric is bounded, as with indices constrained between $X(t)$7 and a maximum value. The model yields bounded, monotonically increasing variance, which is stated to align with some empirical degradation patterns (Chen et al., 18 Aug 2025).

The paper “Unified Modelling of Infrastructure Asset Performance Deterioration -- a bounded gamma process approach” compares a proposed bounded transformed gamma process (BTGP) against a BNGP model from both deterioration modelling and asset management decision-making perspectives (Chen et al., 18 Aug 2025). An empirical study using real-world historical bridge condition data is reported, and in a comparative study with 267 bridge condition profiles, “BTGP was selected as best fit in 42% of cases, BNGP only 5%” (Chen et al., 18 Aug 2025). The same source states that the choice between BNGP and BTGP affects predictions of remaining life, optimal inspection and replacement policies, and cost rates, sometimes substantially, especially at low failure thresholds.

These findings place the BNGP in a specific methodological niche: it is a bounded-moment gamma-process model that preserves a familiar increment structure, but it is not the most flexible bounded deterioration model considered in that comparison.

6. Comparison with related bounded gamma-process models

The most explicit contrast in the supplied literature is between the BNGP and the BTGP proposed in (Chen et al., 18 Aug 2025). The BTGP is constructed by applying a nonlinear transformation to a standard gamma process:

$X(t)$8

This guarantees $X(t)$9 for all paths and all times (Chen et al., 18 Aug 2025), in contrast to the BNGP, whose paths are not strictly bounded.

The comparison reported in the source can be summarized as follows:

Characteristic	BNGP	BTGP
Sample path boundedness	No	Yes
Variance pattern	Bounded, increases with time	Bounded, nonmonotonic (peaks then falls)
Increments	Independent	Not independent

The same comparison states that the BNGP has no transform and obtains boundedness via the shape function, whereas the BTGP uses a nonlinear transform applied to a standard gamma process (Chen et al., 18 Aug 2025). The BNGP yields analytical mean and variance, while the BTGP has closed-form marginal and conditional PDFs for estimation. The source also states that BNGP may allow sample paths exceeding physical limits, whereas BTGP always conforms to the system’s physical limits.

From this comparison, several recurring misconceptions can be addressed precisely. First, boundedness of the BNGP does not mean strict pathwise boundedness; the supplied source explicitly rejects that interpretation (Chen et al., 18 Aug 2025). Second, nonstationarity in the BNGP does not imply loss of independent increments in the infrastructure formulation; independent increments are retained (Chen et al., 18 Aug 2025). Third, the BNGP is not presented as the uniquely appropriate bounded gamma-process model; the same empirical comparison identifies settings in which alternative bounded constructions fit better (Chen et al., 18 Aug 2025).

7. Interpretation, significance, and limitations

The significance of the BNGP lies in the combination of bounded mean and variance with monotone-increasing dynamics and independent increments. In the infrastructure setting, this makes it attractive for condition trajectories that are expected to saturate and for decision systems that rely on tractable stochastic increments (Chen et al., 18 Aug 2025). In the SGLDS setting, the associated gamma-process machinery supports sparse latent-state selection and sparse transition structure through the Bernoulli-Poisson link (Kalantari et al., 2018).

At the same time, the limitations are explicit in the supplied material. In infrastructure deterioration modeling, the BNGP cannot prevent sample paths from occasionally exceeding physical limits, and its variance is bounded but monotonically increasing, which may be unsuitable when observed variability rises and then falls with age (Chen et al., 18 Aug 2025). The empirical comparison further states that these features can yield less realistic predictive intervals and can affect maintenance policy recommendations (Chen et al., 18 Aug 2025).

A plausible synthesis is that “BNGP” names a family resemblance rather than a single universally standardized object. In one line of work, it denotes a bounded and nonstationary gamma-process deterioration model defined through a bounded shape function. In another, it denotes a gamma-process-based latent-structure prior whose boundedness is tied to finiteness of total mass and whose nonstationarity arises from variation in the underlying measure over the domain (Kalantari et al., 2018, Roychowdhury et al., 2014). Across these settings, the common thread is the use of gamma-process structure to regulate complexity or deterioration while preserving tractable stochastic semantics.