A method for predicting remaining useful life (RUL) of an aircraft system based on combined probability density

Abstract

This invention provides a method for predicting the relative safety (RUL) of an aircraft system based on combined probability density. The specific steps are as follows: In the training module, preprocessed training data is input into an FFT model, and different quantile loss functions are set. After the loss functions converge, the predicted values ​​at each quantile are obtained. In the testing module, processed test data is input into a trained QRFFT model to obtain multi-quantile RUL prediction results. These results are then used as input to a KDE (Knowledge-Defined Allocation) model, and the RUL probability density distribution (PDF) is obtained through a Gaussian kernel function and optimal bandwidth. This model combines RUL point prediction, interval prediction, and probability density prediction functions. Experiments using real aircraft flight path data are conducted, and a new probability prediction evaluation index is introduced. Comparison with existing QR models in terms of point prediction accuracy and interval prediction performance shows that this invention has higher effectiveness and superiority.

Description

Technical Field

[0001] This invention pertains to aircraft system lifetime probability prediction, and relates to an aircraft system lifetime prediction method based on combined probability density. Background Technology

[0002] Ideally, lifetime prediction models can accurately predict equipment degradation behavior. However, under various operating conditions and noise environments, multiple uncertainties in the prediction process make it difficult for the model to provide absolutely accurate values. While the proposed Remaining Useful Life (RUL) prediction model based on Feature Fusion Transformer (FFT) has achieved good point prediction results, it only provides point predictions of RUL rather than interval predictions, making it difficult to quantify uncertainties in the prediction process. Furthermore, decisions based on RUL point estimations may be prone to errors and even expose the system to risks, which has limitations in practical engineering applications. Therefore, this paper, based on the FFT model, studies a probabilistic RUL prediction model for civil aircraft systems, quantifies prediction uncertainties, and provides probability distribution estimates and confidence interval estimates of RUL to support more informed maintenance decisions. Summary of the Invention

[0003] 1. The technical problem to be solved:

[0004] How to make life probability prediction deterministic, provide probability distribution estimates and confidence interval estimates for RUL, and support more informed maintenance decisions.

[0005] 2. Technical Solution:

[0006] To address the above problems, this invention provides a method for predicting the RUL (Recovery Limit) of an aircraft system based on combined probability density, comprising the following steps:

[0007] Step S01: In the training module, the preprocessed training set data is input into the FFT model, and different quantile loss functions are set. The network parameters and features in the model are updated as the model is trained and iterated. When the loss function converges, the training ends and the predicted values ​​under different quantiles are obtained.

[0008] Step S02: In the test module, the test set data after prediction processing is sent into the trained QRFFT model to obtain the RUL prediction results under multiple quantiles and use them as input to KDE. The PDF of the RUL prediction value is obtained through the Gaussian kernel function and the optimal bandwidth. Based on the set confidence level, the corresponding upper quantile curve and lower quantile curve are selected to finally obtain the interval prediction results of RUL under the corresponding confidence level.

[0009] Furthermore, the FFT output module is fused with QR by adding quantiles after the fully connected layer to obtain the QRFFT module, and the loss function of the QRFFT model is set as quantile loss.

[0010] Furthermore, let the explanatory variables be X = [X1, X2, …, X…]. N The response variable is Y = [Y1, Y2, … , Y]. N ], where N is the number of samples, the QR model is represented as:

[0011] (1)

[0012] In the formula, τ is the quantile, τ (0,1); It is the output at time t with respect to the τ quantile; This represents the regression coefficient vector of the QR model at the τ quantile. The value is calculated by minimizing the error loss function:

[0013] (2)

[0014] In the formula, L(τ) is the error loss function. for The estimated value, Let be the absolute value function of the tilt, also known as the bouncing loss function. The expression for the bouncing loss function is:

[0015] (3)

[0016] In the formula, , For indicator functions,

[0017] Substituting equation (3) into equation (2), we get:

[0018] get:

[0019] (4).

[0020] Furthermore, the QRFFT model expression is as follows:

[0021] (5)

[0022] In the formula, and These are the weight parameters and network structure bias terms of the QRFFT model, respectively. and By minimizing the quantile loss function, according to equations (4) and (5), we can obtain:

[0023] (6)

[0024] The Adam stochastic gradient descent method is used to solve equation (6), and the equation is solved. , After solving the equation, we can enter equation (5) to obtain the quantile prediction results of RUL.

[0025] Furthermore, using the quantile function obtained from QRFFT as input to KDE, the expression for the resulting quantile function is:

[0026] (7)

[0027] In the formula, T is the number of quantiles.

[0028] Let (Z1, Z2, ..., Z) T ) is a certain estimated density function The derived independent quantile function then gives the expression for its kernel density estimator as follows:

[0029] (8)

[0030] In the formula, b is the bandwidth, which determines... Smoothness; It refers to a kernel function, which can be a Gaussian kernel, an Epanechnikov kernel, or a trigonometric kernel.

[0031] Furthermore, the Gaussian kernel function is:

[0032] (9)

[0033] Substituting equation (9) into equation (8), we get:

[0034] (10)

[0035] The kernel function is determined.

[0036] Furthermore, the optimal asymptotic choice of bandwidth b is achieved by minimizing the value of the average integral squared error:

[0037] (11)

[0038] Let f(Z) be assumed to have a variance of σ. 2 The optimal bandwidth for a normally distributed system is calculated using the following formula:

[0039] (12)

[0040] In the formula, d is the lag number and φ is the standard deviation of the sample.

[0041] 3. Beneficial effects:

[0042] This invention, based on the FFT model, combines the FFT deep learning model with QR and KDE to propose a combined probability density-based RUL prediction method for aircraft systems. The probabilistic prediction model consists of two parts: one for RUL interval prediction and the other for RUL probability density prediction. Experiments were conducted on real flight path data of aircraft systems. A new probabilistic prediction evaluation index was introduced to measure the performance of the proposed model. A comprehensive comparison was made with other QR models in terms of RUL point prediction accuracy and interval prediction capability. The verification results confirm the effectiveness and superiority of the proposed RUL probabilistic prediction model. Attached Figure Description

[0044] Figure 1 This is a flowchart of the present invention.

[0045] Figure 2 This is a schematic diagram of the ball loss function.

[0046] Figure 3 This is a schematic diagram of the gas turbine engine in the embodiment.

[0047] Figure 4 This is a schematic diagram of the general format of APU 13 message. Detailed Implementation

[0049] The present invention will now be described in detail with reference to the accompanying drawings and embodiments.

[0050] A method for predicting the relative safety (RUL) of an aircraft system based on combined probability density, such as... Figure 1 As shown, it includes the following steps: The steps involve inputting the preprocessed training data into the model in the training module, setting different quantile loss functions, updating the network parameters and features in the model as the model iterates through training, and ending training when the loss function converges, thus obtaining the optimal model parameters under different quantiles.

[0051] Step S02: In the testing module, the predicted test set data is fed into the trained QRFFT model to obtain the RUL prediction results under multiple quantiles, which are then used as inputs to the KDE. The PDF of the RUL prediction values ​​is obtained through a Gaussian kernel function and optimal bandwidth. Furthermore, based on the set confidence level, the corresponding upper and lower quantile curves are selected to finally obtain the interval prediction results of RUL at the corresponding confidence level.

[0052] In one embodiment, the FFT output module is fused with QR by adding quantiles after the fully connected layer to obtain the QRFFT module, and the loss function of the QRFFT model is set as quantile loss.

[0053] The QR (Quantitative Regression) study investigates the conditional quantile relationship between a set of explanatory variables and a response variable. It splits the data into multiple quantiles and analyzes the regression influence at different quantiles. QR is an extension of the traditional mean regression model, improving the regression model's ability to fit the data while enhancing the stability and accuracy of data analysis. Let the explanatory variables be X = [X1, X2, …, X…]. N The response variable is Y = [Y1, Y2, … , Y]. N Where N is the number of samples, the QR model can be represented as:

[0054] (1), In the formula, is the quantile, is the time step at the quantile, and is the regression coefficient vector of the model at the quantile.

[0055] In one embodiment, The value of can be estimated by minimizing the error loss function:

[0056] (2)

[0058] In the formula, L(τ) is the error loss function. for The estimated value, This is a tilted absolute value function, also known as the bouncing loss function because it resembles the motion of a bouncing ball. Figure 2 As shown. The expression for the bouncing loss function is:

[0059] (3)

[0061] In the formula, , Let (3) be the indicator function. Substituting equation (3) into equation (2), we get:

[0062] (4)

[0064] According to equation (4), different quantiles τ can yield different results. This allows us to calculate the impact of explanatory variables on response variables at different quantiles.

[0065] The QR model is highly flexible; it can be applied to other models simply by changing the loss function of other models to the bouncing loss.

[0066] The QR model restricts the relationship between explanatory and response variables to a linear one. However, the relationship between RUL and the feature parameters is not a simple linear one. This invention integrates QR with the FFT model to propose the QRFFT model, which can output prediction results under various quantile conditions.

[0067] In one embodiment, the FFT output module achieves fusion with QRFFT by adding quantiles after the fully connected layer, and sets the loss function of the QRFFT model to quantile loss. The QRFFT model expression is as follows:

[0068] (5)

[0070] In the formula, and These represent the weight parameters and network structure bias terms of the QRFFT model, respectively. and By minimizing the quantile loss function, according to equations (4) and (5), we can obtain:

[0071] (6)

[0073] The Adam stochastic gradient descent method is used to solve equation (6), and the equation is solved. , After solving the equation, we can enter equation (5) to obtain the quantile prediction results of RUL.

[0074] The kernel density function of the RUL predictor is estimated from the outputs of multiple QRFFTs, thus the probability interval can be constructed by setting multiple quantiles. Furthermore, the RUL prediction interval requires different combinations of quantiles. If the confidence level is set to 1-α, then the upper quantile is 1-α / 2, and the lower quantile is α / 2. The predicted values ​​corresponding to the upper and lower quantiles are respectively... and The prediction interval is [ For example, the 0.95 quantile curve and the 0.05 quantile curve form a 90% confidence interval. 0.95 means that the actual RUL value has a 95% probability of being below the quantile curve and a 5% probability of being above the quantile curve.

[0075] While QRFFT can obtain the predicted RUL values ​​at each quantile, it cannot obtain the PDF at the predicted points. KDE is used to obtain the PDF of RUL at different times. KDE estimates the characteristics of the overall data by analyzing a limited sample of data. Conditional quantiles are equivalent to conditional density theory.

[0076] In one embodiment, when τ belongs to (0,1), the quantile curve of Q can be regarded as a distribution function. Therefore, the quantile function obtained by QRFFT is used as the input of KDE, and a reasonable PDF is obtained through the kernel density estimator by setting an appropriate window width. The expression of the quantile function is:

[0077] (7)

[0078] In the formula, T is the number of quantiles.

[0079] Let (Z1, Z2, ..., Z) T ) is a certain estimated density function The derived independent quantile function then gives the expression for its kernel density estimator as follows:

[0080] (8)

[0082] In the formula, b is the bandwidth, which determines... Smoothness; These are kernel functions. Commonly used kernel functions include the Gaussian kernel, the Epanechnikov kernel, and the trigonometric kernel.

[0083] The Gaussian kernel function is:

[0084] (9)

[0086] When bandwidth is the optimal choice, the kernel function type has no significant impact on the density estimation result. This paper selects a smooth, continuous Gaussian kernel function for kernel density estimation. Substituting equation (9) into equation (8), we get:

[0087] (10)

[0088] For a given kernel function, the choice of bandwidth b has a significant impact on the density estimation results. Within the allowable range of the data, b should be chosen as small as possible. However, too small a b will lead to greater variability in the estimator, and the resulting noisy estimates will show certain characteristics that the density function does not possess. On the other hand, when b is too large, it will lead to greater bias in the estimator, and the resulting flat estimates will mask certain distributional characteristics of the density function.

[0089] In one embodiment, the optimal asymptotic choice of bandwidth b is achieved by minimizing the value of the Mean Integrated Squared Error (MISE):

[0090] (11)

[0092] Since the form of f(Z) is unknown, equation (21) is not easy to solve. Usually, f(Z) is assumed to have a variance of σ. 2 The optimal bandwidth for a normally distributed system can be calculated using the following formula:

[0093] (12)

[0094] In the formula, d is the lag number and φ is the standard deviation of the sample.

[0095] Example

[0096] The Auxiliary Power Unit (APU) was selected as the verification case. The APU is a complex gas turbine engine composed of components such as an electronic control box, compressor, combustion chamber, turbine, gearbox, and accessories. Its internal structure is as follows: Figure 3 As shown.

[0097] As a crucial subsystem of an aircraft, the APU (Aircraft Power Unit) primarily provides electricity and compressed air. When starting the APU, the pilot presses the start button, activating the APU's starter motor, which drives the gearbox and turbine. Once the turbine reaches a certain speed, the APU controller automatically ignites the fuel mixture, injecting it into the combustion chamber. The igniter then sets the mixture ablaze, and the resulting high-temperature, high-pressure gas further accelerates the turbine until the APU reaches its normal operating speed. Once this speed is reached, the APU generator begins supplying power, and the APU controller takes over regulation to maintain stable operation. Before takeoff, the APU powers or supplies air to start the main engines and power the cabin's air conditioning and lighting, reducing reliance on ground equipment. During takeoff and climb, the APU continues to operate, with engine power allocated to ground acceleration and climb, enhancing takeoff performance. Once the aircraft reaches a specific altitude, the APU shuts down. In the event of a main engine failure during flight, the APU can be activated in an emergency, providing power to restart the engine and improving aircraft safety and flight capability. After landing, the main engines can be shut down in advance, and the APU can be restarted to provide power for cabin lighting and air conditioning, effectively saving fuel and reducing noise pollution. However, occasional problems with the APU often lead to flight delays or cancellations. Monitoring the APU's operational status to achieve accurate RUL prediction is of great significance for improving aircraft reliability and operability.

[0098] During APU operation, data collected by onboard sensors can be transmitted in real-time to the ground base station via the ACARS system in the form of messages, enabling monitoring of the APU's status. Message 13 is the APU's start / idle report, recording the status of numerous APU performance parameters, such as exhaust temperature, start-up time, and lubricating oil temperature, reflecting information on APU anomalies, degradation, faults, and failure trends. The general format of the APU message 13 is as follows: Figure 4 As shown.

[0099] APU Message 13 contains four parts: header, APU history information, APU operating parameters during engine start-up, and APU operating parameters during auto-start. The header consists of the CC-CE segments and includes flight information, the segment from which the message originated, and environmental information. The APU history information, as shown in segment E1, includes the APU serial number, usage hours, and usage cycle number. The engine start-up parameters consist of six segments: N1-N3 and S1-S3. N1 and S1 record the operating parameters when the first APU engine is started, N2 and S2 record the operating parameters when the second engine is started, and N3 and S3 record the APU idle status after engine start-up. Segment V1 records the parameter values ​​during APU auto-start, such as start-up time and peak exhaust temperature. Detailed descriptions of the relevant APU parameters are shown in Table 1.

[0100] APU Message 13 contains four parts: header, APU history information, APU operating parameters during engine start-up, and APU operating parameters during auto-start. The header consists of the CC-CE segments and includes flight information, the segment from which the message originated, and environmental information. The APU history information, as shown in segment E1, includes the APU serial number, usage hours, and usage cycle number. The engine start-up parameters consist of six segments: N1-N3 and S1-S3. N1 and S1 record the operating parameters when the first APU engine is started, N2 and S2 record the operating parameters when the second engine is started, and N3 and S3 record the APU idle status after engine start-up. Segment V1 records the parameter values ​​during APU auto-start, such as start-up time and peak exhaust temperature. Detailed descriptions of the relevant APU parameters are shown in Table 1.

[0101] Table 1. Description of APU-related parameters

[0102]

[0103] This paper collects Message No. 13 from 12 APUs throughout their entire lifecycle from the airline as verification data, i.e., operational data from installation to removal. The number of messages is taken as the APU flight cycle count, and the specific information is shown in Table 2.

[0104] Table 2. Detailed Information on APU Validation Data

[0105]

[0106] In Table 1, the first four parameters and PFAD do not reflect the APU's performance status, and LCDT data is only displayed in the engine ACARS message, so these parameters will be removed from the modeling. ALT and TAT are environmental parameters that reflect the APU's operating environment and should be used as auxiliary parameters for RUL prediction. The three parameter segments N1 and S1, N2 and S2, and N3 and S3 are the same, only recording data under different operating conditions; only one that best reflects the APU's operating status needs to be selected. According to the APU's working principle, when the APU starts the main engine, it is in a full-load operating state with maximum power, which best reflects the APU's performance status. In addition, the parameters in segment V1 when the APU starts also reflect the APU's status. Selecting all parameters reflecting APU performance and modeling using N1, S1, V1, and environmental parameters; N2, S2, V1, and environmental parameters; and N3, S3, V1, and environmental parameters respectively, the model using N1, S1, V1, and environmental parameters shows better prediction results. Because this paper employs a sparse self-attention mechanism, the model can focus on information that contributes more and ignore irrelevant information, so that even if some parameters have little value in predicting RUL, it will not have an impact. Therefore, the environmental parameters, N1 segment, S1 segment, and V1 segment message parameters are finally selected as the APU lifetime prediction modeling parameters, as shown in Table 3.

[0107] Table 3 RUL Prediction Modeling Parameters

[0108]

[0109] The ACARS system transmits airborne sensor data to the ground in real time via an air-to-ground data link. Data loss and anomalies may occur during storage, encoding, transmission, and decoding. Furthermore, sensor data is typically measured under multi-frequency / multi-source excitation, containing significant background noise. Directly using raw monitoring data for calculations may fail to reflect the true operational status of the aircraft system, resulting in numerous false alarms and missed alarms, severely impacting the quality and efficiency of aircraft system life prediction. Before performing life prediction, raw data cleaning is necessary to improve data quality. This includes missing value imputation, outlier removal, data smoothing and standardization, RUL normalization, and sliding time window processing.

[0110] (1) Missing value imputation

[0111] Missing ACARS message data is characterized by random single value loss. To address this characteristic, the nearest neighbor mean interpolation method is used, and the calculation formula is as follows:

[0112] (34)

[0113] In the formula, t represents the missing point, and y t The result after interpolation, y t-i y t+i 2i represents the nearest neighbor value of the missing point, and 2i represents the total number of nearest neighbors.

[0114] (2) Outlier detection

[0115] Outliers in time series refer to monitoring values ​​at a specific timestamp that deviate significantly from other samples. These can be categorized as point-based or clustered anomalies. The operating environment of aircraft systems is complex and variable, and sensors are inevitably subject to interference from various factors, resulting in anomalies. To address the characteristics of anomalous ACARS message data, a density-based spatial clustering of applications with noise (DBSCAN) algorithm is employed. Outliers are identified. The DBSCAN algorithm clusters data based on point density, rather than simply the distance between points. It can identify all dense regions in the data and treat these regions as independent clusters. By setting the minimum number of points (Minpts) and the maximum radius (Eps) within a neighborhood, DBSCAN separates normal and outlier data based on the density of neighborhood samples.

[0116] (3) Data smoothing

[0117] Even after removing outliers, the data remains noisy. Kalman filtering is used to smooth the data, improving its quality and facilitating subsequent feature extraction and analysis. Kalman filtering is an estimation algorithm that uses the dynamic equations of a linear system to make an optimal estimate of the system state based on observed data. It can filter out noise and interference, hence the estimation process can also be called a filtering process. Conventional filtering methods are mostly based on frequency domain processing, while Kalman filtering is a time domain processing method. Its core is prediction and updating. Specifically, it works by taking a weighted average of the measured value at time t, the predicted value at time t-1, and the error, updating the optimal prediction result at time t, and so on, predicting the value at time t+1. The state-space model of the Kalman filter is shown below:

[0118] (13)

[0119] In the formula, Let be the state vector at time t; Let X be the state transition matrix. t-1 To X t The conversion; This is the system observation vector; Let X be the observation matrix. t To Yt The conversion; and These are model noise and observation noise, respectively. Let B... t and U t Using Gaussian white noise, the dynamic prediction formula for Kalman filtering is derived:

[0120] (14)

[0121] In the formula, It is the Kalman gain matrix. and It is to estimate the covariance matrix. and They are U t and B t The covariance matrix, It is the estimated value after filtering.

[0122] This invention selects the Kalman filter algorithm as the data smoothing algorithm and performs smoothing operation on all data that has undergone outlier detection.

[0123] (4) Data standardization

[0124] Differences in the original data on orders of magnitude can introduce errors into the prediction model. The feature parameters are normalized and reduced to [0,1].

[0125] (5) RUL normalization

[0126] The number of packets remaining in the APU is used as the absolute value of the Recovery Limit (RUL). If the number of packets remaining at time t is 100, then the RUL at time t is 100. However, due to differences in operating environments, different APUs have different lifecycles. Directly using the absolute value of RUL as the sample label may affect the convergence speed and generalization ability of the prediction model. To solve this problem, RUL is normalized, and the ratio of the current RUL to the total lifetime is used as the label to represent the degradation state. The calculation formula is as follows:

[0127] (15)

[0128] In the formula, This represents the normalized RUL, with a value between 0 and 1. t represents the total lifetime, and t represents the running time.

[0129] (6) Sliding time window

[0130] Effective sample embedding can improve the predictive performance of a model. If only data collected at a single sampling time step is used as initial input, the model cannot learn the relationship between the current degradation state and previous degradation states. Considering the continuity and causal relationships between input samples, and enabling the model to extract as much valuable information as possible from the time series, a sliding time window is used to segment the data. Data from multiple sampling time steps are concatenated sequentially using a fixed-length time window T, and the resulting whole is then input into the prediction model as an independent sample. The RUL of the last data point in the time window is used as the RUL of that window, and the time window movement step is set to 1.

[0131] The mean of the prediction results of different quantiles of each model is used as the point prediction result. Table 4 shows the RMSE, MAE and SCORE evaluation results of the four QR models and FFT models.

[0132] Table 4. Evaluation results of point prediction for each QR model

[0133]

[0134] As can be clearly observed from Table 4, on the test sets APU-4988 and APU-5408, QRFFT performed best in prediction among the four QR models. Compared with the original FFT model, the RMSE, MAE, and SCORE evaluation results of QRFFT are similar, indicating that QRFFT and FFT have comparable prediction performance. Introducing quantiles does not reduce the accuracy of the original prediction model and provides the possibility of interval prediction.

[0135] This invention proposes a probabilistic prediction model for the Recovery Under Variable (RUL) of complex aircraft systems based on QRFFT-KDE. It introduces QRFFT and KDE methods to quantify the uncertainty of RUL on top of FFT, providing more reliable prediction information for aircraft system state monitoring. QRFFT can be used to obtain RUL prediction values ​​under different quantile conditions. Using the predicted conditional quantiles as input, the KDE method is used to obtain probability density curves at different times. Experiments were conducted on real-world APU flight operation data, yielding the following conclusions:

[0136] (1) Compared with QRSVM, QRF and QRGBDT models, QRFFT model has the best performance and reliability in point prediction, and the RMSE, MAE, SCORE and ACD evaluation results are the best.

[0137] (2) The RUL intervals were predicted at 80%, 85%, 90%, and 95% confidence levels. The results showed that the predicted intervals of QRFFT could cover most of the actual RUL values, and the coverage of the predicted intervals increased with the increase of the confidence level. Compared with other models, the QRFFT model has the highest interval coverage and the narrowest interval width, which verifies the effectiveness of the interval prediction.

[0138] (3) The probability density curve of the predicted RUL value was obtained based on the KDE method. CRPS was used as the evaluation index for probability prediction. The QRFFT-KDE model had the lowest CRPS. Moreover, the actual value of RUL at different times was located near the peak of the probability density curve, indicating that the probability prediction results of the QRFFT-KDE model have high reliability.

[0139] In summary, the QRFFT-KDE model can not only provide accurate RUL point predictions, but also highly reliable RUL range predictions and probability predictions, which facilitates maintenance personnel in formulating maintenance plans.

Claims

1. A method for predicting the relative safety (RUL) of an aircraft system based on combined probability density, characterized in that: The steps include: Step S01: In the training module, the preprocessed training set data is input into the FFT model, and different quantile loss functions are set. The network parameters and features in the model are updated as the model is trained and iterated. When the loss function converges, the training ends and the predicted values ​​under different quantiles are obtained. Step S02: In the test module, the test set data after prediction processing is sent into the trained QRFFT model to obtain the RUL prediction results under multiple quantiles and use them as input to KDE. The PDF of the RUL prediction value is obtained through the Gaussian kernel function and the optimal bandwidth. Based on the set confidence level, the corresponding upper quantile curve and lower quantile curve are selected to finally obtain the interval prediction results of RUL under the corresponding confidence level.

2. The RUL prediction method for aircraft systems based on combined probability density as described in claim 1, characterized in that: The FFT output module is fused with QR by adding quantiles after the fully connected layer to obtain the QRFFT module, and the loss function of the QRFFT model is set as quantile loss.

3. The RUL prediction method for aircraft systems based on combined probability density as described in claim 2, characterized in that: [The following is a partial translation of the original text, which is not possible without further context.] The explanatory variables are X = [X1, X2, …, X…] N The response variable is Y = [Y1, Y2, … , Y]. N ], where N is the number of samples, the QR model is represented as: （1） In the formula, τ is the quantile, τ (0,1); It is the output at time t with respect to the τ quantile; This represents the regression coefficient vector of the QR model at the τ quantile. The value is calculated by minimizing the error loss function: （2） In the formula, L(τ) is the error loss function. for The estimated value, Let be the absolute value function of the tilt, also known as the bouncing loss function. The expression for the bouncing loss function is: （3） In the formula, , For indicator functions, Substituting equation (3) into equation (2), we get: get: （4）。 4. The RUL prediction method for aircraft systems based on combined probability density as described in claim 3, characterized in that: The QRFFT model expression is as follows: （5） In the formula, and These are the weight parameters and network structure bias terms of the QRFFT model, respectively. and By minimizing the quantile loss function, according to equations (4) and (5), we can obtain: （6） The Adam stochastic gradient descent method is used to solve equation (6), and the equation is solved. , After solving the equation, we can enter equation (5) to obtain the quantile prediction results of RUL.

5. The RUL prediction method for aircraft systems based on combined probability density as described in claim 4, characterized in that: Using the quantile function obtained from QRFFT as input to KDE, the expression for the resulting quantile function is: （7） In the formula, T is the number of quantiles. Let (Z1, Z2, ..., Z) T ) is a certain estimated density function The derived independent quantile function then gives the expression for its kernel density estimator as follows: （8） In the formula, b is the bandwidth, which determines... Smoothness; It refers to a kernel function, which can be a Gaussian kernel, an Epanechnikov kernel, or a trigonometric kernel.

6. The RUL prediction method for aircraft systems based on combined probability density as described in claim 5, characterized in that: The Gaussian kernel function is: （9） Substituting equation (9) into equation (8), we get: （10）， The kernel function is determined.

7. The RUL prediction method for aircraft systems based on combined probability density as described in claim 6, characterized in that: The optimal asymptotic choice of bandwidth b is achieved by minimizing the value of the average integral squared error: （11） Let f(Z) be assumed to have a variance of σ. 2 The optimal bandwidth for a normally distributed system is calculated using the following formula: （12） In the formula, d is the lag number and φ is the standard deviation of the sample.

Images