Time sequence prediction method based on adaptive filtering

Abstract

The invention discloses a time sequence prediction method based on adaptive filtering, and the method comprises the steps: S1, obtaining time sequence data for training a filter, and carrying out the preprocessing of the time sequence data, and obtaining a signal feature vector; s2, calculating time sequence prediction output of the kth time point; s3, according to the time sequence prediction output and the expected output of the kth time point, calculating an error between the two; calculating a Gaussian kernel function based on the error and the Gaussian kernel variance sigma; s4, according to the Gaussian kernel function and the error, calculating a gradient term of the kth time point, and then according to the gradient term and the learning rate, updating a weight vector of the kth time point; s5, judging whether the filter meets a convergence condition or not, if so, completing the training of the filter, and entering step S6; otherwise, enabling k to be equal to k + 1, and returning to the step S2; and S6, inputting the historical time sequence and the prediction period into the trained filter for prediction to obtain prediction time sequence data in the prediction period.

Description

Technical Field

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

[0002] Time series forecasting 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 predicted future data trends, thereby optimizing resource usage and strategy deployment.

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

[0004] Although existing kernel methods improve the model's processing of non-Gaussian signals by introducing non-linear kernel functions, they still have many limitations, such as slow convergence and poor performance on non-convex performance surfaces. These limitations have prompted researchers to explore new adaptive filtering algorithms, such as kernel-power mean error-based methods, which improve the robustness and convergence speed of algorithms 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 when a fast response is required in a resource-constrained environment.

[0005] In order to further improve the performance of time series forecasting and solve the above problems, it is necessary to develop a suitable extension of the KMPE criterion (kernel mean square p-power error criterion) so that it can better cope with the interference of non-Gaussian signals and meet higher convergence speed and accuracy requirements. Summary of the invention

[0006] In view of the above-mentioned deficiencies in the prior art, the time series prediction method based on adaptive filtering provided by the present invention solves the problem that when filtering time series, extreme non-Gaussian noise or rapid response is difficult to achieve in a resource-constrained environment.

[0007] In order to achieve the above-mentioned object of the invention, the technical solution adopted by the present invention is:

[0008] A time series prediction method based on adaptive filtering is provided, which comprises the steps of:

[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 kth time point based on the signal feature vector at the kth time point and the weight vector at the k-1th time point;

[0011] 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;

[0012] 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:

[0013] a(1)=ηd(1)

[0014] Where a(k) and a(k―1) are the weight vectors at the kth and k-1th time points, respectively, k ≥ 2, d(1) is the expected output at the first time point; η is the learning step size; g(e(k)) and e(k) are the gradient term and error at the kth time point, respectively; x(υ(k)) is the signal feature vector at the kth time point, and υ(k) is the time series at the kth time point;

[0015] 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;

[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, the method further includes calculating a representation of the signal feature vector in a finite-dimensional reproducing kernel Hilbert space to update the signal feature vector:

[0018] 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;

[0019] A2. Perform eigendecomposition on the kernel Gram matrix to obtain the eigenvalue λ m and the eigenvector matrix X m , and use the eigenvalue λ m and the eigenvector matrix X m Construct the mapping matrix P m :

[0020]

[0021] in, For X m The transpose of

[0022] A3, respectively calculating the data of each time point k in the signal feature vector and the Gaussian kernel function values ​​of all sample points in the dictionary D to form a vector κ(k) at each time point k;

[0023] A4. According to the vector κ(k) and the mapping matrix P m , update the signal feature vector x(v(x))=P m κ(k).

[0024] Furthermore, the expression for calculating the time series forecast output at the kth time point is:

[0025] y(k)=a(k―1) T x(υ(k))

[0026] Among them, y(k) is the time series prediction output at the kth time point; a(k―1) T is the transpose of a(k-1);

[0027] The expression of Gaussian kernel function is:

[0028]

[0029] The expression of the gradient term is:

[0030] g(e(k))=(1-k σ (e(k))) p / 2 (1-k σ (e(k))) p / 2―1 k σ (e(k))·e(k)

[0031] Among them, κ σ (e(k)) is the Gaussian kernel at the kth time point; exp(·) is the exponential function with the base of the natural logarithm being e.

[0032] Furthermore, the convergence condition is:

[0033]

[0034] in, is the expectation operator; ||·|| 2 is the square of the norm of the vector; e p (k) is the prior error at the kth time point.

[0035] Furthermore, the conditions that the learning step should satisfy are:

[0036]

[0037] in, is the expectation operator; ||·|| 2 is the square of the norm of the vector; e p (k) is the prior error at the kth time point; e 2 (k) is the square of the error at the kth time point; κ σ (e(k)) is the Gaussian kernel at the kth time point.

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

[0039]

[0040] in, is the error vector of a(k-1); a 0 is the optimal weight vector.

[0041] Further, the time series is an RSSI value collected by indoor positioning or vibration data collected when estimating the remaining life of a bearing;

[0042] 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;

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

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

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

[0046]

[0047] Where L is the excess average error; is the expectation operator; Tr(υ(k)) is the trace of the covariance matrix of υ(k); g′(μ) and g″(μ) are the first-order derivative and the second-order derivative, respectively; u is the expected value of the error e(k); |·| is the absolute value sign; μ 0 For given prior parameters;

[0048] Plot the time series forecast outputs of all time points in chronological order into an excess mean square curve;

[0049] 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.

[0050] Compared with the prior art, the present invention has the following beneficial effects:

[0051] This scheme uses the q-analogy method to generalize the existing KMPE, so that this scheme can capture high-order statistical characteristics when processing non-Gaussian signals. By introducing the q parameter, this scheme has a more flexible error control capability and can adjust itself in different noise environments to obtain the best filtering effect. Experiments show that this scheme not only outperforms KMPE and traditional MCC 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 signals change dramatically. The fast convergence of this scheme ensures that the system can respond to signal changes in a timely manner, avoiding delays or reduced accuracy.

[0052] In practical applications, signals are often mixed with Gaussian noise and impulse noise, which puts higher robustness requirements on adaptive filtering. This scheme enhances the resistance to impulse noise through q-analogy expansion, so that it can still maintain high accuracy in the face of strong interference. At the same time, this scheme effectively reduces the computational complexity by calculating the representation of the signal feature vector in the finite-dimensional reproducing kernel Hilbert space to update the signal feature vector for sparse operations. While reducing storage and computing requirements, it still maintains good noise resistance and filtering accuracy, which makes this scheme particularly suitable for large-scale data set processing or real-time application scenarios with limited resources. BRIEF DESCRIPTION OF THE DRAWINGS

[0053] Figure 1 Flowchart of the time series prediction method based on adaptive filtering.

[0054] Figure 2 : The 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 The performance graphs of pulse noise plus Gaussian noise and various denoising methods under pulse noise, where (a) is a schematic diagram of pulse noise plus Gaussian noise, and (b) is a performance comparison graph of various denoising methods under pulse noise plus Gaussian noise.

[0056] Figure 4 The following are performance comparisons of various denoising methods under different noises; (a) is a performance comparison of various denoising methods under impulse noise; (b) is a performance comparison of various denoising methods under impulse noise plus Gaussian noise.

[0057] Figure 5 For indoor positioning scenarios, fingerprint establishment nodes and test node distribution maps.

[0058] Figure 6 This is a comparison chart of the x-coordinate prediction curves of multiple algorithms for nodes in a university classroom data set.

[0059] Figure 7 This is a comparison chart of the y-coordinate prediction curves of multiple algorithms for nodes in a university classroom data set.

[0060] Figure 8 This is a comparison chart of the indoor positioning indicator MSE (mean square error) of all algorithms.

[0061] Fig. 9 This is the XJTU-SY physical hardware test configuration diagram.

[0062] Fig.10 This is the prediction result diagram of the XTJU dataset 1-1.

[0063] Fig.11 This is the prediction result diagram of XTJU dataset 2-1.

[0064] Fig.12 This is the prediction result graph of XTJU dataset 3-5.

[0065] Fig.13 This is the prediction result diagram of PRONOSTIA data 1-2.

[0066] Fig.14 This is the prediction result diagram of PRONOSTIA data 2-1. DETAILED DESCRIPTION

[0067] The specific implementation modes of the present invention are described below so that those skilled in the art can understand the present invention. However, it should be clear that the present invention is not limited to the scope of the specific implementation modes. For those of ordinary skill in the art, as long as various changes are within the spirit and scope of the present invention as defined and determined by the attached claims, these changes are obvious, and all inventions and creations utilizing the concept of the present invention are protected.

[0068] refer to Figure 1 , Figure 1 A flowchart of a time series prediction method based on adaptive filtering is shown; Figure 1 As shown, the method S includes steps S1 to S6.

[0069] In step S1, the time series data for training the filter is obtained and 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 of this solution is preferably the RSSI value collected by indoor positioning or the vibration data collected when estimating the remaining life of the bearing;

[0071] 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;

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

[0073] In step S2, the time series prediction output at the kth time point is calculated based on the signal feature vector at the kth time point and the weight vector at the k-1th time point:

[0074] y(k)=a(k―1) T x(υ(k))

[0075] Among them, y(k) is the time series prediction output at the kth time point; a(k―1) T is the transpose of a(k-1).

[0076] In step S3, the error between the predicted output and the expected output of the time series at the kth time point is calculated, e(k)=d(k)-y(k), where d(k) is the expected output at the kth time point; based on the error and the Gaussian kernel variance σ, the Gaussian kernel function is calculated:

[0077]

[0078] In step S4, the gradient term at the kth time point is calculated based on the Gaussian kernel function and the error:

[0079] g(e(k))=(1-k σ (e(k))) p / 2 (1-k σ (e(k))) p / 2―1 k σ (e(k))e(k)

[0080] Among them, κ σ (e(k)) is the Gaussian kernel at the kth time point; exp(·) is the exponential function with the base of the natural logarithm being e.

[0081] Then, according to the gradient term and learning rate, the weight vector at the kth time point is updated:

[0082] a(1)=ηd(1)

[0083] Where a(k) and a(k―1) are the weight vectors at the kth and k-1th time points, respectively, k ≥ 2, d(1) is the expected output at the first time point; η is the learning step size; g(e(k)) and e(k) are the gradient term and error at the kth time point, respectively; x(υ(k)) is the signal feature vector at the kth time point, and υ(k) is the time series at the kth time point;

[0084] In step S5, it is determined whether the filter meets the convergence condition. If so, the training of the filter is completed and the process proceeds to step S6; otherwise, k=k+1 is set and the process returns to step S2;

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

[0086]

[0087] in, is the expectation operator; ||·|| 2 is the square of the norm of the vector; e p (k) is the prior error at the kth 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 scheme is based on the variable step size mechanism of the q parameter, which enables the algorithm to dynamically adjust the learning rate under different q values, thereby improving the convergence efficiency and flexibly adapting to different noise environments. Compared with the traditional KMPE algorithm, the qKMPE algorithm (the algorithm of steps S1 to 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.

[0090] In one embodiment of the present invention, between step S1 and step S2, the step further includes calculating the representation of the signal feature vector in the finite-dimensional reproducing kernel Hilbert space to update the signal feature vector:

[0091] 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;

[0092] A2. Perform eigendecomposition on the kernel Gram matrix to obtain the eigenvalue λ m and the eigenvector matrix X m , and use the eigenvalue λ m and the eigenvector matrix X m Construct the mapping matrix P m :

[0093]

[0094] in, For X m The transpose of

[0095] A3, respectively calculating the data of each time point k in the signal feature vector and the Gaussian kernel function values ​​of all sample points in the dictionary D to form a vector κ(k) at each time point k;

[0096] A4. According to the vector κ(k) and the mapping matrix P m , update the signal feature vector x(v(k))=P m κ(k).

[0097] The sparse qKMPE algorithm (the algorithm consisting of steps S1 to S6 + A1 to A4) can effectively reduce the spatial complexity of the algorithm while maintaining the accuracy of the prediction. The method first selects m sample points from the complete time series to form a dictionary, which is used to approximate the feature space of the entire data set, thereby reducing the complexity of the calculation.

[0098] During implementation, the conditions that the learning step size should meet in this solution are:

[0099]

[0100] in, is the expectation operator; ||·|| 2 is the square of the norm of the vector; e p (k) is the prior error at the kth time point; e 2 (k) is the square of the error at the kth time point; κ σ (e(k)) is the Gaussian kernel at the kth time point.

[0101] The learning step size condition ensures that each weight update does not lead to system instability. In practical applications, η should be selected to ensure that it is small enough to maintain system stability, but large enough to ensure learning efficiency.

[0102] The expression of the prior error is:

[0103]

[0104] in, is the error vector of a(k-1); a 0 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 forecast output at each time point, calculate the excess mean square error corresponding to the current time point:

[0107]

[0108] Where L is the excess average error; is the expectation operator; Tr(υ(k)) is the trace of the covariance matrix of υ(k); g′(μ) and g″(μ) are the first-order derivative and the second-order derivative, respectively; u is the expected value of the error e(k); |·| is the absolute value sign; μ 0 For given prior parameters;

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

[0110] 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.

[0111] Excess mean square error (EMSE) is a key metric for evaluating the performance of adaptive filters. Its analysis is usually based on the following three main assumptions:

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

[0113] 2. Prior error e p (k) is not affected by the noise n(k) and has zero mean.

[0114] 3. Square norm and gain g of input variable v(k) 2 (e(k)) is asymptotically independent.

[0115] The excess mean square error quantifies the average power of the error when the filter reaches a stable state. In theoretical analysis, the EMSE can be approximately calculated by Taylor expansion of the nonlinear function of the error g(e(k)) and assuming independence of the noise and signal.

[0116] Where Tr(υ(k)) reflects the energy distribution of the input signal. The first-order derivative g′(μ) and the second-order derivative g″(μ) can be obtained by Taylor expansion of g(e(k)).

[0117] The following is a concrete example to test the qKMPE (algorithm of steps S1 to S6) and ―Effectiveness of qKMPE (algorithm consisting of steps S1 to S6 + A1 to A4):

[0118] In this simulation experiment, the Mackey-Glass (MG) time series prediction problem and the Lorenz time series prediction problem are used to compare qKMPE and The performance of the qKMPE algorithm is verified. In the Mackey-Glass sequence prediction experiment, the experiment first generates noisy time series data. The training data set contains 2000 noisy samples, while the test data set 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 of the next moment. In the experiment, qKMPE and The ―qKMPE algorithm is compared with other adaptive filtering algorithms (such as KLMS, KMC and KMPE). In order to evaluate the filtering performance of each algorithm, the experiment uses the mean square error (MSE) as the main indicator and records the changes in convergence speed and filtering accuracy.

[0119] In addition to the single impact noise test, the experiment also introduced a mixed environment of Gaussian noise and impact noise to further verify the robustness of the algorithm under complex noise. The Lorenz time series prediction experiment is to further test the performance of the qKMPE algorithm in a chaotic system. The experiment was also conducted in two noise environments: impact noise and a mixture of impact noise and Gaussian noise. In order to eliminate the possible impact of computer performance, all experiments will be run on the same computer, with the following specific configurations:

[0120] Intel(R)Core(TM)i5 8600K CPU @ 2.90 GHz, RAM is 8 GB, the computer environment is Windows 7, and the operating platform is MATLAB R 2020b.

[0121] 1. Mackey-Glass (MG) Time Series Forecasting

[0122] The proposed qKMPE and its sparse version were tested ―qKMPE algorithm uses the chaotic MG model, and its dynamic equation is as follows:

[0123]

[0124] The MG time series is discretized with a sampling interval of 6 seconds. The training data set contains 2000 samples. All samples are Figure 2 The same noise contamination as shown in (a) in the figure. 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 [ω i ,ω i―1 ,…,ω i―6 ] to predict ω i+1.

[0125] The filters compared include the kernel least mean square (KLMS), kernel maximum correlation entropy criterion (KMC), and minimum kernel p-power mean error (KMPE) algorithms, which are selected for their excellent robustness and filtering accuracy. Since the optimal weight vector cannot be known in this experiment, the mean square error (MSE) in the test phase is used to evaluate the filtering performance. The test MSE calculation formula is as follows:

[0126]

[0127] Where d(t) represents the actual value, Represents the estimated value, and N represents the number of test samples. The parameters of each algorithm are carefully selected to ensure the expected filtering accuracy while maintaining almost the same convergence rate.

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

[0129] Table 1 Experimental results of MG prediction by related algorithms under impulse noise conditions

[0130] like Figure 2 As shown in (b), the data of qKMPE and NystromqKMPE both show higher filtering accuracy and convergence speed, and their accuracy performance is better than other powerful LMS-type kernel adaptive filters (KAFs). At the same time, compared with other algorithms, qKMPE and NystromqKMPE algorithms both show faster convergence speed under impulse noise conditions.

[0131] The results showed that qKMPE and The qKMPE algorithm has obvious advantages over other algorithms in filtering accuracy and convergence speed. In addition, the sparse qKMPE algorithm can maintain high accuracy and fast running time while significantly reducing the network size, which makes it more suitable for online implementation.

[0132] As can be seen from Table 1, the qKMPE algorithm has the smallest MSE among all algorithms because it combines the advantages of q-analogy and KMPE error loss. In addition, the time overhead of the proposed qKMPE algorithm is the same as that of other algorithms. As shown in Table 1, The performance accuracy of the qKMPE algorithm is almost the same as that of the qKMPE algorithm, but the neural network size required for implementation is only 1 / 25 of that of the qKMPE algorithm. The ―qKMPE algorithm runs six times faster than other algorithms, making its online implementation more feasible in practical applications.

[0133] To evaluate qKMPE and its sparse version ―qKMPE algorithm robustness under extremely harsh practical conditions, the following experiments use an α-stable process composed of impulse noise and Gaussian noise, and a noise reference composed of impulse noise and Gaussian noise Figure 3 (a) in the figure. The qKLMS, KMC and KMPE algorithms are also included in the comparison and use the same parameter settings as the above experiments. The results are shown in Table 2 and the performance comparison is shown in the figure below. Figure 3 As shown in (b) in .

[0134] Table 1 Experimental results of correlation algorithm for predicting MG under pulse plus Gaussian noise conditions

[0135]

[0136] Referring to Table 2, in the mixed noise environment, the proposed qKMPE algorithm performs best in terms of accuracy, and has significant advantages over the KLMS, KMC and KMPE algorithms. At the same time, the computational overhead of qKMPE is the same as that of other algorithms. The qKMPE algorithm has the most compact structure, so it runs the fastest and greatly reduces the network size while maintaining almost the same accuracy as qKMPE.

[0137] Figure 3 (a) shows a sequence combining impulse noise and Gaussian noise, such as Figure 3 As shown in (b), qKMPE and NystromqKMPE effectively alleviate impulse noise and Gaussian noise, and have better filtering accuracy and convergence speed than KLMS, KMC and KMPE algorithms.

[0138] 2. Lorenz Time Series Forecasting

[0139] The Lorenz attractor was first proposed by Edward Lorenz in 1963. It is a dynamical system that describes the long-term behavior of chaotic flows and is known for its unique "butterfly" shape. The system is characterized by nonlinearity, three-dimensional structure and determinism. In 2001, Warwick Tucker confirmed that under certain parameters, the system exhibits chaotic behavior, leading to the formation of the so-called strange attractor phenomenon.

[0140] 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 consisting of impulse noise and Gaussian noise. The corresponding learning curves are shown in Figure 4 As shown in (a) and (b), the results show that the adaptive filtering algorithm based on q-simulation successfully reduces the impact of a large number of errors caused by α-stable impulse noise. By observing and analyzing Table 3, Table 4 and Figure 4 , we can draw the following conclusions:

[0141] 1. In the impact noise and mixed noise environments, the qKMPE algorithm proposed in this scheme performs best in terms of accuracy and is superior to other algorithms.

[0142] 2. In this scenario, use The method is to sparse the qKMPE ( ―qKMP), can effectively reduce the size of the network, but due to the discarding of some data information, its accuracy will decrease. ―The accuracy of qKMPE is still better than other algorithms except qKMPE.

[0143] 3. In addition, the qKMPE and The convergence speed of the ―qKMPE algorithm shows significant advantages compared with other algorithms.

[0144] Table 2 Experimental results of related algorithms for Lorenz time series prediction under impulse noise conditions

[0145]

[0146] Table 3 Experimental results of Lorenz time series prediction using related algorithms under pulse plus Gaussian noise conditions

[0147]

[0148] In summary, the two algorithms qKMPE (the algorithm of steps S1 to S6) and ―qKMPE (the algorithm consisting of steps S1~S6+A1~A4) can effectively deal with impulse noise and Gaussian noise, while using The proposed method can significantly reduce the network size. Although there will be a certain performance loss, compared with the KLMS, KMC and KMPE algorithms, the proposed algorithm performs better in filtering accuracy.

[0149] The following is a description of the prediction effect of this solution on indoor positioning and remaining bearing life in combination with specific experiments:

[0150] Experiment 1: Time series is the RSSI value of indoor positioning

[0151] 1. Scenario description

[0152] This experiment selected a classroom in a university with an area of ​​about 77m 2 , with a length and width of 10.44m and 7.35m respectively, and 90 tables and chairs and a switch cabinet. In the scene, the Sub-1GHz wireless technology widely used in the Internet of Things was selected. Sub-1GHz signals have the following advantages over WiFi and Bluetooth signals: longer wavelength, stronger signal penetration, and lower energy consumption. In the experiment, the four antennas of LPSTK-CC1352R are used to collect signals. It receives and transmits signals in the 915MHz frequency band of the IEEE802.15.4 communication protocol. The training set consists of 7*10 (position coordinates, RSSI) pairs, marked as blue points. The LPSTK-CC1352R measurement hardware is used to measure the RSSI value at the specified location, so as to establish an RSSI fingerprint library of 7*10 scale for training the proposed algorithm. The specific application of the training set can be arranged by the retention method and the cross-validation method. The test set consists of 20 random red points. Three devices with fixed-position transmitters were used in the experiment. The distribution of training points and test points is as follows Figure 5 shown.

[0153] 2. Experimental results

[0154] In the indoor positioning experiment, the experimental training data set contains 7*10 (position coordinates, RSSI) pairs, and the test data set includes 20 random red dots. The RSSI value is measured at each red dot in the test data set, and then sent to the filter trained by the algorithm proposed in this scheme to predict the position corresponding to the current RSSI value. In the experiment, the qKMPE and Nys_qKMPE algorithms (algorithms composed of steps S1~S6+A1~A4) are compared with other adaptive filtering algorithms (such as KLMS, KRLS, KMPE and KMPE). In order to evaluate the filtering performance of each algorithm, the experiment uses the mean square error (MSE) as the main indicator, and records the changes in convergence speed and filtering accuracy.

[0155] The MSE curve of the experimental results is as follows Figure 6 , Figure 7 and Figure 8 As shown in the figure, in the x-coordinate prediction curve and the y-coordinate prediction curve, the MSE of the qKMPE algorithm and the Nys_qKMPE algorithm are the smallest, and the effect is better than other algorithms. Figure 8 The specific MSE values ​​of several algorithms are given in FIG. 1 . In the X-coordinate prediction and the Y-coordinate prediction, the MSE of the qKMPE algorithm and the Nys_qKMPE algorithm (the algorithm consisting of steps S1 to S6 + A1 to A4) are both the smallest.

[0156] Experiment 2: Prediction of remaining bearing life based on time series vibration signals-1

[0157] 1. Scenario description

[0158] In order to verify the effectiveness of the two algorithms qKMPE (the algorithm of steps S1 to S6) and Nys_qKMPE (the algorithm composed of steps S1 to S6 + A1 to A4) proposed in this scheme, a test system based on XJTU-SY (the physical hardware test configuration of XJTU-SY can be referred to Fig. 9 ) is used to demonstrate the performance of the algorithm.

[0159] Experimental steps: The maximum amplitude (MA) is extracted from the original vibration signal as a characteristic signal, that is, 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 proposed algorithm is applied to estimate the future degradation state of the rolling bearing and 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 safe operation of the bearing. If the failure threshold is exceeded, it indicates that the bearing is damaged and should be replaced in time.

[0160] 2. Experimental results

[0161] The vibration data of bearings 1-1, 2-1, and 3-5 in the XJTU-SY data set were selected for the bearing RUL prediction experiment. The degradation index MA was used to process the vibration signal, and the FPT node was determined by the above-mentioned adaptive method based on the 3σ interval. This scheme compares the proposed qKMPE (algorithm of steps S1 to S6) and Nys_qKMPE algorithm (algorithm corresponding to steps S1 to S6 + steps A1 to A4) with existing algorithms such as traditional KLMS, KRLS, and KMPE_LMS. In order to further verify the robustness of the algorithm, the prediction starting point is continuously adjusted, a new RUL prediction experiment is conducted, and the new prediction results are repeatedly recorded until the entire healthy stage is traversed. Repeat the above operations for each bearing, and organize and summarize all data.

[0162] Fig.10 , Fig.11 , Fig.12The bearing degradation prediction results of each algorithm starting from FPT are recorded. The vertical axis in the figure is the degradation index MA, and the horizontal axis is time. It can be seen from the figure that the prediction results of the qKMPE_LMS algorithm (the algorithm corresponding to steps S1 to S6) and the Nys_qKMPE_LMS algorithm (the algorithm corresponding to steps S1 to S6 + steps A1 to A4) have a very high degree of coincidence with the actual curve. It can be seen that the two algorithms qKMPE_LMS algorithm and Nys_qKMPE_LMS algorithm proposed in this scheme have better prediction effects than the traditional adaptive filtering method.

[0163] In order to quantitatively evaluate the estimation performance of the prediction method, the cumulative relative accuracy (CRA) and root mean square error (RMSE) are used to measure the accuracy of the prediction. The calculation formula is:

[0164]

[0165] Table 5 records the CRA and RMSE values ​​of the experimental results of the improved algorithm and the classic kernel adaptive prediction algorithm.

[0166] Table 5 Performance comparison of algorithms before and after improvement on XTJU dataset

[0167]

[0168]

[0169] 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 best-performing KRLS algorithm, the qKMPE algorithm proposed in this scheme gives a minimum CRA value of 0.7882, while the minimum CRA value given by KRLS is 0.7379, indicating that the RUL estimate given by the qKMPE algorithm is more accurate. At the same time, the RMSE values ​​obtained using qKMPE are smaller 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 estimate. In addition, although the performance of the Nys_qKMPE_LMS algorithm is slightly weaker than that of the qKMPE algorithm, it is still improved compared with the other three non-q algorithms.

[0170] 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 RUL estimation of rolling bearings.

[0171] Experiment 3: Prediction of remaining bearing life based on time series accelerated degradation test-2

[0172] 1. Scenario description

[0173] This dataset contains bearing acceleration degradation data collected from the bearing test rig of PRONOSTIA, a subsidiary of FEMTO-ST Institute. The PRONOSTIA platform is specifically designed for accelerated degradation testing of rolling element bearings under different operating conditions to evaluate their performance and lifespan.

[0174] The core of the test platform is a rotating shaft supported by two bearings, one of which is the test bearing and the other serves as a support bearing. To simulate different operating conditions, an adjustable radial load is applied to the test bearing, which is achieved through a hydraulic actuator. The flexibility of the hydraulic system allows researchers to conduct experiments under a variety of load conditions, thereby obtaining more comprehensive degradation data.

[0175] In addition, the rotation of the shaft is controlled by an electric motor, allowing testing at different rotation speeds. This adjustability allows researchers to explore the performance of the bearing at different speeds and further understand its degradation characteristics. To monitor the condition of the bearing in real time, the platform is equipped with two high-frequency sensors that can accurately capture changes in bearing acceleration, providing rich dynamic information to the dataset.

[0176] The bearings were tested at a constant speed of 1800rpm with 4kN applied until all bearings reached complete failure. Every 10s, individual 0.1s amplitude samples of the vibration signal were acquired at a sampling frequency of 25.6kHz, with each amplitude sample containing 2560 data points. This high sampling frequency allows the capture of high frequency vibration components, which is critical for detecting bearing failures at an early stage of degradation.

[0177] 2. Experimental results

[0178] The experiment was tested using the PRONOSTIA dataset, where the true value of dataset 1-2 was 122 minutes and the true value of dataset 2-1 was 176 minutes. This scheme compares the proposed qKMPE_LMS algorithm and Nys_qKMPE_LMS algorithm with existing algorithms such as traditional KLMS, KRLS, and KMPE_LMS.

[0179] As shown in Table 6, among the RUL (remaining life) predicted by different algorithms through data sets, qKMPE_LMS predicted data set 1-2 is 126min, and predicted data set 2-1 is 176min; Nys_qKMPE_LMS data set 1-2 is 128min, and predicted data set 2-1 is 178min, which are both close to the true value.

[0180] Table 6 Comparison of RUL (remaining useful life) predicted by PRONOSTIA dataset with the true value 1-2: 122min, 2-1: 176min

[0181] RUL(min) PRONOSTIAdataset1-2 PRONOSTIAdataset2-1 KLMS 128 180 KRLS 128 180 KMPE_LMS 126 178 qKMPE_LMS 126 176 Nys_qKMPE_LMS 128 178

[0182] Fig.13 and Fig.14 The visualization shows the prediction results of different algorithms through the data set. The vertical axis in the figure is the degradation index MA, and the horizontal axis is time. It can be seen from the figure that the prediction results of the qKMPE_LMS algorithm and the Nys_qKMPE_LMS algorithm have a high degree of coincidence with the actual curve, and it can be seen that the prediction effect of the qKMPE_LMS algorithm and the Nys_qKMPE_LMS algorithm is good. Table 7 quantifies the evaluation indicators of the prediction process of different algorithms.

[0183] Table 7 Comparison of prediction indicators of PRONOSTIA dataset

[0184]

[0185]

[0186] It can be seen from Table 7 that the qKMPE_LMS algorithm (steps S1 to S6 of this scheme) has the lowest root mean square error (RMSE), and the Nys_qKMPE_LMS algorithm (steps S1 to S6 + A1 to A4 of this scheme) also has a low root mean square error (RMSE); for the cumulative relative accuracy (CRA), the qKMPE_LMS algorithm result is 0.8783, which is higher than other algorithms, and the Nys_qKMPE_LMS algorithm result is 0.8035, which is also relatively high, indicating that the two algorithms proposed in this scheme have better effects.

Claims

1. A time series prediction method based on adaptive filtering, characterized in that: Includes steps: 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; 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: Where a(k) and a(k―1) are the weight vectors at the kth and k-1th time points, respectively, k ≥ 2, d(1) is the expected output at the first time point; η is the learning step size; g(e(k)) and e(k) are the gradient term and error at the kth time point, respectively; x(υ(k)) is the signal feature vector at the kth time point, and υ(k) is the time series at 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 the eigenvalue λ m and the eigenvector matrix X m , and use the eigenvalue λ m and the eigenvector matrix X m Construct the mapping matrix P m : in, For X m The transpose of A3, respectively calculating the data of each time point k in the signal feature vector and the Gaussian kernel function values ​​of all sample points in the dictionary D to form a vector κ(k) at each time point k; A4. According to the vector κ(k) and the mapping matrix P m , update the signal feature vector x(v(k))=P m κ(k).

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: y(k)=a(k―1) T x(υ(k)) Among them, y(k) is the time series prediction output at the kth time point; a(k―1) T is the transpose of a(k-1); The expression of Gaussian kernel function is: The expression of the gradient term is: g(e(k))=(1―k σ (e(k))) p / 2 (1-k σ (e(k))) p / 2―1 k σ (e(k))·e(k) Among them, κ σ (e(k)) is the Gaussian kernel at the kth time point; exp(·) is the exponential function with the base of the natural logarithm being e.

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; ||·|| 2 is the square of the norm of the vector; e p (k) 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; ||·|| 2 is the square of the norm of the vector; e p (k) is the prior error at the kth time point; e 2 (k) is the square of the error at the kth time point; κ σ (e(k)) 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); a0 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: The time series is the RSSI value collected by indoor positioning or the vibration data collected when estimating the remaining life of the bearing; 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 average error; is the expectation operator; Tr(υ(k)) is the trace of the covariance matrix of υ(k); g′(μ) and g″(μ) are the first-order derivative and the second-order derivative, respectively; u is the expected value of the error e(k); |·| is the absolute value sign; μ0 is a given priori parameter; 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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