CN115632633A - Minimum error entropy self-adaptive filtering method based on robust M estimation - Google Patents

Minimum error entropy self-adaptive filtering method based on robust M estimation Download PDF

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CN115632633A
CN115632633A CN202211262408.7A CN202211262408A CN115632633A CN 115632633 A CN115632633 A CN 115632633A CN 202211262408 A CN202211262408 A CN 202211262408A CN 115632633 A CN115632633 A CN 115632633A
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adaptive filtering
estimation
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王子逸
周兴立
周玉正
钟山
张洪斌
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University of Electronic Science and Technology of China
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Abstract

The invention discloses a minimum error entropy self-adaptive filtering method based on robust M estimation, which comprises the following steps: s1, constructing a self-adaptive filtering system and acquiring a prediction error; s2, according to the prediction error, constructing a target function and an optimization target based on a new steady M estimation minimum error entropy criterion, and calculating to obtain an MMEE optimization loss criterion corresponding to the prediction error sample; s3, performing partial differentiation on the error function, and updating the weight of the adaptive filtering system; and S4, iterating the steps S1 to S3 to enable the error value of the adaptive filtering system to be converged, and taking the weight of the adaptive filtering system at the moment as a final weight for filtering the input signal. The adaptive filter has extremely high convergence rate and low steady-state error, can effectively improve the robustness of the adaptive filter under various non-Gaussian noises, has good environment adaptability, and can stably process the sudden change of the noise type in the environment.

Description

Minimum error entropy self-adaptive filtering method based on robust M estimation
Technical Field
The invention relates to the technical research in the field of digital signal processing, in particular to a minimum error entropy self-adaptive filtering method based on robust M estimation.
Background
In recent years, adaptive filtering algorithms are gradually developed on the basis of Wiener (Wiener) and Kalman filtering (Kalman) algorithms. The adaptive filtering algorithm is considered as an optimal filtering technology, and has the outstanding advantages of strong adaptability and improvement, and stronger performance in reducing the influence of noise on the whole signal system, so that the adaptive filtering algorithm is widely applied to engineering practice, namely, some practical signal processing tasks, such as echo cancellation, system identification and the like. The subject of adaptive filtering is a system or information process with uncertainty. "uncertainty" here means that the mathematical model of the process under study and its environment is not completely deterministic. Including unknown factors and random factors.
The conventional adaptive filtering technique is based on Minimum Mean Square Error (MMSE), which is the best criterion for linear filter under gaussian assumption, however, in various practical application scenarios, the signal may be contaminated by some non-gaussian noise, and when the facing system or information process belongs to the non-gaussian condition, the second order statistic of the minimum error criterion will not be enough to fully extract effective information from the data sample, in which case the filtering performance of the method based on the minimum mean square error criterion will be seriously degraded. In this context, information Theory Learning (ITL) -based correlation algorithm criteria have been widely studied, wherein the Minimum Error Entropy (MEE) algorithm, which is extended, can capture the high-order statistics of signal error samples, in contrast to the minimum mean square error criterion, which has a better effect when analyzing signal systems contaminated by non-gaussian noise. In general, the minimum error entropy criterion uses a nonparametric gaussian kernel estimation method based on a Parzen window to obtain an estimated value of the second-order rayleigh entropy of the processed data sample, thereby providing a more robust optimization criterion for non-gaussian signal processes. The minimum error entropy can be applied to the iterative update process of the adaptive system identification, and is considered to be an effective method for processing non-gaussian noise.
However, the above criteria still have a certain limitation, and when a desired signal is interfered by heavy-tail or multi-peak impulse noise (a corresponding extremely large outlier occurs in a data sample), it is difficult to achieve a highly accurate estimation of relegant entropy by directly using an error sample containing the outlier, which may limit the improvement of the performance of the algorithm; in addition, once the iterative parameters (such as kernel width and window length) are not properly selected, the high sensitivity of the gaussian kernel probability density estimation method used by the criterion to the parameters further causes the degradation of the filtering performance.
Rayleigh (Renyi's) entropy is a very important non-linear information content similarity measurement and is a core element of a Minimum Error Entropy (MEE) criterion, and a huge space for improving algorithm performance exists because the traditional Minimum Error Entropy (MEE) criterion cannot realize accurate estimation of the rayleigh entropy, especially when an error sample contains a large number of abnormal values.
Disclosure of Invention
In order to overcome the defects in the prior art, the invention provides a minimum error entropy self-adaptive filtering method based on robust M estimation, which combines a minimum error entropy algorithm and an M estimation numerical statistic theory, and has better filtering performance under the working of a non-Gaussian environment while keeping high convergence rate.
The invention is realized by the following technical scheme: the minimum error entropy self-adaptive filtering method based on the robust M estimation is characterized in that: the method comprises the following steps:
s1, constructing a self-adaptive filtering system, and obtaining an output signal as a predicted value by an input signal through the self-adaptive filtering system at each moment, and subtracting an expected output signal to obtain a prediction error;
the step S1 comprises the following steps:
s101, constructing a self-adaptive filtering system: the weight of the adaptive filter system is w n (ii) a Let n time input signal u n Is Gaussian white noise with mean value of 0 and variance of 1, and the optimal weight of the adaptive filtering system is preset to be w o ；
S102, inputting a signal u n And the optimal weight w expected by the filter o Multiplying, and adding the noise signal v (n) to obtain the desired output signal d (n):
Figure BDA0003891683280000021
wherein v is n Is a mixture of Gaussian noise, and v n ～0.95N(0,0.01)+0.05N(0,10)，
Wherein the input signal u n And a noise signal v n Not related;
s102, inputting a signal u n Weights w associated with an adaptive filter system n Multiplying to obtain a predicted output signal y n ：
Figure BDA0003891683280000022
S103, calculating the prediction error between the expected output signal d (n) and the prediction output signal yn, and recording as:
e n ＝d n -y n 。
s2, according to the prediction error, constructing a target function and an optimization target based on a new steady M estimation minimum error entropy criterion, and calculating to obtain an MMEE optimization loss criterion corresponding to the prediction error sample;
the step S2 includes:
s201, before each iteration, calculating errors in the first L window lengths including the time n according to the step S1: e.g. of a cylinder n-L+1 To e n As an error sample vector required for sliding a Parzen window once, where n is an integer not less than L;
s202, constructing a loss function and an optimization target based on the new robust M estimation minimum error entropy criterion: by utilizing the statistical characteristic of M estimation, on the basis of the traditional minimum error entropy criterion, setting M estimation weight factors to weight or cut off errors, reducing the influence of outliers on the estimation of the self-adaptive system by resetting the distribution of error samples, and further obtaining a minimum error entropy loss function based on steady M estimation, specifically:
Figure BDA0003891683280000031
wherein epsilon n ＝[e n ,e n-1 ,...,e n-L+1 ] T Representing an error sample vector;
the robust M-estimation minimum error entropy criterion aims to maximize the above-mentioned penalty function, thereby minimizing the difference between the prediction output and the expected output of the adaptive filtering system, wherein
Figure BDA0003891683280000032
Represents the M estimated weight factors calculated based on L error samples in the window length, and the value range is limited between (0, 1), wherein, delta 1 、Δ 2 、Δ 3 The M estimation weight functions comprise a Hampel weight function and are recorded as:
Figure BDA0003891683280000033
s3, updating the weight of the self-adaptive filtering system by adopting a random gradient ascent method: firstly, partial differentiation is carried out on an error function, and then the weight of the self-adaptive filtering system is updated;
in the step S3, after the objective function is obtained, the adaptive filtering weight parameter is updated by using a random gradient ascent method, which specifically includes:
firstly, partial differentiation is carried out on a target:
Figure BDA0003891683280000034
wherein
Figure BDA0003891683280000035
Re-pair the filter region weight parameter w n Performing a gradient ascent update algorithm, w n The nth time weighting parameter is shown, the subscript n +1 represents the nth +1 th time weighting parameter of the adaptive filter, and the updating formula is as follows:
Figure BDA0003891683280000036
and S4, iterating the steps S1 to S3 to enable the error value of the adaptive filtering system to be converged, namely the difference between the predicted output and the expected output of the adaptive filtering system is smaller than a set threshold value, and taking the weight of the adaptive filtering system at the moment as a final weight for filtering the input signal.
The invention has the following beneficial effects: the invention utilizes the steady statistical characteristic of M-estimation, combines the minimum error entropy algorithm and the statistical theory of M estimation values, introduces a plurality of M-estimation weight factors to recalculate the statistical distribution of error samples, aims to reduce the influence of large outliers to the maximum extent, keeps high convergence speed, has better filtering performance under the working of non-Gaussian environment, is highly insensitive to parameter change, is flexible and easy to adjust, and adapts to various non-Gaussian environments.
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FIG. 1 is a basic block diagram of a general adaptive filtering algorithm;
FIG. 2 is a schematic block diagram of a minimum error entropy adaptive filtering algorithm based on robust M estimation according to the present invention;
FIG. 3 is a schematic flow chart of the present invention;
FIG. 4 is a schematic diagram of comparison between theoretical analysis and experimental results of steady-state performance of the minimum error entropy adaptive filtering method based on robust M estimation, which varies with step length;
FIG. 5 is a comparison graph of the steady state mean square error effect of the present method and other related robust adaptive filtering methods;
fig. 6 shows the following conditions in case of sudden change of ambient noise: the adaptive system generates a system convergence curve comparison schematic diagram of the sudden change of the noise type at 1000 points and 2000 points respectively.
Detailed Description
The technical solutions of the present invention are further described in detail below with reference to the drawings and examples, but the scope of the present invention is not limited to the following descriptions.
Basic functional block diagram of a basic adaptive filter is shown in fig. 1, with an input signal u n Generating an output signal y after passing through a parameter-tunable adaptive digital filter n Is compared with the desired signal d n Comparing to form an error signal e n And adjusting the filter parameters through a self-adaptive algorithm to finally minimize the loss function. Adaptive filtering may automatically adjust the filter parameters at the current time using the results of the filter parameters obtained at the previous time to adapt to the unknown or time-varying statistical properties of the signal and noise to achieve optimal filtering. Due to different objective functions, the adaptive filtering algorithms under different criteria have different effects.
A functional block diagram of a minimum error entropy adaptive filter based on robust M estimation is shown in FIG. 2, where an input signal u n By finding the unknown system with the best weight, using a mixture of Gaussian noises v n 0.95N (0, 0.01) +0.05N (0, 10) as noise signals, producing the actual desired output signal
Figure BDA0003891683280000041
Generating an output signal by a post-convolution operation of an adaptive digital filter system
Figure BDA0003891683280000042
The two are subtracted to form an error signal e n ＝d n -y n And carrying out iterative operation to obtain a plurality of error samples, recalculating the distribution of the error samples by using an M estimation method, returning to 0 for large errors, truncating medium errors, and not operating small error outliers to finally obtain an MMEE error entropy loss function.
The invention relates to an adaptive filtering method based on M estimation and minimum error entropy, which is an improvement on an adaptive filtering algorithm under the minimum error entropy criterion, and is an overall flow chart of the method in fig. 3 (a), and the first step, the second step and the third step of the method are specifically introduced in fig. 3 (b):
1. input signal u n The vector follows a gaussian distribution with mean 0 and variance 1, the instantaneous measurement noise signal follows a mixed gaussian distribution: v. of n ～0.95N(0,0.01)+0.05N(0,10)；
2. Ideal regression output signal
Figure BDA0003891683280000051
Wherein T is a transposition operator, and the actual output of the filter can be obtained
Figure BDA0003891683280000052
Wherein w 0 For the estimated system weight vector, which is an unknown but actually existing value, any initial value can be assigned to it, but this initial value is assumed to be unknown, and the task of the adaptive filtering is to pass w n Is iteratively updated to approximate the true value w o 。
3. Calculating the difference between the estimated output value and the actual value, i.e. the prediction error e n ＝d n -y n 。
4. Establishing a minimum error entropy criterion (MMEE) based on the robust M estimation, determining an optimized object, and considering an error e n Is a random variable with a Probability Density Function (PDF) of p e Through the information theory learning knowledge, we can know that the quadratic rui entropy can be written as:
Figure BDA0003891683280000053
wherein V 2 (e) For quadratic information potential, it is clear that minimizing quadratic relegate entropy is equivalent to maximizing quadratic information potential. And recalculating the error distribution by using an M estimation method, and simultaneously constructing a target function so as to obtain a practical and feasible non-parametric estimation calculation formula and approach to the theoretical secondary information potential and the secondary Reyle entropy value to the maximum extent.
Figure BDA0003891683280000054
Where L is the number of window length samples, κ σ () In the form of a gaussian kernel function,
Figure BDA0003891683280000055
estimating a weight function for M
Figure BDA0003891683280000056
The estimated weight function is most suitably a Hampel function as described above, but other M estimated weight functions may be chosen depending on the situation to be handled:
Figure BDA0003891683280000057
Figure BDA0003891683280000058
5. for the objective function with respect to w n Derivation of the deviation
Figure BDA0003891683280000059
Due to the need to maximize the objective function, an ideal weight vector under the criterion of minimum error entropy based on robust M estimation can be obtained therefrom
Figure BDA00038916832800000510
Obtaining an iterative updating formula of a filter weight vector according to a gradient rising method:
Figure BDA0003891683280000061
wherein eta is a step size parameter,
Figure BDA0003891683280000062
and n represents the nth step iteration.
In conclusion, the method and the improvement innovation provided by the invention expand and improve the existing minimum error entropy adaptive filtering, reduce the influence of outliers on the Rayleigh entropy estimation by recalculating the statistical distribution of error samples, improve the utilization of the error criterion on probability information, and improve the robustness of the algorithm under the condition of processing various non-Gaussian noises and the flexibility of the algorithm aiming at the actual situation by adopting the steady M estimation.
In addition, the corresponding theoretical analysis and verification of the invented method are carried out, including mean value stability and mean square steady state performance, so that the theoretical basis of the invention is greatly ensured. Compared with some existing robust filters aiming at non-Gaussian noise, the filter has extremely strong superiority and effectiveness, can be suitable for various non-Gaussian systems or signal processes, and simultaneously meets performance indexes of rapid convergence and low steady-state error; the method has good universality and flexible adjustment space, can be suitable for different practical application scenes through flexible adjustment of parameters, has more important research significance and wide engineering application value, and is further explained through data verification simulation and comparison experiments:
as shown in fig. 4, the theoretical basis analysis of the ultimate steady-state performance of the present invention is experimentally verified, and it is proved that the simulation experiment result has good consistency with the theoretical analysis result. Wherein MSD means the steady state mean square error, η represents the step size, the term represents the theoretical value, and the simulation represents the experimental value.
FIG. 5 is a graph comparing the results of the method and other related robust adaptive filtering methods in the literature, with the ordinate representing the steady state error averaged over 100 experiments
Figure BDA0003891683280000063
The abscissa represents the number of iterations: 1500 times of experiment iteration are carried out each time, and mixed Gaussian noise is adopted as noise. We have chosen different parameters for each algorithm to ensure the same initial convergence speed and ideal performance. As can be seen from the simulation results, compared with other algorithms, the method provided by the invention has the advantages of optimal comprehensive performance, high convergence rate and steady-state errorLow, with the MMEE method using the Hampel estimated weight function performing most prominently.
The MMEE algorithm proposed by the method performs well under non-Gaussian noise and is not sensitive to the variation of the kernel width on the whole. Therefore, the MMEE algorithm can be used for further exploring the performance under the unsteady condition by utilizing the characteristic of the MMEE algorithm. Fig. 6 is a comparison graph of mean square error averages obtained by running the adaptive system 3000 times by using the robust M estimation method in the present method and other conventional minimum mean square error and minimum error entropy methods when the 1000 th point and 2000 th point occur time-varying, that is, the noise type of the unknown system suddenly changes from gaussian noise to non-gaussian noise. As can be seen in fig. 6, after the time-varying occurs at 1000 and 2000 points: the method is interfered by the double-tail pulse noise, which is very common in reality, the performance of the traditional fixed step length LMS method and the MEE method for inhibiting the noise is still seriously reduced in the process of 1000 th to 2000 th sampling points, the MMEE-Hampel method can still maintain consistent filtering capability and is not influenced by noise type change in the environment, and the method provided by the invention has the advantages of higher speed of returning to a steady state, stronger tracking performance and better filtering performance when the noise jump occurs in the system environment. This further proves the excellent robustness of the MMEE algorithm proposed by the method under the condition of system environment mutation.
While the foregoing description shows and describes a preferred embodiment of the invention, it is to be understood, as noted above, that the invention is not limited to the form disclosed herein, but is not intended to be exhaustive or to exclude other embodiments and may be used in various other combinations, modifications, and environments and may be modified within the scope of the inventive concept described herein by the above teachings or the skill or knowledge of the relevant art. And that modifications and variations may be effected by those skilled in the art without departing from the spirit and scope of the invention as defined by the appended claims.

Claims (4)

1. A minimum error entropy self-adaptive filtering method based on robust M estimation is characterized in that: the method comprises the following steps:
s1, constructing a self-adaptive filtering system, and obtaining an output signal as a predicted value by an input signal through the self-adaptive filtering system at each moment, and subtracting an expected output signal to obtain a prediction error;
s2, according to the prediction error, constructing a target function and an optimization target based on a new steady M estimation minimum error entropy criterion, and calculating to obtain an MMEE optimization loss criterion corresponding to the prediction error sample;
s3, updating the weight of the self-adaptive filtering system by adopting a random gradient ascending method: firstly, partial differentiation is carried out on an error function, and then the weight of the self-adaptive filtering system is updated;
and S4, iterating the steps S1 to S3 to enable the error value of the adaptive filtering system to be converged, namely the difference between the predicted output and the expected output of the adaptive filtering system is smaller than a set threshold value, and taking the weight of the adaptive filtering system at the moment as a final weight for filtering the input signal.
2. The robust M-estimation based minimum error entropy adaptive filtering method of claim 1, wherein: the step S1 comprises the following steps:
s101, constructing a self-adaptive filtering system: the weight of the adaptive filter system is w n (ii) a Let n time input signal u n Is white Gaussian noise with mean value of 0 and variance of 1, and the optimal weight of the self-adaptive filtering system is preset to be w o ；
S102, inputting a signal u n And the optimal weight w expected by the filter o Multiplying, and adding the noise signal v (n) to obtain the desired output signal d (n):
Figure FDA0003891683270000011
wherein v is n Is a mixture of Gaussian noise, and v n ～0.95N(0,0.01)+0.05N(0,10)，
Wherein the input signal u n And a noise signal v n Not related;
s102, inputting a signal u n And adaptive filter systemWeight w of n Multiplying to obtain a predicted output signal y n ：
Figure FDA0003891683270000012
S103, calculating an expected output signal d (n) and a prediction output signal y n The prediction error between, noted as:
e n ＝d n -y n 。
3. the robust M-estimation based minimum error entropy adaptive filtering method of claim 2, wherein: the step S2 includes:
s201, before each iteration, calculating errors in the first L window lengths including the time n according to the step S1: e.g. of the type n-L+1 To e n As an error sample vector required for sliding a Parzen window once, where n is an integer not less than L;
s202, constructing a loss function and an optimization target based on the new robust M estimation minimum error entropy criterion: by utilizing the statistical characteristic of M estimation, on the basis of the traditional minimum error entropy criterion, M estimation weight factors are set to weight or truncate errors, and the influence of outliers on the estimation of the self-adaptive system is reduced by resetting the distribution of error samples, so that a minimum error entropy loss function based on the steady M estimation is obtained, specifically:
Figure FDA0003891683270000021
wherein epsilon n ＝[e n ,e n-1 ,...,e n-L+1 ] T Representing an error sample vector;
the criterion of the minimum error entropy of the robust M estimation aims to maximize the loss function and further minimize the difference between the prediction output and the expected output of the adaptive filtering system, wherein
Figure FDA0003891683270000022
Represents M estimated weight factors obtained by calculation based on L error samples in the window length, and the value range is limited between (0, 1), wherein, delta 1 、Δ 2 、Δ 3 Is a boundary parameter which is a preset constant, the weight factor is calculated, and a Hampel weight function in an M estimation theory is generally adopted, namely, the order is given
Figure FDA0003891683270000023
The expression is noted as:
Figure FDA0003891683270000024
wherein r is i Is a "normalized" residual indicator,
Figure FDA0003891683270000025
med () is the median, s is e i The corresponding residual measure.
4. A method for minimum error entropy adaptive filtering based on robust M estimation according to claim 3, characterized by: in step S3, after obtaining the objective function, the adaptive filtering weight parameter is updated by using a random gradient ascent method, which specifically includes:
firstly, partial differentiation is carried out on a target:
Figure FDA0003891683270000026
wherein
Figure FDA0003891683270000027
Re-pair the filter region weight parameter w n Performing a gradient ascent update algorithm, w n Represents the weighting parameter at the nth time, the subscript n +1 represents the weighting parameter at the nth +1 th time of the adaptive filter, and the updating formula is as follows：
Figure FDA0003891683270000028
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* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN116473526A (en) * 2023-06-25 2023-07-25 湖南尚医康医疗科技有限公司 Medical information acquisition method and system based on artificial intelligence and Internet of things
CN116473526B (en) * 2023-06-25 2023-09-29 湖南尚医康医疗科技有限公司 Medical information acquisition method and system based on artificial intelligence and Internet of things

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