Gear remaining life prediction method based on relative density kernel estimation

Abstract

The present invention belongs to the field of big data and intelligent manufacturing technology, and the specific technical solution is: a gear remaining life prediction method based on relative density kernel estimation, and the specific steps are as follows: 1. Establishing a kernel density estimation model for estimating unknown variables; 2. Determining an adaptive relative density window width; 3. Calculating the characteristic degradation distribution of gear degradation samples; 4. Establishing a gear remaining life prediction model, constructing a probability density function for remaining life prediction, and predicting the remaining life of the gear in real time; this model can adaptively select a more accurate window width, improve the fitting degree, and effectively eliminate the boundary deviation problem caused by kernel density estimation by introducing a kernel differential homeomorphism transformation method. The real-time updated model effectively avoids repeated calculations of kernel density estimation. As the sample data increases, the prediction of the remaining life becomes closer and closer to the actual value, and the variance becomes smaller and smaller, thereby improving the accuracy and effectiveness of the prediction.

Description

Technical Field

[0001] The present invention belongs to the technical field of big data and intelligent manufacturing, and specifically relates to a gear remaining life prediction method based on relative density kernel estimation. Background Art

[0002] In recent years, as industrial equipment continues to evolve toward greater intelligence, performance, and complexity, the probability of equipment failure has increased due to the influence of internal system factors and external environmental factors. Equipment failure can lead to a certain degree of degradation in performance and health. Major equipment failures not only result in significant financial and resource losses but also pose a risk to public safety. Therefore, real-time monitoring of equipment and prediction of its remaining lifespan are of vital research value and significance.

[0003] At the current stage of research, remaining life prediction methods can be broadly categorized into those based on physical failure models, knowledge representation, and data-driven approaches. Given the complexity of equipment, obtaining physical models is extremely difficult. Knowledge representation methods are more suitable for qualitative reasoning than quantitative calculations, and they struggle to obtain complete knowledge. Therefore, data-driven approaches have become the mainstream approach in remaining life prediction research.

[0004] Si et al. summarized and analyzed data-driven methods for lifespan prediction, including gamma distribution, regression models, Wiener processes, and random filtering models. Zhai et al. proposed a remaining lifespan prediction model based on an adaptive Wiener process. Liu Wenyi et al. proposed a remaining lifespan prediction model based on a high-order hidden semi-Markov model. Most of these data-driven methods require assumptions about degradation models and parameter estimation. Parameter estimation methods have limitations in model selection and rely too heavily on a priori definitions of the probability density function form, thus failing to ensure the accuracy and applicability of the prediction models.

[0005] With the increasing complexity of industrial systems, machine learning has become a data-driven approach. Pei Hong et al. conducted a detailed analysis and summary of machine learning-based prediction methods from both shallow and deep learning perspectives. Deep learning-based prediction methods, with their powerful feature extraction capabilities and the lack of prior assumptions about unknown degradation models, are currently the mainstream approach for RLS prediction. Zhang et al. proposed using LSTM neural networks for equipment prediction; Li et al. utilized deep convolutional neural networks for life prediction; Chen et al. combined the strengths of the methods of Guo et al. and Li et al. to propose a method for constructing mechanical health indicators based on an end-to-end trainable convolutional recurrent neural network; and Zhang Jidong et al. proposed a bearing RLS prediction method based on a fully convolutional variational autoencoder network. These machine learning methods often have limitations in adapting to the complex and nonlinear characteristics of manufacturing systems and processes. Furthermore, their internal "black box" model structure cannot clearly characterize changes in system degradation characteristics. Furthermore, as the network learns, parameter adjustments cannot achieve global optimization as input increases, thus affecting the accuracy of RLS prediction.

[0006] Kernel density estimation is a nonparametric estimation method that makes no assumptions about the data distribution and instead studies the data distribution characteristics from the data itself. This method avoids the model assumptions and parameter estimation requirements of most data-driven methods, as well as the shortcomings of machine learning. Therefore, the application of kernel density estimation in remaining life prediction has attracted considerable attention and attention. Among existing kernel density estimation models, HU et al. used nonparametric kernel estimation to model wind speed and assess system reliability; Sidebe et al. proposed a two-function modeling method to calculate system reliability based on the effects of different operating environments on the degradation state of random systems; Yang Nan et al. proposed a method for modeling the probability density of wind power fluctuations based on nonparametric kernel density estimation and, to address the issue of model bandwidth selection, constructed a constrained bandwidth optimization model with goodness-of-fit tests as constraints; Li Cunhua et al. applied kernel density estimation to the construction of clustering algorithms and proposed a binned kernel estimation approximation method based on the centroid of grid data. In the aforementioned literature, the kernel estimation window width is mostly fixed. Using a fixed value as the window width results in low fit in areas with scattered sample points and high fit in dense areas.

[0007] To address the shortcomings of fixed window widths, Zhao Yuan et al. proposed a method for studying load models using nonparametric multivariate kernel density estimation, which achieves adaptive selection of the kernel estimation window width. Yan Wei et al., recognizing the importance of optimal bandwidth selection in kernel estimation, proposed an improved optimal bandwidth model that is independent of the true population distribution. To improve prediction accuracy, Zhang Weizhen et al. proposed a real-time remaining life prediction method that incorporates the integrated mean squared error method into the kernel estimation window width, achieving adaptive window width selection. These methods can adaptively select the window width and achieve real-time updates of the characteristic degradation distribution and remaining life prediction. However, the rationality of window width selection for samples with uneven data distribution and variable density needs to be improved. Furthermore, the kernel estimator of existing kernel density estimation models on the bounded support [0, +∞] is biased around 0, and a left shift of -h is introduced from the origin of the support, which reduces the accuracy of the remaining life estimation. Summary of the Invention

[0008] In order to address the technical problem of inaccurate prediction when using kernel density estimation for equipment remaining life prediction due to inaccurate window width selected in areas with uneven distribution of monitoring data and boundary deviation caused by the boundedness of random variables in the kernel estimation model, the present invention proposes a real-time remaining life prediction method based on relative density kernel estimation. The predicted remaining life is very close to the actual value, and the accuracy of the prediction is greatly improved.

[0009] To achieve the above objectives, the technical solution adopted by the present invention is: a gear remaining life prediction method based on relative density kernel estimation, the specific steps are as follows:

[0010] 1. Establish a kernel density estimation model to estimate unknown variables;

[0011] 2. Determine the adaptive relative density window width;

[0012] 3. Calculate the characteristic degradation distribution of gear degradation samples;

[0013] 4. Establish a gear remaining life prediction model;

[0014] t n The probability density function of the remaining life prediction of the system at time t is:

[0015]

[0016] Where, is the probability density of remaining life prediction, Indicates t n+t Relative density kernel estimation of nuclear diffeomorphism transformation at the moment n+t convolutions, Δx is the degenerate increment, is a diffeomorphic transformation;

[0017] After obtaining new sample data under the real-time changing system, The next moment can be calculated This enables real-time prediction of the remaining life of gears.

[0018] In step 1, the specific modeling process of the kernel density estimation model is as follows:

[0019] 1.1、Assume Δx1, Δx2, ... Δx i ,…,Δx n are n independent and identically distributed random variables, f(Δx) is its probability density function, then the kernel density estimate of f(Δx) is for:

[0020]

[0021] Where h is the window width, K(·) is the kernel function, and n is the number of samples;

[0022] The Gaussian kernel function is selected in formula (1):

[0023]

[0024] 1.2. Adaptive Window Width: Using formula (3), the minimum value of the integrated mean square error is obtained to obtain the initial optimal window width h. n :

[0025]

[0026] Where Δx is the degradation increment, is the probability density function estimated by n initial degradation increment samples, f n (Δx) is the actual probability density function of n initial degraded incremental samples;

[0027] The window width h determined by n initial samples is obtained by formula (3): n for:

[0028]

[0029] Substituting the Gaussian kernel function into equation (4) yields h n for:

[0030]

[0031] Where, f n ” (Δx) represents the n (Δx) find the second derivative, σ n is the variance of the degradation increment of the n initial sample features;

[0032] In step 2, the process of determining the adaptive relative density window width is:

[0033] The k-nearest neighbor idea is used to calculate the relative density of sample points and the adaptive window width is introduced to improve the accuracy of kernel density estimation;

[0034] 2.1. Establish relative density model:

[0035] Assume x1,x2,...,x i ,...x j ,...,x n There are n sample points and they are represented by data set A. The relative density model building process is as follows:

[0036] 2.1.1、Calculate the sample point x i With the sample point x j The Euclidean distance d(x i ,x j ):

[0037]

[0038] 2.1.2. Calculate the sample point x i k nearest neighbor distance: k_dist(x i )=d(x i ,x j ), and satisfy:

[0039] a) For any positive integer k, there is at least one sample that does not include x. i k points x′ j ∈A\{x i}, then d(x i ,x′ j )≤d(x i ,x j ), where A\{x i} means that the set A does not contain the sample point x i ;

[0040] b) The maximum number of samples that do not include x i k-1 points x′ j ∈A\{x i}, satisfying d(x i ,x′ j )<d(x i ,x j );

[0041] 2.1.3、K_dist(x i ), sample point x i The k-distance neighborhood of can be expressed as:

[0042] Nk (x i )={x′∈A\{x i}|d(x i ,x′ i )≤k_dist(x i )} (7)

[0043] Where N k (x i ) represents the sample point x i The k-distance neighborhood of x i The distance is not greater than k_dist(x i ) all samples, x′ i is called x i k nearest neighbors, d(x i ,x′ i ) is x i and x′ i The Euclidean distance of

[0044] 2.1.4. Calculate the sample point x i Relative to x j The reachable distance reach_dist(x i ,x j ):

[0045] reach_dist(x i ,x j )=max{k_dist(x j ),d(x i ,x j )} (8)

[0046] Where, d(x i ,x j ) is the sample point x i with x j The Euclidean distance between j ) is the sample point x j k nearest neighbor distance, max{·,·} means taking the maximum value;

[0047] 2.1.5. Calculate the sample point x i The local reachable density can be expressed as:

[0048]

[0049] Where, lrd(x i ) represents the sample point x i The k distance neighborhood point to x i The inverse of the average reachable distance, |N k (x i)| represents N k (x i )’s absolute value;

[0050] 2.1.6. Sample point x i The relative density ρ(x i ) can be expressed as:

[0051]

[0052] Among them, lrd(x j ) is the sample point x j The local reachable density of , by substituting formula (9) into formula (10) through the above derivation, the relative density ρ(x i ) is:

[0053]

[0054] The sample point x is selected by random sampling j ;

[0055] Relative density ρ(x i ) represents the sample point x i The local reachability density and sample point x i The sample point N in the k-distance neighborhood k (x i ) is the ratio of the average local reachable density of the i ) is closer to 1, the point x i The density of neighborhood points is relatively uniform; if ρ(x i ) is less than 1, then point x i The density of x is higher than that of its neighboring points. i is a dense point; ρ(x i ) is greater than 1, then point x i The density of x is less than the density of its neighborhood points, i is a sparse point;

[0056] 2.2. Determine the adaptive relative density window width:

[0057] The relative density ρ(x i ) is introduced into the kernel density estimation model as the window width, so as to construct the kernel density estimation expression of relative density as follows:

[0058]

[0059] Among them, ρ(x i ) is the relative density window width of kernel estimation at different sample points, and the calculation formula is shown in formula (11); K(·) is the kernel function;

[0060] 2.3. Real-time update of adaptive relative density window width:

[0061] Real-time update of kernel density estimation uses the kernel estimation of known n samples to derive the kernel density estimation of the n+1th sample. The kernel density estimation of the nth sample point is expressed as:

[0062]

[0063] The kernel density estimate of the n+1th sample point is:

[0064]

[0065] Where, ρ(x n+1 ) is the sample point at x n+1 The relative density window width at ;

[0066] From the above recursion, we can know that when any t n+j When j samples are added at a time, the kernel density estimation of n+j sample data can be recursively deduced as:

[0067]

[0068] In step 3, the feature degradation distribution is calculated as follows:

[0069] The degradation data is collected every unit time, and its degradation increment Δx1, Δx2,…, Δx is calculated. n are n independent and identically distributed degenerate incremental samples, The kernel density estimate of is expressed as:

[0070]

[0071] Where ρ(Δx i )(i=1,2,3,…,n) is the relative density window width at different sample points;

[0072] Kernel density estimation can be obtained [0,t n ]Probability density of random degradation increment Launch [0,t n ]The cumulative characteristic degradation amount x of random degradation increment on n (Let x n =nΔx) It can be obtained by n-fold convolution:

[0073]

[0074] When t n+1 When a new sample is added at any moment, [0,t n+1 The probability density function of the degradation amount of ] is as follows:

[0075]

[0076] When t n+j When j samples are added at time [0,t n+j The probability density function of the degradation amount of ] is:

[0077]

[0078] In step 4, the specific process of establishing the remaining life prediction model is as follows:

[0079] 4.1. Estimation of nuclear diffeomorphism:

[0080] Assume that samples x1, x2, ..., x n As a random variable x has an identity distribution with a probability density function f(x), we can form f n A sequence of estimators of Depends only on the first n observations. Let F be the set of probability density functions of all continuous and bounded random variables whose support range is limited to the real interval [a, b]. Then the kernel diffeomorphism estimator is It can be expressed as:

[0081]

[0082] Where K(·) is the kernel function; h n is bandwidth; is a random variable from the interval [a,b] to The differential homeomorphism of ; when x approaches the lower limit or upper limit, its first-order derivative tends to infinity, where The relationship between and x is as follows:

[0083]

[0084] 4.2. Establishing a remaining life prediction model based on nuclear diffeomorphism estimation:

[0085] Perform a logarithmic kernel diffeomorphism transformation on the degenerate increments of a bounded system:

[0086]

[0087] Assume t n is the current moment, [0,t n ] The current sample degradation data collected during the monitoring time, and the feature increment of its corresponding feature degradation over time can be used as a random variable Δx∈[0,x th ](x th is the failure threshold);

[0088] Adaptive relative density estimation of nuclear diffeomorphism for:

[0089]

[0090] K(·) is the kernel function, ρ(Δx i ) is the relative density window width;

[0091] Probability density distribution function F of remaining life T (t) is:

[0092]

[0093] g(x n+t ) is [0,t n+t ]Probability density of feature degradation amount;

[0094] By estimating the known unit time random degradation feature increment kernel differential homeomorphism, the probability density function of its feature degradation amount is obtained by convolution, and the logarithmic differential homeomorphism is transformed into [0,t n+t ]The probability density of the feature degradation amount is recorded as g d (x n+t ):

[0095]

[0096] The probability density of the remaining life prediction is:

[0097]

[0098] Where, Indicates t n+t Relative density kernel estimation of nuclear diffeomorphism transformation at the moment n+t convolutions, under the update of real-time monitoring data, t n+t The relative density kernel estimate of the differential homeomorphism transformation of the n+t sample kernel at time is:

[0099]

[0100] According to formulas (25) and (27), it can be deduced that:

[0101]

[0102] So we can deduce t n The probability density function of the remaining life prediction of the system at time t is:

[0103]

[0104] After obtaining new sample data under the real-time changing system, The next moment can be calculated This enables real-time remaining life prediction.

[0105] Compared with the prior art, the present invention has the following beneficial effects:

[0106] 1. The model of the present invention uses the relative density of sample points calculated by the k-nearest neighbor idea as the window width of kernel density estimation, which not only solves the problem of insufficient fitting caused by uneven distribution of sample data when the fixed window width is used, but also can adaptively select a more accurate window width for data sets of arbitrary shape and uneven density, thereby improving the fitting degree.

[0107] 2. The present invention introduces the method of kernel differential homeomorphism transformation, which effectively eliminates the boundary deviation problem caused by kernel density estimation.

[0108] 3. The present invention also establishes a real-time update model based on relative density kernel estimation, which effectively avoids repeated calculations of kernel density estimation and improves calculation efficiency.

[0109] 4. With the increase of sample data, the prediction of the remaining life of the present invention becomes closer and closer to the actual value, and the variance becomes smaller and smaller, thereby improving the accuracy and effectiveness of the prediction. BRIEF DESCRIPTION OF THE DRAWINGS

[0110] Figure 1 It is a flowchart of the present invention.

[0111] Figure 2 Schematic diagram of kernel density estimation.

[0112] Figure 3 Schematic diagram of kernel density estimation under fixed window width.

[0113] Figure 4 This is a graph showing the degradation of sample features over time.

[0114] Figure 5 This is the structural diagram of the gear test bench.

[0115] Figure 6 Schematic diagram of the sensor location for the main gearbox.

[0116] Figure 7 Schematic diagram of the characteristic value changing with monitoring time.

[0117] Figure 8 Schematic diagram of the relative density window width selection for the sample point at time t-40h, where "o" is the feature degradation increment sample value and "*" is the relative density window width value.

[0118] Figure 9 Schematic diagram of the relative density window width selection for the sample point at time t-70h, where "o" is the feature degradation increment sample value and "*" is the relative density window width value.

[0119] Figure 10 Schematic diagram of the relative density window width selection for the sample point at time t-77h, where "o" is the feature degradation increment sample value and "*" is the relative density window width value.

[0120] Figure 11 The predicted curve of the probability density function at different monitoring times, where “*” is the estimated value and “o” is the actual value.

[0121] Figure 12 The kernel estimation probability density comparison diagram for t=40h, where “*” is the estimated value and “o” is the actual value.

[0122] Figure 13 The kernel estimation probability density comparison diagram for t=70h, where “*” is the estimated value and “o” is the actual value.

[0123] Figure 14 This is a comparison diagram of the kernel estimation deviation at t=40h, where the left curve is the traditional kernel density estimation curve and the right curve is the kernel density estimation curve of the nuclear differential homeomorphism.

[0124] Figure 15 This is a comparison diagram of the kernel estimation deviation at t=70h, where the left curve is the traditional kernel density estimation curve, and the right curve is the kernel density estimation curve of the nuclear differential homeomorphism.

[0125] Figure 16 This is the predicted remaining life result diagram under the LSTM model when the number of cycles is 420 (10 minutes), that is, 70 hours.

[0126] Figure 17 This is the predicted remaining life result graph under the LSTM model when the number of cycles is 450 (10 minutes), that is, 75 hours.

[0127] Figure 18 The comparison diagram of the model of the present invention and the Gamma process at t=60h, "*" is the predicted value and "o" is the actual value.

[0128] Figure 19 The comparison diagram of the model of the present invention and the Gamma process at t=70h, "*" is the predicted value and "o" is the actual value.

[0129] Figure 20 This is a comparison diagram between the model of the present invention and the Wiener process model when t=60h.

[0130] Figure 21 This is a comparison diagram between the model of the present invention and the Wiener process model when t=70h.

[0131] Figure 22 This is a curve diagram showing the characteristic value changing with monitoring time.

[0132] Figure 23 The figure is a comparison chart of the probability density function predictions at different monitoring times. DETAILED DESCRIPTION

[0133] In order to make the technical problems, technical solutions and beneficial effects to be solved by the present invention more clearly understood, the present invention is further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention and are not intended to limit the present invention.

[0134] like Figure 1 As shown in the figure, the gear remaining life prediction method based on relative density kernel estimation has the following specific steps:

[0135] 1. Establish a kernel density estimation model:

[0136] The kernel density estimation method does not make any assumptions about the data distribution. It is a method used to estimate the probability density function of an unknown variable based on the data itself. It uses N known sample points to obtain N kernel functions by selecting an arbitrary kernel function (such as a Gaussian kernel function) and a window width, and then linearly superimposes them to form an estimation function of the kernel density. For example, for a sample set of:

[0137] The kernel density estimation fitting result of x={x1=-2.1,x2=-1.3,x3=-0.4,x4=1.9,x5=5.1,x6=6.2} is as follows Figure 2 shown.

[0138] Let Δx1, Δx2, … Δx i ,…,x n are n independent and identically distributed random variables, f(Δx) is its probability density function, then the kernel density estimate of f(Δx) is for:

[0139]

[0140] Where h is the window width; K(·) is the kernel function; n is the number of samples; the kernel density estimation depends on the choice of K(·) and h.

[0141] The kernel function is a factor that affects kernel density estimation. Generally, any function can be used as a kernel function. Commonly used kernel functions include the quartic kernel, uniform kernel, triangular kernel, and Gaussian kernel. The choice of kernel function has little effect on the accuracy of kernel density estimation. This paper uses the widely used Gaussian kernel function.

[0142]

[0143] The window width h is a major factor affecting the smoothness of kernel density estimation and the width of the kernel function. When h is small, the kernel density estimation curve is not smooth and tends to be tortuous, revealing more details. When the window width is large, the kernel density estimation curve is smoother but obscures more details. Therefore, choosing an appropriate window width is crucial for kernel density estimation. Existing kernel estimation window widths are categorized as fixed and adaptive. The following describes each type of window width and analyzes the issues they present.

[0144] 1. Fixed window width is to use a fixed value when estimating kernel density. This method of selecting window width generally leads to insufficient fitting of kernel estimation in low-density areas and overfitting in high-density areas. Figure 2 The shortcomings of fixed window width can be clearly analyzed.

[0145] Figure 3 The kernel density estimation diagram for four groups of sample data with different densities is shown in Figure 1. Figure 3 It can be seen that in the interval where there is insufficient data, the kernel estimate is underfitting; in the area in the dataset, the kernel estimate is overfitting.

[0146] 2. Currently, adaptive window width is the mainstream direction of window width selection. Its essence is to be able to adaptively select the window width as the number of sample points increases, thereby improving the estimation accuracy. In the current research stage, the commonly used adaptive window width method is mostly to obtain the initial optimal window width h by minimizing the integrated mean square error through formula (3). n :

[0147]

[0148] Where Δx is the degradation increment, The probability density function estimated for n initial samples, f n (Δx) is the actual probability density function of the n initial samples.

[0149] From formula (3), we can get h n for:

[0150]

[0151] Where: Substituting the Gaussian kernel function into formula (4) can obtain h n for:

[0152]

[0153] Where, f n ” (Δx) represents the n (Δx) find the second derivative, σ n is the variance of the degradation increment of the n initial sample features;

[0154] In practical applications, the adaptive window width method described above solves the over-fitting (or under-fitting) problem caused by selecting a window width that changes in real-time sample data, as well as the problem of fixed window width. This approach is optimal when the sample distribution is close to normal; however, when the actual distribution is asymmetric or multimodal, this method may lead to oversmoothing, and accuracy needs to be improved.

[0155] 2. Determine the relative density window width:

[0156] Adaptive window width selection has become a growing trend. This method can adaptively select a reasonable window width when monitoring data changes in real time, and can also address the problems of underfitting and overfitting of data caused by fixed window widths. By calculating the relative density of sample points using the k-nearest neighbor approach, the sparseness and density of each sample point relative to its surrounding sample points can be determined by the k-nearest neighbor distance range between the sample points, thereby selecting a reasonable relative density. Furthermore, the essence of relative density is to directly analyze and calculate the density of the data, which coincides with the concept of kernel density estimation and can quickly and accurately identify the center of samples in data sets of arbitrary shapes. Therefore, the relative density of sample points calculated using the k-nearest neighbor approach can be introduced into the adaptive window width to improve the accuracy of kernel density estimation.

[0157] 2.1. Establishment of relative density model:

[0158] Assume x1,x2,...,x i ,...x j ,...,x n There are n sample points and they are represented by data set A. The relative density model building process is as follows:

[0159] 2.1.1、Calculate the sample point x i With the sample point x j The Euclidean distance d(x i ,x j ):

[0160]

[0161] 2.2.2. Calculate the sample point x i The k-nearest neighbor distance k_dist(x i ):k_dist(x i )=d(x i ,x j ), and satisfy:

[0162] a) For any positive integer k, there is at least one sample that does not include x. i k points x′ j ∈A\{x i}, then d(xi ,x′ j )≤d(x i ,x j );

[0163] b) The maximum number of samples that do not include x i k-1 points x′ j ∈A\{x i}, satisfying d(x i ,x' j )<d(x i ,x j );

[0164] 2.1.3、K_dist(x i ), sample point x i The k-distance neighborhood of can be expressed as:

[0165] N k (x i )={x′ i ∈A\{x i}|d(x i ,x′ i )≤k_dist(x i )} (7)

[0166] Where N k (x i ) represents the sample point x i The k-distance neighborhood of x i The distance is not greater than k_dist(x i ) all samples, x′ i is called x i k nearest neighbors, d(x i ,x′ i ) is x i and x′ i The Ohl's distance.

[0167] 2.1.4. Calculate the sample point x i Relative to x j The reachable distance reach_dist(x i ,x j ):

[0168] reach_dist(x i ,x j )=max{k_dist(x j ),d(x i ,x j )} (8)

[0169] Where, reach_dist(xi ,x j ) represents the sample point x i Relative to x j The reachable distance; d(x i ,x j ) is the sample point x i with x j The Euclidean distance between j ) is the sample point x j The k-nearest neighbor distance of , max{·,·} indicates the maximum value.

[0170] 2.1.5. Calculate the sample point x i The local reachable density can be expressed as:

[0171]

[0172] Where, lrd(x i ) represents the sample point x i The k distance neighborhood point to x i The inverse of the average reachable distance, |N k (x i )| is the sample point x i All samples in the k-distance neighborhood of |N k (x i )| represents N k (x i )’s absolute value.

[0173] 2.1.6. Sample point x i The relative density ρ(x i ) can be expressed as:

[0174]

[0175] Among them, lrd(x j ) is the sample point x j The local reachable density of , by substituting formula (9) into formula (10) through the above derivation, the relative density ρ(x i ) is:

[0176]

[0177] The present invention adopts random sampling method to select sample points x j .

[0178] It is not difficult to see from the formula that the relative density ρ(x i ) represents the sample point x i The local reachability density and sample point x i The sample points in the k-distance neighborhood|N k(x i )| is the ratio of the average local reachable density. If ρ(x i ) is closer to 1, the point x i The density of neighborhood points is relatively uniform; if ρ(x i ) is less than 1, then point x i The density of x is higher than that of its neighboring points. i is a dense point; ρ(x i ) is greater than 1, then point x i The density of x is less than the density of its neighborhood points, i For sparse points.

[0179] 2.2 Determination of adaptive relative density window width:

[0180] The relative density of the sample points ρ(x i ) can calculate the density of the sample by calculating the distance between points. The larger the distance between points, the lower the density; the smaller the distance between points, the higher the density. i ) to select the window width, which can be based on ρ(x i ) and the value 1 to first determine the density of the sample and then select the window width, so that a large window width is selected in the low-density area and a small window width is selected in the high-density area. Therefore, the relative density ρ(x i ) is introduced into the kernel density estimation model as the window width, so as to construct the kernel density estimation expression of relative density as follows:

[0181]

[0182] Among them, ρ(x i ) is the relative density window width of kernel estimation at different sample points, and the calculation formula is shown in formula (11); K(·) is the kernel function.

[0183] 2.3 Real-time update of adaptive relative density window width

[0184] Since sample data is updated in real time in practical applications, if the calculation is restarted from scratch for each additional sample, the computational complexity will become more complex as the data increases. Therefore, in order to improve the computational performance of kernel estimation, it is essential to implement real-time updates of kernel density estimation.

[0185] Real-time update of kernel density estimation uses the kernel estimation of known n samples to derive the kernel density estimation of the n+1th sample. The derivation process is as follows:

[0186] The kernel density estimate of the nth sample point is expressed as:

[0187]

[0188] The kernel density estimate of the n+1th sample point is:

[0189]

[0190] Where, ρ(x n+1 ) is the sample point at x n+1 The relative density window width at .

[0191] From the above recursion, we can know that when any t n+j Time increase j When the sample n+j The kernel density estimation of sample data can be recursively deduced as:

[0192]

[0193] 3 Calculation of characteristic degradation distribution

[0194] Assume t n is the system degradation state time, we can get [0,t n The time-varying trend of the characteristic degradation of all degraded samples within the time period can be expressed as Figure 4 As shown. Since the degradation increment of the system is monotonically non-negative and independent and identically distributed, the degradation data is collected once every unit time and its degradation increment Δx1, Δx2,…, Δx is calculated. n are n independent and identically distributed degenerate incremental samples, The kernel density estimate of is expressed as:

[0195]

[0196] Where ρ(Δx i )(i=1,2,3,…,n) is the relative density window width at different sample points.

[0197] In order to accurately predict the remaining life of the equipment, the kernel density estimation can be used to obtain [0,t n ]Probability density of random degradation increment Launch [0,t n ]The cumulative characteristic degradation amount x of random degradation increment on n (Let x n =nΔx) It can be obtained by n-fold convolution:

[0198]

[0199] When t n+1 When a new sample is added at any moment, [0,t n+1 The probability density function of the degradation amount of ] is as follows:

[0200]

[0201] When t n+j When j samples are added at time [0,t n+j The probability density function of the degradation amount of ] is:

[0202]

[0203] 4. Remaining Life Prediction Model

[0204] By introducing relative density to establish a kernel-estimated remaining life estimation model, the smoothness and convergence of the kernel estimation are guaranteed. In order to eliminate the bounded bias problem of traditional kernel density estimation, the present invention introduces a kernel differential homeomorphism transformation method based on the kernel density estimation model of adaptive relative density, and transforms the remaining life prediction model through spatial mapping, thereby solving the bias and unboundedness of the kernel function remaining life prediction model in the boundary estimation.

[0205] 4.1 Nuclear Diffeomorphism Estimation

[0206] Assume that samples x1, x2, ..., x n As a random variable x has an identity distribution with a probability density function f(x), we can form f n A sequence of estimators of Depends only on the first n observations. Let F be the set of probability density functions of all continuous and bounded random variables whose support range is limited to the real interval [a, b]. Then the kernel diffeomorphism estimator is It can be expressed as:

[0207]

[0208] Where K(·) is the kernel function; h n is bandwidth; is a random variable from the interval [a,b] to The differential homeomorphism of ; when x approaches the lower limit or upper limit, its first-order derivative tends to infinity. Among them, The relationship between and x is as follows:

[0209]

[0210] 4.2 Remaining Lifetime Prediction Model Based on Nuclear Diffeomorphism Estimation

[0211] Assume t n is the current moment, [0,t n ] The current sample degradation data collected during the monitoring time, and the feature increment of its corresponding feature degradation over time can be used as a random variable Δx∈[0,x th ](x th is the failure threshold).

[0212] In order to eliminate the boundary offset of the kernel estimate and improve its accuracy, the logarithmic kernel differential homeomorphism transformation is performed on the degenerate increment of the bounded system:

[0213]

[0214] By transforming bounded random variables into the real number domain as described above, a differential homeomorphism kernel density estimation model is established through this spatial mapping method, which effectively reduces the problem of independent variable offset at the boundary.

[0215] From formula (22), it can be deduced that the kernel differential homeomorphism estimate of the adaptive relative density is for:

[0216]

[0217] K(·) is the kernel function, ρ(x i ) is the relative density window width.

[0218] Assume t n+t At this moment, the feature degradation reaches x th (like Figure 4 The system fails when the current t n The remaining life of the product can be predicted by the initial time to the current time t n The degradation amount x at the moment 1:n (Note x n =x(t n ), x 1:n ={x1,x2,...,x n}), launch t n+t x at the moment n+t Let T be the remaining life of the equipment, then the probability density distribution function of the remaining life F T (t) is:

[0219]

[0220] Among them, g(x n+t ) is [0,t n+t ]Probability density of feature degradation amount.

[0221] By estimating the known unit time random degradation feature increment kernel differential homeomorphism, the probability density function of its feature degradation amount is obtained by convolution, and the logarithmic differential homeomorphism is transformed into [0,t n+t ]The probability density of the feature degradation amount is recorded as g d (x n+t ):

[0222]

[0223] The probability density of the remaining life prediction is:

[0224]

[0225] In the formula Indicates t n+t Relative density kernel estimation of nuclear diffeomorphism transformation at the moment n+t convolutions, under the update of real-time monitoring data, t n+t The relative density kernel estimate of the differential homeomorphism transformation of the n+t sample kernel at time is:

[0226]

[0227] According to formulas (25) and (27), it can be deduced that:

[0228]

[0229] So we can deduce t n The probability density function of the remaining life prediction of the system at time t is:

[0230]

[0231] As real-time monitoring progresses, the number of monitored sample data continues to increase, and the kernel density estimate of the samples is also constantly updated. When using a non-real-time life prediction model, the kernel density estimate based on each new sample data must be recalculated, which will cause historical samples to be repeatedly calculated, and the amount of calculation will become larger and larger. In order to avoid the problem of repeated calculation of sample kernel density estimate in the real-time monitoring system, a recursive algorithm for real-time updating of the kernel density estimation model is proposed, thereby realizing the continuous real-time update of the feature degradation distribution and the real-time remaining life.

[0232] Therefore, after obtaining new sample data under the real-time changing system, The next moment can be calculated This enables real-time remaining life prediction.

[0233] 5. Analysis of experimental results using this prediction method:

[0234] 5.1 Gear wear test:

[0235] The model of the present invention is verified by taking the gears of the gearbox as the research object. Figure 5 As shown in the figure, the center distance is 150mm and the motor speed is 1200r / min. This experimental process is mainly aimed at monitoring the vibration signal of the acceleration sensor.

[0236] like Figure 6As shown, sensors 1# through 8# are acceleration sensors (with sensor 4# installed radially along the bearing housing); sensors 9# and 10# are acoustic sensors installed 40 cm above the main and auxiliary test chambers. The gearbox temperature is measured using sensor 11# within the main test chamber. Using a rapid measurement method, eight load levels were applied, with tooth breakage occurring at the eighth level. This experiment primarily analyzes the acceleration data recorded by sensor 4# at the eighth level. The sampling frequency is 25.6 kHz, with each level lasting 60 seconds, and data recorded every 9 minutes.

[0237] In order to better show the degradation trend, the mean square amplitude method can be used to extract features from the data to reduce the error in the subsequent prediction. The mean square amplitude of the sampled signal is obtained as:

[0238]

[0239] Where n is the number of sampling points, y j is the initial vibration signal, y i is the root mean square amplitude.

[0240] like Figure 7 As shown in the figure, this feature extraction method can well show the gearbox degradation trend.

[0241] Depend on Figure 7 It is known that at t∈[0,10]h, the gear is in the meshing stage; at t∈[10,68]h, the characteristic value gradually increases, and the gear enters normal wear; at t∈[68,77]h, the gear wear intensifies, and a tooth breaks at 77.17h. At this time, the fault threshold of the gear is y=76.325mm / s 2 .

[0242] 5.1.1 Determination of Adaptive Relative Density Window Width

[0243] The window width of kernel density estimation is selected by adaptive relative density method. Figure 8 、 Figure 9 As shown in Figure 10, the relative density window width selection of sample points at three different times t = 40h, 70h, and 77h is given respectively. Figure 8 、 Figure 9 and Figure 10 It can be seen that as the number of sample data increases, the relative density window width method can adaptively select the window width and can select the appropriate window width under different sample densities.

[0244] 5.1.2 Real-time remaining life prediction of gears

[0245] The model of the present invention is used to predict the remaining life of the gear in real time. Figure 11The figure shows a comparison of the estimated value of the remaining life estimated by the adaptive relative density kernel estimation and the actual value at different monitoring times under the real-time monitoring system. Among them, the remaining life prediction value is obtained by averaging the remaining life:

[0246]

[0247] Depend on Figure 11 Analysis shows that in the initial stage, due to the small amount of sample data, the error between the predicted value of the remaining life and the actual value is large; with the change of monitoring time and the increase of samples, the probability density of the remaining life becomes higher and narrower, the variance gradually becomes smaller, and the predicted value gradually approaches the actual value, indicating that the predicted remaining life is becoming more accurate.

[0248] To more clearly compare the probability density of remaining useful life at different times, Table 1 shows the root mean square error (RMS) between the predicted and actual values of the remaining useful life using the proposed method. The data in this table demonstrates that, as the monitoring system is updated in real time and sample data continues to increase, the error between the predicted and actual values gradually decreases, validating the feasibility of the proposed method.

[0249] Table 1 Comparison of predicted and actual remaining life

[0250]

[0251] 5.1.3 Comparison of the accuracy of three window widths based on fixed, integrated mean square error, and relative density

[0252] In a real-time monitoring system, as the data continues to change, the selected window width must be able to adaptively select a larger window width in a low-density interval and a smaller window width in a high-density interval.

[0253] Figure 12 and Figure 13 The comparison of kernel estimation probability density based on three window widths, namely fixed window width, adaptive rule of thumb window width and adaptive relative density window width, is given when the monitoring time is t=40h and t=70h respectively.

[0254] Depend on Figure 12 and Figure 13 It can be seen that the method based on adaptive relative density window width is closer to the actual value than the other two window width methods. With the change of monitoring time and the continuous increase of sample data, the error between the estimated value of remaining life under the three window widths and the actual value will gradually become smaller, and the error of the prediction result of the relative density window width method of the present invention is relatively smaller than that of other methods. From an overall analysis, the relative density window width proposed by the present invention can better fit the data during kernel estimation and can more accurately estimate the probability density function, thereby further improving the reliability and accuracy of the remaining life prediction.

[0255] 5.1.4 Boundary Shift Problem of Kernel Estimation

[0256] In the remaining life prediction, in order to eliminate the boundary offset problem of kernel estimation, the logarithmic kernel diffeomorphism transformation is performed on the bounded random variable to transform the domain into the real number domain, and then the kernel density estimation is used to perform probability density estimation, thereby improving the accuracy of the prediction. Figure 14 and Figure 15 A comparison of the deviation problem of kernel estimation between the model proposed in the present invention and the traditional kernel density estimation model at two different times t=40h and t=70h is given.

[0257] from Figure 14 and Figure 15 As can be seen from the two different moments, both reflect the boundary bias problem caused by the boundedness of traditional kernel density estimation. However, the method proposed in this paper effectively eliminates the boundary bias of the kernel estimation by transforming the domain of definition to the real number domain through kernel diffeomorphism transformation. Therefore, the model proposed in this paper can effectively solve the boundary bias problem and improve the reliability and accuracy of remaining useful life prediction.

[0258] 5.1.5 Comparison of different models

[0259] Among the current machine learning methods for predicting remaining life, the LSTM neural network method has shown good results for predicting failure time series. The Gamma process and the Wiener process are two of the most commonly used methods for modeling random process degradation and are widely used in life prediction modeling. Therefore, to verify the accuracy of the proposed model, a comparative analysis was conducted using these three methods.

[0260] (1) The LSTM neural network method is used to predict the remaining life of the gear. Figure 16 As shown, when the cycle number is 455 (10 minutes), the threshold value of 76.325 mm / s is reached. 2 , so its predicted remaining life is 455-420=35(10min)=5.83h. Figure 17 As shown, when the cycle number is 456 (10 minutes), the threshold value of 76.325 mm / s is reached. 2 Therefore, the predicted remaining life is 456-450=6(10min)=1h. The comparison with the specific results of the model of the present invention is shown in Table 2.

[0261] (2) Gear degradation is a continuous cumulative degradation process. The Gamma process is widely used in degradation modeling of gradually accumulating damage processes due to its non-negative, growing, and independent increment properties. To further test the accuracy of the proposed model, a Gamma process-based remaining life prediction method was used for comparison under the same conditions. Figure 18 and Figure 19 They are the remaining life probability density of the two models at monitoring time t=60h and t=70h respectively.

[0262] pass Figure 18 and Figure 19 Analysis shows that the variance of the probability density function of the remaining life using the proposed model is increasingly smaller than that of the Gamma process-based method, better reflecting the sample data and resulting in predictions closer to the true values. At t = 70 hours, it is clearly evident that the predicted values of the proposed method are closer to the actual values. Therefore, the proposed model is more accurate and effective in the application of remaining life prediction.

[0263] (3) The Wiener process model is often used to model equipment with non-monotonic degradation processes. Under the same conditions, the Wiener process model is used to predict the remaining life of the gear. Figure 20 and Figure 21 The comparison of the remaining life probability density of the model of the present invention and the Wiener process model when the monitoring time is t=60h and t=70h is given respectively.

[0264] Depend on Figure 20 and Figure 21 It can be seen that compared with the Wiener process model, the variance of the probability density function of the remaining life of the model of the present invention is getting smaller and smaller, the predicted value is closer to the true value, and the accuracy of the prediction is improved.

[0265] In addition, the relative error index is introduced to further quantify the prediction accuracy, and the comparison results of the four methods at five monitoring times of t = 60h, 65h, 70h, 75h, and 77h are given.

[0266] As shown in Table 2, when comparing the relative errors at different times, the remaining life prediction results of the four models gradually approach the true value. The prediction results of the model of the present invention have a smaller relative error than those of the LSTM model, the Gamma process, and the Wiener process-based method, thus verifying the superiority of the model of the present invention.

[0267] Table 2 Comparison results of relative errors at different monitoring times

[0268]

[0269] 5.2 Accelerated life test of rolling bearings

[0270] To verify the effectiveness of the proposed method, the model was validated using full-life bearing data provided by the IEEE PHM2012 database. The analysis used full-life vibration data from Bearing 1-1 under a rotational speed of 1800 rpm and a load of 4000 N. This data was derived from accelerated life testing of rolling bearings conducted on the PRONOSTIA test bench at the FEMTO-ST Research Center. The vibration signal was sampled at a frequency of 25.6 kHz, with a 0.1-second interval every 10 seconds, resulting in 2560 sample points collected.

[0271] The full life vibration data of Bearing 1-1 is processed by the mean square amplitude method, such as Figure 21 The characteristic value shown in the figure changes with the monitoring time. Figure 22 It can be seen that the root mean square shows a monotonically increasing trend over time, which can better reflect its degradation trend. The bearing is 4 s, the wear begins to intensify, and at t=2.803×10 4 The failure threshold of the root mean square is 5.607 mm / s. 2 .

[0272] The model of the present invention is used to predict the remaining life of the bearing. Figure 23 It can be seen that as the system operation time increases, the number of monitoring samples received continues to increase, and the probability density of the remaining life estimated based on the relative density kernel is continuously updated in real time. The probability density of the remaining life becomes narrower and higher, and the variance becomes smaller and smaller, indicating that the accuracy of the prediction continues to improve.

[0273] In order to further evaluate the prediction effect of the method proposed in the present invention, Table 3 shows the root mean square error comparison of the actual remaining life and the average remaining life predicted by the model of the present invention at different monitoring times.

[0274] Table 3 Comparison of predicted and actual remaining life

[0275]

[0276] As can be seen from Table 3, with the increase of monitoring time, the RMSE shows a trend of gradually decreasing, and with the increase of monitoring data, the error between the predicted remaining life and the actual life is smaller, indicating that the remaining life predicted by the model of the present invention is closer to the actual life value.

[0277] This model uses the relative density of sample points calculated using the k-nearest neighbor approach as the window width for kernel density estimation. This not only solves the problem of inadequate fitting of fixed window widths due to uneven distribution of sample data, but also adaptively selects a more accurate window width for data sets of arbitrary shape and uneven density, thereby improving the degree of fit. Regarding the boundary deviation of kernel estimation, the introduction of the kernel differential homeomorphism transformation method effectively eliminates the boundary deviation problem caused by kernel density estimation. Furthermore, as the number of samples increases, a real-time update model based on relative density kernel estimation is established to avoid repeated calculations of kernel density estimation. Example analysis shows that as the number of sample data increases, the prediction of remaining life becomes closer and closer to the actual value, the variance becomes smaller and smaller, and the accuracy of the prediction is improved. Furthermore, by comparing with the LSTM model, prediction methods based on the Wiener process, and prediction methods based on the Gamma process, the accuracy and effectiveness of the proposed model are further verified.

[0278] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions and improvements made within the spirit and principles of the present invention should be included in the scope of the present invention.

Claims

1. A gear remaining life prediction method based on relative density kernel estimation is characterized by the following specific steps: Step 1: Establish a kernel density estimation model to estimate unknown variables. The specific modeling process of the kernel density estimation model is as follows: 1.1、Assume Δx1, Δx2,…, Δx n are n independent and identically distributed degenerate incremental random variables, f(Δx) is its probability density function, then the kernel density estimate of f(Δx) is for: Where h is the window width, K(·) is the kernel function, n is the number of samples, Δx i is any received degraded incremental sample; The Gaussian kernel function is selected in formula (1): 1.

2. Adaptive Window Width: Using formula (3), the minimum value of the integrated mean square error is obtained to obtain the initial optimal window width h. n : in, Δx is the degradation increment, The probability density function estimated for n initial samples, f n (Δx) is the actual probability density function of n initial samples; The window width h determined by n initial samples is obtained by formula (3): n for: Substitute the Gaussian kernel function into equation (4) to obtain h n for: Where, f n ” (Δx) represents the n (Δx) find the second derivative, σ n is the variance of the degradation increment of the n initial sample features; Step 2: Determine the adaptive relative density window width. The specific process is as follows: The k-nearest neighbor idea is used to calculate the relative density of sample points and the adaptive window width is introduced to improve the accuracy of kernel density estimation; 2.

1. Establish relative density model: Assume x1,x2,...,x i ,...x j ,...,x n Using data set A, the process of establishing the relative density model is as follows: 2.1.1、Calculate the sample point x i With the sample point x j The Euclidean distance d(x i ,x j ): 2.1.

2. Calculate the sample point x i k nearest neighbor distance: k_dist(x i )=d(x i ,x j ), and satisfy: a) For any positive integer k, there is at least one sample that does not include x. i k points x′ j ∈A\{x i }, then d(x i ,x′ j )≤d(x i ,x j ), where A\{x i } means that the set A does not contain the sample point x i ; b) The maximum number of samples that do not include x i k-1 points x′ j ∈A\{x i }, satisfying d(x i ,x′ j )<d(x i ,x j ); 2.1.3、K_dist(x i ), sample point x i The k-distance neighborhood of is expressed as: N k (x i )={x′ i ∈A\{x i }|d(x i ,x′ i )≤k_dist(x i )} (7) Where, k_dist(x i ) is the sample point x i k nearest neighbor distance, N k (x i ) represents the sample point x i The k-distance neighborhood of x i The distance is not greater than k_dist(x i ), x′ i is called x i k nearest neighbors, d(x i ,x′ i ) is x i The Euclidean distance from x′; 2.1.

4. Calculate the sample point x i Relative to x j The reachable distance reach_dist(xi,xj): reach_dist(x i ,x j )=max{k_dist(x j ),d(x i ,x j )} (8) Where, d(x i ,x j ) is the sample point x i with x j The Euclidean distance between j ) is the sample point x j k nearest neighbor distance, max{·,·} means taking the maximum value; 2.1.

5. Calculate the sample point x i The local reachability density is expressed as: Where, lrd(x i ) represents the sample point x i The k distance neighborhood point to x i The inverse of the average reachable distance, |N k (x i )| represents N k (x i ), ∑ represents the summation sign; 2.1.

6. Sample point x i The relative density ρ(x i ) is expressed as: Among them, lrd(x j ) is the sample point x j The local reachable density of , through the above derivation, substitute formula (9) into formula (10) to derive the relative density ρ(x i ) is: The sample point x is selected by random sampling j ; Relative density ρ(x i ) represents the sample point x i The local reachability density and sample point x i The sample point N in the k-distance neighborhood k (x i ) is the ratio of the average local reachable density of the i ) is closer to 1, the point x i The density of neighborhood points is relatively uniform; if ρ(x i ) is less than 1, then point x i The density of x is higher than that of its neighboring points. i is a dense point; ρ(x i ) is greater than 1, then point x i The density of x is less than the density of its neighborhood points, i is a sparse point; 2.

2. Determine the adaptive relative density window width: The relative density ρ(x i ) is introduced into the kernel density estimation model as the window width, so as to construct the kernel density estimation expression of relative density as follows: Among them, ρ(x i ) is the relative density window width of kernel estimation at different sample points, and the calculation formula is shown in formula (11); K(·) is the kernel function; 2.

3. Real-time update of adaptive relative density window width: Real-time update of kernel density estimation uses the kernel estimation of known n samples to derive the kernel density estimation of the n+1th sample. The kernel density estimation of the nth sample point is expressed as: The kernel density estimate of the n+1th sample point is: Where, ρ(x n+1 ) is the sample point at x n+1 The relative density window width at ; Through the above recursion, we know that when any t n+j When j samples are added at a time, the kernel density estimation of n+j sample data is recursively calculated as: Step 3: Calculate the characteristic degradation distribution of the gear degradation sample; Step 4: Establish a gear remaining life prediction model; t n The probability density function of the remaining life prediction of the system at time t is: Where, is the distribution function of the remaining life prediction, Indicates t n+t Relative density kernel estimation of nuclear diffeomorphism transformation at the moment n+t convolutions, Δx is the degenerate increment, is a diffeomorphic transformation; After obtaining new sample data under the real-time changing system, Calculate the next moment This enables real-time prediction of the remaining life of gears.

2. The gear remaining life prediction method based on relative density kernel estimation according to claim 1 is characterized in that: In step 3, the feature degradation distribution is calculated as follows: The degradation data is collected every unit time, and its degradation increment Δx1, Δx2,…, Δx is calculated. i ,…Δx n For n independent and identically distributed degradation increment samples, the kernel density estimate of the probability density function of the degradation increment is Expressed as: In the formula, ρ(Δx i )(i=1,2,3,…,n) is the relative density window width at different sample points; Using [0,t n ] Random degenerate incremental sample data, and t is obtained by kernel density estimation method n Probability density of the degradation amount estimated at the moment Depend on Reconvolution gives: where x n =nΔx, when t n+1 When a new sample is added at any moment, the probability density function of the degradation amount is as follows: When t n+j When j samples are added at time [0,t n+j The probability density function of the degradation amount of ] is:

3. The gear remaining life prediction method based on relative density kernel estimation according to claim 1 is characterized in that: In step 4, the specific process of establishing the remaining life prediction model is as follows: 4.

1. Estimation of nuclear diffeomorphism: Assume that samples x1, x2, ..., x n As a random variable x, the probability density function is f(x), forming f n A sequence of estimators of Depends only on the first n observations. Let F be the set of probability density functions of all continuous and bounded random variables whose support range is limited to the real interval [a, b]. Then the kernel diffeomorphism estimator is Expressed as: Where K(·) is the kernel function; h n is bandwidth; is a random variable from the interval [a,b] to The differential homeomorphism of ; when x approaches the lower limit or upper limit, its first-order derivative tends to infinity, where The relationship between and x is as follows: in represents the field of real numbers; 4.

2. Establishing a remaining life prediction model based on nuclear diffeomorphism estimation: Perform a logarithmic kernel diffeomorphism transformation on the degenerate increments of a bounded system: Assume t n is the current moment, [0,t n ] The current sample degradation data collected during the monitoring time, and the feature increment of its corresponding feature degradation over time is used as a random variable Δx∈[0,x th ](x th is the failure threshold); Adaptive relative density estimation of nuclear diffeomorphism for: K(·) is the kernel function, ρ(Δx i ) is the relative density window width; Probability density distribution function F of remaining life T (t) is: g(x n+t ) is [0,t n+t ]Probability density of feature degradation amount; By estimating the known unit time random degradation feature increment kernel differential homeomorphism, the probability density function of its feature degradation amount is obtained by convolution, and the logarithmic differential homeomorphism is transformed into [0,t n+t ]The probability density of the feature degradation amount is recorded as g d (x n+t ): The probability density of the remaining life prediction is: Where, Indicates t n+t Relative density kernel estimation of nuclear diffeomorphism transformation at the moment n+t convolutions, under the update of real-time monitoring data, t n+t The relative density kernel estimate of the differential homeomorphism transformation of the n+t sample kernel at time is: Where * represents convolution, according to formulas (25) and (27): So we can deduce t n The probability density function of the remaining life prediction of the system at time t is: After obtaining new sample data under the real-time changing system, Calculate the next moment This enables real-time remaining life prediction.

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