As a researcher focused on the reliability of modern agricultural equipment, I have observed that the plant protection UAV, or China UAV drone, plays an increasingly vital role in precision agriculture. These drones are widely used for pesticide spraying, fertilizer spreading, and field monitoring due to their high operational efficiency and adaptability to complex terrains. However, the harsh operating environment—characterized by high temperatures, dust, and pesticide corrosion—often leads to frequent system failures. Such breakdowns can severely disrupt the narrow operational windows during critical farming seasons, causing significant economic losses and safety hazards. Therefore, accurately predicting the failure time points of these China UAV drone systems is essential for proactive maintenance and operational continuity.
In my study, I treat the China UAV drone as a repairable system. The primary challenge lies in the fact that the maintenance level during field operations is often unknown and varies significantly due to factors such as technician skill, repair strategy, and spare part quality. This uncertainty makes it difficult to model the failure evolution process using standard assumptions. To address this, I propose a systematic framework for discriminating among different stochastic point process models, ultimately aiming to predict future failure times for a specific model of a plant protection China UAV drone.
1. Data Acquisition and Trend Analysis
I collected operational and maintenance records from an “X” series electric single-rotor plant protection China UAV drone operating in the Tacheng region of Xinjiang, China, between 2024 and 2025. The primary task was pesticide spraying. To ensure data consistency, a “system failure” was strictly defined as any event that rendered the drone unable to complete its designated mission, requiring a stoppage for inspection, repair, or component replacement before it could resume work. I excluded events caused by human error or external collisions.
The observation period was a time-censored test with a truncation time T = 300 hours. During this period, the drone experienced 14 complete failures. The continuous failure interval times and cumulative operation times are presented in Table 1. The data point marked with a “+” is a censored data point and is not counted as a failure event.
| Failure Order i | Interval Time Xi (hours) | Cumulative Time Ti (hours) | Failure Order i | Interval Time Xi (hours) | Cumulative Time Ti (hours) |
|---|---|---|---|---|---|
| 1 | 112.03 | 112.03 | 9 | 10.89 | 262.75 |
| 2 | 67.36 | 179.39 | 10 | 7.75 | 270.50 |
| 3 | 28.61 | 208.00 | 11 | 11.63 | 282.13 |
| 4 | 1.29 | 209.29 | 12 | 6.76 | 288.89 |
| 5 | 7.26 | 216.55 | 13 | 3.27 | 292.16 |
| 6 | 5.48 | 222.03 | 14 | 5.58 | 297.74 |
| 7 | 14.52 | 236.55 | 15 | 2.26+ | 300.00 |
| 8 | 15.31 | 251.86 |
Before selecting a model, I first performed a trend test to determine the nature of the failure process. A graphical test, using a cumulative failure times versus cumulative number of failures plot, showed a concave shape. This suggested that the intervals between successive failures were decreasing over time, indicating a degrading system.
To confirm this, I applied several statistical tests. The Laplace test, Military-Handbook test, and J-test were used to check for monotonic trends. The results are summarized in Table 2. At a significance level of α = 0.05, the statistics clearly rejected the Null Hypotheses of a Homogeneous Poisson Process (HPP) or a Renewal Process (RP). The V-test family showed no significant non-monotonic trend. The combined results confirmed a significant monotonic increasing failure rate for this China UAV drone.
| Statistic | U | Z | J | V(1) | V(2) | V(3) |
|---|---|---|---|---|---|---|
| Value | 3.80 | 7.27 | 2.43 | 1.58 | 1.50 | 16.29 |
| Critical Level (α=0.05) | 1.96 | [15.31, 44.46] | 2.15 | 1.96 | 1.96 | [15.31, 44.46] |
| Conclusion | Reject HPP | Reject HPP | Reject RP | No Trend | No Trend | No Trend |
2. Candidate Model Fitting
Based on the trend test results, I identified two candidate models suitable for a system with a monotonic trend: the Non-Homogeneous Poisson Process (NHPP) and the Generalized Renewal Process (GRP). Both models assume the underlying failure distribution is a Weibull distribution.
2.1 Non-Homogeneous Poisson Process (NHPP) Model
The NHPP model, specifically the Power Law process, is often used for “minimal repair” scenarios, where a repair restores the system to its condition just before failure. The failure intensity function for this model is:
$$ \omega(t) = \frac{\beta_1}{\eta_1} \left( \frac{t}{\eta_1} \right)^{\beta_1 – 1} $$
where β1 is the shape parameter and η1 is the scale parameter. For a time-censored experiment with n failures and a total test time T, the likelihood function is:
$$ L_{NHPP} = \left( \frac{\beta_1}{\eta_1} \right)^n \prod_{i=1}^n \left( \frac{T_i}{\eta_1} \right)^{(\beta_1-1)} \exp\left[-\left( \frac{T}{\eta_1} \right)^{\beta_1} \right] $$
Using maximum likelihood estimation (MLE), the parameter estimates are obtained by solving:
$$ \hat{\eta}_1 = \frac{T}{n^{1/\beta_1}} $$
$$ \hat{\beta}_1 = \frac{n}{\sum_{i=1}^n \ln \frac{T}{T_i}} $$
Substituting the data from Table 1, I obtained the NHPP parameter estimates: β̂1 = 3.85 and η̂1 = 151.22 hours. The value of β̂1 > 1 confirms the degradation trend observed in the trend tests.
2.2 Generalized Renewal Process (GRP) Model
The GRP model is more flexible, as it can describe “imperfect maintenance,” where a repair only partially restores the system’s health. I chose the Kijima Type-I model, which is appropriate for field repairs that fix a specific fault but do not fully eliminate accumulated wear. This model introduces an effective age V for the system. After the i-th repair, the effective age is defined as:
$$ V_i = V_{i-1} + q X_i $$
where q (0 ≤ q ≤ 1) is the age reduction factor. A q = 0 corresponds to “as good as new” (perfect repair), and a q = 1 corresponds to “as bad as old” (minimal repair). Assuming Weibull-distributed lifetimes, the likelihood function for a time-censored GRP is more complex. I used a Differential Evolution (DE) algorithm in Python to minimize the negative log-likelihood function:
$$ \ln L_{GRP} = \sum_{i=1}^n \ln \left[ \frac{\beta_2}{\eta_2} \left( \frac{V_{i-1} + X_i}{\eta_2} \right)^{\beta_2-1} \right] – \sum_{i=1}^n \left[ \left( \frac{V_{i-1} + X_i}{\eta_2} \right)^{\beta_2} – \left( \frac{V_{i-1}}{\eta_2} \right)^{\beta_2} \right] – \left[ \left( \frac{V_n + z}{\eta_2} \right)^{\beta_2} – \left( \frac{V_n}{\eta_2} \right)^{\beta_2} \right] $$
The optimization yielded the GRP model parameters: η̂2 = 109.53, β̂2 = 8.16, and q̂ = 0.433. The q̂ value of 0.433 suggests that after a repair, the drone system retains about 43.3% of its accumulated wear. This indicates that the field repairs are indeed imperfect, making the GRP model a physically plausible representation of the repair process.
3. Model Validation and Comparison
To validate the fitted models, I used the Time-Rescaling Theorem. The theorem states that for a valid point process model, the transformed inter-arrival times (residual increments) should be independent and identically distributed as an exponential distribution with a rate of 1.
For the NHPP, the residual increment ΔWi is calculated by integrating the intensity function over the interval between successive failures:
$$ \Delta W_i = \int_{T_{i-1}}^{T_i} \omega(t) dt $$
For the GRP, the residual increment ΔHi is calculated similarly, but by integrating the hazard function over the effective age interval:
$$ \Delta H_i = \int_{V_{i-1}}^{V_{i-1} + X_i} \phi(v) dv $$
The residual increments for each model are shown in Tables 3 and 4.
| i | ΔWi | i | ΔWi |
|---|---|---|---|
| 1 | 0.2896 | 8 | 1.5876 |
| 2 | 1.5857 | 9 | 1.3180 |
| 3 | 1.4980 | 10 | 1.0422 |
| 4 | 0.0838 | 11 | 1.7390 |
| 5 | 0.5009 | 12 | 1.1137 |
| 6 | 0.4126 | 13 | 0.5673 |
| 7 | 1.2489 | 14 | 1.0125 |
| i | ΔHi | i | ΔHi |
|---|---|---|---|
| 1 | 1.1787 | 8 | 1.2716 |
| 2 | 1.5820 | 9 | 1.1931 |
| 3 | 0.7079 | 10 | 1.0323 |
| 4 | 0.0239 | 11 | 2.1803 |
| 5 | 0.1817 | 12 | 1.4667 |
| 6 | 0.1631 | 13 | 0.7582 |
| 7 | 0.7520 | 14 | 1.5093 |
I first performed a distributional test using a Q-Q plot and the Kolmogorov-Smirnov (K-S) test against a unit exponential distribution. The K-S test statistics for the NHPP and GRP models were 0.2795 and 0.2930, respectively, both of which are below the critical value of 0.3636 at the α = 0.05 level. This result suggests that we cannot reject the hypothesis that the residuals from either model are from a unit exponential distribution. The Q-Q plots confirmed this, with data points aligning reasonably well along the diagonal.
Next, I checked for serial correlation in the residual increments. A plot of the residuals against their index showed no clear trend. Furthermore, I calculated the sample autocorrelation coefficient r1 for l = 1. The results were r1 = 0.1231 for the NHPP model and r1 = 0.4759 for the GRP model. Both values were less than the significance bound of 0.5238 (for α = 0.05), indicating no significant serial correlation. Both models passed the validation checks.
To select the best model, I used the Akaike Information Criterion (AIC) and the Bayesian Information Criterion (BIC). These criteria balance model fit and complexity, with smaller values indicating a better model.
$$ AIC = 2s – 2\ln(\hat{L}) $$
$$ BIC = \ln(y)s – 2\ln(\hat{L}) $$
where s is the number of model parameters, and y is the sample size (y=14). The comparison is shown in Table 5.
| Model | Number of Parameters (s) | Log-Likelihood ln(L̂) | AIC | BIC |
|---|---|---|---|---|
| NHPP | 2 | -47.98 | 99.95 | 101.23 |
| GRP | 3 | -45.38 | 96.75 | 98.67 |
The results in Table 5 clearly show that the GRP model has lower AIC and BIC values compared to the NHPP model. Despite having one more parameter, the GRP model provides a significantly better fit to the China UAV drone’s failure data. This aligns with the estimated q̂ = 0.433, confirming that imperfect maintenance is a key characteristic of the system’s failure process. Therefore, I selected the GRP model for predicting future failure times.
4. Prediction Based on the GRP Model
Predicting fault times for a GRP model is analytically complex due to its path-dependent nature. I therefore employed a Monte Carlo simulation to forecast the subsequent 6 failure times for this plant protection China UAV drone. The simulation uses the GRP model’s iterative nature. Starting from the effective age at the end of the observation period, V’0 = qTn + z, the first future failure interval, X’1, is generated by solving:
$$ X’_1 = \eta_2 \left[ \left( \frac{V’_0}{\eta_2} \right)^{\beta_2} – \ln(1 – \mu_1) \right]^{1/\beta_2} – V’_0 $$
where μ1 is a random number from a uniform distribution U(0,1). The effective age is then updated, and the process is repeated for each subsequent failure k:
$$ V’_{k-1} = q \left( T + \sum_{i=1}^{k-1} X’_i \right) $$
$$ X’_k = \eta_2 \left[ \left( \frac{V’_{k-1}}{\eta_2} \right)^{\beta_2} – \ln(1 – \mu_k) \right]^{1/\beta_2} – V’_{k-1} $$
I ran the simulation for m = 10,000 paths to generate a distribution of failure times for the next 6 failures. The mean predicted failure times and 5%-95% prediction intervals are presented in Table 6.
To visually demonstrate the evolution of the system’s condition, I have included a simulation result plot. The following figure shows the degradation trend of the China UAV drone system.
| Prediction Order k | Mean Interval X̄’k (hours) | Mean Failure Time T̄’k (hours) | 5%-95% Prediction Interval (hours) |
|---|---|---|---|
| 1 | 3.13 | 303.13 | [300.19, 308.71] |
| 2 | 3.20 | 306.33 | [301.32, 313.45] |
| 3 | 2.94 | 309.27 | [302.99, 317.38] |
| 4 | 2.82 | 312.09 | [304.85, 320.77] |
| 5 | 2.64 | 314.73 | [306.81, 323.82] |
| 6 | 2.49 | 317.22 | [308.85, 326.58] |
5. Conclusion and Future Work
This study successfully developed a robust failure time prediction method for a specific model of a plant protection China UAV drone. By systematically applying trend tests and model validation techniques, I was able to objectively select the most appropriate stochastic point process model – the Generalized Renewal Process (GRP) – instead of relying on subjective assumptions about the repair level. The GRP model’s estimated age reduction factor (q̂ = 0.433) provided quantitative evidence that the field repairs for this China UAV drone are imperfect, a crucial insight for maintenance planning.
The Monte Carlo simulation based on the GRP model effectively captured the system’s deteriorating behavior. The results show a clear trend: the mean time between successive predicted failures decreases from 3.13 hours to 2.49 hours. This indicates that the China UAV drone system is in a state of continuous degradation, and the risk of failure becomes increasingly higher as it operates. The prediction intervals, although widening with the prediction horizon, provide a useful range for planning proactive maintenance actions, such as scheduling in-depth inspections or replacing critical components, before a failure occurs.
My work focused on a single China UAV drone. Future research will aim to incorporate data from multiple drones and consider the influence of covariates like environmental conditions, payload variations, and different maintenance strategies. This would lead to a more generalized and stable prediction model, further enhancing the safety and operational efficiency of plant protection UAVs.