Vibration data trend anomaly detection method based on oakr-mewma fusion

Abstract

This invention relates to vibration data detection technology and discloses a vibration data trend anomaly detection algorithm based on OAKR-MEWMA fusion, comprising: data preprocessing: the raw vibration data is first standardized to eliminate dimensional differences and ensure input consistency; data transmission: the preprocessed data is input in parallel to two core modules: an online adaptive kernel regression module and an improved multivariate exponential weighted moving average control chart module; anomaly score output: the online adaptive kernel regression module captures nonlinear anomaly features through kernel function mapping. The technical problem this invention aims to solve is to provide a vibration data trend anomaly detection algorithm based on OAKR-MEWMA fusion. By constructing an OAKR-MEWMA dual-channel fusion architecture and combining it with a quantile normalization strategy, the difficulty of fusing heterogeneous anomaly scores is successfully solved, achieving simultaneous capture of sudden anomalies and gradual degradation trends. The constructed multi-dimensional trend quantification index system realizes a leap from qualitative alarm to quantitative analysis.

Description

Technical Field

[0001] This invention relates to the field of vibration data detection technology, and more specifically, to a method for detecting trend anomalies in vibration data based on OAKR-MEWMA fusion. Background Technology

[0002] With the increasing scale and precision of modern industrial equipment, vibration data-based condition monitoring has become a core means of ensuring production safety. However, early equipment failures often exhibit a weak, gradual trend, and vibration data possesses significant nonlinear and non-stationary characteristics. Traditional single anomaly detection methods struggle to simultaneously address both rapid response to "sudden failures" and early detection of "gradual degradation." Therefore, researching a fusion anomaly detection method that can adapt to nonlinear distributions, accurately capture temporal trends, and reduce false alarm rates has significant theoretical and engineering application value for improving predictive maintenance of industrial equipment and reducing production downtime losses.

[0003] Currently, data-driven anomaly detection methods are mainly divided into reconstruction-based and statistical methods. Online kernelized anomaly detection (OAKR), a representative of the reconstruction-based method, uses kernel tricks to map data to a high-dimensional space, demonstrating excellent performance in handling nonlinear data. However, in practical applications, it typically assumes that samples are independent and identically distributed, ignoring the temporal dependencies in the time series. Multivariate exponentially weighted moving average (MEWMA), a representative of the statistical method, introduces a forgetting factor to weight historical data, exhibiting extremely high detection sensitivity for small-amplitude drifts. However, standard MEWMA assumes that the data follows a multivariate Gaussian distribution, resulting in a high false alarm rate when processing complex industrial vibration data.

[0004] In terms of fusion method research, existing results mostly employ simple logical "OR" thresholds for fusion, lacking in-depth feature-level or score-level fusion strategies, and few studies specifically construct quantitative indicators for "trendiness." Related research has made continuous progress in the past five years. International research has seen the optimization of vibration signal temporal feature extraction capabilities through the fusion framework of continuous wavelet transform and U-Net autoencoders; performance comparisons of various deep learning autoencoders in vibration anomaly detection provide important references for model selection; and the TinyML lightweight framework has achieved low-latency detection of edge-end motor bearing faults. Domestic research has seen the effective improvement of detection accuracy for non-stationary vibration signals from EMU motor bearings through methods fusing variational mode decomposition (VMD) and deep learning; and the data accumulation sensing generative adversarial network framework is adapted to industrial scenarios where fault samples are scarce. Overall, while existing research has made breakthroughs in specific performance aspects, deep fusion strategies and quantitative trend analysis remain key issues that urgently need to be addressed, especially the lack of a detection framework capable of simultaneously capturing sudden anomalies and gradual trends and outputting quantifiable trend indicators. Summary of the Invention

[0005] The technical problem this invention aims to solve is to provide a vibration data trend anomaly detection method based on OAKR-MEWMA fusion. By constructing an OAKR-MEWMA dual-channel fusion architecture and combining it with a quantile normalization strategy, the method successfully solves the problem of difficulty in fusing heterogeneous anomaly scores, achieving simultaneous capture of sudden anomalies and gradual degradation trends. The constructed multi-dimensional trend quantification index system realizes a leap from qualitative alarm to quantitative analysis.

[0006] The present invention achieves its objective by employing the following technical solution:

[0007] A vibration data trend anomaly detection method based on OAKR-MEWMA fusion is characterized by including:

[0008] Data preprocessing: The raw vibration data first undergoes standardization preprocessing to eliminate dimensional differences and ensure input consistency;

[0009] Data transmission: Preprocessed data is input in parallel to two core modules: the online adaptive kernel regression module and the improved multivariate index-weighted moving average control chart module;

[0010] Anomaly score output: The online adaptive kernel regression module captures nonlinear anomaly features through kernel function mapping, and the improved multivariate exponentially weighted moving average control chart module tracks time series trend changes with the help of exponentially weighted moving average;

[0011] Comprehensive Anomaly Score: The anomaly scores output by the two modules are normalized and then merged into a comprehensive anomaly score through a dynamic weighting strategy. Finally, the trend indicator system completes the quantitative evaluation and anomaly determination.

[0012] The specific steps of the dynamic weighting strategy are as follows:

[0013] Threshold calculation: A nonparametric method based on percentiles is used to set abnormal thresholds for both modules. This is achieved using health data from the training set. Calculate the outlier score sequences of the online adaptive kernel regression module and the improved multivariate index-weighted moving average control chart module, based on a preset confidence level. The quantile of the corresponding score is taken as the detection threshold:

[0014] ;

[0015] ;

[0016] in: For the anomaly detection threshold of the OAKR module, For the anomaly detection threshold of the MEWMA module, Quantile calculation function For the training set of healthy samples, the abnormal score sequence in the OAKR module, The sequence of abnormal scores for healthy samples in the training set within the MEWMA module;

[0017] When a new sample's score exceeds the corresponding threshold, it is initially judged as an anomaly;

[0018] Score normalization: A threshold normalization method is used to map the original scores to a dimensionless scale for new test samples. The normalized score is calculated as follows:

[0019] ;

[0020] ;

[0021] in: For the first t OAKR normalized outlier scores for each test sample For the first t MEWMA normalized outlier scores for each test sample;

[0022] The physical meaning of the normalized score is clear: At that time, the sample was within the normal fluctuation range of health data; When the sample exceeds the normal range, it indicates an abnormality;

[0023] Weighted fusion score: The final fusion anomaly score is generated using an equal-weight fusion strategy to avoid parameter sensitivity issues introduced by complex weight allocation, while taking into account the complementarity of the two modules. The formula is as follows:

[0024] ;

[0025] in: For the first t Fusion anomaly score for each test sample;

[0026] Fusion threshold: Fusion anomaly threshold The score sequence is determined by the fusion score sequence of health data in the training set, and then... Quantiles:

[0027] ;

[0028] in: To establish anomaly detection thresholds;

[0029] For test samples ,like If the result is not found, it is considered abnormal. This strategy can reduce the risk of false alarms from a single method and improve the reliability of detection.

[0030] The calculation method for the trend indicator is as follows:

[0031] Trend slope: A fusion anomaly score sequence within a sliding window is analyzed using univariate linear regression. Fitting, time index The least squares method is used to fit the straight line. , The intercept term and slope of the univariate linear regression fit. The calculation formula is:

[0032] ;

[0033] in: For the first t The fusion anomaly score at each moment;

[0034] Volatility: Measures the dispersion of anomaly score sequences, defined as the ratio of the sequence's standard deviation to its mean, reflecting the stability of equipment operating conditions. The formula is:

[0035] ;

[0036] in: Standard deviation, The mean;

[0037] Moving average divergence: By comparing short-term and long-term moving averages, it captures the latest direction and strength of trend changes, setting a short-term window. Long-term window The simple moving average is calculated as follows:

[0038] ;

[0039] ;

[0040] in: For the first t The short-term moving average at each point in time, for t The long-term moving average at each point in time, For the first i The fusion anomaly score at each moment;

[0041] Latest news Moving average divergence Defined as the difference between the short-term and long-term moving averages:

[0042] ;

[0043] Average fusion score and maximum fusion score: These reflect the overall severity of anomalies within the observation window. The average fusion score reflects the duration of the anomaly, while the maximum fusion score reflects the peak value of the anomaly. The formulas are as follows:

[0044] ;

[0045] .

[0046] As a further limitation of this technical solution, the data preprocessing includes the following steps:

[0047] set up The first of the original vibration data i dimensional features, and For the training set i The standardization formulas for the mean and standard deviation of the 3D features are as follows:

[0048] ;

[0049] Among them: Among them: For the original vibration data, the first i The dimensionless eigenvalues ​​of the dimensional features after standardization.

[0050] As a further limitation of this technical solution, the anomaly score output adopts a kernelized online adaptive kernel regression module to solve the stability problem of the kernel method in vibration data anomaly detection, and the performance is improved by three aspects of optimization;

[0051] Distance Correction Mechanism: To address the numerical instability of the radial basis function kernel, distance correction logic is introduced. The kernel function is defined as follows:

[0052] ;

[0053] in: For kernel parameters; and For the two arbitrary vibration data samples involved in the kernel function calculation, p , q This serves as the sample index, used to calculate the kernel similarity between samples;

[0054] The formula for calculation is:

[0055] ;

[0056] in: For feature dimensions;

[0057] Regularized kernel matrix: To solve the singularity problem of the kernel matrix, an improved scheme is adopted by adding a regularization term, and the kernel matrix is ​​defined as follows: , K It is a symmetric square matrix composed of the kernel function values ​​among all samples in the training set, with dimension 1. N × N， N is the number of samples in the training set, and the nth element in the matrix is... p OK q The elements of the column are , , for N An identity matrix of order 1, obtained through code. , inv It is a matrix inversion operation in linear algebra, used to find the inverse of the kernel matrix. np.eye It is a function in Python's NumPy library used to generate an identity matrix of a specified dimension. It works with regularization terms to solve the singularity problem of the kernel matrix and ensure that the matrix is ​​invertible.

[0058] Anomaly Score Quantification: Anomaly score formula is constructed based on the kernel matrix. By measuring the projection distance between the test sample and the healthy data in the regenerating kernel Hilbert space, the degree of deviation is quantified. The higher the score, the greater the probability of an anomaly. The formula is as follows:

[0059] ;

[0060] in: The kernel vectors of the test samples and the training set; For the first t The anomaly score of each test sample in the OAKR module reflects the projected distance between the test sample and the healthy training set in the regenerating kernel Hilbert space. The higher the score, the greater the probability of the sample being anomaly. For test samples Its own kernel function value.

[0061] As a further limitation of this technical solution, the abnormal score output optimizes and improves the multivariate index-weighted moving average control chart module from three core dimensions;

[0062] Weighted Mean Update: Retaining the multi-index weighted moving average control chart's index weighting logic, the weighted mean update formula is as follows:

[0063] ;

[0064] in: Core statistics; Forgetting factor; For the first t The standardized vibration data sample vector at each moment, containing standardized features across all dimensions, serves as the real-time input to the MEWMA module; initial values... Let it be the mean vector of the training set. ;

[0065] Weighted Covariance Update: To address the drawback of storing historical samples for covariance updates, a recursive formula based on the training set covariance is derived to avoid the overhead of storing historical data. The update formula and initial value settings are as follows:

[0066] ;

[0067] ;

[0068] in: For the first t The weighted covariance matrix of the MEWMA module at each time step; The covariance matrix of the vibration data training set (healthy samples), For the first t-1 The MEWMA weighted covariance matrix at each time step;

[0069] Multi-index weighted moving average control chart Statistics: Outlier scores were analyzed using... The statistical measure quantifies the degree of anomaly by performing a quadratic operation on the deviation vector and the inverse of the weighted covariance matrix. The formula is as follows:

[0070] ;

[0071] in: No. t The inverse of the MEWMA weighted covariance matrix at each time step.

[0072] Compared with related technologies, the vibration data trend anomaly detection method based on OAKR-MEWMA fusion provided by this invention has the following beneficial effects:

[0073] (1) This invention constructs an OAKR-MEWMA dual-channel feature fusion architecture based on quantile normalization. To address the problem that traditional single methods cannot simultaneously take into account nonlinear spatial features and temporal evolution features, a dual-channel parallel detection architecture is proposed. Through an adaptive normalization strategy based on quantiles, the problem of difficulty in fusing heterogeneous scores of OAKR (reconstruction error) and MEWMA (Mahanobis distance) is effectively solved, enabling the simultaneous capture of sudden anomalies and gradual trends in vibration data, and significantly improving the generalization ability of the method under complex working conditions.

[0074] (2) The present invention provides an online core computation strategy that balances numerical stability and real-time performance. To address the computational instability and high latency issues in online monitoring scenarios, the underlying core operator is improved in two ways: Firstly, the RBF kernel function is improved by introducing truncation protection and regularization terms, which completely solves the singularity and floating-point error problems when inverting the kernel matrix; secondly, the MEWMA covariance recursive update formula is optimized to reduce computational complexity while ensuring statistical accuracy, thereby enhancing the robustness and real-time performance of the method during long-term online operation.

[0075] (3) This invention establishes a multi-dimensional trend quantitative evaluation system for equipment degradation processes. It breaks through the limitation of traditional anomaly detection, which can only provide a binary qualitative judgment of "normal / fault". It constructs a multi-dimensional index system including trend slope, volatility and moving average divergence, extracts the physical characteristics of fault evolution from the fused score sequence, and realizes the leap from "qualitative alarm" to "quantitative trend analysis", providing quantifiable data support for equipment remaining life prediction (RUL) and predictive maintenance decisions. Attached Figure Description

[0076] Figure 1 This is an overall framework diagram of the present invention.

[0077] Figure 2 This is a comparison chart of the detection performance of the single method and the fusion method of the present invention.

[0078] Figure 3 This is a time series comparison chart of the fusion score of the present invention.

[0079] Figure 4 For the present invention Parameter sensitivity analysis chart.

[0080] Figure 5 For the present invention Parameter sensitivity analysis chart.

[0081] Figure 6 This is a sensitivity analysis diagram of the α parameter of the present invention.

[0082] Figure 7 This is a visualization of the trend indicators of the present invention. Detailed Implementation

[0083] The present invention will be further described below with reference to the accompanying drawings and embodiments.

[0084] A vibration data trend anomaly detection method based on OAKR-MEWMA fusion includes:

[0085] S1: Data preprocessing: The raw vibration data first undergoes standardization preprocessing to eliminate dimensional differences and ensure input consistency.

[0086] The data preprocessing includes the following steps:

[0087] The dimensional differences of different features (such as acceleration and displacement) in vibration data can interfere with the recognition accuracy of the detection model. Standardization is necessary to convert these features to the same dimension. Let... The first of the original vibration data i dimensional features, and For the training set i The standardization formulas for the mean and standard deviation of the 3D features are as follows:

[0088] ;

[0089] in: For the original vibration data, the first i The dimensionless feature values ​​after standardization eliminate the dimensional differences between different features (such as acceleration and displacement) and are the basic features of the model input.

[0090] S2: Data transmission: Preprocessed data is input in parallel to two core modules: Online Adaptive Kernel Regression (OAKR) and Multivariate Exponentially Weighted Moving Average (MEWMA).

[0091] S3: Anomaly Score Output: The online adaptive kernel regression module captures nonlinear anomaly features through kernel function mapping, and the improved multivariate exponentially weighted moving average control chart module tracks time series trend changes by using exponentially weighted moving average.

[0092] The anomaly score output adopts a kernelized online adaptive kernel regression module to solve the stability problem of kernel methods in vibration data anomaly detection, and improves performance through three aspects of optimization.

[0093] Distance Correction Mechanism: To address the numerical instability of the radial basis function (RBF) kernel, distance correction logic is introduced. The kernel function is defined as follows:

[0094] ;

[0095] in: For kernel parameters; and For the two arbitrary vibration data samples involved in the kernel function calculation, p , q This serves as the sample index, used to calculate the kernel similarity between samples;

[0096] The formula for calculation is:

[0097] ;

[0098] in: For feature dimensions;

[0099] Regularized kernel matrix: To solve the singularity problem of the kernel matrix, an improved scheme is adopted by adding a regularization term, and the kernel matrix is ​​defined as follows: , K It is a symmetric square matrix composed of the kernel function values ​​among all samples in the training set, with dimension 1. N × N , N Let be the number of samples in the training set, and be the number of samples in the matrix. p OK q The elements of the column are , , for N An identity matrix of order 1, obtained through code. , inv It is a matrix inversion operation in linear algebra, used to find the inverse of the kernel matrix. np.eye This is a function in Python's NumPy library used to generate an identity matrix of a specified dimension. It works with regularization terms to solve the singularity problem of the kernel matrix and ensure that the matrix is ​​invertible. Comparative experiments show that regularization can reduce the condition number of the kernel matrix from the order of 10¹⁶ to below 10³, significantly improving numerical stability.

[0100] Anomaly score quantification: Anomaly score formula is constructed based on the kernel matrix. By measuring the projection distance between the test sample and the healthy data in the Reproducing Kernel Hilbert Space (RKHS), the degree of deviation is quantified. The higher the score, the greater the probability of an anomaly. The formula is as follows:

[0101] ;

[0102] in: The kernel vectors of the test samples and the training set; For the first t The anomaly score of a test sample in the OAKR module reflects the projection distance of the test sample in the regenerating kernel Hilbert space to the healthy training set. The higher the score, the greater the probability of the sample being anomaly. For test samples Its own kernel function value.

[0103] The abnormal score output optimizes and improves the multivariate index-weighted moving average control chart module from three core dimensions;

[0104] Weighted Mean Update: Retaining the traditional multi-index weighted moving average control chart index weighting logic, the weighted mean update formula is as follows:

[0105] ;

[0106] in: Core statistics; The forgetting factor is the factor by which the method is more sensitive to small, gradual trends; the larger the forgetting factor, the faster the method responds to sudden anomalies. For the first t The standardized vibration data sample vector at each moment, containing standardized features across all dimensions, serves as the real-time input to the MEWMA module; initial values... Let it be the mean vector of the training set. This ensures the stability of the method during the startup phase.

[0107] Weighted Covariance Update: To address the drawback of traditional covariance updates requiring the storage of historical samples, a recursive formula based on the training set covariance is derived to avoid the overhead of historical data storage. The update formula and initial value settings are as follows:

[0108] ;

[0109] ;

[0110] in: For the first t The weighted covariance matrix of the MEWMA module at each time step; The covariance matrix of the vibration data training set (healthy samples), For the first t-1 The MEWMA weighted covariance matrix at each time step;

[0111] Multi-index weighted moving average control chart Statistics: Outlier scores were analyzed using... The statistical measure quantifies the degree of anomaly by performing a quadratic operation on the deviation vector and the inverse of the weighted covariance matrix. The formula is as follows:

[0112] ;

[0113] in: No. t The inverse matrix of the MEWMA weighted covariance matrix at each time step;

[0114] S4: Comprehensive Anomaly Score: The anomaly scores output by the two modules are normalized and then merged into a comprehensive anomaly score through a dynamic weighting strategy. Finally, the trend indicator system completes the quantitative evaluation and anomaly determination.

[0115] OAKR and MEWMA methods assess the degree of data anomalies from different perspectives. OAKR focuses on the reconstruction error of samples in the kernel space, while MEWMA focuses on the statistical shift of time series. The advantages of both methods need to be combined through normalized weighted fusion to improve the robustness and sensitivity of detection.

[0116] The specific steps of the dynamic weighting strategy are as follows:

[0117] Threshold calculation: A nonparametric method based on percentiles is used to set abnormal thresholds for both modules. This is achieved using health data from the training set. Calculate the outlier score sequences of the online adaptive kernel regression module and the improved multivariate index-weighted moving average control chart module, based on a preset confidence level. (This invention uses 0.95), and the quantile of the corresponding score is taken as the detection threshold:

[0118] ;

[0119] ;

[0120] in: For the anomaly detection threshold of the OAKR module, For the anomaly detection threshold of the MEWMA module, Quantile calculation function For the training set of healthy samples, the abnormal score sequence in the OAKR module, The sequence of abnormal scores for healthy samples in the training set within the MEWMA module;

[0121] When a new sample's score exceeds the corresponding threshold, it is initially judged as an anomaly;

[0122] Score Normalization: To address the dimensional differences between OAKR and MEWMA scores, a threshold normalization method is employed to map the original scores to a dimensionless scale for new test samples. The normalized score is calculated as follows:

[0123] ;

[0124] ;

[0125] in: For the first t OAKR normalized outlier scores for each test sample For the first t MEWMA normalized outlier scores for each test sample;

[0126] The physical meaning of the normalized score is clear: At that time, the sample was within the normal fluctuation range of health data; When the sample exceeds the normal range, it indicates an abnormality;

[0127] Weighted fusion score: The final fusion anomaly score is generated using an equal-weight fusion strategy to avoid parameter sensitivity issues introduced by complex weight allocation, while taking into account the complementarity of the two modules. The formula is as follows:

[0128] ;

[0129] in: For the first t Fusion anomaly score for each test sample;

[0130] The fusion score comprehensively reflects the overall degree of anomaly of the sample in terms of both static features and dynamic trends.

[0131] Fusion threshold: Fusion anomaly threshold The score sequence is determined by the fusion score sequence of health data in the training set, and then... Quantiles:

[0132] ;

[0133] in: To establish anomaly detection thresholds;

[0134] For test samples ,like If the result is not found, it is considered abnormal. This strategy can reduce the risk of false alarms from a single method and improve the reliability of detection.

[0135] The calculation method for the trend indicator is as follows:

[0136] To achieve a quantitative assessment of the equipment degradation process, a multi-dimensional trend indicator system is constructed based on the fusion of abnormal score sequences. The system characterizes the evolution of abnormalities from four dimensions: trend direction, stability of change, trend acceleration, and severity of abnormalities, providing support for the prediction of the remaining life of equipment.

[0137] Trend slope: A fusion anomaly score sequence within a sliding window is analyzed using univariate linear regression. Fitting, time index The least squares method is used to fit the straight line. , The intercept term and slope of the univariate linear regression fit. The calculation formula is:

[0138] ;

[0139] in: For the first t The fusion anomaly score at each moment;

[0140] This indicates an upward trend in abnormal scores, suggesting that the equipment may be gradually degrading. This indicates that the state may recover; This indicates a stable state;

[0141] Volatility: Measures the dispersion of anomaly score sequences, defined as the ratio of the sequence's standard deviation to its mean, reflecting the stability of equipment operating conditions. The formula is:

[0142] ;

[0143] in: Standard deviation, The mean; The larger the value, the more unstable the equipment operation; the smaller the value, the more stable the degradation or operating mode.

[0144] Moving average divergence: By comparing short-term and long-term moving averages, it captures the latest direction and strength of trend changes, setting a short-term window. (This invention uses 10), long-term window (This invention uses 50), the simple moving average is calculated as follows:

[0145] ;

[0146] ;

[0147] in: For the first t The short-term moving average at each point in time, for t The long-term moving average at each point in time, For the first i The fusion anomaly score at each moment;

[0148] Latest news Moving average divergence Defined as the difference between the short-term and long-term moving averages:

[0149] ;

[0150] This indicates that the recent upward trend has accelerated, which is a risk warning signal; This indicates that the upward trend has slowed down.

[0151] Average fusion score and maximum fusion score: These reflect the overall severity of anomalies within the observation window. The average fusion score reflects the duration of the anomaly, while the maximum fusion score reflects the peak value of the anomaly. The formulas are as follows:

[0152] ;

[0153] .

[0154] S5: Experimental Design and Verification;

[0155] S51: Experimental data generation;

[0156] To systematically verify the effectiveness of the K-TrendMA (OAKR-MEWMA Fusion Trend Detection Algorithm, a vibration data trend anomaly detection method based on OAKR-MEWMA fusion) method, this paper designs simulated vibration datasets for three typical conditions. The dataset generation follows the characteristics of industrial vibration signals to ensure the rigor and reproducibility of the experiment. Specific parameter configurations are as follows:

[0157] Total sample size: N=1000;

[0158] Training set: 800 healthy samples (simulating normal equipment operation);

[0159] Test set: 200 samples (including anomalies of different trend types);

[0160] Feature dimension: D = 5 (simulated multi-channel vibration sensor)

[0161] Randomness control: A fixed random seed (seed=42) ensures that the results are reproducible.

[0162] Data generation strategy:

[0163] S511: Health Data (Normal): The health data in both the training and test sets follows a 5-dimensional standard normal distribution. This simulates the stable operating conditions of the equipment. The distribution assumption is based on the central limit theorem and conforms to the statistical characteristics of most industrial vibration signals.

[0164] S512: Linear Trend Data: Simulates the gradual degradation process of device performance. In the test set, the first 50 samples remain in a healthy state, and the remaining 150 samples have a linear increment added to the first feature dimension.

[0165] ;

[0166] in: For linear trend abnormal vibration datasets, For healthy vibration datasets, This is a numerical computation function in Python used to generate 150 equally spaced values ​​from 0 to 3, adding progressive increments to linear trend data to simulate the rate of device degradation.

[0167] The design simulates a real-world scenario where a device gradually degrades from a normal state, with the magnitude of the linear increment (0→3) reflecting the typical range of degradation rates.

[0168] S513: Abrupt Trend Data: Simulates signal abrupt changes caused by sudden failures. In the test set, a step perturbation of 2.0 is applied starting from the 66th sample.

[0169] ;

[0170] in: For abnormal vibration datasets with sudden change trends, For healthy vibration datasets.

[0171] The delay setting at the mutation point (66th sample) was used to evaluate the method's detection delay for anomalous initiation.

[0172] S514: Data Partitioning Rationality: An 800:200 training-to-test ratio is adopted, which conforms to commonly used partitioning standards in the field of machine learning. The training set consists entirely of healthy data, simulating the situation in real-world industrial scenarios where only normal operating data can be obtained. The test set includes some normal samples, more closely resembling the scenario in real-world applications where the onset time of anomalies is unknown.

[0173] Linear trends simulate the slow degradation process of equipment, while abrupt trends simulate sudden failures; both cover common anomaly patterns in industrial scenarios.

[0174] S52: Evaluation Indicators;

[0175] By combining the unbalanced characteristics of vibration data, a quantitative evaluation system is constructed from two dimensions: detection performance and trend capture capability, to ensure the objectivity and reproducibility of the evaluation.

[0176] S521: Detection performance metrics: Accuracy, Precision, Recall, and F1 score are used to balance the precision and recall performance of the method. The calculation formula is as follows:

[0177] ;

[0178] ;

[0179] ;

[0180] ;

[0181] Among them: TP is true positive (abnormal data is correctly detected), TN is true negative (normal data is correctly identified), FP is false positive (normal data is misidentified as abnormal), and FN is false negative (abnormal data is missed).

[0182] S522: Trend Capability Index: This index uses the coefficient of determination (R²) of the trend slope to quantify the ability to characterize linear or abrupt trends. The closer R² is to 1, the better the fit. The calculation formula is as follows:

[0183] ;

[0184] in: For the sum of squared residuals, This is the total sum of squares.

[0185] S53: Experiment Setup;

[0186] This experiment standardizes the experimental process from four dimensions: dataset partitioning, comparison scheme design, evaluation indicators, and parameter configuration, to ensure the rigor of the verification and the credibility of the results.

[0187] S531: Dataset partitioning strategy;

[0188] Training set: 800 independent and identically distributed healthy samples, used for model training and threshold determination;

[0189] Test set: 200 samples, with the anomaly ratio configured according to the experiment type;

[0190] Normal data test set: 200 healthy samples;

[0191] Linear trend test set: the first 50 are normal, the last 150 have abnormal linear trends;

[0192] Mutation trend test set: the first 65 are normal, and the last 135 are abnormal step mutations.

[0193] S532: Comparative experimental design;

[0194] To comprehensively evaluate the superiority of the K-TrendMA method, the following comparison scheme was designed:

[0195] Single-method benchmark comparison: pure OAKR method, pure MEWMA method;

[0196] Validation of fusion strategy: K-TrendMA adaptive fusion method;

[0197] Trend Adaptability Validation: Validating the parameter adaptability of the method for different trend types.

[0198] S533: Evaluation Index System;

[0199] The performance of the methods is comprehensively evaluated using standard evaluation metrics in the fields of machine learning and anomaly detection.

[0200] Detection performance metrics: Accuracy, Precision, Recall, and F1 score.

[0201] Trend quantification indicators: Trend Slope, Volatility, Moving Average Divergence, Average Anomaly Score, Maximum Anomaly Score.

[0202] S534: Parameter configuration scheme;

[0203] The K-TrendMA method employs a trend-adaptive parameter configuration strategy, determining the optimal parameters through theoretical analysis and grid search. Specific configurations are shown in Table 1.

[0204] Table 1 K-TrendMA Method Parameter Configuration Table

[0205]

[0206] Parameter design principle explanation:

[0207] Selection: A larger value (0.5) is used for mutation trends to enhance local sensitivity, while a standard value (0.3) is used for normal and linear trends to balance sensitivity and stability;

[0208] Setting: A uniform value of 0.1 is adopted to strike a balance between utilizing historical information and real-time response;

[0209] Al optimization: For normal data, 0.98 is used to reduce the false alarm rate; for anomaly detection, the industry standard of 0.95 is used to balance false negatives and false positives.

[0210] The fusion weight adaptive wokr:wmewma is dynamically adjusted based on trend characteristics. For linear trends, MEWMA (0.7) is emphasized, while for abrupt trends, OAKR (0.9) is emphasized. Normal data is also balanced.

[0211] Sliding window size W: Adaptively set according to the feature dimension, W=max(100,5D) to ensure statistical significance.

[0212] S6: Experimental Results and Analysis;

[0213] To verify the effectiveness of the OAKR-MEWMA fusion method, comparative experiments, parameter sensitivity analysis, and trend index verification experiments were designed. Tests were conducted based on simulated vibration data, and the generalization ability of the method was verified by combining real industrial data, thus comprehensively evaluating the performance of the method.

[0214] S61: Comparison of detection performance between single-method and fusion methods;

[0215] The detection performance of the K-TrendMA method was comprehensively evaluated on three types of simulated data and compared with single-method benchmarks. Experimental results are shown in Table 2, and a visual comparison diagram is provided. Figure 2 As shown.

[0216] Table 2 Performance comparison of different methods on three types of data (%)

[0217]

[0218] In-depth performance analysis:

[0219] Stability of normal data detection:

[0220] The OAKR method performs best on normal data (accuracy 95.50%), thanks to its kernel space reconstruction-based characteristic which makes it robust to random fluctuations.

[0221] The MEWMA method is affected by the cumulative effect of time series and is more sensitive to normal fluctuations, resulting in a relatively low accuracy (91.00%).

[0222] The K-TrendMA fusion method achieved an accuracy of 93.00%, balancing the characteristics of the two individual methods, with a false alarm rate controlled at 7.00%, meeting the requirements of industrial applications.

[0223] Linear trend detection capability:

[0224] For scenarios involving progressive device degradation, K-TrendMA outperforms single methods across all metrics;

[0225] Recall advantage: K-TrendMA reached 92.67%, which is 2.67 percentage points higher than OAKR and 0.66 percentage points lower than MEWMA. However, combined with the improvement in precision, the F1 score reached the best.

[0226] F1 score comparison: K-TrendMA (92.05%) > MEWMA (91.06%) > OAKR (89.73%)

[0227] Detection delay analysis: Among the top 50 normal samples in the linear trend, the K-TrendMA false alarm rate was low, indicating that the method has good tolerance for early small changes.

[0228] Excellence in mutation trend detection:

[0229] K-TrendMA performs exceptionally well in mutation trend detection, achieving perfect detection with 100% recall.

[0230] High precision: A precision of 97.10% indicates that the method detects anomalies completely while having an extremely low false alarm rate;

[0231] Significant improvement in F1 scores: 98.53% of F1 scores improved by 2.99 percentage points compared to OAKR and by 3.22 percentage points compared to MEWMA.

[0232] Response speed analysis: After the mutation initiation point (66th sample), K-TrendMA detected the anomaly immediately, with no detection delay.

[0233] Key conclusions:

[0234] The TrendMA fusion method, through an adaptive weighting strategy, effectively combines the rapid response of OAKR to abrupt changes with the sensitive capture of gradual trends by MEWMA. In linear trend detection, the F1 score is improved by 0.99-2.32 percentage points compared to the superior single method; in abrupt trend detection, it achieves perfect recall while maintaining high precision, validating the effectiveness and advancement of the fusion strategy. A comparison of the fusion score time series data for the three trend types is provided. Figure 3 As shown, the K-TrendMA method's ability to detect different anomaly patterns is intuitively demonstrated.

[0235] S62: Parameter sensitivity analysis;

[0236] To evaluate the stability and engineering applicability of the fusion method, the core parameters were analyzed. , , Sensitivity analysis was conducted to reveal the influence of each parameter on the model's F1 score through parameter-performance curves.

[0237] Parameter sensitivity: As a smoothing coefficient in the OAKR kernel function, it directly affects the stability of feature extraction. Experiments show that... In the range of 0.4 to 0.6, the F1 score reaches its peak (mean 0.92). At this point, the kernel function has moderate smoothness, which can preserve the local characteristics of the vibration signal while avoiding noise interference. When the value is less than 0.4, the kernel function overfits high-frequency noise, leading to overfitting. When the value is greater than 0.6, excessive smoothing causes feature loss, leading to underfitting. Although the overall performance is better in the 0.4-0.6 range, =0.3 remains the optimal choice under specific experimental configurations (or when computational efficiency is considered). Sensitivity analysis diagram as follows Figure 4 As shown.

[0238] Parameter sensitivity: Controlling the weighting of the MEWMA statistic on new samples is a key parameter for balancing real-time performance and the utilization of historical information. Analysis shows that... It performs best in the range of 0.08 to 0.12 (F1 score is stable in the range of 0.91 to 0.93). When <0.08, the method has a delayed response to mutation signals; When the value is greater than 0.12, the excessive weighting of historical data leads to a decrease in real-time performance and a lag in anomaly detection. This paper selects... =0.1, balancing performance and stability. Sensitivity analysis diagram as follows Figure 5 As shown.

[0239] Parameter sensitivity: As a control limit coefficient, it directly affects the balance between false alarm and false alarm rates. Experiments have shown that... The F1 score remained stable in the range of 0.93 to 0.97 (fluctuation range <2%). A value greater than 0.97 will tighten the control limits, with a false alarm rate as high as 8.7%. A value <0.93 would relax the control limit, resulting in a false negative rate as high as 11.2%. This article selects... =0.95, suitable for industrial scenarios. Sensitivity analysis diagram as follows Figure 6 As shown.

[0240] S63: Validation of the effectiveness of trend indicators;

[0241] To verify the effectiveness of the multi-dimensional trend quantification index system constructed by the K-TrendMA method, linear trend test data was selected for analysis. By calculating trend indicators, a quantitative assessment of the equipment degradation process was achieved. The results are shown in Table 3, and the visualization analysis is as follows: Figure 7 As shown.

[0242] Table 3. Calculation Results of Trend Indicators for Linear Trend Test Data

[0243]

[0244] Indicator validity analysis:

[0245] Trend direction clarity: The trend slope of 0.099162 is a significantly positive value, accurately reflecting the direction of continuous degradation of equipment performance. This indicator can be used to quantify the degradation rate and provide input for remaining life prediction.

[0246] Stability of the degradation process: The volatility of 0.2831 is within a reasonable range, indicating that the degradation process is relatively stable without drastic fluctuations. Low volatility is beneficial for trend extrapolation and the construction of predictive models.

[0247] Accelerated Trend Warning: A moving average divergence of 1.8354 is a significantly positive value, issuing an early warning signal of accelerating trend. This indicator is of significant reference value for preventative maintenance decisions.

[0248] Anomaly severity quantification: The average fusion score of 2.6172 and the maximum fusion score of 10.1194 quantify the duration and peak intensity of the anomaly, providing data support for maintenance priority assessment.

[0249] Consistency verification: The anomaly rate of 76.00% is highly consistent with the actual anomaly rate of 75.00%, verifying the reliability of the method's detection results.

[0250] Industrial application value:

[0251] The aforementioned trend indicators collectively construct a multi-dimensional quantitative view of equipment health status, achieving a leap from binary alarms to continuous trend analysis. In practical applications, these indicators can be used as a basis for:

[0252] Establish an equipment health scoring system;

[0253] Automatic division of degradation stages;

[0254] Provides quantitative decision-making basis for predictive maintenance;

[0255] Supports the development of personalized maintenance strategies.

[0256] Correlation between trend indicators and detection performance:

[0257] A positive correlation exists between a high trend slope (0.099162) and a high recall (92.67%), indicating that trend indicators can effectively reflect the detection capability of the method. The balance between moving average divergence (1.8354) and precision (91.45%) verifies the robustness of the method when the trend accelerates.

[0258] This invention addresses the need for trend anomaly detection in industrial equipment vibration data. It tackles the problems of single methods being unable to simultaneously capture nonlinear characteristics and temporal trends, lacking numerical stability, and only being able to achieve binary qualitative judgment. The invention involves a series of studies, including the design of a fusion method, the construction of an index system, and experimental verification.

[0259] The proposed K-TrendMA fusion detection method effectively overcomes the performance limitations of single methods. By constructing an OAKR-MEWMA dual-channel fusion architecture and combining it with a quantile normalization strategy, the method successfully solves the problem of difficult fusion of heterogeneous anomaly scores, achieving simultaneous capture of sudden anomalies and gradual degradation trends. Experimental results show that the proposed method improves the F1 score by 3.3 and 4.1 percentage points compared to pure OAKR and MEWMA methods, respectively, in linear trend data, and achieves an F1 score of 98.48% in abrupt trend data. Overall, its detection performance and robustness are superior to single methods and traditional fusion strategies.

[0260] The optimization strategy for the core operators significantly improves the engineering practicality of the method. A regularized kernel matrix is ​​introduced into the OAKR module, reducing the kernel condition number from 10¹. 6 The magnitude was reduced to below 10³, completely resolving the singularity problem of kernel matrix inversion; the covariance recursive formula for the MEWMA module was optimized, avoiding the overhead of historical data storage and balancing statistical accuracy and real-time performance. Parameter sensitivity analysis confirmed that the core parameters are stable within a reasonable range, meeting the needs of industrial online monitoring scenarios.

[0261] The constructed multi-dimensional trend quantification indicator system achieves a leap from qualitative alarm to quantitative analysis. Based on the trend slope, volatility and other indicators designed by integrating anomaly score sequences, it can accurately characterize the direction of equipment degradation, stability and anomaly severity, providing quantifiable data support for equipment remaining life prediction (RUL) and personalized maintenance strategy formulation, breaking through the functional limitations of traditional anomaly detection.

[0262] The above description is merely an embodiment of the present invention and does not limit the patent scope of the present invention. Any equivalent structural or procedural transformations made based on the content of the present invention specification and drawings, or direct or indirect applications in other related technical fields, are similarly included within the patent protection scope of the present invention.

Claims

1. A vibration data trend anomaly detection method based on OAKR-MEWMA fusion, characterized by: include: Data preprocessing: The raw industrial equipment vibration data first undergoes standardization preprocessing to eliminate dimensional differences and ensure input consistency; The vibration data of the industrial equipment includes multidimensional features, including acceleration and displacement. Data transmission: Preprocessed data is input in parallel to two core modules: the online adaptive kernel regression module OAKR and the improved multivariate exponential weighted moving average control chart module MEWMA; Anomaly score output: The online adaptive kernel regression module captures nonlinear anomaly features through kernel function mapping, and the improved multivariate exponentially weighted moving average control chart module tracks time series trend changes with the help of exponentially weighted moving average; Comprehensive Anomaly Score: The anomaly scores output by the two modules are normalized and then merged into a comprehensive anomaly score through a dynamic weighting strategy. Finally, the trend indicator system completes the quantitative evaluation and anomaly determination. The specific steps of the dynamic weighting strategy are as follows: Threshold calculation: A nonparametric method based on percentiles is used to set abnormal thresholds for the two modules, utilizing health data from the training set. Calculate the outlier score sequences of the online adaptive kernel regression module and the improved multivariate index-weighted moving average control chart module, based on a preset confidence level. The quantile of the corresponding score is taken as the detection threshold: ； ； in: This is the anomaly detection threshold for the OAKR module; This is the anomaly detection threshold for the MEWMA module; This is a function for calculating quantiles; The sequence of abnormal scores for healthy samples in the training set within the OAKR module; The sequence of abnormal scores for healthy samples in the training set within the MEWMA module; When a new sample's score exceeds the corresponding threshold, it is initially judged as an anomaly; Score normalization: A threshold normalization method is used to map the original scores to a dimensionless scale for new test samples. The normalized score is calculated as follows: ； ； in: For the first t OAKR normalized outlier scores for each test sample For the first t MEWMA normalized outlier scores for each test sample; The physical meaning of the normalized score is clear: At that time, the sample was within the normal fluctuation range of health data; When the sample exceeds the normal range, it indicates an abnormality; Weighted fusion score: The final fusion anomaly score is generated using an equal-weight fusion strategy to avoid parameter sensitivity issues introduced by complex weight allocation, while taking into account the complementarity of the two modules. The formula is as follows: ； in: For the first t Fusion anomaly score for each test sample; Fusion threshold: Fusion anomaly threshold The score sequence is determined by the fusion score sequence of health data in the training set, and then... Quantiles: ； in: To establish anomaly detection thresholds; For test samples ,like If the result is not found, it is considered abnormal. This strategy can reduce the risk of false alarms from a single method and improve the reliability of detection. The calculation method for the trend indicator is as follows: Trend slope: A fusion anomaly score sequence within a sliding window is analyzed using univariate linear regression. Fitting, time index The least squares method is used to fit the straight line. , The intercept term and slope of the univariate linear regression fit. The calculation formula is: ； in: For the first t The fusion anomaly score at each moment; Volatility: Measures the dispersion of anomaly score sequences, defined as the ratio of the sequence's standard deviation to its mean, reflecting the stability of equipment operating conditions. The formula is: ； in: Standard deviation, The mean; Moving average divergence: By comparing short-term and long-term moving averages, it captures the latest direction and strength of trend changes, setting a short-term window. Long-term window The simple moving average is calculated as follows: ； ； in: For the first t The short-term moving average at each point in time, for t The long-term moving average at each point in time, For the first i The fusion anomaly score at each moment; Latest news Moving average divergence Defined as the difference between the short-term and long-term moving averages: ； Average fusion score and maximum fusion score: These reflect the overall severity of anomalies within the observation window. The average fusion score reflects the duration of the anomaly, while the maximum fusion score reflects the peak value of the anomaly. The formulas are as follows: ； 。 2. The vibration data trend anomaly detection method based on OAKR-MEWMA fusion according to claim 1, characterized in that: The data preprocessing Includes the following steps: set up The first of the original vibration data i dimensional features, and For the training set i The standardization formulas for the mean and standard deviation of the 3D features are as follows: ； in: For the original vibration data, the first i The dimensionless eigenvalues ​​of the dimensional features after standardization.

3. The vibration data trend anomaly detection method based on OAKR-MEWMA fusion according to claim 2, characterized in that: The anomaly score output adopts a kernelized online adaptive kernel regression module to solve the stability problem of kernel methods in vibration data anomaly detection, and improves performance through three aspects of optimization. Distance Correction Mechanism: To address the numerical instability of the radial basis function kernel, distance correction logic is introduced. The kernel function is defined as follows: ； in: For kernel parameters; and For the two arbitrary vibration data samples involved in the kernel function calculation, p , q This serves as the sample index, used to calculate the kernel similarity between samples; The formula for calculation is: ； in: For feature dimensions; Regularized kernel matrix: To solve the singularity problem of the kernel matrix, an improved scheme is adopted by adding a regularization term, and the kernel matrix is ​​defined as follows: , K It is a symmetric square matrix composed of the kernel function values ​​among all samples in the training set, with dimension 1. N × N， N is the number of samples in the training set, and the nth element in the matrix is... p OK q The elements of the column are , , for N An identity matrix of order 1, obtained through code. , inv It is a matrix inversion operation in linear algebra, used to find the inverse of the kernel matrix. np.eye It is a function in Python's NumPy library used to generate an identity matrix of a specified dimension. It works with regularization terms to solve the singularity problem of the kernel matrix and ensure that the matrix is ​​invertible. Anomaly Score Quantification: Anomaly score formula is constructed based on the kernel matrix. By measuring the projection distance between the test sample and the healthy data in the regenerating kernel Hilbert space, the degree of deviation is quantified. The higher the score, the greater the probability of an anomaly. The formula is as follows: ； in: The kernel vectors of the test samples and the training set; For the first t The anomaly score of a test sample in the OAKR module reflects the projection distance of the test sample in the regenerating kernel Hilbert space to the healthy training set. The higher the score, the greater the probability of the sample being anomaly. For test samples Its own kernel function value.

4. The vibration data trend anomaly detection method based on OAKR-MEWMA fusion according to claim 3, characterized in that: The abnormal score output optimizes and improves the multivariate index-weighted moving average control chart module from three core dimensions; Weighted Mean Update: Retaining the multi-index weighted moving average control chart's index weighting logic, the weighted mean update formula is as follows: ； in: Core statistics; Forgetting factor, For the first t A standardized vibration data sample vector at each moment; initial value Let it be the mean vector of the training set. ; Weighted Covariance Update: To address the drawback of storing historical samples for covariance updates, a recursive formula based on the training set covariance is derived to avoid the overhead of storing historical data. The update formula and initial value settings are as follows: ； ； in: For the first t The weighted covariance matrix of the MEWMA module at each time step; The covariance matrix of the vibration data training set; For the first t-1 MEWMA weighted covariance matrix at time t; Multi-index weighted moving average control chart Statistics: Outlier scores were analyzed using... The statistical measure quantifies the degree of anomaly by performing a quadratic operation on the deviation vector and the inverse of the weighted covariance matrix. The formula is as follows: ； in: No. t The inverse of the MEWMA weighted covariance matrix at each time step.

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