A Noise-Driven Robust Sequential Adaptive Estimation Method

Abstract

The present invention discloses a noise-driven robust sequential adaptive estimation method, which belongs to the fields of signal estimation, filtering, and system identification. The noise excitation effect of the present invention is reflected in: in a dynamic and heavy-tailed noise environment, a random noise variable is introduced into the score function of the robust M-estimation, and the score function is convolved with the noise density function to transform the form of the original M-estimation score function, and a parameter estimation adaptive update algorithm is recursively obtained. This method is based on the relevant principles of stochastic resonance and can reduce the mean square error (MSE) of the robust estimation with the help of noise. The present invention utilizes the relevant theory of stochastic resonance and enhances the anti-interference ability and convergence efficiency of the original estimation algorithm by dynamically adjusting the adaptive noise injection mechanism. The algorithm can asymptotically achieve the maximum likelihood estimator performance of system parameter estimation under finite samples, providing an efficient and robust solution for parameter estimation in complex noise environments.

Description

Technical Field

[0001] The present invention belongs to the fields of signal estimation, filtering and system identification, and particularly relates to a noise-driven robust sequential adaptive estimation method. Background Art

[0002] In many statistical models, optimal estimators or detectors are derived based on strong assumptions, such as that the data follows a specific distribution (such as a Gaussian distribution) or that the model is precisely specified. However, these idealized assumptions may not always hold true in the real world. Outliers, heavy-tailed noise, and other factors can significantly degrade the performance of traditional methods. The M-estimation method is an effective robust estimation method that addresses these issues.

[0003] However, in many contemporary applications such as real-time signal processing, sensor networks, and streaming data analytics, data is not available all at once, but arrives sequentially over time. In these dynamic environments, traditional batch estimation techniques are inapplicable due to their lack of real-time performance, as they typically require access to the entire dataset. This necessitates the use of sequential estimation techniques, which incrementally update parameter estimates as new data arrive, thereby enabling real-time estimation of the target state or system parameters. While sequential estimation methods such as recursive least squares (RLS), Kalman filtering, or stochastic gradient descent are widely used, they lack sufficient robustness to outliers or distribution shifts in the data stream, making it difficult to achieve a balance between robustness and effectiveness in robust sequence estimation.

[0004] Therefore, the sequential M-estimation pursues the unity of robustness and effectiveness from a practical perspective. In order to further improve the performance of the estimator, utilizing noise characteristics to improve the estimation performance is the main technical idea of ​​the present invention. Summary of the Invention

[0005] In view of the above technical problems existing in the prior art, the present invention proposes a noise-driven robust sequential adaptive estimation method, which has a reasonable design, overcomes the shortcomings of the prior art, and has good effects.

[0006] In order to achieve the above object, the present invention adopts the following technical solution: a robust sequential adaptive estimation method based on noise driving, comprising the following steps: Step 1: Initialize parameters: record sample observation data x={x_1, x_2, ..., x_N} containing outliers and their length N; Step 2: Establish a data linear model; Step 3: Determine the score function ; Step 4: Parameters Perform adaptive sequential estimation so that the estimated amount is updated iteratively as the data arrives, and obtain Parameter estimates at time ; Step 5: Calculate The average error between the parameter estimate and the true value at the moment is the mean square error of the estimator; Step 6: Use the adaptive optimization algorithm to optimize the noise parameters and obtain the minimum mean square error under the optimal noise parameters.

[0007] Preferably, in step 2, the position parameter model is as shown in formula (1): (1); among them, The probability density function is Background noise with heavy tail distribution; is the observation model coefficient vector; To set the estimated reference parameters.

[0008] Preferably, step 3 specifically includes the following steps: Step 3.1: Determine background noise Distribution type; Step 3.2: Choose a bounded score function The corresponding M-estimator; Step 3.3: Introduce noise intensity Noise , its probability density function Estimation score function with M Perform convolution; get a new score function , as shown in formula (2): (2); where η is the injected random noise variable; It is about random variables The mathematical expectation of .

[0009] Preferably, in step 4, the parameter estimate The expression of is shown in formula (3): (3); among them, is the gain matrix, which is expressed as follows: ; ; is the prediction error, which is expressed as follows: ; is the covariance matrix, and its update rule is .

[0010] Preferably, in step 5, the expression of the mean square error is as shown in formula (4): (4).

[0011] Preferably, the scoring function is a bounded function, and its bound s satisfies the robustness constraint to heavy-tailed noise.

[0012] Preferably, background noise Obey Cauchy distribution or symmetric alpha stable distribution, inject noise It has a symmetric distribution, and its distribution parameters are estimated offline or adjusted online adaptively.

[0013] The beneficial technical effects of this invention include leveraging stochastic resonance theory and dynamically adjusting an adaptive noise injection mechanism to enhance the original estimation algorithm's anti-interference capabilities and convergence efficiency. This algorithm can asymptotically achieve maximum likelihood estimator performance for system parameter estimation under finite samples, providing an efficient and robust solution for parameter estimation in complex noisy environments. BRIEF DESCRIPTION OF THE DRAWINGS

[0014] Figure 1 This is a curve diagram showing how the mean square error of the estimated value varies with noise intensity.

[0015] Figure 2 This is a convergence curve diagram of the mean square error versus the number of iterations of the present invention.

[0016] Figure 3 This is a graph showing the change of mean square error with input signal-to-noise ratio under Cauchy background noise.

[0017] Figure 4 for A plot of mean square error versus input signal-to-noise ratio in the presence of background noise. DETAILED DESCRIPTION

[0018] The present invention is further described in detail below with reference to the accompanying drawings and specific implementation methods: In order to further improve the robustness and effectiveness of the estimator in a heavy-tailed noise environment, the theory of signal stochastic resonance is applied to the fields of signal estimation and filtering. By adaptively injecting noise and modifying the scoring function, the noise is fully utilized rather than eliminated, so as to further improve the performance of the estimator.

[0019] The method of using noise to improve robust estimation is as follows: 1. Initialize parameters: record sample observation data containing outliers and its length , estimated reference parameters .

[0020] 2. Build a linear model for the data: , It has a probability density function of Background noise of heavy-tail distribution; 3. Choose a bounded score function The corresponding M-estimator, The function is bounded by ;4. Set the noise intensity to Noise The probability density function of With the score function Perform convolution to obtain a new score function ;5.Parameters Make an estimate that is updated incrementally as data arrives: ; Among them: the gain matrix is , where the covariance matrix The update rule is .

[0021] 6. Using adaptive optimization algorithm, minimize Mean square error at time , optimize the noise parameters and obtain the minimum mean square error under the optimal noise parameters.

[0022] Experimental results: Considering the heavy-tailed Cauchy background noise, the introduced noise intensity is Gaussian noise excitation, a sequential adaptive estimation algorithm based on Gaussian noise drive is obtained. Figure 1 The mean square error of the algorithm is given as the noise intensity As can be seen from the figure, as the noise intensity The mean square error (MSE) shows a non-monotonic change that first decreases and then increases, that is, there is an optimal noise intensity. , which minimizes the mean square error of the system estimate and is much smaller than the mean square error of the traditional sequential estimation algorithm.

[0023] Considering two types of heavy-tailed noise distribution environments, Cauchy noise and symmetric alpha-stable noise (SαS), the performance of the sequential algorithm of adaptive Gaussian noise injection and adaptive optimal noise injection is verified. Figure 2 Gaussian noise with optimal noise intensity and approximately optimal density distribution are given. The mean square error convergence curve of the noise-driven adaptive sequential M algorithm under the optimal noise is shown, and is compared with the mean square error of the estimators of traditional least squares and traditional sequential M estimation. Figure 3 The MSE variation curves of several algorithms under different input signal-to-noise ratios are given under Cauchy background noise. Figure 4 The MSE curves of several algorithms under different input signal-to-noise ratios are given under SαS background noise. Figure 3 and Figure 4 It can be seen that whether in the sequential estimation algorithm with fixed noise intensity or adaptively tuned noise intensity or noise distribution, compared with the two algorithms without noise injection, noise injection can effectively reduce the mean square error of the estimation.

[0024] Of course, the above description is not a limitation of the present invention, and the present invention is not limited to the above examples. Changes, modifications, additions or substitutions made by technicians in this technical field within the essential scope of the present invention should also fall within the scope of protection of the present invention.

Claims

1. A noise-driven robust sequential adaptive estimation method, characterized in that: The following steps are involved: Step 1: Initialize parameters: record the sample observation data x = {x_1, x_2, ..., x_N} containing outliers and its length N; Step 2: Build a linear model for the data; Step 3: Determine the scoring function Step 4: Perform adaptive sequential estimation on the parameter θ so that it is updated iteratively as the data arrives, and obtain the parameter estimate at time n Step 5: Calculate the average error between the parameter estimate and the true value at time n, that is, the mean square error of the estimator; Step 6: Use the adaptive optimization algorithm to optimize the noise parameters and obtain the minimum mean square error under the optimal noise parameters; In step 2, the linear model is shown in formula (1): Among them, ω n The probability density function is f ω The background noise of heavy-tail distribution; h n is the observation model coefficient vector; θ is the set estimation reference parameter; Step 3 specifically includes the following steps: Step 3.1: Determine the background noise ω n The distribution type of Step 3.2: Select the M-estimator corresponding to the bounded score function ψ(x); Step 3.3: Introduce noise intensity σ η The noise η, its probability density function f η (x) is convolved with the M estimated score function ψ(x); a new score function is obtained As shown in formula (2): Where η is the injected random noise variable; E η [] is the mathematical expectation of the random variable η; In step 4, the parameter estimates The expression of is shown in formula (3): Where K[n] is the gain matrix, which is expressed as follows: e n is the prediction error, which is expressed as follows: ∑ is the covariance matrix, and its update rule is In step 5, the expression of mean square error is shown in formula (4):

2. The noise-driven robust sequential adaptive estimation method according to claim 1, characterized in that: The score function ψ(x) is a bounded function, and its bound s satisfies the robustness constraint to heavy-tailed noise.

3. The noise-driven robust sequential adaptive estimation method according to claim 1, characterized in that: Background noise ω n Obeying the Cauchy distribution or symmetric alpha stable distribution, the injected noise η has a symmetric distribution, and its distribution parameters are estimated offline or adjusted online adaptively.

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