Time Series Prediction Method Based on Adaptive Filtering

Abstract

The present invention discloses a time series prediction method based on adaptive filtering, which includes: S1 obtaining time series data for training the filter, and preprocessing it to obtain a signal feature vector; S2 calculating the time series prediction output at the k-th time point; S3 calculating the error between the time series prediction output and the desired output at the k-th time point; calculating a Gaussian kernel function based on the error and the Gaussian kernel variance σ; S4 calculating the gradient term at the k-th time point according to the Gaussian kernel function and the error, and then updating the weight vector at the k-th time point according to the gradient term and the learning rate; S5 determining whether the filter meets the convergence condition, if so, completing the training of the filter and entering step S6; otherwise, setting k = k + 1 and returning to step S2; S6 inputting the historical time series and the prediction period into the trained filter for prediction to obtain the predicted time series data within the prediction period.

Description

Technical Field

[0001] The present invention relates to time series prediction technology, and particularly to a time series prediction method based on adaptive filtering. Background Art

[0002] Time series prediction is a method of using historical data sequences to predict future values, which is extremely important in many fields, such as indoor positioning, equipment remaining life estimation, economic forecasting, stock market analysis, resource allocation, and weather forecasting. This predictive ability enables decision-makers to make more informed choices based on the predicted future data trends, thereby optimizing the use of resources and the deployment of strategies.

[0003] In time series prediction, traditional adaptive filtering methods widely use the mean square error as a cost function to adjust the prediction model in an attempt to minimize the prediction error. These methods are effective in dealing with Gaussian noise and linear problems, but when faced with non-Gaussian or non-linear complex data structures, they often struggle to capture the high-order statistical characteristics in the data, thus affecting the prediction accuracy and the adaptability of the model.

[0004] Although existing kernel methods improve the model's handling of non-Gaussian signals by introducing non-linear kernel functions, there are still many limitations, such as slow convergence speed and poor performance on non-convex performance surfaces. These limitations have prompted researchers to explore new adaptive filtering algorithms, such as methods based on kernel-power mean error, which improve the robustness and convergence speed of the algorithm in non-Gaussian noise environments through more complex cost function designs. Nevertheless, existing methods are still insufficient when faced with extreme non-Gaussian noise or the need for rapid response in resource-constrained environments.

[0005] To further improve the performance of time series prediction and solve the above problems, it is necessary to develop a suitable extension of the KMPE criterion (Kernel Mean Square p-th Power Error criterion) to enable it to better cope with the interference of non-Gaussian signals and meet higher requirements for convergence speed and accuracy. Summary of the Invention

[0006] Aiming at the above deficiencies in the prior art, the time series prediction method based on adaptive filtering provided by the present invention solves the problem that it is difficult to achieve rapid response when the time series is faced with extreme non-Gaussian noise or in a resource-constrained environment during the filtering process.

[0007] To achieve the above invention purpose, the technical solution adopted by the present invention is as follows:

[0008] Provide a time series prediction method based on adaptive filtering, which includes the steps:

[0009] S1. Obtain the time series data for training the filter, and preprocess it to obtain the signal feature vector; set the learning step size of the filter, the Gaussian kernel variance σ, and the parameters p and q of the weight vector.

[0010] S2. Calculate the time series prediction output at the k-th time point according to the signal feature vector at the k-th time point and the weight vector at the (k - 1)-th time point.

[0011] S3. Calculate the error between the time series prediction output at the k-th time point and the expected output; calculate the Gaussian kernel function based on the error and the Gaussian kernel variance σ.

[0012] S4. Calculate the gradient term at the k-th time point according to the Gaussian kernel function and the error, and then update the weight vector at the k-th time point according to the gradient term and the learning rate:

[0013] ,

[0014] where and are the weight vectors at the k-th and (k - 1)-th time points respectively, k ≥ 2, is the expected output at the first time point; η is the learning step size; and are the gradient term and the error at the k-th time point respectively; is the signal feature vector at the k-th time point, is the time series at the k-th time point;

[0015] S5. Determine whether the filter meets the convergence condition. If so, complete the training of the filter and enter step S6; otherwise, let k = k + 1 and return to step S2.

[0016] S6. Input the historical time series and the prediction period into the trained filter for prediction to obtain the predicted time series data within the prediction period.

[0017] Furthermore, between step S1 and step S2, it also includes calculating the representation of the signal feature vector in the finite-dimensional reproducing kernel Hilbert space to update the signal feature vector:

[0018] A1. Randomly select m sample points from the signal feature vector to form a dictionary D; calculate the Gaussian kernel function of the selected sample points to construct the kernel Gram matrix.

[0019] A2. Perform eigenvalue decomposition on the kernel Gram matrix to obtain the eigenvalues and the eigenvector matrix ,and use the eigenvalue and the eigenvector matrix Construct a mapping matrix :

[0020]

[0021] wherein, is transpose;

[0022] A3. Calculate the Gaussian kernel function values of the data at each time point k in the signal feature vector and all sample points in the dictionary D to form a vector at each time point k ;

[0023] A4. According to the vector and the mapping matrix , update the signal feature vector .

[0024] Furthermore, the expression for calculating the time series prediction output at the k-th time point is:

[0025]

[0026] wherein, is the time series prediction output at the k-th time point; is the transpose of a(k - 1);

[0027] The expression of the Gaussian kernel function is:

[0028]

[0029] The expression of the gradient term is:

[0030]

[0031] wherein, is the Gaussian kernel at the k-th time point; is the exponential function with the natural logarithm base .

[0032] Furthermore, the convergence condition is:

[0033]

[0034] wherein, is the expectation operator; is the square of the norm of the vector; is the prior error at the k-th time point.

[0035] Furthermore, the condition that the learning step size should satisfy is:

[0036]

[0037] Among them, is the expectation operator; is the square of the norm of the vector; is the prior error at the k-th time point; is the square of the error at the k-th time point; is the Gaussian kernel at the k-th time point.

[0038] Furthermore, the expression of the prior error is:

[0039] ,

[0040] Among them, is the error vector of a(k - 1); is the optimal weight vector.

[0041] Furthermore, the time series is the RSSI value collected for indoor positioning or the vibration data collected for estimating the remaining life of the bearing;

[0042] When the time series is the RSSI value, the preprocessing thereof includes eliminating the RSSI values greater than the preset error in the RSSI values and performing a normalization operation on the remaining RSSI values;

[0043] When the time series is the vibration data, the preprocessing thereof is to extract features from the vibration data.

[0044] Furthermore, the time series prediction method based on adaptive filtering further includes evaluating the performance of the filter:

[0045] When obtaining the time series prediction output at each time point, calculate the excess mean square error corresponding to the current time point:

[0046]

[0047] Among them, L is the excess mean square error; is the expectation operator; is the trace of the covariance matrix of; and are the first derivative and the second derivative respectively; is the error the expected value of; is the absolute value symbol; is the given prior parameter;

[0048] Plot the time series prediction outputs at all time points in chronological order to form an excess mean square curve;

[0049] Determine whether the excess mean square curve is a decreasing curve. If so, the performance of the filter meets the requirements; otherwise, the performance of the filter does not meet the requirements.

[0050] Compared with the prior art, the beneficial effects of the present invention are as follows:

[0051] This solution uses - an analogical method to generalize the existing so that when this solution processes non-Gaussian signals, it is capable of capturing higher-order statistical characteristics. By introducing parameters, this solution has a more flexible error control ability and can adjust itself in different noise environments to obtain the best filtering effect. Experiments show that this solution is not only superior to and traditional algorithms in terms of accuracy, but also significantly improves the convergence speed of filtering. This is crucial for the performance of adaptive filters in real-time applications. Especially in scenarios where the signal changes violently, the fast convergence of this solution ensures that the system can respond to signal changes in a timely manner, avoiding problems such as delay or reduced accuracy.

[0052] In practical applications, signals are often mixed with Gaussian noise and impulse noise, which poses higher requirements for the robustness of adaptive filtering. This solution enhances the resistance to impulse noise through - analogical extension, enabling it to maintain high accuracy even in the face of strong interference. At the same time, this solution calculates the representation of the signal eigenvector in a finite-dimensional reproducing kernel Hilbert space and updates the signal eigenvector for sparsification operations, effectively reducing the computational complexity. While reducing the storage and operation requirements, it still maintains good anti-noise performance and filtering accuracy, making this solution particularly suitable for processing large-scale data sets or real-time application scenarios with limited resources. BRIEF DESCRIPTION OF THE DRAWINGS

[0053] Figure 1 is a flowchart of a time series prediction method based on adaptive filtering.

[0054] Figure 2 is a performance comparison diagram of impulse noise and various denoising methods under impulse noise, where (a) is a schematic diagram of impulse noise, and (b) is a performance comparison diagram of various denoising methods under impulse noise.

[0055] Figure 3 is a performance diagram of impulse noise plus Gaussian noise and various denoising methods under impulse noise, where (a) is a schematic diagram of impulse noise plus Gaussian noise, and (b) is a performance comparison diagram of various denoising methods under impulse noise plus Gaussian noise.

[0056] Figure 4Performance comparison diagrams of multiple denoising methods under different noises; among them, (a) is the performance comparison diagram of multiple denoising methods under impulse noise; (b) is the performance comparison diagram of multiple denoising methods under impulse noise plus Gaussian noise.

[0057] Figure 5 It is a distribution diagram of fingerprint establishment nodes and test nodes for the indoor positioning scenario.

[0058] Figure 6 It is a comparison diagram of the x - coordinate prediction curves of nodes in the dataset of a certain university classroom by multiple algorithms.

[0059] Figure 7 It is a comparison diagram of the y - coordinate prediction curves of nodes in the dataset of a certain university classroom by multiple algorithms.

[0060] Figure 8 It is a comparison value diagram of the indoor positioning index MSE (mean square error) of all algorithms.

[0061] Figure 9 It is a test configuration diagram of XJTU - SY physical hardware.

[0062] Figure 10 It is a prediction result diagram of XTJU dataset 1 - 1.

[0063] Figure 11 It is a prediction result diagram of XTJU dataset 2 - 1.

[0064] Figure 12 It is a prediction result diagram of XTJU dataset 3 - 5.

[0065] Figure 13 It is a prediction result diagram of PRONOSTIA data 1 - 2.

[0066] Figure 14 It is a prediction result diagram of PRONOSTIA data 2 - 1. Specific implementation manners

[0067] The specific implementation manners of the present invention will be described below to facilitate those skilled in the art of the present technology to understand the present invention. However, it should be clear that the present invention is not limited to the scope of the specific implementation manners. For those ordinary skilled in the art of the present technology, as long as various changes are within the spirit and scope of the present invention defined and determined by the appended claims, these changes are obvious, and all inventions created using the concept of the present invention are within the scope of protection.

[0068] Reference Figure 1 , Figure 1 shows a flowchart of a time - series prediction method based on adaptive filtering; as Figure 1 shown, this method S includes steps S1 to S6.

[0069] In step S1, time series data for training the filter is obtained, and it is preprocessed to obtain a signal feature vector; the learning step size of the filter, the Gaussian kernel variance σ, and the parameters p and q of the weight vector are set.

[0070] The time series in this solution is preferably the RSSI value collected for indoor positioning or the vibration data collected for estimating the remaining life of a bearing;

[0071] When the time series is the RSSI value, the preprocessing thereof includes eliminating the RSSI values greater than the preset error in the RSSI values, and performing a normalization operation on the remaining RSSI values;

[0072] When the time series is vibration data, the preprocessing thereof is to extract features from the vibration data.

[0073] In step S2, according to the signal feature vector at the k-th time point and the weight vector at the (k - 1)-th time point, the time series prediction output at the k-th time point is calculated:

[0074]

[0075] Wherein, is the time series prediction output at the k-th time point; is the transpose of a(k - 1).

[0076] In step S3, according to the time series prediction output and the desired output at the k-th time point, the error between the two is calculated , is the desired output at the k-th time point; based on the error and the Gaussian kernel variance σ, the Gaussian kernel function is calculated:

[0077]

[0078] In step S4, according to the Gaussian kernel function and the error, the gradient term at the k-th time point is calculated:

[0079]

[0080] Wherein, is the Gaussian kernel at the k-th time point; is the exponential function with the natural logarithm base being e.

[0081] After that, according to the gradient term and the learning rate, the weight vector at the k-th time point is updated:

[0082] ,

[0083] Wherein, and The weight vectors at the k-th and (k - 1)-th time points respectively, where k ≥ 2, is the expected output at the first time point; η is the learning step size; and are the gradient term and the error at the k-th time point respectively; is the signal feature vector at the k-th time point, is the time series at the k-th time point;

[0084] In step S5, it is judged whether the filter satisfies the convergence condition. If so, the training of the filter is completed and step S6 is entered; otherwise, let k = k + 1 and return to step S2;

[0085] In implementation, preferably, the convergence condition of this solution is:

[0086]

[0087] where, is the expectation operator; is the square of the norm of the vector; is the prior error at the k-th time point.

[0088] In step S6, the historical time series and the prediction period are input into the trained filter for prediction to obtain the predicted time series data within the prediction period.

[0089] The prediction method of this solution is based on the variable step size mechanism of the parameter, enabling the algorithm to dynamically adjust the learning rate under different values, thereby improving the convergence efficiency and flexibly adapting to different noise environments. Compared with the traditional algorithm, the

[0090] algorithm (the algorithm of steps S1~S6) can provide smoother behavior when facing large errors, enabling it to effectively handle impulse noise and other complex noises, thereby improving the overall filtering performance.

[0091] A1. Randomly select m sample points from the signal feature vector to form a dictionary D; calculate the Gaussian kernel function of the selected sample points to construct a kernel Gram matrix;

[0092] A2. Perform eigenvalue decomposition on the kernel Gram matrix to obtain the eigenvalues and the eigenvector matrix , and use the eigenvalue and the eigenvector matrix Construct a mapping matrix :

[0093]

[0094] wherein, is the transpose of;

[0095] A3. Calculate the Gaussian kernel function values of the data at each time point k in the signal feature vector and all sample points in the dictionary D respectively to form a vector for each time point k ;

[0096] A4. Update the signal feature vector according to the vector and the mapping matrix .

[0097] Sparse algorithm (the algorithm composed of steps S1~S6+A1~A4) can effectively reduce the space complexity of the algorithm while maintaining the prediction accuracy. This method first selects m sample points from the complete time series to form a dictionary, and these sample points are used to approximate the feature space of the entire data set, thereby reducing the computational complexity.

[0098] During implementation, the condition that the learning step size of this solution preferably satisfies is:

[0099]

[0100] wherein, is the expectation operator; is the square of the norm of the vector; is the prior error at the k-th time point; is the square of the error at the k-th time point; is the Gaussian kernel at the k-th time point.

[0101] This condition of the learning step size ensures that each weight update will not cause the instability of the system. In practical applications, when selecting η, it is necessary to ensure that it is small enough to maintain the system stability, and at the same time, it is large enough to ensure the learning efficiency.

[0102] The expression of the prior error is:

[0103] ,

[0104] wherein, is the error vector of a(k-1); is the optimal weight vector.

[0105] In one embodiment of the present invention, the time series prediction method based on adaptive filtering further includes evaluating the performance of the filter:

[0106] When obtaining the time series prediction output at each time point, calculate the excess mean square error corresponding to the current time point:

[0107]

[0108] where L is the excess mean square error; is the expectation operator; is the trace of the covariance matrix of and are the first derivative and the second derivative respectively; is the error expectation value of; is the absolute value symbol; is the given prior parameter;

[0109] Plot the time series prediction outputs of all time points in chronological order to form an excess mean square curve;

[0110] Judge whether the excess mean square curve is a decreasing curve. If so, the performance of the filter meets the requirements; otherwise, the performance of the filter does not meet the requirements.

[0111] The excess mean square error ( ) is a key indicator for evaluating the performance of an adaptive filter, and its analysis is usually based on the following three main assumptions:

[0112] 1. The noise perturbation is zero-mean, independent and identically distributed, and uncorrelated with the input signal.

[0113] 2. The prior error is not affected by the noise and has zero mean.

[0114] 3. The squared norm of the input variable and the gain are asymptotically independent.

[0115] The excess mean square error quantifies the average power of the error when the filter reaches a steady state. In theoretical analysis, by performing a Taylor expansion on the non-linear function of the error and assuming the independence of the noise and the signal, can be approximately calculated.

[0116] where reflects the distribution of the energy of the input signal. For the first derivative and the second derivative can be obtained by taking the derivative of Obtained by performing Taylor expansion.

[0117] The following combines specific examples to verify the (algorithms of steps S1 - S6) and (algorithms composed of steps S1 - S6 + A1 - A4):

[0118] In this simulation experiment, time series prediction problems and time series prediction problems are used to and verify the performance of the algorithms. In the sequence prediction experiment, the experiment first generated time series data with noise. The training dataset contains 2000 noisy samples, while the test dataset includes 200 clean samples. The sampling interval of the sequence is set to 6 seconds, and the length of the filter is set to 7, that is, 7 historical samples are used to predict the value at the next moment. In the experiment, and the algorithms are compared with other adaptive filtering algorithms (such as , and ). To evaluate the filtering performance of each algorithm, the experiment uses the mean square error ( ) as the main index and records the changes in the convergence speed and filtering accuracy.

[0119] In addition to the single impulse noise test, the experiment also introduces a mixed environment of Gaussian noise and impulse noise to further verify the robustness of the algorithms under complex noise. The time series prediction experiment is to further test the performance of the algorithms in chaotic systems. The experiment is also carried out in two noise environments: impulse noise and the mixture of impulse noise and Gaussian noise. And to eliminate the possible influence brought by computer performance, all experiments will be run on the same computer, with the specific configuration as:

[0120] Intel(R) Core(TM) i5¬8600K CPU @ 2.90GHz, 8GB of RAM, computer environment is Windows 7, and the running platform is MATLAB R2020b.

[0121] 1. Time series prediction

[0122] Tested the proposed and its sparse version algorithms, using the chaotic MG model, whose dynamic equation is as follows:

[0123]

[0124] The time series is discretized with a sampling interval of 6 seconds. The training dataset contains 2000 samples, and all samples are contaminated by the same noise as shown in (a) below. The test dataset includes an additional 200 clean samples. The length of the filter is set to L = 7 and is designed to use the sequence Figure 2 to predict .

[0125] The filters for comparison include Kernel Least Mean Square ( ), Kernel Maximum Correntropy Criterion ( ), and Minimum Kernel -Power Mean Error ( ) algorithms, which are selected for their excellent robustness and filtering accuracy. Since the optimal weight vector is unknown in this experiment, the Mean Square Error ( ) in the test stage is used to evaluate the filtering performance. The formula for calculating the test MSE is as follows:

[0126]

[0127] where represents the actual value, represents the estimated value, and represents the number of test samples. The parameters of each algorithm are carefully selected to ensure the expected filtering accuracy while maintaining an almost identical convergence rate.

[0128] Figure 2 Figure (b) below shows the learning curves of the KLMS, KMC, KMPE, qKMPE, and NystromqKMPE algorithms in an impulse noise environment. Figure 2 The detailed experimental results corresponding to Figure (b) are shown in Table 1.

[0129] Table 1 Experimental results of relevant algorithms for prediction under impulse noise conditions

[0130]

[0131] As shown in Figure 2 Figure (b) below, the data of qKMPE and NystromqKMPE both show higher filtering accuracy and convergence speed, and their accuracy performance is better than that of other powerful LMS-type Kernel Adaptive Filters (KAFs). At the same time, compared with other algorithms, the qKMPE algorithm and the NystromqKMPE algorithm both show a faster convergence speed under impulse noise conditions.

[0132] The results show that ​and The algorithm has obvious advantages over other algorithms in terms of filtering accuracy and convergence speed. In addition, the sparse

[0133] algorithm, while greatly reducing the network scale, can still maintain high accuracy and fast running time, making it more suitable for online implementation. From Table 1, it can be seen that since the algorithm combines the advantages of -analogy and KMPE error loss, the algorithm has the smallest among all algorithms. In addition, the time overhead of the proposed algorithm is the same as that of other algorithms. As shown in Table 1, the performance accuracy of the algorithm is almost the same as that of the algorithm, but the required neural network scale for implementation is only 1 / 25 of that of the algorithm. At the same time, it can be found that the running speed of the proposed

[0134] algorithm is six times that of other algorithms, making its online implementation in practical applications more feasible. and its sparse version algorithm under extremely harsh actual conditions, the following experiment uses a stable process composed of impulse noise and Gaussian noise, and the noise reference Figure 3 in (a) of , and algorithms are also included in the comparison and the same parameter settings as the above experiment are used. The results are shown in Table 2, and the performance comparison graph is as shown in Figure 3 in (b).

[0135] Table 2 Experimental results of relevant algorithms predicting under impulse plus Gaussian noise conditions

[0136]

[0137] Referring to Table 2, in the mixed noise environment, the proposed algorithm performs best in terms of accuracy and has significant advantages over the , and algorithms. At the same time, the has the same computational overhead as other algorithms. The proposed algorithm has the most compact structure, so it runs the fastest and greatly reduces the network scale while maintaining almost the same accuracy performance as the algorithm.

[0138] Figure 3 (a) gives the sequence of combined impulse noise and Gaussian noise, as Figure 3 (b) shows. qKMPE and NystromqKMPE effectively mitigate impulse noise and Gaussian noise, and have better filtering accuracy and convergence speed than the KLMS, KMC, and KMPE algorithms.

[0139] 2. Time series prediction

[0140] The attractor was first proposed by in 1963. It is a dynamic system that describes the long-term behavior of chaotic flows and is famous for its unique "butterfly" shape. The characteristics of this system are non-linearity, three-dimensional structure, and determinism. In 2001, confirmed that under specific parameters, this system exhibits chaotic behavior, leading to the formation of the so-called strange attractor phenomenon.

[0141] The learning curves generated by the adaptive filtering algorithm are evaluated using two different noise signals: an impulse noise signal and a composite noise signal composed of impulse noise and Gaussian noise. The corresponding learning curves are as shown in Figure 4 (a) and (b) in the figure. The results show that the adaptive filtering algorithm based on q - simulation successfully reduces the influence of a large number of errors caused by α - stable impulse noise. By observing and analyzing Table 3, Table 4, and Figure 4 , the following conclusions can be drawn:

[0142] 1. In the environments of impulse noise and mixed noise, the algorithm proposed in this scheme performs best in terms of accuracy, superior to other algorithms.

[0143] 2. In this scenario, using the method to sparsify ( ) can effectively reduce the scale of the implemented network. Due to discarding the information of a part of the data, its accuracy will decrease. In the case of the decrease, the accuracy of is still superior to other algorithms except .

[0144] 3. In addition, the and algorithms proposed in this scheme show significant advantages in terms of convergence speed compared with other algorithms.

[0145] Table 3 Experimental results of relevant algorithms for time series prediction under impulse noise conditions

[0146]

[0147] Table 4 Experimental results of the relevant algorithms for time series prediction under impulse plus Gaussian noise

[0148]

[0149] In summary, the two algorithms proposed in this scheme (the algorithm in steps S1 - S6) and (the algorithm composed of steps S1 - S6 + A1 - A4) can effectively cope with impulse noise and Gaussian noise. At the same time, the method can significantly reduce the network scale. Although there will be a certain performance loss, compared with , and algorithms, the proposed algorithm shows better performance in filtering accuracy.

[0150] The following combines specific experiments to illustrate the prediction effects of this scheme on indoor positioning and bearing remaining life:

[0151] Experiment 1: The time series is the RSSI value for indoor positioning

[0152] 1. Scenario description

[0153] In this experiment, a classroom in a certain university was selected. Its area is about 77 , with a length of 10.44 m and a width of 7.35 m. There are 90 desks and chairs and a switch cabinet. In the scenario, the Sub - 1GHz wireless technology widely used in the Internet of Things was selected. Compared with WiFi and Bluetooth signals, Sub - 1GHz signals have the following advantages: longer wavelength, stronger signal penetration, and lower energy consumption. In the experiment, four antennas of LPSTK - CC1352R were used to collect signals, and it receives and transmits signals in the 915 MHz frequency band of the IEEE802.15.4 communication protocol. The training set consists of 7 * 10 (position coordinates, RSSI) pairs, marked as blue dots. The LPSTK - CC1352R measurement hardware was used to measure the RSSI value at the specified position, thereby establishing an RSSI fingerprint database with a scale of 7 * 10 for training the proposed algorithm. For the application of the specific training set, the leave - one - out method or cross - validation method can be used for arrangement. The test set consists of 20 random red dots. Three devices with fixed - position transmitters were used in the experiment. The distribution of training points and test points is as follows Figure 5 as shown.

[0154] 2. Experimental results

[0155] ​In the indoor positioning experiment, the experimental training dataset contains 7 * 10 (position coordinates, RSSI) pairs, while the test dataset includes 20 randomly placed red dots. At each red dot in the test dataset, the RSSI value is measured and then fed into the filter trained by the algorithm proposed in this solution, enabling prediction, that is, giving the position corresponding to the current RSSI value. In the experiment, and (the algorithm composed of steps S1~S6 + A1~A4) and other adaptive filtering algorithms (such as , KRLS, and ) were compared. To evaluate the filtering performance of each algorithm, the mean square error ( ) was used as the main metric in the experiment, and the changes in the convergence rate and filtering accuracy were recorded.

[0156] The MSE curves of the experimental results are shown in Figure 6 , Figure 7 and Figure 8 . In the x - coordinate prediction curve and y - coordinate prediction curve, the MSE of the qKMPE algorithm and the Nys_qKMPE algorithm is the smallest, and the effect is better than that of other algorithms. The specific MSE values of several algorithms are given in Figure 8 . In the X - coordinate prediction and Y - coordinate prediction, the MSE of the qKMPE algorithm and the Nys_qKMPE algorithm (the algorithm composed of steps S1~S6 + A1~A4) is the smallest.

[0157] Experiment 2: Prediction of Bearing Remaining Life Based on Time - Series Vibration Signals - 1

[0158] 1. Scenario Description

[0159] To verify the effectiveness of the two algorithms proposed in this solution (the algorithm of steps S1~S6) and the Nys_qKMPE algorithm (the algorithm composed of steps S1~S6 + A1~A4), a bearing dataset from the rolling - element bearing life dataset based on XJTU - SY (the physical hardware test configuration of XJTU - SY can be referred to Figure 9 ) was used to demonstrate and verify the performance of the algorithms.

[0160] Experimental procedures: The maximum amplitude (MA) is extracted from the original vibration signal as the feature signal, i.e., the maximum sample obtained at each sampling, to monitor the fault progression of the bearing under test. When a fault is detected, the prediction process is triggered, and the algorithm proposed in this scheme is applied to estimate the future degradation state of the rolling bearing and the RUL. In addition, through careful observation of each bearing degradation mode and repeated trial-and-error experiments, combined with expert knowledge, all failure thresholds are set to 20g. These failure thresholds correspond to the maximum allowable amplitude for the safe operation of the bearing. If the failure threshold is exceeded, it indicates that the bearing is damaged and should be replaced in a timely manner.

[0161] 2. Experimental Results

[0162] The vibration data of bearings 11, 21, and 35 in the XJTUSY dataset are selected for the bearing RUL prediction experiment. The degradation index MA is used to process the vibration signal, and the FPT nodes are determined by the above-mentioned adaptive method based on the 3σ interval. This scheme compares the (algorithms in steps S1 - S6) and the Nys_qKMPE algorithm (the algorithm corresponding to steps S1 - S6 + steps A1 - A4) with existing algorithms such as traditional KLMS, KRLS, and KMPE_LMS. To further verify the robustness of the algorithm, the prediction starting point is continuously adjusted, and a new RUL prediction experiment is carried out. The new prediction results are repeatedly recorded until the entire healthy stage is traversed. The above operations are repeated for each bearing, and all the data are sorted and summarized.

[0163] Figure 10 、 Figure 11 、 Figure 12 Record the bearing degradation prediction results of each algorithm starting from FPT. The ordinate in the figure is the degradation index MA, and the abscissa is time. From the figure, it can be seen that the prediction results of the qKMPE_LMS algorithm (the algorithm corresponding to steps S1 - S6) and the Nys_qKMPE_LMS algorithm (the algorithm corresponding to steps S1 - S6 + steps A1 - A4) coincide highly with the actual curve. It can be seen that the two algorithms qKMPE_LMS and Nys_qKMPE_LMS proposed in this scheme have better prediction effects than traditional adaptive filtering methods.

[0164] To quantitatively evaluate the estimation performance of the prediction method, the cumulative relative accuracy (CRA) and the root mean square error (RMSE) are used to measure the prediction accuracy. The calculation formulas are as follows:

[0165]

[0166]

[0167]

[0168] The CRA values and RMSE values of the experimental results of the improved algorithm and the classical kernel adaptive prediction algorithm are recorded in Table 5.

[0169] Table 5 Performance Comparison of Algorithms before and after Improvement on XTJU Dataset

[0170]

[0171] As shown in Table 5, the CRA value of the qKMPE algorithm is not lower than that of the other four adaptive methods. Compared with the traditional KRLS algorithm with the best performance, the minimum CRA value given by the qKMPE algorithm proposed in this scheme is 0.7882, while the minimum CRA value given by KRLS is 0.7379, indicating that the RUL estimation given by the qKMPE algorithm is more accurate. At the same time, the RMSE values obtained using qKMPE are less than those obtained by the other four kernel adaptive filtering algorithms except for Bearing 1-1, which means that the qKMPE proposed in this scheme provides a more robust RUL estimation. In addition, although the performance of the Nys_qKMPE_LMS algorithm is slightly weaker than that of the qKMPE algorithm, it still has performance improvement compared with the other three unquantized algorithms.

[0172] In summary, compared with the traditional kernel adaptive filtering algorithm, the proposed qKMPE algorithm and Nys_qKMPE_LMS algorithm have advantages in the accuracy of rolling bearing RUL estimation.

[0173] Experiment 3: Prediction of Bearing Remaining Life Based on Time-Series Accelerated Degradation Test - 2

[0174] 1. Scenario Description

[0175] This dataset contains bearing acceleration degradation data collected from the bearing test bench of PRONOSTIA, a sub-institute of FEMTO-ST Research Institute. The PRONOSTIA platform is specifically designed to conduct accelerated degradation tests on rolling bearings under different operating conditions to evaluate their performance and lifespan.

[0176] The core of the test platform is a rotating shaft supported by two bearings, one of which is the test bearing and the other is the support bearing. To simulate different working conditions, an adjustable radial load is applied to the test bearing, which is achieved through a hydraulic actuator. The flexibility of the hydraulic system enables researchers to conduct experiments under various load conditions, thereby obtaining more comprehensive degradation data.

[0177] In addition, the rotation of the shaft is controlled by a motor, allowing tests to be carried out at different rotational speeds. This adjustability enables researchers to explore the performance of the bearings at different speeds and further understand their degradation characteristics. To monitor the condition of the bearings in real time, the platform is equipped with two high-frequency sensors that can precisely capture changes in the bearing acceleration, thus providing rich dynamic information for the dataset.

[0178] These bearings were tested at a constant rotational speed of 1800 rpm with a 4 kN load applied until all bearings reached a complete failure state. Every 10 s, amplitude samples of the individual 0.1 s vibration signals were acquired at a sampling frequency of 25.6 kHz, and each amplitude sample of the signal contained 2560 data points. This high sampling frequency allows the capture of high-frequency vibration components, which is crucial for detecting bearing faults in the early stages of degradation.

[0179] 2. Experimental Results

[0180] The experiments were tested using the PRONOSTIA dataset, where the true values of Dataset 1-2 were 122 min and those of Dataset 2-1 were 176 min. This scheme compared the proposed qKMPE_LMS algorithm and Nys_qKMPE_LMS algorithm with existing algorithms such as traditional KLMS, KRLS, and KMPE_LMS.

[0181] As shown in Table 6, among the RUL (Remaining Useful Life) predicted by different algorithms through the dataset, the qKMPE_LMS predicted 126 min for Dataset 1-2 and 176 min for Dataset 2-1; the Nys_qKMPE_LMS predicted 128 min for Dataset 1-2 and 178 min for Dataset 2-1, both of which are relatively close to the true values.

[0182] Table 6 Comparison of RUL (Remaining Useful Life) Predicted by the PRONOSTIA Dataset with True Values 1-2: 122 min, 2-1: 176 min

[0183]

[0184] Figure 13 and Figure 14 The visualization shows the prediction results of different algorithms through the dataset. The ordinate in the figure is the degradation index MA, and the abscissa is time. It can be seen from the figure that the prediction results of the qKMPE_LMS algorithm and Nys_qKMPE_LMS algorithm coincide highly with the actual curve, indicating that the prediction effects of the qKMPE_LMS algorithm and Nys_qKMPE_LMS algorithm are good. Table 7 quantitatively shows the evaluation indicators in the prediction process of different algorithms.

[0185] Table 7 Comparison List of Prediction Indicators for the PRONOSTIA Dataset

[0186]

[0187] As can be seen from Table 7, the root mean square error (RMSE) of the qKMPE_LMS algorithm (Steps S1 - S6 of this solution) is the lowest, and the root mean square error (RMSE) of the Nys_qKMPE_LMS algorithm (Steps S1 - S6 + A1 - A4 of this solution) is also relatively low; for the cumulative relative accuracy (CRA), the result of the qKMPE_LMS algorithm is 0.8783, which is higher than that of other algorithms, and the result of the Nys_qKMPE_LMS algorithm is 0.8035, which is also relatively high, indicating that the two algorithms proposed in this solution have good effects.

Claims

1. A time series prediction method based on adaptive filtering, characterized in that: Includes steps: S1. Obtain time series data for training the filter, and preprocess it to obtain a signal feature vector; set the learning step size of the filter, the Gaussian kernel variance σ, and the parameters p and q of the weight vector; the time series is the RSSI value collected by indoor positioning or the vibration data collected when estimating the remaining life of the bearing; S2. Calculate the time series prediction output at the kth time point based on the signal feature vector at the kth time point and the weight vector at the k-1th time point; S3, according to the time series prediction output and the expected output at the kth time point, calculate the error between the two; based on the error and the Gaussian kernel variance σ, calculate the Gaussian kernel function; S4. Calculate the gradient term at the kth time point based on the Gaussian kernel function and the error, and then update the weight vector at the kth time point based on the gradient term and the learning rate: ， in, and are the weight vectors for the kth and k-1th time points, respectively, k ≥ 2, is the expected output at the first time point; η is the learning step length; and are the gradient term and error at the kth time point respectively; is the signal feature vector at the kth time point, is the time series of the kth time point; S5, determine whether the filter meets the convergence condition, if so, complete the filter training and go to step S6; otherwise, set k=k+1 and return to step S2; S6. Input the historical time series and the prediction period into the trained filter for prediction to obtain the predicted time series data within the prediction period.

2. The time series prediction method based on adaptive filtering according to claim 1, characterized in that: The method between step S1 and step S2 also includes calculating the representation of the signal feature vector in the finite-dimensional reproducing kernel Hilbert space to update the signal feature vector: A1. Randomly select m sample points in the signal feature vector to form a dictionary D; calculate the Gaussian kernel function of the selected sample points to construct a kernel Gram matrix; A2. Perform eigendecomposition on the kernel Gram matrix to obtain eigenvalues and the eigenvector matrix , and use the eigenvalue and the eigenvector matrix Constructing the mapping matrix : in, for The transpose of A3. Calculate the data of each time point k in the signal feature vector and the Gaussian kernel function value of all sample points in the dictionary D to form the vector of each time point k. ; A4. According to the vector and the mapping matrix , update the signal feature vector .

3. The time series prediction method based on adaptive filtering according to claim 1, characterized in that: The expression for calculating the time series forecast output at the kth time point is: in, The time series prediction output for the kth time point; is the transpose of a(k-1); The expression of Gaussian kernel function is: The expression of the gradient term is: in, is the Gaussian kernel at the kth time point; The base of the natural logarithm is The exponential function of .

4. The time series prediction method based on adaptive filtering according to claim 1, characterized in that: The convergence condition is: in, is the expectation operator; is the square of the norm of the vector; is the prior error at the kth time point.

5. The time series prediction method based on adaptive filtering according to claim 1, characterized in that: The conditions that the learning step should satisfy are: in, is the expectation operator; is the square of the norm of the vector; is the prior error at the kth time point; is the square of the error at the kth time point; is the Gaussian kernel at the kth time point.

6. The time series prediction method based on adaptive filtering according to claim 4 or 5, characterized in that: The expression of the prior error is: ， in, is the error vector of a(k-1); is the optimal weight vector.

7. The time series prediction method based on adaptive filtering according to any one of claims 1 to 5, characterized in that: When the time series is an RSSI value, preprocessing is performed on it, including removing RSSI values ​​greater than a preset error from the RSSI values, and normalizing the retained RSSI values; When the time series is vibration data, it is preprocessed to extract features from the vibration data.

8. The time series prediction method based on adaptive filtering according to any one of claims 1 to 5, characterized in that: It also includes an evaluation of the filter's performance: When obtaining the time series forecast output at each time point, calculate the excess mean square error corresponding to the current time point: Where L is the excess mean square error; is the expectation operator; for The trace of the covariance matrix of ; and are the first and second order derivatives respectively; Error Expected value; is the absolute value symbol; For given prior parameters; Plot the time series forecast outputs of all time points in chronological order as excess mean square curves; It is determined whether the excess mean square curve is a decreasing curve. If so, the performance of the filter meets the requirements, otherwise the performance of the filter does not meet the requirements.

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