CN109657273B - Bayesian parameter estimation method based on noise enhancement - Google Patents

Bayesian parameter estimation method based on noise enhancement Download PDF

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CN109657273B
CN109657273B CN201811365349.XA CN201811365349A CN109657273B CN 109657273 B CN109657273 B CN 109657273B CN 201811365349 A CN201811365349 A CN 201811365349A CN 109657273 B CN109657273 B CN 109657273B
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刘书君
杨婷
周喜川
张奎
李东
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Chongqing Nanao Technology Co ltd
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Abstract

The invention discloses a Bayesian parameter estimation method based on noise enhancement. Belonging to the field of signal processing. The method is a method for improving parameter estimation performance under the Bayesian rule by utilizing noise enhancement characteristics. Adding independent additive noise to a nonlinear system input signal, establishing a noise enhancement Bayesian parameter estimation model, carrying out Bayesian estimation on the input parameter by utilizing the nonlinear system output signal, finally solving the optimal additive noise under the model, and obtaining the optimal noise enhancement Bayesian estimation performance. The invention improves the Bayesian parameter estimation performance by utilizing noise enhancement, and can further reduce the corresponding mean square error when carrying out Bayesian estimation on the input parameters based on the nonlinear system output signals by adding proper noise to the nonlinear system input signals.

Description

Bayesian parameter estimation method based on noise enhancement
Technical Field
The invention belongs to the field of signal processing, and particularly relates to noise enhancement and Bayesian parameter estimation.
Background
In conventional signal processing, noise is often considered as an unwanted signal, which is not only of no benefit but can cause interference to the system. In fact, noise always exists with the useful signal, and excessive noise in the system usually causes the capacity of the transmission channel to be reduced, and the signal detection accuracy is reduced, and simultaneously, the performance of parameter estimation is also worsened, so that the operation of the system is seriously affected. To improve system performance, noise is typically removed or separated from the useful signal as much as possible. However, the effect of noise on the system is not all negative. Under certain conditions, noise can generate positive enhancement effect on signals through a nonlinear system, and the phenomenon is noise enhancement. Studies have shown that adding noise to the input of some nonlinear systems or adjusting the background noise level can result in significant improvements in system detection and/or estimation performance. And enabling the estimation with the minimum average cost to be Bayesian estimation, and obtaining the estimated quantity which is the Bayesian estimated quantity. The square error between the parameter estimate and the true value is one of the most common cost functions for bayesian estimation. When the nonlinear system output signal is utilized to carry out Bayesian estimation on unknown parameters in the input signal, the theory of noise enhancement is combined, and under certain conditions, the mean square error of the optimal Bayesian estimation corresponding to the nonlinear system input signal when noise is added is possibly smaller than the mean square error of the original optimal Bayesian estimation corresponding to the nonlinear system input signal when no noise is added.
Disclosure of Invention
The invention aims to provide a Bayesian parameter estimation method based on noise enhancement by combining a noise enhancement principle on the basis of Bayesian estimation by taking a square error between a parameter estimation value and a true value as a cost function. After adding proper noise to the nonlinear system input signal, when the nonlinear system output signal is used for carrying out Bayesian estimation on the input parameter, the mean square error between the parameter estimation value and the true value is further reduced.
The method specifically comprises the following steps:
1) Constructing a noise enhancement nonlinear system:
the nonlinear system includes three parts: a nonlinear system input signal, a nonlinear system, and a nonlinear system output signal; the nonlinear system input signal x is closely related to the parameter θ, and the value of θ is determined by its probability density function p θ (θ) determination; adding independent additive noise n to the nonlinear system input signal x, and obtaining a noise-corrected nonlinear system output signal y=t (x+n) after the nonlinear system is passed, wherein T (·) represents a transfer function of the nonlinear system;
2) Establishing a noise enhancement Bayes parameter estimation model:
estimating an input parameter theta by using the nonlinear system output signal y; when the value of y is constant, the input parameter theta and the estimated quantity thereof are made
Figure BDA0001868375210000021
Bayes estimation with minimum mean square error
Figure BDA0001868375210000022
The corresponding mean square error is
ε MMSE (y)＝E(θ 2 |y)-E 2 (θ|y) (2)
Wherein E (θ|y) and E (θ) 2 Y) represents the timing θ and θ of the y value, respectively 2 Is not limited to the desired one; further, the method comprises the steps of,
Figure BDA0001868375210000023
also, an estimate that minimizes the average mean squared error, the corresponding minimum mean squared error is
Figure BDA0001868375210000024
Wherein p is y (y) is a probability density function of the nonlinear system output signal y; p is p y (y), E (θ|y) and E (θ) 2 Y) is calculated as:
Figure BDA0001868375210000025
Figure BDA0001868375210000026
Figure BDA0001868375210000027
wherein the method comprises the steps of
Figure BDA0001868375210000028
Figure BDA0001868375210000029
p n (n) is a probability density function of additive noise n, and p x (x|θ) represents a conditional probability density function of the value of θ to time the nonlinear system input signal x;
3) Solving for additive noise required to minimize bayesian costs:
substituting the expression (4), the expression (5) and the expression (6) into the expression (3) can know that the probability density function is added to the nonlinear system input signal x to be p n In the case of additive noise n of (n), when Bayesian estimation is performed on the input parameter θ by using the noise-corrected nonlinear system output signal y, the corresponding average mean square error is
Figure BDA0001868375210000031
To obtain the minimum MMSE (p) in the formula (7) n (n)) corresponding optimal additive noise, constructing the following noise enhancement optimization problem:
Figure BDA0001868375210000032
bonding of
Figure BDA0001868375210000033
The characteristics of (1) are known to be +.>
Figure BDA0001868375210000034
Figure BDA0001868375210000035
The problem of extremum solving of the multiple function in the model of the formula (8) can be equivalent to the problem of extremum solving of the single function of the parameter n in the formula (9):
Figure BDA0001868375210000036
wherein the method comprises the steps of
Figure BDA0001868375210000037
Representing the corresponding average minimum mean square error when the added noise n is a constant vector; after the optimized solution of the above unitary function is obtained, the additive noise n required for minimizing the mean square error can be obtained opt ；
4) Optimal noise-enhanced bayesian estimation:
nonlinear system output signal y=t (x+n) with noise enhancement opt ) When the input parameter theta is estimated, the Bayesian estimation with the minimum mean square error is as follows:
Figure BDA0001868375210000038
input parameter θ and bayesian estimator thereof
Figure BDA0001868375210000039
The mean square error between them is:
Figure BDA00018683752100000310
the invention combines the noise enhancement with the Bayesian estimation method, and when the nonlinear system output signal is utilized to carry out Bayesian estimation on the input parameters after adding noise to the nonlinear system input signal, the minimum mean square error between the parameter estimation value and the true value can be further reduced.
The invention is mainly verified by adopting a simulation experiment method, and all steps and conclusions are verified to be correct on MATLAB R2016 a.
Drawings
Fig. 1 is a block diagram of the workflow of the present invention.
FIG. 2 shows the mean square error of the noise enhancement and the original optimal Bayes estimation corresponding to different t values in the simulation of the present invention.
FIG. 3 shows the mean square error of the noise enhancement and the original optimal Bayes estimation corresponding to different A values in the simulation of the present invention.
FIG. 4 shows the different μ's in the simulation of the present invention θ The noise enhancement corresponding to the value and the original optimal Bayes estimate the mean square error.
FIG. 5 is a graph of the different sigma in the simulation of the present invention θ The noise enhancement corresponding to the value and the original optimal Bayes estimate the mean square error.
Detailed Description
The present invention is further described below with reference to examples, but it should not be construed that the scope of the above subject matter of the present invention is limited to the following examples. Various substitutions and alterations are made according to the ordinary skill and familiar means of the art without departing from the technical spirit of the invention, and all such substitutions and alterations are intended to be included in the scope of the invention.
The embodiment discloses a Bayesian parameter estimation method based on noise enhancement, which comprises the following steps:
1) Constructing a noise enhancement nonlinear system:
the nonlinear system includes three parts: a nonlinear system input signal, a nonlinear system, and a nonlinear system output signal; the nonlinear system input signal x is closely related to the parameter θ, and the value of θ is determined by its probability density function p θ (θ) determination; adding independent additive noise n to the nonlinear system input signal x, and obtaining a noise-corrected nonlinear system output signal y=t (x+n) after the nonlinear system is passed, wherein T (·) represents a transfer function of the nonlinear system;
2) Establishing a noise enhancement Bayes parameter estimation model:
and estimating an input parameter theta by using the nonlinear system output signal y. When the value of y is constant, the input parameter θ and its estimated value
Figure BDA0001868375210000041
The mean square error epsilon (y) between them is
Figure BDA0001868375210000042
Wherein the method comprises the steps of
Figure BDA0001868375210000051
And->
Figure BDA0001868375210000052
Respectively representing the conditional mean and the conditional variance of the input parameter θ in case the nonlinear system output signal y is constant. Further, p (θ|y) represents the posterior probability of the input parameter θ, and is obtained by the expression (13)
Figure BDA0001868375210000053
Wherein p is y (y) represents the probability density function of the nonlinear system output signal y, and p y (y|θ) represents the case where the value of the parameter θ is constantThe nonlinear system outputs a conditional probability density function of the signal y.
From equation (12), since var (θ|y) is non-negative and is equal to the estimated amount
Figure BDA0001868375210000054
Irrelevant, the Bayes estimate that minimizes the mean square error ε (y) when the y value is constant is
Figure BDA0001868375210000055
That is, when the y value is constant, the corresponding minimum mean square error epsilon is estimated by Bayes MMSE (y) is the conditional variance of the input parameter θ:
Figure BDA0001868375210000056
wherein the method comprises the steps of
Figure BDA0001868375210000057
Represents θ in the case where the nonlinear system output signal y is constant 2 Is not limited to the above-described embodiments. Further, for any nonlinear system output signal y, the Bayesian estimate that minimizes the mean square error is +.>
Figure BDA0001868375210000058
Therefore, the estimate that minimizes the average mean square error of all possible nonlinear system output signals y is likewise +.>
Figure BDA0001868375210000059
I.e. < ->
Figure BDA00018683752100000510
Is such that ≡ε (y) p y (y) dy minimum estimation, corresponding mean square error of
Figure BDA00018683752100000511
Adding a probability density function p to a nonlinear system input signal x n When the additive noise n of (n) is applied, the conditional probability density function of the nonlinear system output signal y can be calculated as
Figure BDA00018683752100000512
Further, the probability density function of the nonlinear system output signal y is calculated as follows
Figure BDA0001868375210000061
Wherein the method comprises the steps of
Figure BDA0001868375210000062
Can be regarded as a probability density function of the nonlinear system output signal y when the added noise n is a constant vector, and R is equal to or less than 0 and is known by definition of the probability density function y (n)≤1。
Further, E (θ|y) and E (θ) 2 Y) is calculated as:
Figure BDA0001868375210000063
Figure BDA0001868375210000064
wherein the method comprises the steps of
Figure BDA0001868375210000065
Figure BDA0001868375210000066
3) Solving for additive noise required to minimize bayesian costs:
substituting (18), (20) and (21) into (16) results in the addition of a probability density function p to the nonlinear system input signal x n In the case of additive noise n of (n), when Bayesian estimation is performed on input parameter θ by using noise-corrected nonlinear system output signal y, the corresponding minimum mean square error is ε MMSE ＝MMSE(p n (n)), wherein
Figure BDA0001868375210000071
(24) In E n The representation is based on p n (n) the desire.
To obtain (24) the noise enhancement minimum mean square error MMSE (p) n (n)) corresponding optimal additive noise, constructing the following noise enhancement optimization model:
Figure BDA0001868375210000072
to solve the optimization problem in equation (25), a function is first introduced
Figure BDA0001868375210000073
When z 2 And (3) when the black plug moment of the function F (z) is equal to or more than 0, the black plug moment of the function F (z) is semi-positive, and therefore the F (z) can be obtained to be a convex function. Conversely, when z 2 When not less than 0, function->
Figure BDA0001868375210000074
Is a concave function, then there is (26) established
Figure BDA0001868375210000075
Let z 1 ＝G y (n)，z 2 ＝R y (n) and z 3 ＝J y (n). Because ofR y (n). Gtoreq.0, so that the probability density function p for any nonlinear system output signal y and any possible noise n (n) all can be seen
Figure BDA0001868375210000076
The two-sided integration of the inequality of (27) has the following result
Figure BDA0001868375210000081
Wherein the method comprises the steps of
Figure BDA0001868375210000082
Indicating the corresponding average minimum mean square error when the added noise n is a constant vector. Due to->
Figure BDA0001868375210000083
Thus have +.>
Figure BDA0001868375210000084
Figure BDA0001868375210000085
Further, the method comprises the steps of,
Figure BDA0001868375210000086
to sum up, the problem of extremum solving of the multiple function in the formula (25) can be equivalent to the problem of extremum solving of the single function with respect to the parameter n in the formula (29):
Figure BDA0001868375210000087
after the optimized solution of the above unitary function is obtained, the additive noise n minimizing the mean square error can be obtained opt ；
4) Optimal noise-enhanced bayesian estimation:
nonlinear system output signal y=t (x+n) with noise enhancement opt ) The input parameter θ is estimated such that the bayesian estimation with the minimum mean square error is:
Figure BDA0001868375210000088
input parameter θ and bayesian estimator thereof
Figure BDA0001868375210000089
The mean square error between them is:
Figure BDA00018683752100000810
the effect of the invention can be further illustrated by the following simulation experiments:
let x=θ+v be the nonlinear system input signal, where θ is the unknown parameter, and the corresponding probability density function is
Figure BDA00018683752100000811
I.e. θ obeys a mean value of μ θ Variance is->
Figure BDA0001868375210000091
Is a gaussian distribution of (c). Furthermore, v is zero-mean asymmetric Gaussian mixture background noise, whose probability density function is expressed as p v (v)＝tγ(v；(1-t)μ b ,σ b 2 )+(1-t)γ(v；-tμ b ,σ b 2 ) Wherein 0 < t < 1. When the value of theta is fixed, the conditional probability density function of the nonlinear system input signal x is p x (x|θ)＝p v (x- θ). Assuming that the nonlinear system is a amplitude limiting system, when a constant n is added to an input signal x of the nonlinear system as noise, the corresponding output signal y of the nonlinear system is:
Figure BDA0001868375210000092
implementation of n using MATLAB language programming opt Is used for solving the problem of (1),
Figure BDA0001868375210000093
and LMMSE (n) opt ) Is performed in the first step. At t=0.75, a=3, μ θ ＝3、σ θ ＝1、μ b =3 and σ b For example, =1, when no noise is added to the nonlinear system input signal x, the corresponding minimum mean square error of the original bayesian estimate is 0.8067 by adding a constant n to the nonlinear system input signal x opt The minimum mean square error of the Bayesian estimation can be reduced to 0.7003 by the= -2.75, and the performance is improved by 13.2% compared with the original Bayesian estimation.
Table 1 shows when a=3, μ θ ＝3、σ θ ＝1、μ b =3 and σ b When=1, the constant n is added to the nonlinear system input signal at t values of 0.075, 0.75 and 0.9, respectively opt And in addition, the corresponding minimum mean square error is estimated by Bayes before and after noise addition.
TABLE 1 minimum mean square error for Bayes estimation before and after noise addition
Figure BDA0001868375210000094
As can be seen from table 1, adding a proper constant to the nonlinear system input signal under certain conditions can further improve the performance of the optimal bayesian estimation.
To further study the estimation performance achieved by adding different noise to the nonlinear system input signal, the background noise parameter t, nonlinear system threshold A, and the average μ of the input parameters θ are then successively changed θ And standard deviation sigma θ The mean square error of the optimal Bayes estimation before and after noise addition under different conditions is compared, and the mean square error is specifically as follows:
hold a=3, μ θ ＝3、σ θ ＝1、μ b =3 and σ b =1, increasing t from 0 to 1 at 0.05 intervals. NeedleFor each value of t, adding the corresponding optimal additive noise n to the nonlinear system input signal x opt The minimum mean square error of the corresponding noise enhanced Bayes estimation after noise addition is obtained, and compared with the minimum mean square error of the original Bayes estimation when no noise addition is carried out, and the result is shown in figure 2. As the value of t increases, the minimum mean square error corresponding to both the noise enhanced bayesian estimation and the original bayesian estimation increases and decreases, and the former is symmetrical about t=0.5. In addition, for any possible value of t, the minimum mean square error corresponding to the noise enhanced bayesian estimate is smaller than the minimum mean square error corresponding to the Yu Yuanbei leaf estimate.
Hold t=0.75, μ θ ＝3、σ θ ＝1、μ b =3 and σ b =1, increasing a from 0 to 10 at 0.5 intervals, solving the mean square error of the corresponding noise enhanced optimal bayesian estimate for each a value, and comparing with the case without noise addition, the result is shown in fig. 3. The original bayesian estimate minimum mean square error gradually decreases from 0.9358 as the a value increases from zero, and remains constant 0.6895 as the a value is greater than 5.75. When 0 < A < 7.5, the noise enhanced Bayesian estimation minimum mean square error is smaller than the Yu Yuanbei Bayesian estimation minimum mean square error, and when A > 7.5, the mean square error corresponding to the Bayesian estimation cannot be reduced no matter any noise is added.
Hold t=0.75, a=3, σ θ ＝1μ b =3 and σ b =1 unchanged, mu θ Increasing from 0 to 5 at 0.25 intervals for each μ θ The mean square error of the corresponding noise enhanced optimal Bayes estimation is solved, and compared with the situation of no noise addition, and the result is shown in figure 4. The corresponding minimum mean square error of the original Bayes estimation is along with mu θ The increase in value increases, while the minimum mean square error corresponding to the optimal noise-enhanced bayesian estimation remains constant 0.7003 at all times. This indicates mu θ Performance of a > 0 noise enhanced Bayes estimation, mu θ The performance of the original bayesian estimate is the same when=0. In addition, the degree of improvement of the minimum mean square error corresponding to the Bayes estimation is along with mu θ The increase in value increases.
Hold t=0.75, a=3, μ θ ＝3、μ b =3 and σ b =1 unchanged, will σ θ Increasing from 0 to 2 at intervals of 0.1 for each sigma θ The mean square error of the corresponding noise enhanced optimal Bayes estimation is solved, and compared with the situation of no noise addition, and the result is shown in figure 5. The minimum mean square error corresponding to the noise enhanced Bayes estimation and the original Bayes estimation is about sigma θ Monotonically increasing functions. When sigma is θ The value approaches zero, and no noise is added, so that the Bayesian estimation performance is not likely to be improved. When sigma is θ When the value is increased to a certain degree, the improvement degree of the minimum mean square error corresponding to Bayesian estimation through noise addition is along with sigma θ Is increased by an increase in (a).

Claims (1)

1. The Bayesian parameter estimation method based on noise enhancement is characterized by comprising the following steps of:
1) Constructing a noise enhancement nonlinear system:
the nonlinear system includes three parts: a nonlinear system input signal, a nonlinear system, and a nonlinear system output signal; the nonlinear system input signal x is closely related to the input parameter θ, and the value of θ is determined by its probability density function p θ (θ) determination; adding independent additive noise n to the nonlinear system input signal x, and obtaining a noise-corrected nonlinear system output signal y=t (x+n) after the nonlinear system is passed, wherein T (·) represents a transfer function of the nonlinear system;
2) Establishing a noise enhancement Bayes parameter estimation model:
estimating an input parameter theta by using the nonlinear system output signal y; when the value of y is constant, the input parameter theta and the estimated quantity thereof are made
Figure FDA0004243007420000011
Bayes estimation with minimum mean square error
Figure FDA0004243007420000012
Corresponding mean square errorThe difference is
ε MMSE (y)＝E(θ 2 |y)-E 2 (θ|y) (2)
Wherein E (θ|y) and E (θ) 2 Y) represents the timing θ and θ of the y value, respectively 2 Is not limited to the desired one; further, the method comprises the steps of,
Figure FDA0004243007420000013
also, an estimate that minimizes the average mean squared error, the corresponding minimum mean squared error is
Figure FDA0004243007420000014
And
Wherein p is y (y) is a probability density function of the nonlinear system output signal y; p is p y (y), E (θ|y) and E (θ) 2 Y) is calculated as:
Figure FDA0004243007420000015
Figure FDA00042430074200000110
Figure FDA0004243007420000017
wherein the method comprises the steps of
Figure FDA0004243007420000018
Figure FDA0004243007420000019
p n (n) is a probability density function of additive noise n, and p x (x|θ) represents a conditional probability density function of the value of θ to time the nonlinear system input signal x;
3) Solving for additive noise required to minimize bayesian costs:
substituting the expression (4), the expression (5) and the expression (6) into the expression (3) can know that the probability density function is added to the nonlinear system input signal x to be p n In the case of additive noise n of (n), when Bayesian estimation is performed on the input parameter θ by using the noise-corrected nonlinear system output signal y, the corresponding average mean square error is
Figure FDA00042430074200000211
To obtain the minimum MMSE (p) in the formula (7) n (n)) corresponding optimal additive noise, constructing the following noise enhancement optimization problem:
Figure FDA00042430074200000212
bonding of
Figure FDA0004243007420000023
The characteristics of (1) are known to be +.>
Figure FDA0004243007420000024
Figure FDA0004243007420000025
The problem of extremum solving of the multiple function in the model of the formula (8) can be equivalent to the problem of extremum solving of the single function of the parameter n in the formula (9):
Figure FDA00042430074200000213
wherein the method comprises the steps of
Figure FDA0004243007420000027
Representing the corresponding average minimum mean square error when the additive noise n is a constant vector; obtaining the optimized solution of the unitary function to obtain the maximum mean square errorAdditive noise n required for hours opt ；
4) Optimal noise-enhanced bayesian estimation:
nonlinear system output signal y=t (x+n) with noise enhancement opt ) When the input parameter theta is estimated, the Bayesian estimation with the minimum mean square error is as follows:
Figure FDA00042430074200000214
input parameter θ and bayesian estimator thereof
Figure FDA0004243007420000029
The mean square error between them is:
Figure FDA00042430074200000215
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