Relationship Analysis Between Helicopter Gearbox Bearing Condition Indicators and Oil Temperature Through Dynamic ARDL and Wavelet Coherence Techniques

Abstract

This study investigates the dynamic relationship between bearing gearbox condition indicators (BGCIs) and the lubrication oil temperature within the framework of health and usage monitoring system (HUMS) applications. Using the dynamic autoregressive distributed lag (DARDL) simulation model, we quantified both the short- and long-term responses of condition indicators to shocks in oil temperature, offering a robust framework for a counterfactual analysis. To complement the time-domain perspective, we applied a wavelet coherence analysis (WCA) to explore time–frequency co-movements and phase relationships between the condition indicators under varying operational regimes. The DARDL results revealed that the ball energy, cage energy, and inner and outer race indicators significantly increased in response to the oil temperature in the long run. The WCA results further confirmed the positive association between oil temperature and the condition indicators under examination, aligning with the DARDL estimations. The DARDL model revealed that the ball energy and the inner race energy have statistically significant long-term effects on the oil temperature, with p-values < 0.01. The adjusted R² of 0.785 and the root mean square error (MSE) of 0.008 confirm the model’s robustness. The wavelet coherence analysis showed strong time–frequency correlations, especially in the 8–16 scale range, while the frequency-domain causality (FDC) tests confirmed a bidirectional influence between the oil temperature and several condition indicators. The FDC analysis showed that the oil temperature significantly affected the BGCIs, with evidence of feedback effects, suggesting a mutual dependency. These findings contribute to the advancement of predictive maintenance frameworks in HUMSs by providing practical insights for enhancing system reliability and optimizing maintenance schedules. The integration of dynamic econometric approaches demonstrates a robust methodology for monitoring critical mechanical components and encourages further research in broader aerospace and industrial contexts.

1. Introduction

Helicopter gearbox systems are critical mechanical components that operate under high stress and varying operational conditions, playing a central role in power transmission. Due to the demanding nature of their function, ensuring the health and reliability of these gearboxes is vital, especially during critical flight phases such as takeoff, landing, and adverse weather conditions [1,2,3,4,5]. The gearbox performance directly influences the flight safety, mission success, and operational costs. In this context, advanced predictive maintenance systems are increasingly replacing traditional maintenance strategies, which often rely on fixed schedules or failure-based models. These systems are integrated into HUMSs, which continuously monitor the status of critical components in real time, enabling more effective and timely maintenance decisions [4,6,7,8].

The analysis of the relationship between BGCIs and the oil temperature (OT) is of critical importance for enhancing the health monitoring, diagnostic precision, and predictive maintenance of helicopter gearbox systems [9,10]. While the OT serves as a thermal health indicator—signaling issues such as overheating, poor lubrication, or internal friction—BGCIs reflect mechanical degradation, particularly of components like the bearing inner race, outer race, cage, and rolling elements [11,12]. Studying these two condition indicators (CIs) in conjunction provides complementary insights into both thermal and mechanical health, allowing for early fault detection, degradation pattern recognition, and a more accurate estimation of the remaining useful life (RUL) [5,6,7,8,9,10,11,12]. Moreover, understanding the dynamic interaction between BGCIs and the OT is essential for developing robust prognostic models, especially in real-world environments where systems exhibit nonlinear and non-stationary behavior. This paper makes several original contributions to the field of gearbox diagnostics and health monitoring. While the individual analytical techniques employed—such as the DARDL model, the WCA, and FDC testing—are established in the literature, their combined application to real helicopter HUMS data for analyzing the interplay between the OT and BGCIs is, to our knowledge, novel. Specifically, this study is among the first to apply probabilistic DARDL simulations to model both positive and negative shocks in CIs, providing a more realistic and interpretable forecast of system behavior under stress. Moreover, the integration of both time-domain (DARDL) and frequency-domain frameworks enables a deeper understanding of how mechanical and thermal factors influence each other over multiple time scales. This dual perspective enhances the diagnostic sensitivity and system-level insights, extending the practical utility of condition-monitoring models in aerospace HUMS platforms.

Compared to traditional condition-monitoring approaches—which often rely on manual inspection, high-volume continuous data acquisition, or reactive diagnostics—HUMSs offer a more integrated and efficient alternative. Systems like GPMS enable automated, periodic in-flight monitoring that reduces data overhead while maintaining diagnostic sensitivity. By embedding sensors and analytics onboard, HUMSs facilitate real-time health tracking, early fault detection, and proactive maintenance planning, in contrast to the offline analysis or post-failure investigations typical of conventional systems.

One key challenge in gearbox health monitoring is the integration of multiple performance indicators that reflect different aspects of the system condition [9,12,13]. Among these, two critical parameters are the lubrication OT and BGCIs, both of which provide essential diagnostic information [14]. An elevated OT can indicate potential issues such as a lack of lubrication, overheating, or mechanical faults. BGCIs, on the other hand, directly reflect bearing wear and degradation, which are among the most failure-prone components in a gearbox. Understanding the relationship between the OT and BGCIs is therefore crucial for maintaining optimal gearbox performance and ensuring safe, efficient operations.

Although earlier studies have explored the individual effects of various CIs on the OT—such as the outer race energy (OR_E), inner race energy (IR_E), spectral kurtosis (SK), cage energy (C_E), and ball energy (B_E)—many treat these CIs in isolation. For instance, prior research has employed ARDL models to quantify the impact of condition indicators on the OT trends in Bell 407 helicopters [9,13]. However, such approaches often assume linearity and stationarity and overlook the interaction effects across time scales. Recent work has proposed the use of wavelet-based methods, such as the autocorrelation of Morlet wavelet transforms [15] and continuous wavelet transforms [16], for early fault detection under non-stationary conditions. Additionally, a machine learning-assisted wavelet analysis has demonstrated strong potential for improving the health state classification in complex mechanical systems [17,18,19].

Furthermore, studies in data-driven HUMS strategies have shown how integrating mechanical and thermal indicators—through dynamic modeling and spectral causality—enhances the prognostic accuracy in rotating systems. These include the application of spectral entropy, signal decomposition, and hybrid signal-processing techniques for gearbox diagnostics under real flight conditions [10,12,20].

Despite these advances, there remains a gap in the literature concerning the integrated use of dynamic econometric models (e.g., DARDL), a wavelet coherence analysis (WCA), and frequency-domain Granger causality to assess the coupled dynamics between the OT and BGCIs. Our work addresses this gap by combining these methods into a unified framework for analyzing the temporal, spectral, and causal dependencies in helicopter gearboxes.

Recent developments in fault detection frameworks further emphasize combining wavelet methods with advanced analytics such as adaptive filtering and deep learning [21,22]. These studies highlight the growing integration of data-driven and hybrid approaches to improve the early detection of complex mechanical failures.

Recent advancements in intelligent monitoring systems have also explored the use of federated learning for distributed fault diagnoses. For example, adaptive federated approaches have been proposed to address challenges like label heterogeneity and communication redundancy in collaborative industrial environments [23,24]. Additionally, client-level imbalances have been tackled to enhance model generalization across diverse operational conditions [25]. These techniques support the growing trend toward decentralized, privacy-preserving fault diagnosis frameworks, complementing the predictive analytics used in HUMS applications.

Building on these analytical advancements, a broader bibliometric perspective reveals how such methodologies align with the prevailing research trends in vertical flight HUMSs. Specifically, the integration of CIs like the OT and BGCIs within advanced diagnostic frameworks corresponds with the growing emphasis on data-driven techniques such as fault diagnoses, RUL estimation, and predictive maintenance, as illustrated in the keyword co-occurrence and thematic clusters shown in Figure 1.

Figure 1 provides a bibliometric overview supporting the relevance of this study’s methodology. Subfigure (a) groups research themes based on keyword co-occurrence, showing a dense cluster around “condition-based maintenance” and related terms. Subfigure (b) highlights the global distribution of research, with darker regions representing a higher level of publication activity. Subfigure (c) identifies the leading journals contributing to the HUMS literature, such as Reliability Engineering & System Safety and IEEE Access, while (d) maps the most cited authors, with the node size reflecting the citation count. The added legends clarify the visual encoding used in each network diagram.

The central theme, as evidenced by the prominence of the “condition-based maintenance” node, underscores its foundational role in the field. Closely associated concepts include “fault-diagnosis”, “deep learning”, “remaining useful life estimation”, and “prognostics and health management” (PHM), indicating a strong research focus on advanced diagnostic and prognostic methodologies. “Vibration monitoring” and “bearing” were also significant, highlighting the importance of specific condition-monitoring techniques and critical components in these systems. While “oil debris analysis” was not explicitly a large node, its relevance is implied through the connections to “bearing” and “diagnosis”, which are integral to a comprehensive condition assessment. The field extensively leverages “artificial intelligence” and “neural networks” for developing robust “models” for “degradation” and “system” analyses, ultimately contributing to “predictive maintenance” strategies.

In terms of publication outlets, Reliability Engineering & System Safety emerged as a highly influential journal, serving as a central hub for research dissemination. Other notable journals and proceedings included Sensors, IEEE Access, Mathematics, Journal of Intelligent Manufacturing, Journal of Quality in Maintenance Engineering, Proceedings of the Institution of Mechanical Engineers, IEEE Transactions on Reliability, and Ocean Engineering, reflecting the interdisciplinary nature of the research, which encompasses engineering, computer science, and applied sciences. Geographically, research contributions were predominantly led by the “Peoples Republic of China” and the “USA”, demonstrating their significant impact and investment in this area. Other active countries include Italy, France, England, Germany, Canada, and India, indicating a global collaborative effort. Prominent authors such as A.K.S. Jardine, J. Lee, and Z. Li were identified as key contributors, whose work likely shapes the theoretical and practical advancements in HUMSs, CBM, and the broader predictive maintenance landscape.

Building on this foundation, this study employs the DARDL model and a WCA to investigate the dynamic relationship between the OT and BGCIs in the Bell 407 helicopter model. The DARDL model, introduced by Jordan and Philips [19], is well-suited for helicopter gearbox data due to its robustness in capturing both short- and long-term dynamics, even with small sample sizes. Additionally, a WCA enables a time–frequency perspective, revealing how the relationship between BGCIs and the OT evolves over time and across various frequency bands. This is particularly valuable for detecting both short-term fluctuations and long-term trends that conventional time-domain models may overlook. Together, these methods provide a robust analytical framework for understanding the interactions that influence gearbox performance and contribute to system reliability and maintenance outcomes.

This study contributes to the literature in several key ways. First, while previous works—such as [9]—have utilized ARDL models to explore the relationships between CIs, this study employs the more advanced DARDL model, which accounts for actual variation in the independent CIs. This approach enables a more accurate assessment of how both positive and negative changes in the OT affect BGCIs and allows these effects to be simulated and visualized across different time horizons.

Second, most existing studies rely on time-domain Granger causality tests, which may miss causal interactions that vary across frequencies. Since the direction and strength of causality between the OT and BGCIs can change depending on the time scale—such as short-term operational shocks versus long-term wear patterns—a purely time-domain approach provides only a partial view of system behavior. To address this, we incorporated the FDC framework developed by Breitung and Candelon [20], which enables a more nuanced examination of causality across time scales, offering deeper insights into the interaction between lubrication performance and gearbox health.

The remaining structure of this paper is as follows: Section 2 provides a comprehensive review of the relevant literature. Section 3 outlines the materials and methodological approach employed in this study. Section 4 reports and analyzes the empirical findings. Lastly, Section 5 summarizes the main conclusions and discusses their implications for predictive maintenance practices, along with suggestions for future research.

2. Material and Methods

2.1. HUMS Data Collection

The HUMS provides a comprehensive aircraft health assessment by automating flight data monitoring, the rotor track and balance, the engine performance, and drivetrain diagnostics [26]. It uses ten smart accelerometers to gather vibration data, with an accessory drive sensor monitoring the duplex bearing at 4450 RPM. Data are collected at 46.875 samples per second over 2 s every four minutes during straight and level flight.

This 2 s sampling window at 46 Hz, taken every 4 min, reflects a common trade-off in HUMSs between a detailed resolution and a manageable data size for long-duration missions. It is designed to capture vibration under stable flight conditions, minimizing transient interference and ensuring consistent baseline monitoring. This approach is implemented in the GPMS Foresight MX system to optimize the diagnostic reliability while maintaining data efficiency.

Figure 1, Figure 2 and Figure 3 show the GPMS Foresight MX Bell 407 kit, the variable reluctance sensors (VR sensors) and main rotor [26,27,28], and the OT and BGCIs for a Bell 407 helicopter, respectively.

More information can be found in references [26,27,28].

Figure 4 illustrates the temporal evolution of key vibration-based indicators (C_E, B_E, IR_E, OR_E, and SK) alongside the oil temperature (T). Each subplot captures the sensor output over time during straight and level flight. Notably, B_E and IR_E exhibited multiple peaks that aligned with temperature spikes, suggesting strong thermal–mechanical coupling. C_E and OR_E also showed fluctuations that preceded or followed thermal events. These patterns support the hypothesis that gearbox heat buildup and mechanical wear are interrelated. SK, representing impulsiveness, fluctuated less predictably, but still showed minor co-variation. Understanding these trends is essential for HUMSs: elevated vibration levels paired with a rising OT can provide early warnings of bearing degradation or lubrication failure.

2.2. Model Specification

The mathematical formulations used in this study originate from both the established literature and adaptations to suit our empirical framework. Specifically, Equations (1)–(3) are adapted from the ARDL modeling approach described by Pesaran et al. [29]. Equation (4) applies the dynamic ARDL (DARDL) simulation model introduced by Jordan and Philips [19]. Equations (5)–(9) represent standard continuous wavelet and coherence formulations, as developed by Goupillaud, Grossmann, and Morlet [29,30,31,32,33], and later operationalized by Torrence and Compo [34]. Where relevant, we have included direct citations with each formula to ensure clarity of origin.

This study analyzed the dynamic impacts of the C_E, B_E, IR_E, and OR_E, and SK on the OT, using the model presented in Equation (1):

$$O{T}_t={\alpha }_0+{\alpha }_1{\mathrm{C}}_{Et}+{\alpha }_2{\mathrm{B}}_{Et}+{\alpha }_3\mathrm{I}{\mathrm{R}}_{Et}+{\alpha }_4\mathrm{O}{\mathrm{R}}_{Et}+{\alpha }_5\mathrm{S}{\mathrm{K}}_t+{\epsilon }_t$$

(1)

where T is the OT and t denotes the acquisition time.

α₀ is the constant term. α₁, α₂, α₃, α₄, and α₅ represent the long-run elasticities of the explanatory CIs. ε_t is the regression error term.

The econometric model in Equation (1) is not linear, which could lead to inconsistent results and complicate decision-making. To address this, we applied a natural logarithm (Ln) transformation to convert the CIs into comparable units, improving the results and interpretation. The log-linear form is shown in Equation (2):

$$LnO{T}_t={\alpha }_0+{\alpha }_1\mathrm{Ln}{\mathrm{C}}_{Et}+{\alpha }_2\mathrm{Ln}{\mathrm{B}}_{Et}+{\alpha }_3\mathrm{LnI}{\mathrm{R}}_{Et}+{\alpha }_4\mathrm{LnO}{\mathrm{R}}_{Et}+{\alpha }_5\mathrm{LnS}{\mathrm{K}}_t+{\epsilon }_t$$

(2)

Equations (1) and (2) define the foundational regression structure between the OT and selected CIs, using a log-linear form to ensure the comparability and interpretability of the coefficients. This formulation sets the base for assessing the long-term elasticity and informs subsequent modeling steps.

As mentioned above, this study addresses the challenges of using the ARDL model for analyzing the short- and long-term effects of independent CIs, including cointegration testing, lag complexities, and software limitations. To overcome these, a dynamic simulation approach incorporating an error correction term is proposed. This innovative method enhances the traditional ARDL model, enabling a more effective analysis of exogenous CI impacts. Figure 5 outlines this methodology.

2.3. Algorithmic Procedure for Experimental Analysis

To improve the clarity and reproducibility of our methodological framework, we outline here the key steps followed in the experimental analysis. The overall procedure integrates DARDL simulations, a WCA, and FDC testing. These steps are illustrated in Figure 5 and can be summarized as follows:

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Data Acquisition: Vibration and OT data are collected from HUMS-equipped Bell 407 helicopters under steady-state flight conditions.

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Preprocessing: Raw signals are log-transformed and smoothed to ensure the stationarity and comparability of CIs.

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Unit Root Testing: The stationarity of the time series is assessed using the ADF and KPSS tests.

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ARDL Bounds Testing: Cointegration relationships between the OT and BGCIs are verified.

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DARDL Simulation: The DARDL model is applied to simulate the short- and long-term impacts of the BGCIs on the OT.

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Wavelet coherence analysis: A WCA is conducted to examine the time–frequency correlations between variables.

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Frequency-domain causality: Spectral Granger causality is tested across short-, medium-, and long-term bands.

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Diagnostics and validation: Statistical tests (LM, RESET, Jarque-Bera) validate the model robustness.

This structured approach enhances the transparency of our analysis and supports replication in future HUMS-related research.

All variable notations (e.g., frequency ω, lag terms, and log-transformed variables such as LnOT) have been standardized throughout the manuscript to ensure typographic and semantic consistency.

3. Empirical Analysis Procedure

To test for the presence of unit roots in the study, we applied the augmented Dickey–Fuller (ADF) test [30] and the Kwiatkowski–Phillips–Schmidt–Shin (KPSS) test [31]. Both tests assess the null hypothesis of a unit root against the alternative hypothesis of stationarity. The DARDL approach requires the dependent CI to be integrated at I(1).

To examine the long-term relationships between CIs, we employed the ARDL cointegration methodology proposed by Pesaran et al. [29]. This approach offers several advantages over the other traditional cointegration techniques.

Equation (3) shows the ARDL bounds testing method applied to analyze the long-run relationship between the CIs. This equation presents the ARDL bounds testing framework, specifying both lagged and differenced terms to estimate short- and long-term interactions between the CIs and the OT. This model forms the core of the empirical estimation strategy.

$$\begin{array}{ll}\hfill \Delta LnO{T}_t=& {\alpha }_0+{\displaystyle \sum _{k=1}^n{\alpha }_{1k}}\mathrm{\Delta }LnO{T}_{t-k}+{\displaystyle \sum _{k=1}^n{\alpha }_{2k}}\mathrm{\Delta }Ln{\mathrm{C}}_{Et-k}+{\displaystyle \sum _{k=1}^n{\alpha }_{3k}}\mathrm{\Delta }Ln{\mathrm{B}}_{Et-k}+{\displaystyle \sum _{k=1}^n{\alpha }_{4k}}\mathrm{\Delta }Ln\mathrm{I}{\mathrm{R}}_{Et-k}+{\displaystyle \sum _{k=1}^n{\alpha }_{5k}}\mathrm{\Delta }Ln\mathrm{O}{\mathrm{R}}_{Et-k}+\hfill \\ & +{\displaystyle \sum _{k=1}^n{\alpha }_{6k}}\mathrm{\Delta }Ln\mathrm{S}{\mathrm{K}}_t+{\delta }_{1}LnO{T}_{t-1}+{\delta }_{2}Ln{\mathrm{C}}_{Et-1}+{\delta }_{3}Ln{\mathrm{B}}_{Et-1}+{\delta }_{4}Ln\mathrm{I}{\mathrm{R}}_{Et-1}+{\delta }_{5}LnO{\mathrm{R}}_{Et-1}\hfill \\ & +{\delta }_{6}LnS{K}_{t-1}+{\epsilon }_t\hfill \end{array}$$

(3)

These ARDL formulations follow the structure proposed by Pesaran et al. [29] and are adapted for our multi-indicator context.

Here, ∆ represents the first difference operator, while t−1 refers to the optimal lag length determined using the Akaike information criterion (AIC). The parameters α and δ capture the short-run and long-run dynamics, respectively. When a long-run relationship among the CIs is established, the ARDL model is used to estimate both the short-term and long-term coefficients. The H₀ suggests no cointegration (H₀ = δ₁ = δ₂ = δ₃ = δ₄ = δ₅ = δ₆ = 0), while the alternative hypothesis indicates cointegration (H₁ ≠ δ₁ ≠ δ₂ ≠ δ₃ ≠ δ₄ ≠ δ₅ ≠ δ₆ ≠ 0).

This hypothesis is tested using the F-statistic [32]. Cointegration is indicated if the F-value exceeds the upper bound, while no cointegration is present if it falls below the lower bound. If the F-value is between the bounds, the decision is inconclusive. A significant error correction term suggests long-term associations between the CIs.

Jordan and Philips [19] introduced a DARDL simulation approach to simplify the analysis of short- and long-run impacts in ARDL models. This method simulates and plots probabilistic shifts in the dependent CI due to changes in a single independent CI, with other regressors held constant. It requires the data to be integrated of order one and cointegrated. Using 5000 parameter vector simulations from a multivariate normal distribution, the DARDL error correction algorithm ensures a robust analysis. The ARDL bounds test for error correction follows the methodology outlined in [19] and is given in Equation (4). Equation (4) introduces the DARDL simulation, where shocks in individual CIs are simulated to project the probabilistic responses of the OT over time. This technique enhances the ARDL estimation by allowing Monte Carlo simulations and graphical scenario analyses.

Equation (4) implements the DARDL simulation model as outlined by Jordan and Philips [19].

$$\begin{array}{cc}\hfill \Delta Ln{T}_t=& {\alpha }_0+{\theta }_{0}Ln{T}_{t-1}+{\beta }_1\Delta Ln{C}_{Et}+{\theta }_{1}Ln{C}_{Et-1}+{\beta }_2\Delta Ln{B}_{Et}+{\theta }_{2}Ln{B}_{Et-1}+{\beta }_3\Delta LnI{R}_{Et}\\ & +{\theta }_{3}LnI{R}_{Et-1}+{\beta }_4\Delta LnO{R}_{Et}+{\theta }_{4}LnO{R}_{Et-1}+{\beta }_5\Delta LnS{K}_t+{\theta }_{5}LnS{K}_{t-1}+{\epsilon }_t\end{array}$$

(4)

This equation illustrates the DARDL simulation model, where β_i (i = 1 to 5) represents the long-run coefficients and θ_i (i = 1 to 5) corresponds to the short-run coefficients. The error correction term (ECT) captures the speed at which the system returns to equilibrium following a disturbance.

To ensure the robustness of the findings, several diagnostic tests were performed. These included the Breusch–Godfrey Lagrange multiplier (LM) test to detect serial correlations, the Cameron and Trivedi decomposition of the Im–Pesaran–Shin (IM) test for heteroscedasticity, skewness and kurtosis tests for normality, and the cumulative sum of recursive residuals (CUSUM) and cumulative sum of recursive residual squares (CUSUMSQ) tests to evaluate the model stability [34].

This investigation employed the wavelet approach to explore the time–frequency dependency among the selected CIs. Specifically, this research identified the temporal and frequency-based interactions between inclusive growth and its explanatory factors. To achieve this, we applied a WCA that included phase differences developed by [33,35]. This method reveals the identification of spectral properties in the time series, especially in a manner in which various periodic characteristics of the time-series change over time. Equation (5) presents the mathematical representation of the Morlet wavelet function:

$$\omega \left(t\right)={\pi }^{-{\displaystyle \frac{1}{4}}}{e}^{-j\omega t}{e}^{-{\displaystyle \frac{1}{2}}{t}^2}$$

(5)

where ω represents the frequency used for a limited time series and j is the imaginary unit.

Wavelets are characterized by two fundamental factors: time (denoted by k) and frequency (denoted by f). The CWT, using time-series data p(t), is given by Equation (6):

$${\omega }_p(k,f)=\underset{-\infty }{\overset{\infty }{{\displaystyle \int }}}p\left(t\right){\displaystyle \frac{1}{\sqrt{f}}}\omega \left({\displaystyle \frac{\overline{t-k}}{f}}\right)dt$$

(6)

The wavelet power spectrum (WPS) is used to illustrate the sensitivity of both the dependent and explanatory CIs, as demonstrated in the following expression:

$$WP{S}_p\left(k,f\right)|W_p(k,f){|}^2$$

(7)

The CWT method was used to convert the time-series CI in Equation (7) into the form presented in Equation (7). In order to explore any relationship between two time series (p(t) and q(t)) in the context of time–frequency-based causalities, we utilized the WCA. Equation (8) describes the application of the CWT to the time series:

$$Wpq\left(k,f\right)=W_p\left(k,f\right)\overline{W_q\left(k,f\right)}$$

(8)

The CWT of the OT and the BGCIs are represented by W_p(k,f) and W_q(k,f), respectively. The squared wavelet coherence is presented in Equation (9):

$$R^2\left(k,f\right)={\displaystyle \frac{{\left|S\left(f^{-1}{W}_{pq}\left(k,f\right)\right)\right|}^2}{S\left(f^{-1}{\left|W_p\left(k,f\right)\right|}^2\right)S(f^{-1}{\left|W_q\left(k,f\right)\right|}^2}}$$

(9)

A value of R²(k,f) approaching one suggests a strong connection between the time-series components, potentially indicating a causal link between the CIs at a specific frequency. This relationship is typically represented by a black contour outlined in red.

Torrence and Compo [34] introduced a technique to detect discrepancies in wavelet coherence by employing deferral indicators to analyze the fluctuations between two time series. The corresponding formulation for the wavelet coherence difference is presented in Equation (10) below:

$${\varphi }_{pq}\left(k,f\right)={\tan }^{-1}\left({\displaystyle \frac{L\left\{S\left(f^{-1}{W}_{pj}\left(k,f\right)\right)\right\}}{O\left\{S\left(f^{-1}{W}_{pj}\left(k,f\right)\right)\right\}}}\right)$$

(10)

Equations (5)–(9) are standard in the wavelet coherence literature and are based on the formulations of Goupillaud et al. [33] and Torrence and Compo [34].

These equations introduce the wavelet coherence methodology, which complements the time-domain analysis by identifying co-movement and directional dependence between the OT and CIs across various frequency bands. This dual-domain approach strengthens the robustness of the experimental framework shown in Figure 5.

We applied the FDC method by Breitung and Candelon [20] to examine the spectral Granger causality between the OT and BGCI values. This approach predicts CI values at specific time frequencies, identifying historical patterns to guide maintenance and operations. Unlike conventional tests, it decomposes causality into short-term (high-frequency) and long-term (low-frequency) components, providing insights into how factors like a fluctuating OT impact the gearbox condition over varying time horizons. For example, it reveals whether short-term temperature changes affect BGCIs differently from long-term thermal trends. The equation from [20] is expressed as follows:

$$x_t={\alpha }_1{X}_{t-1}+\dots +{\alpha }_p{X}_{t-p}+{\beta }_1{Y}_{t-1}+\dots +{\beta }_p{Y}_{t-p}+{\epsilon }_t$$

(11)

The null hypothesis of Equation (8) is M_y→x (ω) = 0. However, α and β represent the assessed parameters at time t, with a lag p, and ε_t denotes the error term.

4. Empirical Results and Discussion

4.1. Unit Root Test

Prior to analyzing the interrelationships among the CIs, it is crucial to assess their stationarity.

Table 1. Results of ADF and KPSS unit root tests.

CIs	ADF		KPSS
	Level	First Difference	Level	First Difference
LnOT	−0.460	−32.875 ***	−0.013	−23.324 ***
LnCE	−3.619 ***	−32.923 ***	−9.183 ***	−32.202 ***
LnBE	−5.725 ***	−30.881 ***	−8.252 ***	−35.617 ***
LnIRE	−0.703	−35.604 ***	−1.356	−31.267 ***
LnORE	−4.548 ***	−30.650 ***	−15.142 ***	−30.262 ***
LnSK	−3.817 ***	−34.203 ***	−10.461 ***	−30.166 ***

Note: *** p < 0.01.

presents the outcomes of the unit root tests employed in this study. The KPSS test, which assumes stationarity as the null hypothesis, and the ADF test, which assumes the presence of a unit root, were both applied. The findings revealed that most CIs—specifically, LnC_E, LnB_E, LnOR_E, and LnSK—are stationary at level, while LnIR_E and LnOT become stationary only after first differencing. Consequently, the CIs exhibited a mixed integration order, with some being I(0) and others I(1), as consistently indicated by the results of both tests.

4.2. ARDL Cointegration Test

Following the confirmation of stationarity among the CIs, this study applied the ARDL bounds testing approach to assess potential cointegration relationships among the CIs. The results of the bounds test are summarized in

Table 2. Results of estimated ARDL model for cointegration.

Estimated Model	F-Statistics
LnOTt = f(LnCEt, LnBEt, LnIREt, LnOREt, LnSKt)	5.534 *
Significance level	Lower bound	Upper bound
1%	3.09	3.86
5%	2.93	3.83
10%	2.101	3.869

Note: * p < 0.05.

. Specifically, when LnOT was treated as the dependent variable in the ARDL framework, the analysis revealed evidence of cointegration. This is supported by the fact that the computed F-statistic (5.534) exceeded both the lower and upper critical bounds, indicating a long-run relationship among the variables.

4.3. Dynamic Autoregressive Distributed Lag Simulations

The results of the DARDL simulations are presented in

Table 3. DARDL simulations results.

CIs	Coefficient	St. Error	t-Value
ECT	−0.778	0.163	−2.96 ***
LnORE	0.205	−5.287	0.000 ***
LnOREt−1	0.122	2.366	0.000 ***
LnSK	0.025	0.636	0.250
LnSKt−1	0.020	0.211	0.704
LnIRE	0.262	3.139	0.000 ***
LnIREt−1	0.292	3.766	0.000 ***
LnCE	0.374	0.712	0.010
LnCEt−1	−0.020	−0.110	0.028
LnBE	0.120	4.448	0.000 ***
LnBEt−1	0.266	4.883	0.000 ***
Cons	3.281	0.069	0.000 ***
Adj R-squared	0.7850	Root MSE	0.008
R2	0.772
Simulation	5000

Note: *** denote statistical significance at the 10% level.

.

The DARDL method can simulate, predict, and plot projections of counterfactual shifts in one regressor and their effects on the dependent CI while maintaining the stability of the model. A key advantage of the DARDL simulated model is its ability to provide instant forecasts, robust estimations, and highly accurate predictions. This approach also enables the visualization of actual changes in explanatory and dependent CIs, offering valuable insights through dynamic estimations.

In terms of computational complexity, the DARDL simulation involves 5000 iterations of a Monte Carlo process using a multivariate normal distribution. The estimated time complexity is approximately O(n × p × s), where n is the number of regressors, p is the lag order, and s is the number of simulation draws (typically 5000). The WCA has a computational complexity of O(N log N) for each pair of time series, where N is the length of the signal. These operations remain tractable and suitable for embedded HUMS platforms with appropriate data preprocessing and hardware acceleration.

Graphical illustrations of the DARDL were employed to depict the precise relationships between the explanatory CIs and their impacts on the dependent CI. The DARDL simulations illustrate the forecasted effects of changes in the actual regressors on the LnOT.

Specifically, the impacts of LnC_Et, LnB_Et, LnIR_Et, LnOR_Et, and LnSK_t on LnT were predicted under scenarios of a 10% increase and a 10% decrease in these CIs.

It is also important to consider the role of potential sensor defects or measurement errors. Although the GPMS smart accelerometers and VR sensors are STC-certified and undergo regular calibration, they are not immune to noise, drift, or minor malfunctioning. These imperfections may propagate through the signal-processing pipeline and affect the accuracy of BGCI calculations. Since our model uses log-transformed and smoothed data to reduce the noise impact, some robustness is retained. However, we acknowledge that such sensor-induced uncertainty could influence DARDL outputs, especially over long prediction horizons. Future work may include incorporating uncertainty quantification techniques, such as bootstrapping or Bayesian inference, to formally assess how sensor variability affects model confidence.

The graphical results of these simulations are presented in Figure 6, Figure 7, Figure 8, Figure 9 and Figure 10.

Figure 6 illustrates the effect of both positive and negative OR_E variations on the OT. Figure 6a shows the impact of a 10% positive change in the OR_E on the OT in both the short and long term. Figure 6b depicts the effect of a 10% negative change in the OR_E on the OT in the short and long term. Figure 7 presents the impulse response plot illustrating the nexus between the IR_E and the OT. Graph (a) demonstrates that a 10% rise in the IR_E positively affects the OT in the short and long run. Figure 8a indicates that a 10% increase in the C_E positively impacts the OT in the short and long term. Conversely, a 10% decrease in the C_E is associated with an increased OT in both the short and long run (Figure 8b). Figure 9b indicates that a 10% decrease in the SK negatively influences the OT in both the short and long run.

This result may appear counterintuitive, but suggests that a reduction in the B_E, typically indicative of less vibration or frictional excitation, may signal sensor underperformance or hidden mechanical anomalies, leading to a compensatory rise in the OT.

Figure 10 provides insight into the dynamic behavior of the OT in response to SK variation. In Figure 10a, a 10% increase in the SK led to a marginal, but delayed, rise in the OT, while Figure 10b reveals that a 10% decrease in the SK resulted in a faster and more pronounced increase in the OT. This implies that a lower SK, which typically represents less impulsiveness in the signal, could mask early-stage faults, leading to delayed detection and increased thermal buildup over time.

These findings emphasize the importance of jointly monitoring the SK and B_E in addition to traditional energy-based indicators.

Table 4. Diagnostic tests.

Diagnostic Test	X2 (p-Value)	Result
Breusch–Godfrey LM	0.16 (0.863)	No evidence of serial correlations
Breusch–Pagan–Godfrey	0.388 (0.960)	No evidence of heteroscedasticity
Ramsey RESET test	2.839 (0.213)	Model specified correctly
Jarque–Bera test	0.144 (0.930)	Residuals are normally estimated

summarizes the outcomes of various diagnostic tests performed in this study. To assess the serial correlation among the selected CIs, the LM test was applied, and the findings confirmed the absence of a serial correlation in the model. The Breusch–Pagan–Godfrey test was used to evaluate the heteroscedasticity, revealing that the model does not suffer from any heteroscedasticity issues. Furthermore, the Ramsey RESET test was conducted to verify the correctness of the DARDL model specification, and the results suggest that the model is appropriately specified. Lastly, the Jarque–Bera test confirmed that the residuals follow a normal distribution.

4.4. Wavelet Coherence Analysis

The WCA simultaneously captures the correlation and potential causality between two CIs across both the time and frequency domains [33,34,35,36,37,38,39]. In this study, we applied a WCA to examine the time-varying relationships and directional causality between the BGCIs and the lubrication OT.

Figure 11 illustrates the analysis over short (0–4 scale), medium (4–8 scale), and long (8–16 scale) periods. The strength of the association between the two time series is represented by color: blue (cool) indicates a weak or no correlation (a correlation close to zero), while red (warm) indicates a strong and significant relationship within the cone of influence. Areas outside the cone are not considered reliable for interpretation. In the wavelet coherence diagrams, arrows within the thick black contour lines indicate the direction of causality and the phase relationship between the CIs.

Figure 11a presents the WCA of the relationship between the OT and the OR_E. The presence of upward arrows indicates a positive relationship at low frequencies (short-term), which becomes stronger as the frequency increases (longer-term). This suggests that the OR_E influences the OT in the short, medium, and long run. Figure 11b provides evidence of a positive relationship between the OT and the SK, with most arrows pointing upward or upward-right, especially at high frequencies in the long run. The relationship also appears more stable in the long run compared to the short and medium term.

Figure 11c,d illustrate the positive relationship between the OT and the IR_E. In Figure 11c, the rightward-pointing arrows indicate that the IR_E influenced the OT. Similarly, Figure 11d,e show that a positive relationship exists between the OT and the C_E and B_E, respectively, particularly at higher frequencies.

4.5. Frequency-Domain Causality

This study employed the FDC test introduced by [20] to examine potential links between the OT and the BGCI values in the Bell 407 helicopter. The results of the FDC test provide valuable insights into the dynamics of the causal relationships between the CIs across different temporal frequencies. By analyzing causality across long-term, medium-term, and short-term frequencies, this method offers a more detailed understanding of interrelationships that may not be captured by traditional causality tests. The null hypothesis posits that there is no Granger causality between the independent and dependent CIs. This hypothesis can be rejected at specific frequencies: ω_i = 0→1 for the long term, ω_i = 1→2 for the medium term, and ω_i = 2→3 for the short term. The results of the FDC test are presented in Figure 12, Figure 13, Figure 14, Figure 15 and Figure 16.

In Figure 12, Figure 13, Figure 14, Figure 15 and Figure 16, the x-axis represents the frequency (ω) in Hertz [Hz] and the y-axis displays the corresponding F-statistic. The horizontal dashed line indicates the 5% critical value for significance. F-statistic values above this threshold indicate the rejection of the null hypothesis of no Granger causality from the CIs to the OT at that frequency.

Figure 12 indicates a statistically significant causal relationship between the OT and the OR_E in the long term. However, as the frequency increases, the test statistic decreases, falling below the critical values. This implies that the causal relationship from the OT to the OR_E is not significant in the short term. Figure 13 demonstrates the presence of causality in the medium and short run, confirming the interdependence between the B_E and the OT. Figure 14 illustrates that there is causality from the C_E to the OT in the short and medium terms at the 5% significance level. However, in the long term, the causal relationship is significant at the frequency range ω ∈ [0.8, 1]. Figure 15 indicates a significant causal relationship from the OT to the IR_E in the long term. Figure 16 demonstrates the presence of causality in both the medium and short run, confirming the causality relationship from the SK to the OT.

5. Conclusions and Policy Implications

This study offers valuable insights into the relationship between the OT and the BGCIs in the Bell 407 helicopter. It effectively applies the DARDL model to examine both the short- and long-term relationships between the OT and the BGCIs, improving the understanding of how changes in condition indicators influence the OT. This addresses gaps in earlier research, which mainly focused on the effects of individual CIs. In addition, the wavelet coherence method provides a time–frequency analysis that captures how the OT–BGCI relationship evolves over time and across different frequencies. This is especially useful for detecting short-term fluctuations and long-term trends that traditional time-domain models may overlook. The WCA also proved effective in identifying lead–lag relationships and structural shifts, offering deeper insight into the dynamic interactions between the OT and the BGCIs. By revealing localized time–frequency dependencies, this method is particularly well-suited for assessing how variations in the OT impact the mechanical performance under different operational conditions. Moreover, this study also explored the causality between the OT and the BGCIs using a frequency-domain approach. This method reveals that the direction of causality may vary across different frequencies, providing a more comprehensive understanding of the interactions between these CIs compared to traditional time-domain analyses.

The DARDL model results indicate that the condition indicators B_E, C_E, IR_E, and ORE have a significant positive impact on the OT in the long term. Additionally, the WCA highlights the dynamic interactions between these CIs and the OT across various time scales and frequencies. The overall WCA findings support the conclusion that the B_E, C_E, IR_E, and OR_E drive changes in the OT in both the short and long run, aligning with the DARDL estimates and confirming a positive relationship. Furthermore, the FDC analysis revealed that the OT has a significant influence on the BGCIs, with clear evidence of bidirectional causality, indicating mutual interdependence.

The study reinforces how HUMS frameworks differ from and improve upon conventional condition monitoring. Unlike threshold-based systems that generate alerts post-deviation, HUMSs continuously track mechanical and thermal health parameters like BGCIs and the OT during normal operations. By applying advanced models such as DARDL and wavelet coherence, we can move beyond descriptive monitoring to predictive diagnostics, enabling the simulation of failure scenarios and deeper causality analyses—features not achievable with traditional methods. This evolution is crucial for minimizing downtime and optimizing maintenance in aerospace applications.

This study contributes valuable insights for predictive maintenance strategies. For maintenance engineers and operational managers, this shows the importance of closely monitoring these CIs as part of predictive maintenance strategies. Elevated values in these parameters can serve as early warning signs of a rising OT, which may compromise lubrication effectiveness and lead to increased wear or the potential failure of gearbox components. Implementing real-time monitoring systems that track these CIs alongside the OT can improve the diagnostic accuracy and maintenance scheduling. Additionally, integrating these insights into maintenance protocols can help reduce unplanned downtime, extend equipment life, and enhance the overall operational safety and efficiency in aviation and similar high-performance mechanical systems [40,41,42,43,44,45].

However, the snapshot-based data acquisition method—where 2 s samples are taken every 4 min—introduces a known tradeoff. While it offers efficiency in terms of the data volume and transmission bandwidth, it may miss short-lived transient fault signatures that occur outside the sampling window. This limitation is particularly relevant in dynamic systems such as helicopters, where rapid load changes or short-duration events may provide early signs of faults. To partially mitigate this, the HUMS employs long-term repeated snapshots under steady-state conditions, and our use of a wavelet-based time–frequency analysis enhances the sensitivity to transient patterns within those windows. Nonetheless, we acknowledge that continuous monitoring or adaptive, higher-frequency snapshot intervals could further improve the system’s diagnostic resolution and responsiveness to intermittent faults.