CN103984843A - Bayesian approximation method for fuzzy hypotheses testing of fuzzy data in radar detection - Google Patents

Abstract

The invention provides a Bayesian solution for hypotheses testing in which some concepts lack precision due to ambiguity and when hypotheses and data have ambiguity in statistical decision in radar signals. Based on the fuzzy hypotheses testing of clear data, for fuzzy hypotheses and clear data, a radar detection criterion is determined based on the Bayesian method. For the fuzzy data, the Bayesian fuzzy hypotheses testing method is provided. Finally, the comparison with other existing testing methods is carried out.

Description

In radar detecting, for fuzzy data, carry out Bayes's approximatioss of Fuzzy Hypothesis Testing

Technical field

The present invention is to provide a kind ofly in radar signal statistical decision, because the existence of ambiguity makes some concept, lack accuracy, and Bayes's solving method of the test of hypothesis when supposing and data all have ambiguity.

Take the Fuzzy Hypothesis Testing of clear data as basis, for fuzzy hypothesis and clear data, based on bayes method, determined radar detecting criterion.Then for the data with ambiguity, Bayes's Fuzzy Hypothesis Testing method has been proposed.Last and other existing method of inspection contrasts.

Background technology

Bayesian theory provides Fundamentals of Mathematics for processing the sample data of new generation, and these data are through becoming sequence data after a while.It is a kind of when having increasing sample data available that bayesian theory provides, and constantly updates the method for θ.All there are two kinds of uncertainties in data.Be a uncertainty of bringing due to the error that the randomness of data produces, these data are represented by probability density function conventionally.Another is the uncertainty of bringing due to the ambiguity of data, and these data are represented by subordinate function conventionally.The data of real world are fuzzy (coarse), and randomness and ambiguity are also deposited.One of basic object of statistical reasoning is exactly test of hypothesis.

In traditional radar detection system, for the signal receiving, judge that it is target or noise, employing be a kind of method of hard decision, this signal only has two kinds of possibilities: be not that target is exactly noise.This judgement mode is applied in some false judgment of the middle inevitably generation of radar (as tracking radar) of some kind.

This patent has proposed Bayes's solving method of the test of hypothesis when supposing and data all have ambiguity, and it can, according to the fuzzy hypothesis that a certain degree is accepted or refusal is predetermined, can be used for different types of radar receiving trap.

Summary of the invention

The object of the present invention is to provide a kind of given assumed condition of working as, the bayes decision method when data that radar receives all have ambiguity.

Make X ₁..., X ₅be the energy signal of the independent sample with same distribution that receives from radar receiver, its distribution is that to obey average μ unknown, variances sigma ²=9 known normal probability density functions.Their value is respectively: x ₁=0.05, x ₂=0.25, x ₃=1.35, x ₄=2.1, x ₅=2.65.

First define: H ₀table driftlessness, H ₁table has target;

" false alarm rate "=P _fa=β=I type error probability=P (refusal H ₀| H ₀be true);

" false dismissed rate "=λ=II type error probability=P (accepts H ₀| H ₁be true);

" verification and measurement ratio "=1-λ;

Dualism hypothesis: (noise)

(target)

The priori probability density that makes μ is N (0,1). it is also normal state that the posterior probability of well-known normal density function distributes.Test of hypothesis is:

H ₀(μ): μ≤2micro watt (noise or driftlessness)

H ₁(μ): μ >2micro watt (signal+noise=have target)

Its subordinate function as shown in Figure 1, is respectively:

$H_0\left(\mathrm{\&mu;}\right)\left\{\begin{array}{c}\mathrm{1,0}\&le;\mathrm{\&mu;}<1.5\\ 2-\frac{2}{3}\mathrm{\&mu;},1.5\&le;\mathrm{\&mu;}<\&infin;\end{array}\right.$ $H_1\left(\mathrm{\&mu;}\right)=\left\{\begin{array}{c}\frac{1}{2}\mathrm{\&mu;}-\frac{1}{1},1\&le;\mathrm{\&mu;}<3\\ \mathrm{1,3}\&le;\mathrm{\&mu;}<\&infin;\end{array}\right.$

Computing formula is:

$\underset{\mathrm{\&theta;}\&Element;{\mathrm{\&Theta;}}_0}{\&Integral;}K(\mathrm{\&theta;}/x)H_0\left(\mathrm{\&theta;}\right)\mathrm{d\&theta;}=\frac{1}{3\&CenterDot;\sqrt{2\mathrm{\&pi;}}}[\underset{0}{\overset{15}{\&Integral;}}{e}^{\frac{-{(x-\mathrm{\&mu;})}^2}{18}}\mathrm{d\&mu;}+\underset{1.5}{\overset{3}{\&Integral;}}{e}^{\frac{-{(x-\mathrm{\&mu;})}^2}{18}}\&CenterDot;(2-\frac{2}{3}\mathrm{\&mu;})\mathrm{d\&mu;}]$

$\underset{\mathrm{\&theta;}\&Element;{\mathrm{\&Theta;}}_1}{\&Integral;}K(\mathrm{\&theta;}/x)H_1\left(\mathrm{\&theta;}\right)\mathrm{d\&theta;}=\frac{1}{3\&CenterDot;\sqrt{2\mathrm{\&pi;}}}[\underset{3}{\overset{\&infin;}{\&Integral;}}{e}^{\frac{-{(x-\mathrm{\&mu;})}^2}{18}}\mathrm{d\&mu;}+\underset{1}{\overset{3}{\&Integral;}}{e}^{\frac{-{(x-\mathrm{\&mu;})}^2}{18}}\&CenterDot;(\frac{1}{2}\mathrm{\&mu;}-\frac{1}{2})\mathrm{d\&mu;}]$

By MATLAB software for calculation, obtain table 1 result.

Table 1 shows when the increase of x value, refusal hypothesis H ₀fuzzy value (being decision-making degree) reduce, accept hypothesis H ₁fuzzy value (being decision-making degree) increase.This just means the power that increases radar receiving trap, the decision-making degree that will to increase signal decision be target.

The decision table of the different x values of table 1

The new algorithm that fuzzy data is carried out to Fuzzy Hypothesis Testing with bayes method is as follows:

When sample data is fuzzy data,

The 1st) step: determine the subordinate function of the data x (δ) receiving, δ shows degree of membership, δ ∈ [0,1] x (δ)=[x ^l(δ), x ^u(δ)] (due to data produce fuzzy)

The 2nd) step: determine H ₀(θ) and H ₁(θ) subordinate function (fuzzy due to what suppose to produce)

The 3rd) step: to each δ ₁, (i=1 ..., n, here δ ₁=0 until δ _n=1), calculate x ^l(δ ₁) and x ^u(δ ₁)

The 4th) step: to x ^l(δ ₁) and x ^u(δ ₁), calculate

$A\left[\mathrm{\&delta;}\right]=\underset{\mathrm{\&theta;}\&Element;{\mathrm{\&Theta;}}_0}{\&Integral;}K(\mathrm{\&theta;}/x)H_0\left(\mathrm{\&theta;}\right)\mathrm{d\&theta;}=[a_1\left(\mathrm{\&delta;}\right),a_2\left(\mathrm{\&delta;}\right)]B\left[\mathrm{\&delta;}\right]=\underset{\mathrm{\&theta;}\&Element;{\mathrm{\&Theta;}}_1}{\&Integral;}K(\mathrm{\&theta;}/x)H_1\left(\mathrm{\&theta;}\right)\mathrm{d\&theta;}=[b_1\left(\mathrm{\&delta;}\right),b_2\left(\mathrm{\&delta;}\right)]$

The 5th) step: draw subordinate function curve

$A\left[\mathrm{\&delta;}\right]=\underset{\mathrm{\&theta;}\&Element;{\mathrm{\&Theta;}}_0}{\&Integral;}K(\mathrm{\&theta;}/x)H_0\left(\mathrm{\&theta;}\right)\mathrm{d\&theta;},B\left[\mathrm{\&delta;}\right]=\underset{\mathrm{\&theta;}\&Element;{\mathrm{\&Theta;}}_1}{\&Integral;}K(\mathrm{\&theta;}/x)H_1\left(\mathrm{\&theta;}\right)\mathrm{d\&theta;}$

The 6th) step: accept hypothesis $H_0=\frac{\underset{\mathrm{\&delta;}=0}{\overset{\mathrm{\&delta;}<1}{\mathrm{\&Sigma;}}}[\left(A\right[\mathrm{\&delta;}]>B[\mathrm{\&delta;}\left]\right)\&CenterDot;(b_2\left(\mathrm{\&delta;}\right)-b_1\left(\mathrm{\&delta;}\right))\&CenterDot;(a_2\left(\mathrm{\&delta;}\right)-a_1\left(\mathrm{\&delta;}\right))]}{\underset{\mathrm{\&delta;}=0}{\overset{\mathrm{\&delta;}<1}{\mathrm{\&Sigma;}}}[(b_2\left(\mathrm{\&delta;}\right)-b_1\left(\mathrm{\&delta;}\right))\&CenterDot;(a_2\left(\mathrm{\&delta;}\right)-a_1\left(\mathrm{\&delta;}\right))]}$ Here,

$A\left[\mathrm{\&delta;}\right]>B\left[\mathrm{\&delta;}\right]=\frac{a_2\left(\mathrm{\&delta;}\right)-b_1\left(\mathrm{\&delta;}\right)}{(b_2\left(\mathrm{\&delta;}\right)-b_1\left(\mathrm{\&delta;}\right)+(a_2\left(\mathrm{\&delta;}\right)-a_1\left(\mathrm{\&delta;}\right))}.$

Accompanying drawing explanation

Fig. 1 H ₀(μ) and H ₁(μ) subordinate function

The subordinate function of Fig. 2 x (δ)

The fuzzy hypothesis of Fig. 3 H ₀and H ₁subordinate function curve

Embodiment

Radar detection process based on new algorithm.Due to the complicacy of algorithm, x only gets a value, supposes x=0.25, and its subordinate function is x (δ).

The 1st) step: determine the subordinate function of x (δ) as shown in Figure 2.

Be embodied as: $x\left(\mathrm{\&delta;}\right)=[x^L\left(\mathrm{\&delta;}\right),x^U\left(\mathrm{\&delta;}\right)]=[\frac{\mathrm{\&delta;}}{4},\frac{1}{2}-\frac{1}{4}\mathrm{\&delta;}]$

The 2nd) step: determine H ₀(θ) and H ₁(θ) subordinate function, as shown in Figure 1

The 3rd) the, the 4th) step: utilize MATLAB, calculate table 2 data:

The A[δ that the different δ values of table 2 are corresponding] and B[δ]

?	a 1(δ)	a 2(δ)	b 1(δ)	b 2(δ)
					δ＝0	15.3397	16.0917	14.4982	17.6288
δ＝0.1	15.3856	16.0632	14.6471	17.4656
					δ＝0.2	15.4314	16.0338	14.7968	17.3031
δ＝0.3	15.4758	16.003	14.9479	17.1408
					δ＝0.4	15.5194	15.9721	15.0998	16.9798
δ＝0.5	15.5623	15.9398	15.2533	16.8188
					δ＝0.6	15.6036	15.9067	15.4059	16.6595
δ＝0.7	15.6458	15.8714	15.56	16.4993
					δ＝0.8	15.6856	15.8368	15.7150	16.3413
δ＝0.9	15.7239	15.7999	15.8706	16.1835

That table 2 is listed is A[δ] and B[δ] membership function value.The interval of δ value is less, and their membership function value is more accurate, but also can make calculating more complicated, extends the time that radar produces the result of decision.

The 5th) step: draw subordinate function curve as shown in Figure 3.

$A\left[\mathrm{\&delta;}\right]=\underset{\mathrm{\&theta;}\&Element;{\mathrm{\&Theta;}}_0}{\&Integral;}K(\mathrm{\&theta;}/x)H_0\left(\mathrm{\&theta;}\right)\mathrm{d\&theta;}$

$B\left[\mathrm{\&delta;}\right]=\underset{\mathrm{\&theta;}\&Element;{\mathrm{\&Theta;}}_1}{\&Integral;}K(\mathrm{\&theta;}/x)H_1\left(\mathrm{\&theta;}\right)\mathrm{d\&theta;}$

The 6th) step: accept H ₀=38.7995%, accept H ₁=61.2005%

Table 3 has been enumerated common hypothesis (CHCD), has the fuzzy hypothesis (FHCD) of precise information and has the difference between the fuzzy hypothesis (FHFD) of fuzzy data.First, for CHCD, refusal hypothesis H ₀decision-making without any acceptance or refusal degree.For FHCD, with 50.4024% refusal hypothesis H ₀, for FHFD, with 61.2005% refusal hypothesis H ₀

Comparison between the different hypothesis of table 3 kind

。

Claims (1)

1. be applied in radar detecting, for Bayes's Fuzzy Hypothesis Testing method of fuzzy data, said method comprising the steps of:

1), first define: H ₀table driftlessness, H ₁table has target;

" false alarm rate "=P _fa=β=I type error probability=P (refusal H ₀| H ₀be true);

" false dismissed rate "=λ=II type error probability=P (accepts H ₀| H ₁be true);

" verification and measurement ratio "=1-λ;

Dualism hypothesis: (noise)

(target)

The priori probability density that makes μ is N (0,1). it is also normal state that the posterior probability of well-known normal density function distributes.Test of hypothesis is:

H ₀(μ): μ≤2micro watt (noise or driftlessness)

H ₁(μ): μ >2microwatt (signal+noise=have target)

Its subordinate function is respectively:

$H_0\left(\mathrm{\&mu;}\right)\left\{\begin{array}{c}\mathrm{1,0}\&le;\mathrm{\&mu;}<1.5\\ 2-\frac{2}{3}\mathrm{\&mu;},1.5\&le;\mathrm{\&mu;}<\&infin;\end{array}\right.$ $H_1\left(\mathrm{\&mu;}\right)=\left\{\begin{array}{c}\frac{1}{2}\mathrm{\&mu;}-\frac{1}{1},1\&le;\mathrm{\&mu;}<3\\ \mathrm{1,3}\&le;\mathrm{\&mu;}<\&infin;\end{array}\right.$

2), when sample data is fuzzy data, determine the subordinate function of the data x (δ) receive, δ shows degree of membership, δ ∈ [0,1] x (δ)=[x ^l(δ), x ^u(δ)] (due to data produce fuzzy)

3), determine H ₀(θ) and H ₁(θ) subordinate function (fuzzy due to what suppose to produce)

4), to each δ ₁, (i=1 ..., n, here δ ₁=0 until δ _n=1), calculate x ^l(δ ₁) and x ^u(δ ₁)

5), to x ^l(δ ₁) and x ^u(δ ₁), calculate

$A\left[\mathrm{\&delta;}\right]=\underset{\mathrm{\&theta;}\&Element;{\mathrm{\&Theta;}}_0}{\&Integral;}K(\mathrm{\&theta;}/x)H_0\left(\mathrm{\&theta;}\right)\mathrm{d\&theta;}=[a_1\left(\mathrm{\&delta;}\right),a_2\left(\mathrm{\&delta;}\right)]B\left[\mathrm{\&delta;}\right]=\underset{\mathrm{\&theta;}\&Element;{\mathrm{\&Theta;}}_1}{\&Integral;}K(\mathrm{\&theta;}/x)H_1\left(\mathrm{\&theta;}\right)\mathrm{d\&theta;}=[b_1\left(\mathrm{\&delta;}\right),b_2\left(\mathrm{\&delta;}\right)]$

6), draw subordinate function curve

$A\left[\mathrm{\&delta;}\right]=\underset{\mathrm{\&theta;}\&Element;{\mathrm{\&Theta;}}_0}{\&Integral;}K(\mathrm{\&theta;}/x)H_0\left(\mathrm{\&theta;}\right)\mathrm{d\&theta;},B\left[\mathrm{\&delta;}\right]=\underset{\mathrm{\&theta;}\&Element;{\mathrm{\&Theta;}}_1}{\&Integral;}K(\mathrm{\&theta;}/x)H_1\left(\mathrm{\&theta;}\right)\mathrm{d\&theta;}$

7), accept hypothesis $H_0=\frac{\underset{\mathrm{\&delta;}=0}{\overset{\mathrm{\&delta;}<1}{\mathrm{\&Sigma;}}}[\left(A\right[\mathrm{\&delta;}]>B[\mathrm{\&delta;}\left]\right)\&CenterDot;(b_2\left(\mathrm{\&delta;}\right)-b_1\left(\mathrm{\&delta;}\right))\&CenterDot;(a_2\left(\mathrm{\&delta;}\right)-a_1\left(\mathrm{\&delta;}\right))]}{\underset{\mathrm{\&delta;}=0}{\overset{\mathrm{\&delta;}<1}{\mathrm{\&Sigma;}}}[(b_2\left(\mathrm{\&delta;}\right)-b_1\left(\mathrm{\&delta;}\right))\&CenterDot;(a_2\left(\mathrm{\&delta;}\right)-a_1\left(\mathrm{\&delta;}\right))]}$

Here, $A\left[\mathrm{\&delta;}\right]>B\left[\mathrm{\&delta;}\right]=\frac{a_2\left(\mathrm{\&delta;}\right)-b_1\left(\mathrm{\&delta;}\right)}{(b_2\left(\mathrm{\&delta;}\right)-b_1\left(\mathrm{\&delta;}\right)+(a_2\left(\mathrm{\&delta;}\right)-a_1\left(\mathrm{\&delta;}\right))}.$