Reduction Gaussian kernel adaptive filtering method and system based on nearest center estimation

Abstract

The invention provides a reduced Gaussian kernel adaptive filtering method and system based on nearest center estimation, and relates to the field of data processing, and the method comprises the steps: obtaining an initial parameter and a training set, and constructing an adaptive filtering system based on the initial parameter; performing cluster division on an input signal, obtaining a feature vector of the input signal by calculating a reduced Gaussian kernel, and selecting a corresponding sub-filter according to a cluster for prediction to obtain prediction output; constructing a target optimization problem according to a maximum correlation entropy criterion based on an error between prediction output and expected output, and converting the target optimization problem into a convex optimization problem; solving a convex optimization problem by hybridizing three conjugate gradients, and updating the weight of a sub-filter; and repeating the steps to converge the error of the adaptive filtering system, and completing the adaptive filtering process. According to the method, the problem of dimension disasters existing in a traditional feature mapping method is solved by reducing the Gaussian kernel, the calculation complexity is remarkably reduced, and meanwhile unnecessary calculation overhead is greatly reduced through nearest center estimation.

Description

Technical Field

[0001] The present invention relates to the technical field, and in particular to a reduced Gaussian kernel adaptive filtering method and system based on nearest center estimation. Background Art

[0002] Adaptive filter theory is an important component of modern digital signal processing and has a broad and significant impact on the development of signal and information processing technology. The ideal operating logic of an adaptive filter (AF) is to design a reasonable filtering algorithm so that the output result is fed back to the filter structure to adaptively adjust the filter parameters to achieve the ideal filtering effect.

[0003] Kernel adaptive filtering (KAF) is a commonly used method for dealing with complex nonlinear problems. The most classic KAF algorithms include the kernel least mean square (KLMS) algorithm and the kernel recursive least mean square (KRLS) algorithm based on mean square error (MSE). These filtering algorithms have the following major drawbacks in practical applications:

[0004] As the number of iterations increases, the network size of the algorithm grows nearly linearly. This growth leads to higher computational complexity when processing large-scale data, and occupies a large amount of computing and storage resources. This makes KAF difficult to apply to online adaptive filtering applications with limited computing resources. The KLMS algorithm uses the stochastic gradient descent (SGD) method to search in the negative gradient direction, which is prone to falling into local optimality and has a slow convergence speed. The KRLS algorithm uses matrix update recursion to solve, and although it has good convergence speed and filtering accuracy, its main difficulty lies in the estimation of the inverse matrix of the Hessian matrix. The inversion process greatly increases the computational complexity and may lead to numerical instability. When the background noise is non-Gaussian, the use of MSE as the error metric often causes the algorithm to perform poorly, specifically manifested in low accuracy and poor stability. Summary of the Invention

[0005] The purpose of the present invention is to provide a reduced Gaussian kernel adaptive filtering method and system based on nearest center estimation to improve the above problems. In order to achieve the above object, the technical solution adopted by the present invention is as follows:

[0006] In a first aspect, the present invention provides a reduced Gaussian kernel adaptive filtering method based on nearest center estimation, comprising:

[0007] Acquiring initial parameters and a training set, and constructing an adaptive filtering system based on the initial parameters, wherein the training set includes an input signal and a desired output;

[0008] The input signal is clustered and the eigenvector of the input signal is obtained by calculating the reduced Gaussian kernel. The corresponding sub-filter is selected according to the cluster for prediction to obtain the predicted output.

[0009] Based on the error between the predicted output and the expected output, the target optimization problem is constructed according to the maximum relevant entropy criterion and converted into a convex optimization problem.

[0010] The weights of the sub-filters are updated by solving the convex optimization problem through hybrid three-term conjugate gradient;

[0011] Repeat the above steps to make the error of the adaptive filtering system converge and complete the adaptive filtering process.

[0012] In a second aspect, the present invention provides a reduced Gaussian kernel adaptive filtering system based on nearest center estimation, comprising:

[0013] An acquisition module is used to acquire initial parameters and a training set, and construct an adaptive filtering system based on the initial parameters, wherein the training set includes an input signal and a desired output;

[0014] The prediction module is used to cluster the input signal and obtain the feature vector of the input signal by calculating the reduced Gaussian kernel. The corresponding sub-filter is selected according to the cluster for prediction to obtain the predicted output;

[0015] A building module is used to construct a target optimization problem based on the error between the predicted output and the expected output according to the maximum relevant entropy criterion, and transform the target optimization problem into a convex optimization problem;

[0016] The update module is used to solve the convex optimization problem by hybridizing three conjugate gradients and update the weights of the sub-filters;

[0017] The loop module is used to repeat the above steps to make the error of the adaptive filtering system converge and complete the adaptive filtering process.

[0018] The beneficial effects of the present invention are:

[0019] The present invention uses the low-order terms of the Taylor expansion of the Gaussian kernel to approximate the traditional feature mapping method, thereby effectively solving the dimensionality curse problem existing in the traditional feature mapping method, thereby significantly reducing the computational complexity of the nonlinear filter algorithm.

[0020] Using the nearest center estimation theory to construct multiple sub-filters, the RGKHTCG algorithm is only executed on the sub-filter closest to the input data and its filter weights are updated. The remaining sub-filters remain static and do not participate in the calculation. This dynamic update strategy further reduces unnecessary computational overhead and achieves a quadratic optimization of computational complexity.

[0021] When faced with complex scenarios with non-Gaussian noise interference, the introduction of the maximum correlation entropy criterion significantly enhances the algorithm's noise resistance, ensuring that the algorithm can still run stably in harsh noise environments and avoiding a sharp decline in performance.

[0022] Other features and advantages of the present invention will be set forth in the following description, and in part will be apparent from the description, or may be learned by practicing embodiments of the present invention. BRIEF DESCRIPTION OF THE DRAWINGS

[0023] In order to more clearly illustrate the technical solutions of the embodiments of the present invention, the following briefly introduces the drawings required for use in the embodiments. It should be understood that the following drawings only illustrate certain embodiments of the present invention and therefore should not be regarded as limiting the scope. For ordinary technicians in this field, other relevant drawings can be obtained based on these drawings without paying any creative work.

[0024] Figure 1 This is a structural diagram of the adaptive filter system in an embodiment of the present application;

[0025] Figure 2 This is a schematic diagram of the kernel method in the embodiment of this application;

[0026] Figure 3 This is a flow chart of a reduced Gaussian kernel adaptive filtering method based on nearest center estimation according to an embodiment of the present application;

[0027] Figure 4 This is a performance comparison of the RGKHTCG algorithm of the embodiment of the present application under different expansion orders (Gaussian noise environment);

[0028] Figure 5 This is a performance comparison between the RGKHTCG algorithm and the RFFCG algorithm in the MG sequence prediction of the embodiment of this application under the same dimension (Gaussian noise environment);

[0029] Figure 6 This is the MSE convergence curve of different algorithms in the MG sequence prediction of the embodiment of this application: (non-Gaussian noise environment);

[0030] Figure 7 Schematic diagram of the structure of the reduced Gaussian kernel adaptive filtering system based on nearest center estimation in an embodiment of the present application.

[0031] Markings in the figure: 100-acquisition module; 200-prediction module; 210-first calculation unit; 220-first update unit; 221-judgment subunit; 230-mapping unit; 240-prediction unit; 300-construction module; 310-second calculation unit; 320-first construction unit; 330-conversion unit; 400-update module; 410-second construction unit; 420-second update unit; 430-third update unit; 500-loop module. DETAILED DESCRIPTION

[0032] In order to make the purpose, technical solutions and advantages of the embodiments of the present invention clearer, the technical solutions in the embodiments of the present invention will be clearly and completely described below in conjunction with the drawings in the embodiments of the present invention. Obviously, the described embodiments are part of the embodiments of the present invention, not all of the embodiments. The components of the embodiments of the present invention generally described and shown in the drawings herein can be arranged and designed in various different configurations. Therefore, the following detailed description of the embodiments of the present invention provided in the drawings is not intended to limit the scope of the claimed invention, but merely represents selected embodiments of the present invention. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without making creative work are within the scope of protection of the present invention.

[0033] It should be noted that similar reference numerals and letters denote similar items in the following drawings, and therefore, once an item is defined in one drawing, it does not need to be further defined or explained in subsequent drawings.

[0034] First, the relevant technologies used in the method of this application are introduced.

[0035] Adaptive filtering system:

[0036] Adaptive filter (AF) is a type of "self-optimizing" filter based on error correction learning mechanism. According to the linear separability of input data, AF can be divided into two categories: linear and nonlinear. Among them, the structure of linear AF is relatively simple and can be regarded as a special case of nonlinear AF. The classic linear AF structure is as follows: Figure 1 As shown. Figure 1 As can be seen from the figure, the classic AF system structure is mainly composed of two parts: filter and filtering algorithm. The filter provides the structural framework, and the filtering algorithm implements the core computing function. At discrete time i, the AF workflow can be divided into the following steps: (1) Input signal After AF processing, the actual output y is generated i =ω T x i ,in represents the coefficient vector of AF, m is the dimension of the input signal; (2) the input signal xi passing through unknown systems and being subjected to noise After interference, the expected output d is obtained i , the expected output The actual output y i The error signal e is obtained by subtracting i =d i -y i , in the initial stage, this error is usually large; (3) The AF algorithm will iteratively adjust the AF coefficient ω through the optimization method according to the selected error criterion to minimize the error e i When the optimization process converges, the output of AF will be as close as possible to the expected output, thus achieving the optimal match (based on the error criterion).

[0037] Kernel methods:

[0038] When faced with nonlinear systems, traditional linear AF often fails to achieve ideal performance, resulting in a significant decrease in filtering effect. The core idea of ​​the kernel method, as a powerful machine learning tool, is to use a nonlinear mapping function to map data from the original space to a high-dimensional RKHS (Reproducing Kernel Hilbert Space). This mapping transforms the originally complex low-dimensional nonlinear problem into a high-dimensional linear problem, making the solution process simpler. Figure 2 The uniqueness of the kernel method is that it does not require an explicit mapping function. Instead of using the specific form of , the inner product of the data in the kernel space is directly calculated using the Mercer kernel function. This approach not only effectively solves the dimensionality difference between the high-dimensional feature space and the original data space, but also gives the kernel method stronger modeling capabilities than traditional nonlinear extended networks.

[0039] The Mercer kernel is a continuous, symmetric and positive definite kernel function, usually expressed as in is the input space. Among the common Mercer kernels, the Gaussian kernel function is often preferred due to its excellent approximation ability and numerical stability. Its mathematical form is

[0040]

[0041] Where σ>0 represents the Gaussian kernel width, and ||·|| represents the Euclidean norm. According to Mercer's theorem, there exists a feature space H and a mapping It is worth noting that the feature space H may be infinite-dimensional, so it is straightforward to calculate the mapping is not feasible. By using the "kernel trick", we can avoid directly calculating Right now:

[0042]

[0043] Furthermore, the KAF model based on Mercer kernel mapping can be expressed as

[0044]

[0045] Where f(·) represents the functional mapping relationship between input and output, α k Represents the weight vector corresponding to the k-th input data.

[0046] As the above formula shows, the dimension of the feature space H is determined by the amount of input data. As the input data increases, the value k in the summation symbol accumulates all data from the initial moment to the current moment. Therefore, the dimension of this feature space H is greater than that of the original space. The significance of mapping is that after mapping the input data to a higher-dimensional space, nonlinear problems are transformed into linear problems in that higher-dimensional space, greatly simplifying the solution process.

[0047] Reduced Gaussian kernel:

[0048] Represents the feature mapping function that maps data in the original space to a high-dimensional feature space. In kernel methods, the dimensionality of this mapping can be infinite, and its complexity increases significantly with the amount of data. Since explicitly representing such high-dimensional feature mappings is generally not feasible in practical applications, a Gaussian kernel function is required to perform a proper decomposition to obtain a workable feature mapping expression. Specifically, the decomposition process is as follows:

[0049]

[0050] Among them, σ 2 is the variance of the Gaussian distribution, It can be regarded as the inner product of the function of exp(·), but the third term in the formula cannot be decomposed into the form of the inner product of the function. Therefore, the Taylor expansion method is used to approximately expand the third term. For the convenience of expression, let Right now:

[0051]

[0052] Where, represents the Lagrange remainder in the Taylor expansion. It is used to quantify the function approximation error. By substituting equation (1.5) into equation (1.4), the Gaussian kernel function can be approximated.

[0053]

[0054] Where p represents the order of the Gaussian kernel through Taylor expansion, and o(n) represents the remainder. Therefore, the explicit feature map vector It can be expressed as Where P represents the order, and the p-th order eigenvector is:

[0055]

[0056] Therefore, the Gaussian kernel function of formula (1.1) can be approximately rewritten as

[0057]

[0058] In order to prove the approximate kernel function κ given in formula (1.8) A (x i ,x j ) is a kernel function at any order p and satisfies Mercer's theory, which can be proved by the following steps:

[0059] First, suppose there is an arbitrary kernel function K q and K p ,and Let f(·) be a real-valued function in space X, and assume that According to the properties of the kernel function, the following proposition holds:

[0060] Proposition 1: If K q and K p are all kernel functions, then K = K q +K p It is also the kernel function.

[0061] Proposition 2: If K q is a kernel function, then according to the properties of the kernel function, K = αK q (where α>0) is also a kernel function.

[0062] Proposition 3: There exist functions f(x1),f(x2) such that K=f(x1)f(x2) which are also kernel functions.

[0063] Assumption 1: For any p-order reduced Gaussian kernel, it is a kernel function.

[0064] If Assumption 1 holds, then according to Proposition 1, for all values ​​of order, the approximate Gaussian kernel function is a kernel function. Specifically, for any p = 1, 2, ..., P, the function is the kernel function. Therefore, we only need to take an approximate Gaussian function of a certain order as the kernel function, and the sum of all orders will also meet the definition of the kernel function.

[0065] Assumption 2: Assume that the kernel parameter a is greater than 0.

[0066] If assumption 1 is satisfied, the coefficients of each order are are all greater than 0. If Assumption 2 also holds, based on Proposition 2, for any order p=1,2,...,P, Established.

[0067] Furthermore, according to Proposition 3, we can derive the following equation:

[0068]

[0069] Therefore, for any order value, the reduced Gaussian kernel function given in formula (1.8) is a kernel function.

[0070] Hybrid Three-term Conjugate Gradient (HTCG):

[0071] In this invention, in order to make full use of the information of the target optimization problem in the current iteration and construct an effective algorithm for solving the unconstrained optimization problem, a hybrid three-term conjugate gradient (HTCG) optimization method based on convex combination technology is introduced. Its search direction d k and its conjugate parameter are defined as follows:

[0072]

[0073] Where -g k is the negative gradient direction vector, y k-1 =g k -g k-1 Is the difference between the current gradient direction vector and the previous gradient direction vector. Convex combination coefficient λ k ∈[0,1], represents the weighted proportion in the direction update. Step size and are from the Dai-Yuan (DY) method and the Polak-Ribière-Polak (PRP) method, respectively, and It is a hybrid of these two methods, which determines the step size by taking the minimum of the two and ensuring its non-negativity. k Used to adjust the search direction, η∈[0,1]. By changing the value of η, the adjustment amplitude of the search direction can be controlled; the search direction vector d k By adding the negative gradient direction vector -g k , the search direction vector of the previous iteration and the gradient vector θ k g k-1 The hybrid three-term conjugate gradient method effectively integrates more information about the target optimization problem in the current iteration and can make better use of historical gradient information, thereby improving the search accuracy and convergence speed.

[0074] Nearest Instance Centroid Estimation (NICE):

[0075] In the KAF algorithm, the number of "dictionaries" grows dynamically with the input. KAF using a sparse approach requires calculating the Euclidean distance between each incoming input data point and all previous data points. This results in a significant time consumption, even exceeding the time saved by the sparse KAF algorithm. To address this issue, clustering is introduced into the sample selection process. Clustering divides data into clusters, with the centroid of each cluster representing the entire cluster. Therefore, the distance between the current input data point and an element in the cluster can be approximately expressed as the distance between the current data point and the cluster centroid. Therefore, if the distance between the current data point and the cluster centroid exceeds a preset distance threshold, it can be roughly considered that the distance between the current data point and every element in the cluster is also greater than the threshold. When learning from the current data point, the influence of all elements in that cluster can be ignored, thereby improving the algorithm's computational efficiency. In the clustering process, the input data point is typically directly assigned to the cluster closest to it. This cluster then serves as the optimal "dictionary" for the input data point.

[0076] The NICE algorithm uses clustering within the framework of subspace decomposition to divide sample points into multiple clusters, where different clusters can be considered quasi-orthogonal regions. Each cluster in the NICE algorithm is considered an independent sub-filter, responsible for filtering different parts of the feature space. This involves calculating the distance between the input data and the centroid of each cluster, applying the standard KAF algorithm to the cluster closest to the input data, and updating the corresponding filter weights.

[0077] Maximum Correntropy Criterion (MCC):

[0078] In the Information Theoretic Learning (ITL) method, as a similarity measure between two random variables X and Y, the correlation entropy is defined in the kernel space as follows

[0079]

[0080] Where F XY (x,y) is the joint distribution function of (X,Y), κ σ (XY) is a time-invariant Mercer kernel with a kernel width σ> 0. The Gaussian kernel is usually used to define the correlation entropy, which is expressed as:

[0081]

[0082] Correlation Entropy Semi-quadratic Optimization:

[0083] For the correlation entropy, when |e i |≤σ, its convexity cannot be guaranteed because its Hessian matrix is ​​not positive definite. Therefore, correlation entropy cannot be directly used as an error criterion for convex optimization problems. However, semi-quadratic optimization can be effectively applied to non-convex optimization problems. It transforms the original objective optimization problem into a semi-square function by introducing intermediate variables. To this end, a semi-quadratic optimization method is used to transform the maximum correlation entropy problem into a weighted least squares problem, thereby establishing a correlation entropy semi-quadratic optimization method.

[0084] Since the correlation entropy is an exponential function, consider the following general form of the exponential function: f(x) = exp(-x), According to the conjugate function theorem, the conjugate function of f(x) is g(v) = -vln(-v) + v, where v < 0. Clearly, g(v) is a convex function. Based on this auxiliary function g(v), we can derive the following proposition.

[0085] Proposition: The exponential function in the correlation entropy is the conjugate function of the convex function g(v) = -vln(-v) + v, where v < 0; that is:

[0086]

[0087] Where e i =y i -f(x i ), and when The upper bound of the formula is obtained.

[0088] Proof: According to the definition of conjugate function, the conjugate function of g(v) is as follows:

[0089]

[0090] Let f(v) = uv + v ln(-v) - v, where v < 0. Therefore, when v = -exp(-u), f(v) reaches its maximum value, that is

[0091]

[0092] Where v = -exp(-u). Then, let We can get

[0093]

[0094] Its supremum is The proof of this proposition is complete.

[0095] Therefore, the problem of solving the optimal solution of the relevant entropy can be equivalent to solving the following problem:

[0096]

[0097] In the formula It's about v i independent functions. The above problem can be solved by the alternating optimization method. Therefore, first use (1.13) to find (1.17) with respect to v i The analytical solution is Herev i <0. Next, given v i , the solution of the optimization problem (1.17) can be derived into the following weighted least squares problem:

[0098]

[0099] Where N is the total number of data;

[0100]

[0101] From (1.18) and (1.19), it can be seen that the weighted least squares problem is a global convex optimization problem. In this application, the above method for solving the correlation entropy is called the correlation entropy semi-quadratic optimization method.

[0102] Example 1

[0103] See also Figure 3 , this embodiment provides a reduced Gaussian kernel adaptive filtering method based on nearest center estimation, including steps S100, S200, S300, S400 and S500;

[0104] S100, obtaining initial parameters and a training set, and constructing an adaptive filtering system based on the initial parameters, wherein the training set includes an input signal and a desired output;

[0105] S110, obtain initial parameters and initialize the adaptive filtering system: the maximum number of iterations of the filtering system is N (i.e. the number of data, one data is iterated once), the order of the filtering system is m, the dimension of the Gaussian kernel is reduced to P, the cluster C1 = {x1} is initialized, the cluster centroid c1 = x1 is initialized, and the clustering threshold parameter r is used. c , the number of initial clusters K = 1, the size of the initial cluster s1 = 1, the weight of the initial adaptive filtering system

[0106] S120 , obtaining a data set, where the data set includes an input signal vector and an original expected output signal vector.

[0107] Among them, the input signal vector is x1,x2,...,x N , xi represents the input signal vector of the filter at the i-th moment, represents the original expected output signal vector of the filter at the i-th moment; x i The corresponding expected output of the input filter is d i ; d1,d2,...,d N ; i=1,2,3,...,N. q i =(1-b i )V A,i +b i V B,i is the mixed noise generated by the system, b i Generated by a Bernoulli random process, satisfying Pr{b i =1}=h,Pr{b i =0}=1-h, 0≤h≤1 controls Gaussian noise V A,i and non-Gaussian noise V B,i The probability of occurrence. After adding noise, the expected output is obtained.

[0108] S200, clustering the input signal and obtaining a feature vector of the input signal by calculating a reduced Gaussian kernel, selecting a corresponding sub-filter according to the cluster for prediction, and obtaining a prediction output;

[0109] S210, calculate input signal x i The minimum distance to the centroid of each cluster; the initial centroid of the cluster is the position of the first input signal;

[0110] Let the current cluster centroid set C={c1,c2,...,c K};

[0111] r min =min 1≤j≤K ‖x i -c j ‖ 2 ;

[0112] r min Represents the minimum distance; this calculation evaluates the relationship between the current input data and the centroid of each cluster, thus providing a basis for subsequent cluster assignment decisions.

[0113] S220, allocating the input signal according to the relationship between the minimum distance and the preset threshold, and updating the clusters; constructing a sub-filter for each cluster;

[0114] The input signal distribution strategy is as follows:

[0115] Determine whether the minimum distance is less than or equal to the preset threshold. If so, assign the input signal to the cluster corresponding to the nearest centroid and update the centroid of the cluster; set the preset threshold to ≤r c ;

[0116] That is, when r min ≤r c , x i It is assigned to the cluster with the closest centroid, that is:

[0117]

[0118] where j * To x i Nearest Cluster The index of , which satisfies:

[0119]

[0120] definition For the jth * The number of input data in the clusters, and is the data set of the cluster. When new input data arrives, the jth * The centroid of the cluster Will be adjusted to:

[0121]

[0122] This is a linearly weighted incremental smoothing process that proportionally adjusts the positions of historical centroids and new data. In this way, the cluster centroid gradually moves toward the center of the new input data distribution, allowing the cluster to better adapt to changes in dynamic data streams.

[0123] If the minimum distance is greater than the preset threshold, a new cluster is generated with the input signal as the initial centroid, and all data in the cluster closest to the input signal are migrated to the new cluster, and the centroid of the new cluster remains unchanged as the input signal;

[0124] That is r min >r c When , a new cluster is initialized to accommodate the input data x i , and x i As the initial centroid. In order to ensure a smooth transition in the classification process and avoid significant discontinuities, all data and related parameters of the nearest cluster need to be migrated to the new cluster. These migrated data will not be used to update the centroid of the new cluster. At this time, the total number of clusters K and the centroid of the new cluster The updates are as follows:

[0125] K=K+1

[0126]

[0127] S230, constructing a mapping function by reducing the Gaussian kernel, inputting the input data into the mapping function, and obtaining a feature vector;

[0128] After completing the cluster assignment, we need to process the high-dimensional features of the input data. To efficiently process these features, this application uses a reduced Gaussian kernel (RGK) to approximate the Gaussian kernel function. Specifically, RGK converts the calculation of the Gaussian kernel function into an explicit calculation of the input data.

[0129]

[0130] in Denotes the explicit mapping function, and P denotes the order. Therefore, for an approximate kernel mapping network with a finite order P, the network is no longer a growing weighted network, thus avoiding the computational burden brought by the corresponding complex operations.

[0131] S240 , inputting the feature vector of the input signal into the corresponding sub-filter according to the cluster to which the input signal belongs, and obtaining a prediction output.

[0132] After completing cluster assignment and feature mapping, the prediction output is calculated through the sub-filter. i Assigned to When the system selects the corresponding sub-filter j * To calculate the predicted output. Specifically, the output of the sub-filter is given by the following formula:

[0133]

[0134] in is the sub-filter j * The weight vector of .

[0135] S300, based on the error between the predicted output and the expected output, construct a target optimization problem according to the maximum relevant entropy criterion, and transform the target optimization problem into a convex optimization problem;

[0136] S310, calculating a prediction error based on the predicted output and the expected output;

[0137]

[0138] Among them, e i is the prediction error;

[0139] S320: Based on the prediction error, construct a target optimization problem according to a maximum relevant entropy criterion;

[0140] Replace y in formula (1.18) i and f(x i) are replaced by d i and get:

[0141]

[0142] in,

[0143] The above formula can be further rewritten as:

[0144] Where, Λ=diag(v1,v2,…,v N ) is a weighted diagonal matrix, d=[d1,...,d N ] is the expected vector, Represents the coefficient vector of the adaptive filter; ω T X represents the transpose of ω; X=[x1,…,x N ] is the input matrix. Solve This is the target optimization problem.

[0145] S330. Based on the semi-quadratic optimization method, an intermediate variable is introduced to convert the target optimization problem into a positive definite quadratic function, thereby obtaining a convex optimization problem.

[0146] Let R = XΛX T ,b=XΛd T , we get the quadratic form corresponding to the above formula:

[0147]

[0148] Where R = R T >0, so the above formula is a positive definite quadratic problem, that is, a convex optimization problem.

[0149] After converting the minimization problem of the target optimization problem into a convex optimization problem, the convex optimization method can be used to solve it. Next, the hybrid three-term conjugate gradient method is used to solve the problem.

[0150] S400, solving the convex optimization problem by hybridizing three conjugate gradients, updating the weights of the sub-filters until the maximum number of iterations is reached, and obtaining the final weights;

[0151] S410, based on the minimization error criterion, transform the convex optimization problem into a positive definite quadratic optimization problem;

[0152] In order to minimize the error, sub-filter j * The corresponding positive definite quadratic optimization problem is:

[0153]

[0154] in, is the weighted autocorrelation matrix, is x i with d i The weighted cross-correlation vector between .

[0155] S420, iteratively updating the autocorrelation matrix and cross-correlation vector in the positive definite quadratic optimization problem based on an exponentially decaying data window method;

[0156] To enhance the adaptability of the algorithm and suppress the influence of historical data on the current estimation, this method introduces an exponential decay data window method. According to this method, matrices and vectors can be iteratively updated in the following way:

[0157]

[0158] Where ρ∈(0,1) is the forgetting factor. Weighting function The forgetting factor ρ directly affects the adaptability of the algorithm and its utilization of historical data. Typically, ρ is initially set to a value close to 1, such as 0.9 or 0.99.

[0159] S430 iteratively updates the weights in the positive definite quadratic optimization problem based on the hybrid three-term conjugate gradient method;

[0160] First, sub-filter j * The corresponding search direction and conjugate parameters of the hybrid three-term conjugate gradient optimization method are defined as follows:

[0161]

[0162] Subfilter j * The weight vector The iterative update formula is:

[0163]

[0164] Among them, the step size parameter α i It can be expressed as follows:

[0165]

[0166] where the residual vector represents the gradient vector, The calculation method is:

[0167]

[0168] The iterative steps S100 to S400 allow the error value of the adaptive filtering system to converge, that is, the adaptive filtering system reaches the maximum number of iterations.

[0169] S500 , repeating the above steps to allow the error of the adaptive filtering system to converge, thus completing the adaptive filtering process.

[0170] The present invention further verifies the performance of the filtering method through Mackey-Glass chaotic time series prediction experiments. The experiments are divided into two parts: Gaussian noise environment and non-Gaussian noise environment to comprehensively evaluate the robustness and adaptability of the algorithm.

[0171] Non-Gaussian noise is generated by simulating a mixture model:

[0172] q i =(1-b i )V A,i +b i V B,i

[0173] b i Generated by a Bernoulli random process, satisfying Pr{b i =1}=h,Pr{b i =0}=1-h, 0≤h≤1 controls Gaussian noise V A,i and non-Gaussian noise V B,i Probability of occurrence.

[0174] In terms of data sets, the training set contains 2000 noise-added samples, and the test set consists of 200 noise-free samples. The mean square error (MSE) curve in dB is defined as the performance indicator of the evaluation algorithm. The specific form is as follows

[0175]

[0176] Where L represents the length of the test dataset, y i Denotes the expected output d i In all simulation experiments in this paper, the convergence curve is taken from the average results of 50 independent Monte Carlo (MC) simulations, and the steady-state MSE value is calculated by averaging the last 100 iterations.

[0177] The Mackey-Glass chaotic time series (MG series) is a series with chaotic dynamic characteristics, predictability, and periodicity. It is widely used to evaluate the performance of algorithms in nonlinear environments. The series can be generated by the following delay differential equation:

[0178]

[0179] The time series value obtained in the formula is the discretized result obtained by sampling once every 6 seconds. The goal of the MG time series prediction task is to predict the time series value x based on the first 7 time series values.i =[x i-7 ,x i-6 ,…,x i-2 ,x i-1 ] T Predict the current time series value d i =x i . Where d i Disturbed by noise.

[0180] Gaussian noise: in, The mean is 0 and the variance is Gaussian noise.

[0181] Non-Gaussian noise: Among them, S[0.8,0,0.1,0] represents an α-stable distribution with a characteristic index of 0.8, a symmetry parameter of 0, a dispersion parameter of 0.1, and a location parameter of 0.

[0182] See also Figure 4 , showing the performance of the RGKHTCG (Reduced Gaussian Kernel-Hybrid Triterm Conjugate Gradient) algorithm under different expansion orders P (increasing from 0 to 3). The observation results show that as the expansion order P increases, the steady-state MSE of the RGKHTCG algorithm decreases significantly. This phenomenon can be attributed to the fact that higher-order expansions can capture more complex nonlinear relationships, thereby enhancing the predictive ability of the algorithm. However, when P>2, the effect of further increasing the order on performance improvement becomes less significant. This indicates that after a certain point, the benefits of increasing the expansion order will gradually decrease. Therefore, in order to achieve a reasonable balance between computational complexity and filtering performance, P=2 was selected as the optimal expansion order of the RGKHTCG algorithm.

[0183] Figure 5 It can be seen that under the same dimension, the prediction performance of the RGKHTCG algorithm is better than the RFFCG (Random Fourier-Conjugate Gradient) algorithm, which illustrates the significant advantages of RGK over Random Fourier Features (RFF) in nonlinear environments.

[0184] Figure 6 This is a comparison chart of the effects of this method and other related robust adaptive filtering methods in a non-Gaussian noise environment. The specific comparison results are shown in Table 1. Figure 6 According to the results in Table 1, NICE-RGKHTCG-MCC demonstrates excellent robustness and filtering performance with the lowest storage cost, highlighting its significant advantages in complex noisy environments.

[0185] Table 1 Detailed comparison results of different algorithms

[0186] algorithm dictionary Runtime MSE NICE-KMSLMS 883.3 18.79 -20.23 NICE-QKMSLMS 140 2.90 -20.01 NICE-KLMP 883.3 18.61 -20.97 NICE-QKLMP 140 2.92 -20.52 RGKHTCG-MCC 21 0.02 -26.57 NICE-RGKHTCG-MCC 21 0.007 -26.70

[0187] In Table 1:

[0188] NICE-KMSLMS: Nearest Center Estimation-Kernel Corrected Signed Least Mean Squares algorithm;

[0189] ICE-QKMSLMS: Closest Center Estimation-Quantization Kernel Corrected Signed Least Mean Squares Algorithm

[0190] NICE-KLMP: Nearest Center Estimation-Kernel Minimum Mean p-Norm Algorithm;

[0191] NICE-QKLMP: Nearest Center Estimation-Quantized Kernel Minimum Mean p-Norm Algorithm;

[0192] RGKHTCG-MCC: Reduced Gaussian kernel hybridization three-term conjugate gradient-maximum correlation entropy algorithm;

[0193] NICE-RGKHTCG-MCC: Nearest Center Estimation-Reduced Gaussian Kernel Hybrid Three-Term Conjugate Gradient-Maximum Correlation Entropy Algorithm, that is, the reduced Gaussian kernel adaptive filtering method based on nearest center estimation in this application.

[0194] Example 2

[0195] This embodiment provides a reduced Gaussian kernel adaptive filtering system based on nearest center estimation, including:

[0196] An acquisition module 100 is used to acquire initial parameters and a training set, and construct an adaptive filtering system based on the initial parameters, wherein the training set includes an input signal and a desired output;

[0197] The prediction module 200 is used to cluster the input signal and obtain the feature vector of the input signal by calculating the reduced Gaussian kernel, and select the corresponding sub-filter according to the cluster to perform prediction and obtain the prediction output;

[0198] A construction module 300 is used to construct a target optimization problem based on the error between the predicted output and the expected output according to the maximum relevant entropy criterion, and convert the target optimization problem into a convex optimization problem;

[0199] An updating module 400 is used to solve the convex optimization problem by hybridizing three conjugate gradients, and update the weights of the sub-filters until the maximum number of iterations is reached to obtain the final weights;

[0200] The loop module 500 is used to repeat the above steps to converge the error of the adaptive filtering system and complete the adaptive filtering process.

[0201] As an optional implementation, the prediction module 200 includes:

[0202] A first calculation unit 210 is configured to calculate the minimum distance between the input signal and the centroid of each cluster; the initial centroid of the cluster is the position of the first input signal;

[0203] The first updating unit 220 is configured to distribute the input signal according to the relationship between the minimum distance and the preset threshold value, and update the clusters; and construct a sub-filter for each cluster;

[0204] A mapping unit 230 is configured to construct a mapping function by using a reduced Gaussian kernel, input the input data into the mapping function, and obtain a feature vector;

[0205] The prediction unit 240 is configured to input a feature vector of an input signal into a corresponding sub-filter according to the cluster to which the input signal belongs, to obtain a prediction output.

[0206] As an optional implementation manner, the first updating unit 220 includes:

[0207] The judging subunit 221 is configured to judge whether the minimum distance is less than or equal to a preset threshold value. If so, the input signal is assigned to the cluster corresponding to the nearest centroid and the centroid of the cluster is updated.

[0208] If not, a new cluster is generated with the input signal as the initial centroid, and all data in the cluster closest to the input signal are migrated to the new cluster, with the centroid of the new cluster remaining unchanged as the input signal.

[0209] As an optional implementation, the building module 300 includes:

[0210] A second calculation unit 310 is configured to calculate a prediction error based on the predicted output and the expected output;

[0211] A first constructing unit 320 is configured to construct a target optimization problem based on the prediction error and according to a maximum relevant entropy criterion;

[0212] The conversion unit 330 is used to introduce intermediate variables based on the semi-quadratic optimization method to convert the target optimization problem into a positive definite quadratic function to obtain a convex optimization problem.

[0213] As an optional implementation, the updating module 400 includes:

[0214] A second construction unit 410 is configured to establish an optimization problem for the sub-filter based on a minimization error criterion, wherein the optimization problem includes matrix terms and vector terms;

[0215] A second updating unit 420 is configured to iteratively update the matrix terms and vector terms in the optimization problem based on an exponential decay data window method;

[0216] The third updating unit 430 is configured to iteratively update the weights in the optimization problem based on the hybrid three-term conjugate gradient.

[0217] It should be noted that, in this document, relational terms such as "first" and "second" are used only to distinguish one entity or operation from another entity or operation, and do not necessarily require or imply any actual relationship or order between these entities or operations. Moreover, the terms "comprises," "comprising," or any other variations thereof are intended to cover non-exclusive inclusion, so that a process, method, article, or device comprising a series of elements includes not only those elements, but also other elements not explicitly listed, or elements inherent to such process, method, article, or device. In the absence of further limitations, an element defined by the phrase "comprising a ..." does not exclude the presence of other identical elements in the process, method, article, or device comprising the element.

[0218] The above description is merely a specific embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any modifications or substitutions that can be easily conceived by a person skilled in the art within the technical scope disclosed in the present invention should be included within the scope of protection of the present invention. Therefore, the scope of protection of the present invention should be based on the scope of protection of the claims.

Claims

1. A reduced Gaussian kernel adaptive filtering method based on nearest center estimation, characterized in that: include: Acquiring initial parameters and a training set, and constructing an adaptive filtering system based on the initial parameters, wherein the training set includes an input signal and a desired output; The input signal is clustered and the eigenvector of the input signal is obtained by calculating the reduced Gaussian kernel. The corresponding sub-filter is selected according to the cluster for prediction to obtain the predicted output. Based on the error between the predicted output and the expected output, the target optimization problem is constructed according to the maximum relevant entropy criterion and converted into a convex optimization problem. The weights of the sub-filters are updated by solving the convex optimization problem through hybrid three-term conjugate gradient; Repeat the above steps to make the error of the adaptive filtering system converge and complete the adaptive filtering process.

2. The reduced Gaussian kernel adaptive filtering method based on nearest center estimation according to claim 1, characterized in that: The input signal is clustered and the eigenvector of the input signal is obtained by calculating the reduced Gaussian kernel. The corresponding sub-filter is selected according to the cluster for prediction to obtain the predicted output, including: Calculate the minimum distance between the input signal and the centroid of each cluster; the initial centroid of the cluster is the position of the first input signal; The input signal is distributed according to the relationship between the minimum distance and the preset threshold, and the clusters are updated; a sub-filter is constructed for each cluster; A mapping function is constructed by reducing the Gaussian kernel, and the input data is input into the mapping function to obtain a feature vector; According to the cluster where the input signal belongs, its feature vector is input into the corresponding sub-filter to obtain the predicted output.

3. The reduced Gaussian kernel adaptive filtering method based on nearest center estimation according to claim 2, characterized in that: The input signal is distributed according to the relationship between the minimum distance and the preset threshold, and the cluster is updated, including: Determine whether the minimum distance is less than or equal to a preset threshold. If so, assign the input signal to the cluster corresponding to the nearest centroid and update the centroid of the cluster. If not, a new cluster is generated with the input signal as the initial centroid, and all data in the cluster closest to the input signal are migrated to the new cluster, with the centroid of the new cluster remaining unchanged as the input signal.

4. The reduced Gaussian kernel adaptive filtering method based on nearest center estimation according to claim 1, characterized in that: Based on the error between the predicted output and the expected output, the target optimization problem is constructed according to the maximum relevant entropy criterion and converted into a convex optimization problem, including: Based on the predicted output and the expected output, the prediction error is calculated; Based on the prediction error, constructing a target optimization problem according to a maximum relevant entropy criterion; Based on the semi-quadratic optimization method, intermediate variables are introduced to transform the target optimization problem into a positive definite quadratic function, thus obtaining a convex optimization problem.

5. The reduced Gaussian kernel adaptive filtering method based on nearest center estimation according to claim 1, characterized in that: The convex optimization problem is solved by hybridizing three conjugate gradients to update the weights of the sub-filters, including: Based on the minimization error criterion, the convex optimization problem is transformed into a positive definite quadratic optimization problem; Based on the exponential decay data window method, the autocorrelation matrix and cross-correlation vector in the positive definite quadratic optimization problem are iteratively updated; The weights in the positive definite quadratic optimization problem are iteratively updated based on the hybrid three-term conjugate gradient method.

6. A reduced Gaussian kernel adaptive filtering system based on nearest center estimation, characterized in that: include: An acquisition module is used to acquire initial parameters and a training set, and construct an adaptive filtering system based on the initial parameters, wherein the training set includes an input signal and a desired output; The prediction module is used to cluster the input signal and obtain the feature vector of the input signal by calculating the reduced Gaussian kernel. The corresponding sub-filter is selected according to the cluster for prediction to obtain the predicted output; A building module is used to construct a target optimization problem based on the error between the predicted output and the expected output according to the maximum relevant entropy criterion, and transform the target optimization problem into a convex optimization problem; The update module is used to solve the convex optimization problem by hybridizing three conjugate gradients and update the weights of the sub-filters; The loop module is used to repeat the above steps to make the error of the adaptive filtering system converge and complete the adaptive filtering process.

7. The reduced Gaussian kernel adaptive filtering system based on nearest center estimation according to claim 6, characterized in that: The prediction module includes: A first calculation unit is used to calculate the minimum distance between the input signal and the centroid of each cluster; the initial centroid of the cluster is the position of the first input signal; The first updating unit is used to distribute the input signal according to the relationship between the minimum distance and the preset threshold value, and update the clusters; and construct a sub-filter for each cluster; A mapping unit, configured to construct a mapping function by reducing a Gaussian kernel, input the input data into the mapping function, and obtain a feature vector; The prediction unit is used to input the feature vector of the input signal into the corresponding sub-filter according to the cluster to which the input signal belongs, and obtain the predicted output.

8. The reduced Gaussian kernel adaptive filtering system based on nearest center estimation according to claim 7, characterized in that: The first updating unit includes: A judgment subunit is used to judge whether the minimum distance is less than or equal to a preset threshold. If so, the input signal is assigned to the cluster corresponding to the nearest centroid and the centroid of the cluster is updated; If not, a new cluster is generated with the input signal as the initial centroid, and all data in the cluster closest to the input signal are migrated to the new cluster, with the centroid of the new cluster remaining unchanged as the input signal.

9. The reduced Gaussian kernel adaptive filtering system based on nearest center estimation according to claim 6, characterized in that: The building blocks include: A second calculation unit is used to calculate a prediction error based on the predicted output and the expected output; A first construction unit is configured to construct a target optimization problem based on the prediction error and a maximum relevant entropy criterion; The conversion unit is used to introduce intermediate variables based on the semi-quadratic optimization method to convert the target optimization problem into a positive definite quadratic function to obtain a convex optimization problem.

10. The reduced Gaussian kernel adaptive filtering system based on nearest center estimation according to claim 6, characterized in that: The update module includes: The second building block is used to transform the convex optimization problem into a positive definite quadratic optimization problem based on a minimization error criterion; The second updating unit iteratively updates the autocorrelation matrix and cross-correlation vector in the positive definite quadratic optimization problem based on the exponential decay data window method; The third updating unit iteratively updates the weights in the positive definite quadratic optimization problem based on the hybrid three-term conjugate gradient method.

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