Time-varying nonlinear system rapid identification method based on beta wavelet basis function expansion

Abstract

The invention provides a time-varying parameter model identification method based on beta wavelet basis function expansion. As shown in the flow in figure 1, the method comprises the following steps: firstly, establishing a nonlinear autoregression time-varying parameter model by using a non-stationary time-varying signal, and determining the order of the model by using an FPE order determination criterion; performing linear weighted expansion on the model time-varying parameters by utilizing a first-order beta wavelet basis function, and converting a model time-varying parameter identification problem into a time-invariant parameter estimation and model effective item selection problem, namely obtaining a beta wavelet expansion time-invariant parameter model; selecting effective items of the model by adopting a classical orthogonal forward regression algorithm, determining the number of the effective items of the model in combination with an APRESS cross validation standard, eliminating redundant items in the expansion model, estimating corresponding time-invariant parameters of the model, and establishing a sparse model structure; and finally, in combination with the obtained effective items and corresponding parameters, obtaining a sparse nonlinear autoregression model and time-varying parameters thereof through reverse solution. Compared with an existing time-varying parameter identification method based on main B-spline multi-wavelet basis function expansion, the time complexity of the identification process can be effectively reduced, the identification precision of a sparse model is improved, and a new thought and a new calculation framework are provided for identification analysis of nonlinear time-varying parameter signals.

Description

Technical Field

[0001] The invention proposes a time-varying nonlinear system fast identification algorithm based on beta wavelet basis function expansion, which provides a new solution for time-varying nonlinear system parameter modeling and belongs to the technical field of signal analysis and processing. Background Art

[0002] Time-varying systems are widely present in current scientific research and engineering practice. The identification problem of time-varying systems can be solved by establishing an effective mathematical model for time-varying systems. The time-varying parameter modeling method is the main research method for non-stationary system analysis. It can be specifically divided into the following three modeling ideas: The first idea is to use a finite number of sliding time intervals of a certain length to divide the time-varying signal into several signal segments, and treat the signal segments as stationary signals in each interval. In this type of method, the identification result of the time-varying signal is often limited by the length of the time interval division, and it is difficult to give a unified time window division standard for different time-varying systems. Therefore, this method has great limitations in universality; the second idea is to use a classical adaptive algorithm, that is, to treat the time-varying parameters in the model as random variables. This type of method mainly includes the least mean squares algorithm (least-mean squares, LMS) and the recursive least squares method (recursive least squares). leastsquares, RLS), etc. Although such algorithms have the advantages of small computational complexity and high stability, for time-varying signals with fast parameter changes, such methods often fail to accurately capture the instantaneous signal characteristics of fast time-varying signals due to their slow convergence speed; the third idea is to use the basis function expansion method to establish a time-varying parameter model, that is, to express the time-varying parameters as a linear weighted combination of a set of known basis functions, thereby transforming the time-varying parameter solution problem into the structural identification and parameter estimation problem of the time-invariant parameter system, and using the time-invariant parameter model sparse algorithm to obtain the system's sparse model structure and corresponding time-invariant parameters, and then obtain the time-varying parameters of the initial system through inverse solution. This type of method can use the good approximation ability of the basis function to quickly track the time-varying system.

[0003] However, in the process of establishing a time-varying system model using the basis function expansion method, it is often necessary to select different basis functions for approximation according to the approximation characteristics of different basis functions and the characteristics of different time-varying signals. For example, Fourier basis functions and Legendre polynomial basis functions are suitable for the identification of time-varying parameters of time-varying systems with relatively gentle changes, while wavelet basis functions, with their excellent linear and nonlinear approximation characteristics, can simultaneously identify smooth and drastic time-varying systems, and are therefore widely used in the identification of non-stationary systems. Among the expansion methods using wavelets as basis functions, the most widely used one is the multi-wavelet basis function expansion method based on principal B-spline wavelets. Since the wave structure of single-order principal B-spline wavelets is relatively monotonous, they often cannot be used alone as basis functions to expand complex time-varying parameters. Therefore, in practical applications, it is necessary to use principal B-spline wavelets of multiple orders to form a multi-wavelet family to expand the complex time-varying system. On the one hand, this increases the number of basis functions and improves the computational complexity and structural complexity of the model. On the other hand, it results in a large number of redundant items in the expanded model, which poses a great challenge to the subsequent model sparse process and easily causes model overfitting, thereby affecting the time-varying system identification accuracy of the sparse model.

[0004] The present invention introduces for the first time the beta wavelet with diversified waveform structure as the basis function for time-varying parameter expansion. The beta wavelet waveform is similar to the waveform of the neural pulse signal and is a single-cycle waveform. Its waveform is similar to a "bell shape" in half a cycle, showing a smooth and steep feature, and the waveform in the other half a cycle is more gentle. At the same time, the beta wavelet has a pair of characteristic control parameters (α, β), so that different optimal parameter combinations can be set according to the characteristics of different time-varying signals to adjust the wavelet waveform. Therefore, the beta wavelet has a good advantage in identifying time-varying systems that are both smooth and drastically changing. The beta wavelet basis function expansion method proposed in the present invention is different from the multi-wavelet basis function expansion method based on multi-order main B-spline wavelets in previous studies. The introduced beta wavelet can realize the rapid identification of complex time-varying systems in combination with different parameter combinations due to the inherent characteristics of its waveform structure. Therefore, the beta wavelet basis function expansion method proposed in the present invention can reduce the number of basis functions to a certain extent, thereby reducing the number of redundant items in the extended model and improving the effectiveness and scope of application of the model sparse algorithm. Compared with the optimization algorithm of the model sparse process, the present invention fundamentally solves the problem of too many redundant items in the expanded model, which not only reduces the structural complexity and computational complexity of the time-varying model, but also effectively alleviates the overfitting problem of the model sparse process and improves the recognition accuracy of the sparse model.

[0005] Based on the beta wavelet basis function expansion method and the classical orthogonal forward regression sparse algorithm (OFR), the present invention proposes a new time-varying nonlinear system identification method, which provides a new solution for simplifying the model structure and improving the identification accuracy and computational efficiency of the time-varying parameter model. Summary of the invention

[0006] The invention provides a method for constructing a time-varying nonlinear system based on the expansion of beta wavelet basis functions. The time-varying parameters in the model are approximated by a linear weighted combination of beta wavelet basis functions, and the rapid identification problem of the time-varying system in the initial model is converted into the selection of effective terms in the expansion model and the estimation problem of the corresponding time-invariant parameters. The expanded time-invariant parameter model is effectively tested by using a classical orthogonal forward regression algorithm, thereby obtaining an effective sparse model structure and the corresponding time-invariant parameter estimation, and then the effective sparse time-varying parameter model is obtained by reverse solution. Among them, the beta wavelet basis function is a new type of single-order wavelet basis function. Compared with the multi-wavelet basis function based on the combination of multi-order main B-spline wavelets in previous studies, the main advantages are: on the one hand, it overcomes the problem of large amount of redundant terms caused by too many basis functions in the expansion of multi-wavelet basis functions, and on the other hand, it effectively improves the identification accuracy of the model. According to the simulation verification results, the beta wavelet basis function expansion method proposed by the invention not only reduces the computational complexity of the model, but also improves the identification accuracy of the sparse model algorithm, providing a new idea for the rapid identification of time-varying nonlinear systems.

[0007] The specific steps of the time-varying nonlinear system rapid identification method based on beta wavelet basis function expansion proposed by the present invention are as follows:

[0008] 1. Time-varying nonlinear model: The nonlinear combination of the system output signal and the input signal is used as the input of the system model to construct a time-varying nonlinear model;

[0009] 2. Beta wavelet expansion model: The beta wavelet basis function is used to perform linear weighted expansion on the time-varying parameters in the time-varying nonlinear model to obtain the beta wavelet expanded time-invariant nonlinear model;

[0010] 3. Effective model structure detection: Use the classic orthogonal forward regression algorithm to select effective items for the time-invariant parameter model after the expansion of the beta wavelet basis function, remove the redundant items in the expanded time-invariant model, establish an effective sparse model structure, and estimate the corresponding time-invariant parameters;

[0011] 4. Model reconstruction: Combining the effective sparse model structure and the estimated time-invariant parameter results, the time-varying parameters of the initial time-varying nonlinear model are obtained through inverse solution, and the sparse time-varying nonlinear parameter model is reconstructed.

[0012] Wherein, in the step 1, the optimal order of the time-varying nonlinear model is determined by using the FPE (Final Prediction Error criterion) order determination criterion.

[0013] In step 2, the time-varying parameters of the nonlinear model are expanded using beta wavelet basis functions, and the time-varying parameters are expressed as a set of basis function linear weighted forms, thereby converting the time-varying parameter identification problem of the initial model into a time-invariant parameter identification problem about the basis function polynomial.

[0014] In step 3, the redundant items in the expanded model can be removed by using the classical OFR algorithm, and the model structure can be further simplified on the basis of the expansion of the beta wavelet basis function, the effective items of the model are selected, the corresponding time-invariant parameters are estimated, and the number of effective items of the sparse model is determined by the adjustable prediction error sum of squares (APRESS) cross-validation criterion, thereby establishing an effective sparse model structure. As mentioned above, after the expansion of the beta wavelet basis function, the number of redundant items in the expanded model has been reduced, that is, the model options that need to be sparsely processed have been relatively reduced, but a large amount of modeling experience shows that there are still a large number of redundant items in the expanded model at this time, and the model structure needs to be further simplified by using the model sparse algorithm. It is worth noting that since the model options after the expansion of the beta wavelet basis function are significantly reduced compared with the expansion of the multi-wavelet basis function, the problem of model overfitting in the sparse process is alleviated to a certain extent.

[0015] The advantages of the time-varying nonlinear system fast identification method based on beta wavelet basis function expansion and orthogonal forward regression algorithm proposed in the present invention include:

[0016] 1. The beta wavelet waveform is controlled by two characteristic parameters. Its waveform forms are diverse and can be adjusted according to the characteristics of different time-varying systems. It has good adaptability and can effectively track different types of smooth and drastically changing complex time-varying nonlinear systems.

[0017] 2. The model structure obtained by expanding the beta wavelet basis function is simple, which greatly reduces the number of redundant items contained in the expanded model to a certain extent, thereby effectively reducing the computational complexity of the model;

[0018] 3. The sparse model obtained by the classic OFR algorithm has a simple structure, accurate parameter estimation, and high model identification accuracy;

[0019] 4. The model construction process is simple, the computational complexity is low, and the APRESS cross-validation standard is used to effectively avoid the overfitting problem of the model. BRIEF DESCRIPTION OF THE DRAWINGS

[0020] Figure 1 A schematic diagram of the flow chart of the time-varying nonlinear system identification method proposed by the present invention;

[0021] Figure 2 For the verification of the fast identification result of the time-varying nonlinear system in simulation example 1, the fast identification result of the time-varying nonlinear system obtained by the beta wavelet basis function expansion method proposed in the present invention is compared with the identification result obtained by the existing principal B-spline multi-wavelet basis function expansion method; wherein, Figure 2 (a) is the time-varying parameter identification effect diagram obtained by two basis function expansion methods. Figure 2 (b) is a comparison diagram of the prediction output effects of non-stationary signals estimated by two basis function expansion methods;

[0022] Figure 3 For the second simulation example: verification of the rapid identification results of time-varying nonlinear systems, the identification results obtained by the above two basis function expansion methods are compared; among them, Figure 3 (a) is the time-varying parameter tracking effect diagram obtained by two basis function expansion methods. Figure 3 (b) is a comparison diagram of the prediction output effects of non-stationary signals estimated by two basis function expansion methods;

[0023] Figure 4 A comparison diagram of the predicted output of the time-varying nonlinear system modeling of the real EEG signal and the tracking effect of the real EEG signal using the time-varying nonlinear parameter modeling method based on the expansion of the beta wavelet basis function proposed in the present invention. DETAILED DESCRIPTION

[0024] In order to better illustrate the specific implementation mode of the present invention, the present invention is further described in detail below with reference to the accompanying drawings.

[0025] The present invention aims to provide a fast identification method for time-varying nonlinear systems based on beta wavelet basis function expansion, so as to solve the problems of complex model structure and high computational complexity caused by the existing method based on principal B-spline multi-wavelet basis function expansion, namely, by reducing the number of basis functions, removing redundant items in the expanded model, improving the effectiveness and computational speed of the sparse model algorithm, realizing model structure optimization, and further improving the system identification accuracy of the wavelet basis function expansion method.

[0026] Figure 1 The flowchart of the time-varying nonlinear system identification method proposed by the present invention is shown, including:

[0027] Firstly, a time-varying nonlinear system modeling is performed on a non-stationary signal with an external source input, that is, a nonlinear combination of the output signal feedback and the input signal is used as the model input, a corresponding time-varying nonlinear model is established, and the optimal order of the model is determined according to the FPE model order determination criterion (step 1); then the beta wavelet basis function introduced by the present invention is used to expand the time-varying parameters of the nonlinear model to obtain a beta wavelet expanded time-invariant parameter model (step 2); then, the classic OFR algorithm is used to select the effective terms of the expanded model, the corresponding time-invariant parameters are estimated, and the number of effective terms is determined according to the APRESS cross-validation criterion, thereby establishing an effective sparse model structure (step 3); finally, the initial time-varying nonlinear model and the basis function linear weighted expressions of the corresponding time-varying parameters are obtained by inversely solving the sparse model structure (step 4).

[0028] The following specifically describes the time-varying nonlinear system rapid identification method based on the beta wavelet basis function expansion method and the OFR algorithm provided by the present invention, and the specific steps include:

[0029] 1. Time-varying nonlinear model: The nonlinear combination of the system output feedback signal and the input signal is used as the system input to construct a time-varying nonlinear model. The expression of the time-invariant nonlinear model is as follows:

[0030] y(t)=f(y(t-1),…,y(tp),u(t-1),…,u(tq))+e(t) (1)

[0031] Among them, y(t) represents the output of the nonlinear system; y(tp) and u(tq) represent the delay of the output and input signals; p is the order of the output signal, q is the order of the input signal; e(t) represents the mean value of 0 and the variance of is a Gaussian white noise sequence; f(·) represents a suitable nonlinear function.

[0032] For time-varying nonlinear systems, the nonlinear function f(·) can be expressed in the form of time-varying parameters, that is:

[0033]

[0034] in, is the nonlinear combination vector of input and output regression terms, and θ(t) is the corresponding time-varying parameter.

[0035] The input items of the time-varying nonlinear model are x1(t),x2(t),…,x M (t), then The specific expression of θ(t) is as follows:

[0036]

[0037] θ(t)=[θ1(t),θ2(t),…,θ M (t)] (4)

[0038] Where M is the number of time-varying nonlinear model terms.

[0039] Taking the second-order nonlinear time-varying system as an example, It can be expressed as:

[0040]

[0041] In order to determine the correct model order (p, q), the present invention adopts the FPE order determination criterion to determine the optimal model order. The FPE order determination criterion expression is as follows:

[0042]

[0043] Where N is the data sample length, m is the number of model terms, γ is the adjustment parameter, (p,q) is the model order, is the prediction variance.

[0044] Note that when determining the model order (p, q) according to the FPE order determination criterion, the single variable principle should be followed, such as first determining p and q, then determining q and p, and repeating this process to ultimately determine the optimal order of the model.

[0045] 2. Beta wavelet expansion model: The beta wavelet basis function is used to perform linear weighted expansion on the time-varying parameters in the time-varying nonlinear model to obtain a time-invariant model based on beta wavelet expansion;

[0046] As mentioned above, because the beta wavelet waveform changes similarly to the neural pulse signal, it contains both a peak half-wave that adapts to rapidly changing signals and a smooth half-wave that adapts to slowly changing signals. Compared with a single-order primary B-spline wavelet that contains only one waveform feature, the beta wavelet waveform feature is more diversified and can accurately track time-varying nonlinear systems. Therefore, the present invention introduces beta wavelet basis functions instead of B-spline multi-wavelet basis functions to expand the time-varying parameters of the nonlinear model.

[0047] Wavelet function has a strong correlation with probability distribution function. Beta wavelet is a new type of compactly supported wavelet function derived from the transformation of beta distribution function. Due to the unimodal nature of beta distribution function, the beta wavelet generated by its first-order derivative has the characteristic of single period. The beta wavelet function obtained by a set of characteristic control parameters (α, β) is:

[0048]

[0049] in, is the characteristic constant, is the support interval length of the beta wavelet function, The support range.

[0050] After translation transformation, the beta wavelet function can obtain a set of 2 (R) The inner square integrable beta wavelet basis function is shown in the following formula:

[0051]

[0052] Among them, x is a function variable defined on the closed interval [0, 1], j is a scaling factor, which can be used to adjust the scale of the basis function; k is a translation factor, which is used to adjust the specific position of the basis function. Represents the set of positive integers.

[0053] For a given scaling factor j0, equation (7) should satisfy the following condition: From this we can further obtain That is, the value range of the translation factor k.

[0054] According to the above process, the beta wavelet function is translated and transformed to obtain a set of beta wavelet basis functions with characteristic parameters (α, β) By changing the values ​​of the characteristic parameters (α, β), multiple groups of beta wavelet basis functions can be obtained.

[0055] According to wavelet theory, the time-varying parameters in the nonlinear model can be expressed as a linear weighted combination of one or more beta wavelet basis functions, as shown in the following formula:

[0056]

[0057] Where i = 1, 2, ..., M is the number of time-varying parameters, is the time-invariant parameter of the linear weighted combination, is the beta wavelet basis function, and is the set of translation factor positions, and l is the number of beta wavelet basis function groups.

[0058] The set of one or more beta wavelet basis functions is recorded as:

[0059]

[0060] Where L is the total number of basis functions in the set.

[0061] Then formula (8) can be simplified as:

[0062]

[0063] Substituting formula (9) into formula (2), we get:

[0064]

[0065] remember:

[0066] X(t)=[π1(t)x1(t),π2(t)x1(t),…,π L (t)x M (t)]

[0067] c=[c 1,1 ,c 1,2 ,…,c M,L ]

[0068] Then formula (10) can be expressed as:

[0069] y(t)=X(t)c T +e(t) (11)

[0070] This formula is the time-invariant parameter model obtained by expanding the beta wavelet basis function.

[0071] From the above process, it can be seen that the beta wavelet basis function expansion method can be used to transform the initial time-varying nonlinear system rapid identification problem into the sparse model structure selection of the time-invariant expansion model and the accurate estimation problem of the corresponding time-invariant parameters. A large amount of modeling experience shows that in the actual modeling process, generally one or two groups of beta wavelet basis functions are used to achieve good system identification accuracy. Therefore, the number of beta wavelet basis functions used in the present invention is reduced relative to the multi-wavelet basis function, thereby effectively reducing the number of redundant items in the expanded model, and fundamentally reducing the computational complexity of the subsequent sparse steps.

[0072] 3. Effective model structure detection: Use the classic OFR algorithm to select effective items for the time-invariant parameter model after the expansion of the beta wavelet basis function, remove redundant items in the time-invariant model, establish an effective sparse model structure, and estimate the corresponding time-invariant model parameters;

[0073] As described in the above step 2, relative to the time-invariant parameter model obtained by the multi-wavelet basis function expansion method, the number of alternatives for model structure identification in the time-invariant model obtained by the beta wavelet basis function expansion has been reduced, which is of great significance for improving the identification accuracy of the OFR algorithm, reducing the complexity of model sparse operations, improving the operation speed, and effectively avoiding model overfitting and ill-conditioned problems. Experience shows that there are still a large number of redundant terms in the above-mentioned time-invariant expansion model, and it is necessary to further select effective terms of the model structure, estimate the corresponding time-invariant parameters, and then establish an effective sparse model structure. The present invention uses the classic OFR algorithm to identify the effective terms of the model and estimate the corresponding time-invariant parameters at the same time, so as to achieve the purpose of removing redundant terms in the model and establishing an effective sparse model structure.

[0074] The OFR algorithm first calculates the alternatives X in the model. i (i=1,2,…,M×L) is orthogonalized to obtain the corresponding orthogonalized candidate wi(i=1,2,…,M×L). The effective items of the model are selected in turn by using the Error Reduction Ratio (ERR) criterion. The ERR expression is as follows:

[0075]

[0076] Among them, Y is the observed signal output sequence, wi is the orthogonalized candidate sequence, M N =M×L is the number of alternatives, and <·,·> represents the vector inner product.

[0077] In the process of selecting valid items, each step is performed by comparing the ERR corresponding to the orthogonalized backup option sequence in that step i The value is used to determine the valid items to be selected. For example, in the first step of selection, let wi = X i , the ERR corresponding to each item is calculated by formula (12) i , take {ERR i} max The corresponding item is option p1 selected in step 1; in the kth step, the k-1 items {p1, p2, …, p k -1} as the remaining options for orthogonal base pairs {X i :i=1,2,…,M N}\{p1,p2,…,p k -1} to orthogonalize, and calculate the ERR values ​​corresponding to the remaining options, and select {ERR} max The corresponding alternative is the kth valid item p kAs can be seen from formula (12), the ERR standard selects valid items by measuring the correlation between the orthogonal backup options and the initial output sequence, that is, the item with the highest correlation in each step is preferentially selected as the valid item obtained in that step.

[0078] The present invention adopts the APRESS cross-validation standard to determine the number of valid items selected in the OFR algorithm. Generalized cross-validation is widely used because of its effectiveness in avoiding model overfitting. The APRESS cross-validation standard is obtained by adding an adjustable factor to the PRESS standard, and its definition is as follows:

[0079]

[0080] Where m is the number of model terms, σ is the adjustment factor, and MSE[m] is the mean sum of squared errors.

[0081] After completing the selection of valid items, the selected model valid items {p1, p2, …, p m}(m <M N ), and use the generalized linear fitting algorithm to estimate the time-invariant parameters {η1,η2,…,η m}.

[0082] 4. Model reconstruction: Combining the effective terms of the sparse model structure and the estimated time-invariant parameters, the time-varying parameters of the initial nonlinear model are obtained by inverse solution, and the sparse time-varying nonlinear parameter model is reconstructed.

[0083] According to the effective items and corresponding time-invariant parameters selected in step 3, they are respectively compared with the alternative items {π n (t)x i (t)} and the time-invariant parameter {c i,n}, and the time-invariant expansion parameters of the sparse model are obtained by reverse solution. Substituting the estimated time-invariant parameters into equation (2), the sparse model of the time-varying nonlinear system can be obtained.

[0084] In order to measure the identification effect of time-varying nonlinear systems, the present invention uses three measurement criteria for evaluation: Mean Absolute Error (MAE), normalized Root Mean Squared Error (RMSE) and standard deviation (Std). The specific expressions of the above criteria are as follows:

[0085]

[0086]

[0087]

[0088] in, is the model prediction output, y(t) is the sample observation value, and N is the sample sequence length.

[0089] It can be seen from the above formula that the smaller the calculated MAE, RMSE, and Std are, the higher the model identification accuracy is and the better the performance of the identified time-varying nonlinear model is.

[0090] The following two nonlinear simulation examples verify the time-varying nonlinear system identification accuracy based on the beta wavelet basis function expansion method proposed in the present invention, and compare the identification effect with the existing B-spline multi-wavelet basis function expansion method:

[0091] Simulation Example 1 builds a time-varying nonlinear system:

[0092] y(t)=a1(t)y(t-1)+a2(t)y(t-1)y(t-2)+b1(t)u(t-1)

[0093] +b2(t)u(t-1)u(t-2)+e(t) (17)

[0094] The time-varying parameters are:

[0095]

[0096] a2(t)=0.4cos(4πt / N),1≤t≤N

[0097]

[0098] b2(t)=0.7,1≤t≤N

[0099] Among them, the sample sequence length N=512, the model input u(t) is a pseudo-random binary sequence, and e(t) is a Gaussian white noise with a mean of 0 and a variance of 0.01.

[0100] The signal-to-noise ratio obtained according to the above simulation model is about 20dB. The beta wavelet basis function expansion method proposed in the present invention and the existing B-spline multi-wavelet basis function expansion method are used to construct a time-varying nonlinear model. A large number of modeling experiments show that the combination of multi-wavelet B-spline basis functions of order 3, 4, and 5 can obtain better parameter identification results. Therefore, in the present invention, according to the model performance evaluation standard formulas (13), (14) and (15), a combination of 3rd to 5th order B-spline functions is used to identify the time-varying nonlinear system. The beta wavelet basis function control parameter selection {(α1=3, β1=7), (α2=5, β2=6)} combination achieves better identification results. According to the APRESS cross-validation standard, the effective model items are 80 items. The time-varying parameter estimation effects obtained by the two methods are shown as follows. Figure 2 (a) shows the predicted output and the actual output obtained by the two basis function identification methods. Figure 2 (b), and the comparison results of time-varying parameter identification accuracy are listed in Table 1.

[0101] Table 1 Comparison of identification accuracy of time-varying nonlinear system of simulation example 1

[0102]

[0103] Simulation Example 2

[0104] In order to further verify the effectiveness of the method proposed in this invention, a nonlinear time-varying system with colored noise is considered:

[0105]

[0106] The time-varying parameters are:

[0107]

[0108] a2(t)=-0.05+0.5cos(4πt / N),1≤t≤N

[0109]

[0110]

[0111] The input u(t) is:

[0112]

[0113] Among them, the sample sequence length N = 1000, v(t) is a Gaussian distribution sequence with a mean of 0 and a variance of 1; e(t) is a Gaussian white noise with a mean of 0 and a standard deviation of 0.02.

[0114] Similar to simulation example 1, according to the APRESS cross-validation standard, the effective model items are 70 items. The identification results obtained by expanding the two basis functions are as follows: Figure 3 The comparison results of time-varying parameter identification accuracy are shown in Table 2.

[0115] Table 2 Comparison of time-varying parameter identification accuracy of simulation example 2

[0116]

[0117] From the comparison results of the time-varying parameter identification accuracy of simulation examples 1 and 2, it can be seen that the time-varying parameter identification effect obtained by using the beta wavelet basis function expansion method proposed in the present invention is significantly better than the identification effect of the B-spline expansion method under the same conditions. Figure 2 and Figure 3 It can be seen that for time-varying parameters with sudden changes, the identification effect of the beta wavelet basis function expansion method is more significant. In addition, in terms of the computational complexity of model identification, the operation time of the beta wavelet basis function expansion method is also significantly less than the operation time required by the B-spline expansion method, and the time computational complexity is lower.

[0118] The time-varying nonlinear system identification method based on beta wavelet basis function expansion proposed in the present invention is applied to the time-varying nonlinear modeling of real EEG signals, wherein the EEG signal data used in the present invention is derived from the public database of the University of Bonn, the data sampling frequency is 173.61Hz, the sampling time is 23.6s, and the epilepsy signal band (6s) is selected for time-varying nonlinear parameter modeling. The model parameter setting is the same as the above simulation example. According to the APRESS cross-validation standard, 18 effective model items are obtained. According to the constructed model, the comparison between the model prediction output and the real EEG signal tracking effect is obtained as follows Figure 4 It can be seen that the time-varying nonlinear fast identification method based on the beta wavelet basis function expansion proposed in the present invention also has a good tracking effect on the real EEG signal, which verifies the applicability of the method proposed in the present invention.

Claims

1. A sparse nonlinear time-varying parameter model reconstruction method based on beta wavelet basis function expansion for EEG signal recognition, characterized by include: Step 1: Take the nonlinear combination of the system output signal regression term and the exogenous input signal regression term as the input term of the system model to construct a nonlinear time-varying autoregressive model with exogenous input; Step 2: Use the first-order beta wavelet basis function to perform linear weighted expansion on the time-varying parameters in the nonlinear time-varying autoregressive model to obtain the beta wavelet expansion model; Step 3: Use the classical orthogonal forward regression algorithm to select effective items from the time-invariant parameter model after the beta wavelet basis function is expanded, remove the redundant items in the expanded model, establish a sparse model structure, and estimate the corresponding time-invariant parameters; Step 4: Combine the effective terms of the sparse model structure and the estimated time-invariant parameters, obtain the time-varying parameters of the initial nonlinear autoregressive model through inverse solution, and reconstruct the sparse nonlinear time-varying parameter model. in: The step 2 comprises: using the beta wavelet basis function to expand the time-varying parameters of the nonlinear autoregressive model containing exogenous input, expressing it as a linear weighted form of the beta wavelet basis function, and then establishing a beta wavelet extended time-invariant parameter model, that is, the time-dependent time-varying parameters θ(t) = [θ1(t), θ2(t), I, θ M (t)] is transformed into the polynomial form of beta wavelet basis function where c i，n is the corresponding time-invariant parameter, π n (t) is the beta wavelet basis function, M is the number of time-varying parameters, and L is the set of beta wavelet basis functions. The total number of basis functions in , The step 3 includes: using the classical forward regression algorithm to establish a sparse model structure, that is, removing redundant items from the beta wavelet expansion model and selecting effective model items; using the APRESS cross-validation standard to determine the number of effective items, that is, as a criterion for judging the complexity of the model structure.

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