CN107315918B - Method for improving steady estimation by using noise - Google Patents

Method for improving steady estimation by using noise Download PDF

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CN107315918B
CN107315918B CN201710546499.XA CN201710546499A CN107315918B CN 107315918 B CN107315918 B CN 107315918B CN 201710546499 A CN201710546499 A CN 201710546499A CN 107315918 B CN107315918 B CN 107315918B
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潘燕
段法兵
任昱昊
许丽艳
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Qingdao University
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Abstract

The invention relates to the field of signal and parameter estimation, in particular to a method for improving the performance of a steady estimation quantity by using noise. The method is based on the correlation principle of stochastic resonance, and can reduce the Mean Square Error (MSE) of the robust estimator. The effect of noise is concentrated on two points: 1) when the background noise is symmetrically distributed with a thick tail, the L steady estimators are connected in parallel, after the noise which is independent and identically distributed is added in the input observation data of each estimator unit, the output of the L estimators is calculated to be the average of statistics and is used as the estimator of the system, the asymptotic efficiency of the estimator relative to the maximum likelihood estimator is calculated, and the asymptotic efficiency of the estimator of the system can be improved after the additive noise with a specific level is added. 2) Under an asymmetric noise model of the pollution distribution, adding noise to the observed data can reduce the maximum asymptotic deviation of the robust estimator. The method further improves the estimation precision and robustness of the robust estimator in the presence of abnormal data, and can be used as an important means for improving the robust estimation.

Description

Method for improving steady estimation by using noise
Technical Field
The invention relates to the field of signal and parameter estimation, in particular to a method for improving the performance of a steady estimation quantity by using noise.
Background
In the signal estimation and detection problem, we often assume that the population obeys a certain distribution or a certain type of distribution, and select the optimal criterion to design the corresponding estimator and test statistic on the basis of the distribution or the distribution. In practice, however, during the measurement activity, the influence of impulse noise may cause abnormal values to be generated in the received or observed data. For example, in an outdoor wireless mobile communication channel, the wireless mobile communication channel is interfered by transient noise of a switch in a power line or impulse noise of automobile ignition; in radar and sonar systems, they are subject to natural or artificial electromagnetic and acoustic pulse interference; in the estimation and tracking of the geographic position, abnormal values are generated in the measurement data due to obstacles such as buildings and trees. In signal estimation, such impulse noise causes a large amount of observation data to have partial "abnormal data", so that a supposed ideal distribution model generates deviation, in such a case, the estimation amount under the optimal criterion is difficult to realize, and the traditional statistical method is greatly influenced by the individual abnormal data, and the robust estimation technology is generated under the background. Robust estimator requirements: under the assumed observation distribution model, the estimation should be the same as that of the classical estimation, or slightly different; when the assumed distribution model has small difference with the actual theoretical distribution model, the estimation value of the distribution model still needs to be close to the optimal value; the estimate is available even if the assumed model has a large deviation from the actual model. Therefore, robust estimation pursues efficient estimation from a practical standpoint. In order to further improve the performance of the robust estimator, it is the main technical idea of the present invention to improve the estimation performance by using the noise characteristics.
Disclosure of Invention
The technical problem to be solved by the present invention is how to overcome the deficiencies of the prior art, and to provide a method for improving the performance of a robust estimator by using noise.
The technical scheme adopted for realizing the purpose is as follows: random noise is added into a sample containing abnormal data, so that the performance of the robust estimator can be improved, and the method comprises the following steps:
1. initializing parameters: recording sample observation data x ═ { x ] containing abnormal values1，x2，...，xnAnd its length n, the reference parameter theta is estimated.
2. Establishing a position parameter model: x is the number ofi＝θ+wi,i＝1,2,...,n，wiIs a function of probability density of fwThick tail distribution or background noise of pollution distribution;
determination of background noise wiSelecting a suitable robust M-estimator
Figure GDA0002417337420000021
Adding an additive noise sequence η with level d to the observed data x { η ═1,η2,...,ηnAfter that, the mean square error MSE of the estimator at this time is calculated:
Figure GDA0002417337420000022
3. suppose background noise wiThe distribution of (A) is in accordance with the symmetrical distribution model of the thick tail
3.1 select unbiased robust M-estimator with corresponding score function ψ;
3.2 connect L selected M-estimators with the scoring function ψ in parallel, forming a parallel subsystem,
each estimation unit has the same input observed data x, and independent and identically distributed noise sequences η are added to the input sequence xl＝{η1l,η2l,...,ηnlGet a new set of sample data yl＝{y1l,y2l,...,ynlIn which y isil＝xi+ηil＝θ+zil，zil＝wi+ηil，i＝1,2,...,n，zilIs the addition of additive noise ηilThe obtained composite noise variable has a probability density function fzFor the output of L estimators
Figure GDA0002417337420000023
Calculating a statistical average, and establishing a final estimation statistic:
Figure GDA0002417337420000024
3.3 the MSE index deviation of this system estimator is zero, the ratio of the variance of this estimator to the variance of the maximum likelihood estimator without noise is calculated as the asymptotic efficiency of the estimator:
Figure GDA0002417337420000025
wherein:
n is the sample length, J (f)w) As a function of the probability density fwL is the number of parallel estimators, Eω、Eη、EzRespectively for background noise wiProbability ofDegree function fwAdditive noise ηilProbability density function fηAnd composite noise zilProbability density function fzMean value of (i)
Figure GDA0002417337420000031
Figure GDA0002417337420000032
3.4 utilization of
Figure GDA0002417337420000033
To obtain an optimal additive noise level doptMaximum asymptotic efficiency value of lower.
4. Suppose background noise wiIs an asymmetric pollution distribution model with a probability density function defined as:
fw(x)＝(1-ε)f0(x；θ)+εq(x)，
wherein: f. of0(x) For a given central probability density function, q (x) is an asymmetric pollution probability density function, and the pollution proportion is epsilon.
4.1 select M-estimator corresponding to bounded scoring function ψ, which is bounded by s, set sample length n large enough, estimate MSE variance asymptotically zero, consider maximum asymptotic deviation b of estimateε；
4.2 Add additive noise η to each observed datailSince 3.2 then considers the case of an M-estimator node, i.e. the number of parallel estimators L is 1 and the index L takes on the value 1, the additive noise η in this case is addedilAbbreviated as ηiComplex noise zilAbbreviated as ziCalculating the composite noise ziIs still a random variable with an epsilon-contamination distribution having a probability density function fz(x)＝(1-ε)fz0(x)+q(x)，fz0As noise ziCorresponding central probability density function, calculating its central probability density function
Figure GDA0002417337420000034
The maximum asymptotic deviation b of the M-estimator is then calculatedεIs a solution of the following equation for b:
Figure GDA0002417337420000035
the maximum asymptotic deviation for the median estimator (with the scoring function ψ (x) ═ sgn (x)) in particular is:
Figure GDA0002417337420000036
wherein
Figure GDA0002417337420000037
For complex noise ziCumulative distribution of
Figure GDA0002417337420000041
The inverse function of (c);
4.3 optimal additive noise level dopt＝argminbεThe minimum value of the maximum asymptotic deviation can be obtained at this noise level.
The invention has the beneficial effects that: random noise with specific distribution is added in observation data by utilizing a stochastic resonance correlation theory, and the asymptotic efficiency of the parallel robust estimation system can be improved under the background noise with symmetrical distribution; under the background noise of the asymmetrically distributed pollution model, the maximum asymptotic deviation of the robust estimator can be effectively reduced, and the performance of the robust estimator containing abnormal data is further improved.
Drawings
Fig. 1 is a schematic diagram of a parallel system of robust estimators.
Figure 2 is a graph of the asymptotic efficiency of the estimator of the present invention as a function of the additive noise level for different numbers of arrays in parallel.
Fig. 3 is a graph of the asymptotic efficiency of the estimator with the parameters of the estimator under optimum additive noise levels and without noise according to the present invention.
Fig. 4 is a graph of the maximum asymptotic deviation of the estimator of the present invention as a function of the additive noise level.
Detailed Description
In order to further improve the performance of the robust estimator under the condition of containing abnormal data, the theory of signal stochastic resonance is applied to the field of signal estimation, noise is fully utilized instead of being eliminated, and the performance of the robust estimator is further improved by adding the noise into the observed data. This is the main technical idea of the present invention.
The invention will be described in further detail below with reference to the accompanying drawings:
the method for improving the robust estimation by using the noise comprises the following specific steps:
(1) recording sample observation data containing abnormal values and the length n thereof, setting a parameter theta to be estimated and the number n of sample points, and establishing the following model
xi＝θ+wi,i＝1,2,...,n
Wherein the content of the first and second substances,
wiis background noise independent of each other and having the same thick tail distribution or pollution distribution, and has a probability density function fw。
(2) At fwUnder the assumption of symmetrical distribution, unbiased M-estimators are selected to form parallel subsystems as shown in FIG. 1, at each observation xiAdding separately identically distributed additive noise ηilAnd then to each unit M-estimator, the input data of the unit estimator becomes:
yil＝θ+wi+ηil＝θ+zil，l＝1,2,…,L，i＝1,2,…,n。
(3) obtaining the outputs of L M-estimators
Figure GDA0002417337420000051
All output is
Figure GDA0002417337420000052
And summing and averaging to obtain an estimation quantity of the whole system:
Figure GDA0002417337420000053
(4) added noise ηilProbability density distribution fηWhen distributed symmetrically, the system estimates the quantity
Figure GDA0002417337420000054
The asymptotic efficiency of the estimator is calculated relative to the maximum likelihood estimator in the absence of additive noise:
Figure GDA0002417337420000055
it is a function of the additive noise level d;
wherein:
n is the sample length, J (f)w) As a function of the probability density fwL is the number of parallel estimators, Eω、Eη、EzRespectively for background noise wiProbability density function fwAdditive noise ηilProbability density function fηAnd composite noise zilProbability density function fzMean value of (i)
Figure GDA0002417337420000056
Figure GDA0002417337420000057
(5) By using
Figure GDA0002417337420000058
To obtain an optimal additive noise level doptAnd calculating to obtain the optimal additive noise level doptLower asymptotic efficiency value
Figure GDA0002417337420000059
(6) Under the assumption that the background noise is asymmetrically distributed, the probability density function is assumed to be:
fw(x)＝(1-ε)f0(x；θ)+εq(x)，
wherein: f. of0(x) For a given central probability density function, q (x) is an asymmetric pollution probability density function, and the pollution ratio is ε. And the sample length n is large enough, the M-estimator for the bounded scoring function ψ is chosen, bounded by s.
(7) Random additive noise η is added to each observationiHaving a probability density fηCalculating the composite noise zi＝wi+ηiCentral probability density function of (1):
Figure GDA0002417337420000061
the maximum asymptotic deviation b of the M-estimator is then calculatedεIs a solution of the following equation for b:
Figure GDA0002417337420000062
(8) by dopt＝argminbεObtaining the optimal additive noise level d with the minimum value of the maximum asymptotic deviationopt. Adding with optimal additive noise level doptThe maximum asymptotic deviation of the estimator is greatly reduced.
(9) Wherein the additive noise ηil(additive noise injected by node L when L nodes are connected in parallel), additive noise ηi(there is no parallel node, i.e. no sign of L), these two noises are a kind of noise, both additive, one in the parallel L configuration and one in the single estimator case.
Results of the experiment
Background noise is symmetrically distributed
(1) Setting the probability density function of background noise as the Cauchy distribution of thick tail
Figure GDA0002417337420000063
It is assumed here that the position parameter of the cauchy distribution is zero and the scale parameter σ is 1. The M-estimator was chosen to be the bisquare estimator, with a scoring function as follows:
Figure GDA0002417337420000071
here, the parameter γ of the estimator is selected to be 1.
Adding symmetrically evenly distributed noise η with noise level dilFIG. 2 then shows the asymptotic efficiency of the estimator for the parallel system
Figure GDA0002417337420000072
The variation with additive noise level d at different number of parallels L. It can be seen from the figure that in the case of a single estimate L ═ 1, the efficiency of the estimate is reduced by adding noise, but that in the case of L > 1, there is an optimum additive noise level doptThe asymptotic efficiency of the system estimator is maximized and is much greater than the efficiency of the estimator without the addition of noise.
(2) The number of fixed parallel estimators L is 1000, and fig. 3 gives the noise level d at an optimum uniform distributionoptLower, asymptotic efficiency
Figure GDA0002417337420000073
Curve when the parameter γ of the estimator changes. As can be seen from the figure, when the value of the estimator parameter γ is within a certain range, the noise addition can effectively improve the efficiency of the estimator, compared with the asymptotic efficiency of the estimator when no noise is added.
Background noise is asymmetrically distributed
(1) The background noise is assumed to be an asymmetric pollution distribution model, wherein the distribution of the central parameters of the subject is a symmetric gaussian mixture distribution:
Figure GDA0002417337420000074
the contamination distribution q (x) is a point mass distribution, i.e.
Figure GDA0002417337420000075
Only when x is equal to x0The probability of time is 1, where a worst limit case is set: x is the number of0→ ∞. Setting center distribution f0Is given by the parameter mu-3 and sigma-1, a median estimator is selected in the form of a simple scoring function, which is a sign function
ψ(x)＝sgn(x)；
(2) Adding bifurcation noise
Figure GDA0002417337420000076
Figure 4 then shows the maximum asymptotic deviation b of the median estimateεCurve with additive noise level d. At an optimum additive noise level doptWith 3, the minimum value of the maximum asymptotic deviation can be reached.
The above embodiments are only for illustrating the technical concept and features of the present invention, and the purpose thereof is to enable those skilled in the art to understand the contents of the present invention and implement the present invention accordingly, and not to limit the protection scope of the present invention accordingly. All equivalent changes or modifications made in accordance with the spirit of the present disclosure are intended to be covered by the scope of the present disclosure.

Claims (1)

1. A method for improving robust estimation using noise, comprising the steps of:
step 1: recording sample observation data x ═ { x ] containing abnormal values1，x2，...，xnAnd the length n thereof, establishing a position parameter model:
xi＝θ+wi,i＝1,2,...,n
wherein:
background noise wiN are random variables independent of each other and have the same thick tail or contamination distribution, wiHas a probability density function of fwSetting a pre-estimated reference parameter theta;
step 2: determination of background noise wiSelecting a suitable robust M-estimator
Figure FDA0002229320050000018
Adding a noise sequence η ═ η with a noise level d to the observed data x1,η2,...,ηnAfter that, the mean square error MSE of the estimator at this time is calculated:
Figure FDA0002229320050000011
wherein:
Figure FDA0002229320050000012
to estimate
Figure FDA0002229320050000013
The term of the variance of (a) is,
Figure FDA0002229320050000014
is a deviation term;
and step 3: based on step 2, an assumption is made about the noise distribution:
(1) suppose background noise wiProbability density function fwIs symmetrical and has thick tail characteristics;
1) selecting an unbiased robust M-estimator with a corresponding score function of ψ;
2) connecting L selected M-estimators having a scoring function psi in parallel to form a parallel subsystem, each estimation unit having the same input observation x, adding an independent and identically distributed noise sequence η to the input sequence x of each estimation unitl＝{η1l,η2l,...,ηnlGet a new set of sample data yl＝{y1l,y2l,...,ynlIn which y isil＝xi+ηil＝θ+zil，zil＝wi+ηil，i＝1,2,...,n，zilIs the addition of additive noise ηilThe obtained composite noise random variable has a probability density function fz(ii) a Obtain LOutput of M-estimator
Figure FDA0002229320050000015
All output is
Figure FDA0002229320050000016
Calculating statistical average, and establishing estimation statistics of the system:
Figure FDA0002229320050000017
3) the MSE bias term of the system estimator is zero and the ratio of the estimator variance to the variance of the maximum likelihood estimator without noise is calculated and referred to as the relative asymptotic efficiency of the estimator:
Figure FDA0002229320050000021
；
wherein:
n is the sample length, J (f)w) As a function of the probability density fwL is the number of parallel estimators, Eω、Eη、EzRespectively for background noise wiProbability density function fwAdditive noise ηilProbability density function fηAnd composite noise zilProbability density function fzMean value of (i)
Figure FDA0002229320050000022
Figure FDA0002229320050000023
4) Asymptotic efficiency
Figure FDA0002229320050000024
The method is a non-monotonic function about the noise level d, and the asymptotic efficiency is improved by adjusting the noise level d to a certain specific value;
(2) suppose background noise wiIs an asymmetric epsilon-pollution distribution model, wherein the pollution distribution model means that the data of 1-epsilon proportion is derived from the overall distribution of F0Called central distribution, the data of the epsilon ratio are generated from a contamination distribution Q of unknown type, called contamination distribution, f0(x) And q (x) respectively a central distribution F0Probability density function, f, corresponding to pollution distribution Q0(x) Called central probability density function, q (x) called pollution probability density function, such that the noise w with the pollution distribution modeliHas a probability density function of fw(x)＝(1-ε)f0(x) + ε q (x), ε is the proportion of contamination;
1) selecting M-estimator corresponding to bounded scoring function psi with boundary s, setting sample length n large enough, MSE variance approaching zero, and considering worst estimation deviation, i.e. maximum asymptotic deviation bε；
2) Additive noise η is added to each observationilThereafter, since step 3 (2) considers the case of one M-estimator node, i.e. the number L of parallel estimators is 1 and the index L takes on the value of only 1, the additive noise η in this case is addedilAbbreviated as ηiComplex noise zilAbbreviated as ziCalculating the composite noise ziIs still a random variable with an epsilon-contamination distribution having a probability density function fz(x)＝(1-ε)fz0(x)+q(x)，fz0As noise ziCorresponding central probability density function, calculating its central probability density function
Figure FDA0002229320050000031
The maximum asymptotic deviation b of the M-estimator is then calculatedεIs a solution of the following equation for b:
Figure FDA0002229320050000032
the maximum asymptotic deviation for a median estimate with a scoring function ψ (x) ═ sgn (x) is specified as:
Figure FDA0002229320050000033
wherein
Figure FDA0002229320050000034
Is cumulatively distributed
Figure FDA0002229320050000035
The inverse function of (c);
3) additive noise ηiThere is an optimum level dopt＝arg minbεThe minimum value of the maximum asymptotic deviation can be obtained at this noise level, which greatly reduces the maximum asymptotic deviation of the estimator.
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