Semi-Physical Gamma Process Degradation Model

• The semi-physical non-homogeneous Gamma process is a stochastic model that combines empirical physical laws with statistical uncertainty to capture monotonic degradation phenomena.
• It employs an exponential shape function and temperature-dependent rate via the Arrhenius law, ensuring parameters are physically interpretable and empirically calibratable.
• The model enables robust risk analysis and maintenance optimization, as demonstrated in LED-package lumen depreciation and extended to non-stationary degradation applications.

A semi-physical non-homogeneous Gamma process is a class of stochastic models used to describe monotonic degradation phenomena in systems where observed behavior is governed by underlying physical mechanisms and substantial randomness. This process combines physical law—often reflecting known time or temperature dependencies of degradation—with statistical uncertainty propagation. In prominent applications such as LED-package lumen depreciation, the semi-physical non-homogeneous Gamma process offers a rigorous framework that both conforms to empirical maintenance laws and enables uncertainty quantification, risk analysis, and @@@@1@@@@ under varying environmental stressors (Shi et al., 14 Jan 2026).

1. Mathematical Structure and Definition

The non-homogeneous Gamma process $\{X(t), t \ge 0\}$ is defined by:

• $X(0) = 0$ almost surely.
• Independent, nonnegative increments.
• For any $0 \le s < t$, the increment $X(t) - X(s)$ is distributed as a Gamma random variable:

$$X(t) - X(s) \sim \mathrm{Gamma}\left(\alpha(t) - \alpha(s),\, \beta\right)$$

where $\alpha: [0,\infty) \to [0, \infty)$ is a nondecreasing shape function and $\beta>0$ is a rate (inverse-scale) parameter. The marginal at time $t$ is

$$f_{X(t)}(x) = \frac{\beta^{\alpha(t)}}{\Gamma(\alpha(t))}x^{\alpha(t)-1}\exp(-\beta x),\qquad x>0.$$

A key distinguishing feature of the "semi-physical" subclass is the construction of $\alpha(t)$ and $\beta$ to be physically interpretable and empirically calibratable, reflecting the observed macro-kinetics of degradation and environmental acceleration (Shi et al., 14 Jan 2026).

2. Semi-Physical Parameterization

Shape Function and Mean Trend

For LED-package degradation, the shape function is taken as

$\alpha(t) = A\, e^{b t},\qquad A > 0,\, b \ge 0$

yielding a process mean

$$m(t) = \mathbb{E}[X(t)] = \alpha(t)/\beta = (A/\beta)e^{b t}$$

which matches the industry-standard TM-21 exponential lumen-maintenance law governing LED light output: $$\mathbb{E}[P_\mathrm{out}(t)] = 1 - m(t) \equiv m_0 e^{-\lambda t}$$ with $m_0 \approx 1$ and $\lambda = b$.

Temperature-Dependent Rate (Arrhenius Law)

Environmental acceleration, such as that due to elevated temperature in accelerated degradation tests, is incorporated via an Arrhenius-type rate parameter: $\beta(T) = C \exp \left(\frac{E_a}{k_B T}\right)$ with $E_a$ the activation energy, $k_B \approx 8.62 \times 10^{-5}~\mathrm{eV/K}$, and $C > 0$. In the linear acceleration regime, $A$ and $b$ are held constant across temperatures.

3. Stochastic Properties and Degradation Rate

The process exhibits the following properties:

• PDF of any increment:

$$f_{X(t)-X(s)}(x) = \frac{\beta^{\alpha(t)-\alpha(s)}}{\Gamma(\alpha(t)-\alpha(s))} \, x^{\alpha(t)-\alpha(s)-1} \exp(-\beta x)$$

• Instantaneous degradation (intensity) rate:

$$\lambda_\mathrm{GP}(t) = \frac{d}{dt} \mathbb{E}[X(t)] = \frac{A b}{\beta} e^{b t}$$

This formalism allows both process and parameter uncertainty to be captured by sampling across the relevant posteriors and propagating via Monte Carlo simulation (Shi et al., 14 Jan 2026).

4. Physical and Statistical Interpretations

"Physical" refers to the direct grounding of $\alpha(t)$ and $\beta(T)$ in empirical and/or mechanistic laws: the exponential $\alpha(t)$ is motivated by the physical chemistry of lumen degradation, and $\beta(T)$'s temperature dependence is Arrhenius-type, as observed in acceleration testing.

"Statistical" denotes the fully stochastic nature of the process: increments are random variables, not merely noisy deviations along a deterministic path. All parameter and process uncertainties propagate to outputs, enabling robust risk analysis and performance optimization (Shi et al., 14 Jan 2026).

The "semi-physical" descriptor thus reflects hybridization—physical structure guides parameterization, but parameter estimation and uncertainty quantification are statistical and data-driven.

5. Parameter Estimation and Uncertainty Quantification

Parameters are typically estimated using LM-80 accelerated-degradation data. The likelihood is constructed from sequential increments: $$L(\boldsymbol\theta; \mathscr{D}) = \prod_{k=1}^M f_{\Gamma}\left(x_k - x_{k-1} \mid \alpha(t_k; \boldsymbol\theta) - \alpha(t_{k-1}; \boldsymbol\theta), \, \beta(T_k; \boldsymbol\theta)\right)$$ with parameter vector $\boldsymbol\theta = (\ln A, b, \ln C, E_a)$.

Diffuse or weakly informative priors are employed:

• $\ln A \sim \mathcal N(0, 10^2)$
• $b \sim \mathcal N_+ (0, 10^6)$ (half-normal)
• $\ln C \sim \mathcal N(0, 10^2)$
• $E_a \sim \mathcal N_+ (0, 10^6)$

Posterior inference is performed via MCMC (e.g., CmdStanPy). Uncertainty is propagated via joint sampling over parameter posteriors and process paths:

• For each posterior sample $\boldsymbol\theta^{(l)}$, simulate $n_\mathrm{ps}$ Gamma-process paths.
• The ensemble of $n_\mathrm{path} \times n_\mathrm{ps}$ trajectories spans both parameter and stochastic path uncertainty (Shi et al., 14 Jan 2026).

6. Integration with Performance-Driven Maintenance Optimization

In the LED system framework:

• Package degradation $X_j(t)$ evolves via the semi-physical Gamma process.
• Driver failures follow a Weibull model, and overall luminaire state is $L_j(t) = \max\{X_j(t), S_j(t)\}$, where $S_j(t)$ is an indicator of driver failure.
• Working-plane illuminance and static performance indices (average illuminance $E_\mathrm{avg}(t)$, uniformity $U(t)$) are mapped via surrogate models.
• Dynamic performance deficiency is quantified via the deficiency duration ratio $R_\mathrm{DR}$, computed from event-wise intervals below illuminance or uniformity standards.

Maintenance policies are optimized in a multi-objective setting, balancing $R_\mathrm{DR}$, site visit count $N_\mathrm{tv}$, and number of replacements $N_\mathrm{tr}$ (Shi et al., 14 Jan 2026).

7. Connections to the Non-Stationary Feller Process and Generalizations

Masoliver (2015) (Masoliver, 2015) develops a non-stationary Feller process characterized by the SDE

$$dX(t) = [B(t) - a(t) X(t)] dt + k(t) \sqrt{X(t)} dW(t)$$

where $a(t)>0$, $B(t)\ge0$, and $k(t)>0$ are smooth functions of time. In the long-time limit, the process approaches a Gamma distribution with time-varying shape and scale, aligning with the non-homogeneous Gamma process framework: $$p(x,t) \simeq \frac{x^{\theta(t)-1}}{\Gamma(\theta(t)) D(t)^{\theta(t)}}\,\exp\left(-\frac{x}{D(t)}\right)$$ where $\theta(t)=2B(t)/k^2(t)$ and $D(t)=k^2(t)/[2a(t)]$. Accessibility of the origin, governed by the power-law exponent at $x=0$, is a dynamic property depending on $\theta(t)$.

This generalization permits the modeling of diverse non-stationary phenomena, including financial volatility and anomalous diffusion, through appropriate choices of time-varying coefficients (Masoliver, 2015). A plausible implication is that the semi-physical non-homogeneous Gamma process used in LED degradation is a particular reduction of the non-stationary Feller framework, specialized to monotone, non-negative increments and exponential mean trends.