Bounded Transformed Gamma Process

• Bounded Transformed Gamma Process is a stochastic model that transforms a latent gamma process via a Weibull-type function to yield bounded, monotonic degradation trajectories.
• It employs a latent stationary gamma process paired with a deterministic Weibull transformation to ensure the observable process remains within physical bounds.
• The framework preserves monotonicity and a Markovian structure while embedding stochastic temporal uncertainty, making it effective for infrastructure asset deterioration modeling.

Bounded transformed gamma process (BTGP) denotes a class of bounded stochastic deterioration models in which an observable degradation or condition trajectory is obtained by applying a deterministic transform to a latent gamma process, thereby combining monotone sample paths with an almost-sure upper or lower bound. In the infrastructure-asset formulation developed in “Unified Modelling of Infrastructure Asset Performance Deterioration -- a bounded gamma process approach” (Chen et al., 18 Aug 2025), the latent process is a standard stationary gamma process and the transform is of Weibull type, yielding increasing or decreasing bounded trajectories suitable for indices such as bridge condition index, pavement condition index, and facility condition index. The acronym is not uniform across arXiv usage: in “Data-Driven Abstractions via Binary-Tree Gaussian Processes for Formal Verification,” BTGP instead denotes a Binary-Tree Gaussian Process, not a bounded transformed gamma process (Schön et al., 2024).

1. Scope and modelling motivation

The BTGP arises from a specific modelling problem in infrastructure asset management: many performance indicators deteriorate stochastically over time but are physically or managerially bounded. The cited application domain includes bounded scales such as $0$–$100$, and deterioration forecasting is used to support inspection scheduling, condition-based maintenance, and age-based replacement (Chen et al., 18 Aug 2025).

The classical gamma process is widely used for degradation modelling because of its monotonic sample paths, independent increments, and tractability. If $X(t)$ is a conventional gamma process, then

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

Its principal limitation in this setting is structural rather than computational: the process is unbounded in sample path, mean, and variance. For bounded infrastructure indicators, the paper stresses that a bounded mean alone is insufficient; the desired property is that the deterioration process be bounded almost surely, not merely in expectation. The same discussion motivates a distinction between a model whose first two moments converge and a model whose trajectories themselves cannot exceed the admissible state range.

As a benchmark, the paper introduces a bounded nonstationary gamma process (BNGP) by using the bounded shape function

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

This makes the mean and standard deviation converge to finite limits, but the process remains an ordinary gamma process, so its sample paths can still exceed $x_{\lim}$. The BTGP is designed precisely to remove that mismatch.

2. Latent-gamma construction

The proposed BTGP is built from a latent standard stationary gamma process $G(t)$ with shape function $a(t)=\alpha t$ and scale $\beta=1$. The observable bounded process is defined by a deterministic Weibull-type transformation (Chen et al., 18 Aug 2025):

$X(t)=T[G(t)],$

with increasing-form transform

$100$0

Here $100$1 is the shape-rate parameter of the latent stationary gamma process, while $100$2 and $100$3 are transformation parameters. The support of the increasing-form process is

$100$4

Because $100$5 is monotone increasing and $100$6 is monotone increasing, every sample path of $100$7 is monotone increasing and bounded almost surely.

The same framework also admits a decreasing version, intended for condition indices that decline with age:

$100$8

Under this form, each path starts at $100$9 and tends to $X(t)$0 as $X(t)$1. In the bridge-condition application, this decreasing specification is used to model BCI trajectories.

A central interpretive device is the latent process $X(t)$2 as an “internal aging clock.” Existing infrastructure software often uses a deterministic nonlinear regression curve of Weibull form with chronological time $X(t)$3 as the argument. The BTGP replaces deterministic age by stochastic latent age $X(t)$4. This preserves the practical interpretability of regression-style deterioration curves while embedding stochastic temporal uncertainty directly into the model. The paper describes this as grounding the BTGP in the traditional regression modelling tradition of infrastructure management systems.

3. Process properties and probabilistic structure

Before transformation, $X(t)$5 has the usual gamma-process properties: monotone paths, independent increments, and stationarity. After transformation, the BTGP remains monotone and Markovian, but independent increments are lost (Chen et al., 18 Aug 2025). The paper derives the conditional increment density for

$X(t)$6

through the latent increment

$X(t)$7

Conditional