CN106156496A - The maximum Likelihood of the sea clutter amplitude model parameter of inverse Gauss texture - Google Patents

Abstract

The invention discloses the maximum Likelihood of the sea clutter amplitude model parameter of a kind of inverse Gauss texture.Mainly solve the problem that prior art is inaccurate to sea clutter amplitude model parameter estimation.Implementation step is: 1) choose N number of clutter amplitude data, calculates moments estimation value and the moments estimation value of form parameter of scale parameter；2) probability density function utilizing sea clutter Amplitude Distributed Model calculates log-likelihood function；3) log-likelihood function is utilized to obtain the two-parameter iterative formula of maximal possibility estimation；4) utilize 1) in two moments estimation values respectively as the iterative initial value of in maximal possibility estimation two parameters；5) according to 3) in two-parameter iterative formula iteration try to achieve the maximum likelihood estimator of scale parameterMaximum likelihood estimator with form parameterThe present invention can make full use of sample information and carry out effectively, estimating accurately, can be used for the target detection under sea clutter background to sea clutter Amplitude Distributed Model parameter based on inverse Gauss texture.

Description

The maximum Likelihood of the sea clutter amplitude model parameter of inverse Gauss texture

Technical field

The invention belongs to signal processing technology field, be specifically related to the sea clutter Amplitude Distributed Model of a kind of inverse Gauss texture Method for parameter estimation, can be used for the target detection under sea clutter background.

Background technology

Surface, sea is referred to as sea clutter or sea surface echo to the back scattering of radar emission signal.Relative to land clutter or Meteorological clutter, the characteristic of sea clutter is much more complex and the existence of sea clutter to the target detection of radar, locating and tracking performance all To produce serious influence.Optimal objective detection method under sea clutter background depends on the model of sea clutter Amplitude Distributed Model Parameter, the Amplitude Distributed Model of sea clutter changes along with radar resolution and the change of sea situation, the most effectively estimates to go to sea miscellaneous The model parameter of wave amplitude distributed model is the key of sea-surface target test problems.Under the conditions of low-resolution radar, multiple Gaussian mode Type can simulate sea clutter distribution, the general Rayleigh distributed of amplitude of sea clutter well.And in low angle of incidence, high-resolution Under the conditions of radar, due to the appearance of sea spike, the amplitude distribution of sea clutter there will be long " hangover " compared with rayleigh distributed, sea Clutter presents stronger non-Gaussian system.To this, miscellaneous compared to the classics that texture component is gamma distribution and inverse gamma distribution Wave pattern, a kind of new Clutter Model that texture component is dead wind area proposed in recent years, i.e. texture is dead wind area Sea clutter amplitude model (Compound-Gaussian model with the inverse Gaussian texture) is permissible The hangover characteristic of clutter is preferably described.Based on this model practicality in actually detected, corresponding adaptive targets The research of detector also yields good result.Document " Adaptive Signal Detection in Compound- Gaussian Clutter with Inverse Gaussian Texture " in propose the clutter being applicable to inverse Gauss texture Self-adapting signal detecting method.Wherein the structure of detector depends on the form parameter in Clutter Model and scale parameter, because of Target detection tool is had very important significance by this method of estimation improving Model Parameter.

For sea clutter Amplitude Distributed Model parameter based on inverse Gauss texture, moments estimation method is common.Amplitude distribution Square can be expressed as the nonlinear function of model parameter, square therefore can be utilized to calculate form parameter and the chi of sea clutter model Degree parameter.Have been achieved with at present utilizing the second moment of clutter amplitude and Fourth-order moment to carry out biparametric estimation.But moment estimation method, Required exponent number is high, wherein for the utilization of amplitude Fourth-order moment, when sample size deficiency, and the estimation essence of moment estimation method Spend the lowest, affect the realization of sea-surface target detection.

Summary of the invention

It is an object of the invention to propose the maximal possibility estimation of the sea clutter amplitude model parameter of a kind of inverse Gauss texture Method, to improve when sample size data deficiencies, the estimation of sea clutter Amplitude Distributed Model form parameter and scale parameter is accurate Degree.

For realizing above-mentioned technical purpose, technical scheme comprises the steps:

(1) produce sea clutter data R based on inverse Gauss texture by emulation, choose from R and only comprise clutter data Distance unit, chooses N number of clutter amplitude data: x altogether₁,x₂,...,x_n,....,x_N, utilize this clutter amplitude data, according to Moment estimation method obtains the moments estimation value of the scale parameter of sea clutter data RMoments estimation value with form parameter

(2) utilize the probability density function f (r, μ, β) of sea clutter Amplitude Distributed Model based on inverse Gauss texture, calculate N Individual sea clutter model amplitude r₁,r₂,...,r_n,....,r_NLog-likelihood function Ψ (r₁,r₂,...,r_n,....,r_N| μ, β):

$\begin{array}{c}\mathrm{\Ψ}\left(r_1,r_2,...,r_n,\mathrm{....},r_N|\mathrm{\μ},\mathrm{\β}\right)=N\left(\ln 2+\ln \mathrm{\β}+{\mathrm{\β}}^{-1}-\ln \mathrm{\μ}\right)\\ -3\underset{n=1}{\overset{N}{\mathrm{\Σ}}}\ln \left(a_n\right)+\underset{n=1}{\overset{N}{\mathrm{\Σ}}}\ln \left(r_n\right)+\underset{n=1}{\overset{N}{\mathrm{\Σ}}}\ln \left(1+{\mathrm{\β}}^{-1}{a}_n\right)-{\mathrm{\β}}^{-1}\underset{n=1}{\overset{N}{\mathrm{\Σ}}}{a}_n,\end{array}$

Wherein,Representing iteration intermediate variable, μ represents the scale parameter of this Amplitude Distributed Model, β represents the form parameter of this Amplitude Distributed Model, and r represents sea clutter amplitude, r_nRepresent the n-th sea clutter range value, n=1, 2 ..., N, N represent sea clutter data amount check；

(3) N number of sea clutter model amplitude r is calculated₁,r₂,...,r_n,....,r_NLog-likelihood function Ψ (r₁,r₂,..., r_n,....,r_N| μ, β) about form parameter μ and the partial derivative of scale parameter β, and to make it be 0:

$\left\{\begin{array}{c}\frac{\∂\mathrm{\Ψ}(r_1,r_2,...,r_n,\mathrm{....},r_N|\mathrm{\μ},\mathrm{\β})}{\∂\mathrm{\μ}}=0\\ \frac{\∂\mathrm{\Ψ}(r_1,r_2,...,r_n,\mathrm{....},r_N|\mathrm{\μ},\mathrm{\β})}{\∂\mathrm{\β}}=0\end{array}\right.$

Obtain maximal possibility estimation about scale parameter μ and the two-parameter iterative formula of form parameter β:

${\widehat{\mathrm{\μ}}}_k=\frac{1}{N}\underset{n=1}{\overset{N}{\Σ}}\frac{a_n^2\left(k\right)+3{\widehat{\mathrm{\β}}}_{k-1}({\widehat{\mathrm{\β}}}_{k-1}+a_n(k\left)\right)}{a_n^2\left(k\right)({\widehat{\mathrm{\β}}}_{k-1}+a_n(k\left)\right)}{r}_n^2,$

${\widehat{\mathrm{\β}}}_k=\frac{1}{N}\underset{n=1}{\overset{N}{\Σ}}(\frac{{\widehat{\mathrm{\β}}}_{k-1}^2}{{\widehat{\mathrm{\β}}}_{k-1}+a_n\left(k\right)}+(a_n\left(k\right)-1\left)\right).$

Wherein,Represent kth time iteration intermediate variable,Represent yardstick ginseng The kth time iterative value of number,Represent the kth time iterative value of form parameter,Represent-1 iterative value of kth of scale parameter,Represent-1 iterative value of kth of form parameter, r_nRepresent the n-th sea clutter range value；

(4) the moments estimation value of the scale parameter of sea clutter data R is utilizedMoments estimation value with form parameterPoint Not as the iterative initial value of maximal possibility estimation mesoscale parameterIterative initial value with form parameter

$\begin{array}{c}{\widehat{\mathrm{\μ}}}_0={\widehat{\mathrm{\μ}}}_{MoM}\\ {\widehat{\mathrm{\β}}}_0={\widehat{\mathrm{\β}}}_{MoM}\end{array};$

(5) according to the two-parameter iterative formula of the maximal possibility estimation in step (3), to scale parameter μ and form parameter β It is iterated, until iterations arrives maximum iteration time L=100, stops iteration, try to achieve the yardstick ginseng of sea clutter data R The maximum likelihood estimator of numberMaximum likelihood estimator with form parameter

The present invention compared with prior art has the advantage that

1) present invention is in making log-likelihood function obtain pole by the method for scale parameter and form parameter iterative numerical Big value, compares with high order method, and especially when sample number deficiency, error is less, the form parameter obtained and scale parameter More accurate.

2) the two-parameter iterative formula of maximal possibility estimation is utilized to be iterated, with moment estimation method phase due to the present invention Ratio, the present invention calculates simpler, solves the time short；

Accompanying drawing explanation

Fig. 1 is the flowchart of the present invention；

When Fig. 2 is scale parameter μ=1, the sea clutter width based on inverse Gauss texture obtained by the present invention and existing method The RMSE root-mean-square error result figure that degree distributed model form parameter is estimated；

When Fig. 3 is scale parameter μ=10, the sea clutter width based on inverse Gauss texture obtained by the present invention and existing method The RMSE root-mean-square error result figure that degree distributed model form parameter is estimated；

The sea clutter width based on inverse Gauss texture that when Fig. 4 is form parameter β=0.5, the present invention and existing method obtain The RMSE root-mean-square error result figure that degree distributed model scale parameter is estimated；

When Fig. 5 is form parameter β=10, the sea clutter width based on inverse Gauss texture obtained by the present invention and existing method The RMSE root-mean-square error result figure that degree distributed model scale parameter is estimated.

Detailed description of the invention

The invention will be further described below in conjunction with the accompanying drawings:

The present invention utilizes moment estimation method to estimate the scale parameter of sea clutter Amplitude Distributed Model based on inverse Gauss texture MeterEstimate with form parameterIterative initial value respectively as scale parameterIterative initial value with form parameter It is iterated according to the two-parameter iterative formula of maximal possibility estimation, finally gives the maximum likelihood estimator of scale parameterWith The maximum likelihood estimator of form parameter

With reference to Fig. 1, the present invention to realize step as follows:

Step 1, chooses N number of clutter amplitude data: x₁,x₂,...,x_n,....,x_N, calculate the yardstick ginseng of sea clutter data R The moments estimation value of numberMoments estimation value with form parameter

In prior art, calculate the estimated value of the scale parameter of sea clutter data R and the method for the estimated value of form parameter It is typically moment estimation method, sees Ollila E, Tyler D E, Koivunen V, et al.. " Compound gaussian clutter modeling with an inverse gaussian texture distribution[J]”.IEEE Signal Processing Letters, 2012,19 (12): 876-879, this example is also adopted by existing moment estimation method, its Step is as follows:

(1.1) produce sea clutter data R based on inverse Gauss texture by emulation, choose from R and only comprise clutter data Distance unit, altogether choose N number of clutter amplitude data: x₁,x₂,...,x_n,....,x_N, x_nRepresent the n-th clutter amplitude number According to, n=1,2 ..., N；

(1.2) this clutter amplitude data x are utilized₁,x₂,...,x_n,....,x_N, the second moment calculating sea clutter data R is estimated EvaluationWith Fourth-order moment estimated value

${\hat{m}}_2=\frac{1}{N}\underset{n=1}{\overset{N}{\Σ}}{x}_n^2$

${\hat{m}}_4=\frac{1}{N}\underset{n=1}{\overset{N}{\&Sigma;}}{x}_n^4---<1>$

(1.2) according to the second moment estimated value of sea clutter data RWith Fourth-order moment estimated valueCalculate sea clutter data R The moments estimation value of scale parameterMoments estimation value with form parameter

$\begin{array}{c}{\widehat{\mathrm{\&mu;}}}_{MoM}={\hat{m}}_2\\ {\widehat{\mathrm{\&beta;}}}_{MoM}=\frac{{\hat{m}}_4}{2{\hat{m}}_2^2}-1\end{array}.---<2>$

Step 2, utilizes the probability density function f (r, μ, β) of sea clutter Amplitude Distributed Model based on inverse Gauss texture, meter Calculate N number of sea clutter model amplitude r₁,r₂,...,r_n,....,r_NLog-likelihood function Ψ (r₁,r₂,...,r_n,....,r_N|μ, β)。

(2.1) N number of sea clutter model amplitude r is calculated respectively₁,r₂,...,r_n,....,r_NProbability density function f (r_n,μ, β):

$f\left(r_n,\mathrm{\&mu;},\mathrm{\&beta;}\right)=\frac{2{\mathrm{\&beta;e}}^{1/\mathrm{\&beta;}}{r}_n}{\mathrm{\&mu;}}{\left(1+\frac{2{{\mathrm{\&beta;r}}_n}^2}{\mathrm{\&mu;}}\right)}^{-3/2}\left(1+\frac{1}{\mathrm{\&beta;}}\sqrt{1+\frac{2{{\mathrm{\&beta;r}}_n}^2}{\mathrm{\&mu;}}}\right)\exp \left(-\frac{1}{\mathrm{\&beta;}}\sqrt{1+\frac{2{{\mathrm{\&beta;r}}_n}^2}{\mathrm{\&mu;}}}\right)---<3>$

Wherein, μ represents the scale parameter of this Amplitude Distributed Model, and β represents the form parameter of this Amplitude Distributed Model, r_nTable Show the n-th sea clutter model amplitude, n=1,2 ..., N；

(2.2) N number of sea clutter model amplitude r is calculated₁,r₂,...,r_n,....,r_NProbability density function f (r_n, μ, β) Product, obtains likelihood function Ω (r₁,r₂,...,r_n,....,r_N| μ, β):

$\begin{array}{c}\mathrm{\&Omega;}\left(r_1,r_2,...,r_n,\mathrm{....},r_N|\mathrm{\&mu;},\mathrm{\&beta;}\right)=\underset{n=1}{\overset{N}{\mathrm{\&Pi;}}}f\left(r_n,\mathrm{\&mu;},\mathrm{\&beta;}\right)\\ ={\left(\frac{2{\mathrm{\&beta;e}}^{1/\mathrm{\&beta;}}}{\mathrm{\&mu;}}\right)}^N\underset{n=1}{\overset{N}{\mathrm{\&Pi;}}}{r}_n{\left(1+\frac{2{{\mathrm{\&beta;r}}_n}^2}{\mathrm{\&mu;}}\right)}^{-3/2}\left(1+\frac{1}{\mathrm{\&beta;}}\sqrt{1+\frac{2{{\mathrm{\&beta;r}}_n}^2}{\mathrm{\&mu;}}}\right)\exp \left(-\frac{1}{\mathrm{\&beta;}}\sqrt{1+\frac{2{{\mathrm{\&beta;r}}_n}^2}{\mathrm{\&mu;}}}\right),\end{array}---<4>$

Wherein, μ represents the scale parameter of this Amplitude Distributed Model, and β represents the form parameter of this Amplitude Distributed Model, r_n, n =1,2 ..., N, represent the n-th sea clutter model amplitude, N represents sea clutter data amount check；

(2.3) by likelihood function Ω (r₁,r₂,...,r_n,....,r_N| μ, β) take the logarithm, obtain log-likelihood function Ψ (r₁,r₂,...,r_n,....,r_N| μ, β):

$\begin{array}{c}\mathrm{\&Psi;}\left(r_1,r_2,...,r_n,\mathrm{....},r_N|\mathrm{\&mu;},\mathrm{\&beta;}\right)=\ln \&lsqb;\mathrm{\&Omega;}\left(r_1,r_2,...,r_n,\mathrm{....},r_N|\mathrm{\&mu;},\mathrm{\&beta;}\right)\&rsqb;\\ =N\left(\ln 2+\ln \mathrm{\&beta;}+{\mathrm{\&beta;}}^{-1}-\ln \mathrm{\&mu;}\right)\\ -3\underset{n=1}{\overset{N}{\mathrm{\&Sigma;}}}\ln \left(a_n\right)+\underset{n=1}{\overset{N}{\mathrm{\&Sigma;}}}\ln \left(r_n\right)+\underset{n=1}{\overset{N}{\mathrm{\&Sigma;}}}\ln \left(1+{\mathrm{\&beta;}}^{-1}{a}_n\right)-{\mathrm{\&beta;}}^{-1}\underset{n=1}{\overset{N}{\mathrm{\&Sigma;}}}{a}_n,\end{array}---<5>$

Wherein,Represent iteration intermediate variable, r_nRepresent the n-th sea clutter model amplitude, μ table Showing the scale parameter of this Amplitude Distributed Model, β represents the form parameter of this Amplitude Distributed Model.

Step 3, utilizes N number of sea clutter model amplitude r₁,r₂,...,r_n,....,r_NLog-likelihood function Ψ (r₁, r₂,...,r_n,....,r_N| μ, β), obtain maximal possibility estimation public about the two-parameter iteration of scale parameter μ and form parameter β Formula；

(3.1) log-likelihood function Ψ (r is calculated respectively₁,r₂,...,r_n,....,r_N| μ, β) about form parameter μ and chi The partial derivative of degree parameter beta, and to make it be 0,

$\left\{\begin{array}{c}\frac{\&part;\mathrm{\&Psi;}(r_1,r_2,...,r_n,\mathrm{....},r_N|\mathrm{\&mu;},\mathrm{\&beta;})}{\&part;\mathrm{\&mu;}}=0\\ \frac{\&part;\mathrm{\&Psi;}(r_1,r_2,...,r_n,\mathrm{....},r_N|\mathrm{\&mu;},\mathrm{\&beta;})}{\&part;\mathrm{\&beta;}}=0\end{array}\right.---<6>$

Obtain the two-parameter formula that is worth most:

$\begin{array}{c}\mathrm{\&mu;}=\frac{1}{N}\underset{n=1}{\overset{N}{\mathrm{\&Sigma;}}}\frac{a_n^2+3\mathrm{\&beta;}(\mathrm{\&beta;}+a_n)}{a_n^2(\mathrm{\&beta;}+a_n)}{r}_n^2\\ \mathrm{\&beta;}=\frac{1}{N}\underset{n=1}{\overset{N}{\&Sigma;}}(\frac{\mathrm{\&beta;}}{\mathrm{\&beta;}+a_n}+(a_n-1\left)\right)\end{array}---<7>$

Wherein,Representing iteration intermediate variable, μ represents the scale parameter of this Amplitude Distributed Model, β represents the form parameter of this Amplitude Distributed Model, r_nRepresent the n-th sea clutter model amplitude, n=1,2 ..., N, N represent sea Clutter data number；

(3.2) utilize and two-parameter in step (3.1) be most worth formula, obtain the two-parameter iterative formula of maximal possibility estimation such as Under:

$\begin{array}{c}{\widehat{\mathrm{\&mu;}}}_k=\frac{1}{N}\underset{n=1}{\overset{N}{\&Sigma;}}\frac{a_n^2\left(k\right)+3{\widehat{\mathrm{\&beta;}}}_{k-1}({\widehat{\mathrm{\&beta;}}}_{k-1}+a_n(k\left)\right)}{a_n^2\left(k\right)({\widehat{\mathrm{\&beta;}}}_{k-1}+a_n(k\left)\right)}{r}_n^2,\\ {\widehat{\mathrm{\&beta;}}}_k=\frac{1}{N}\underset{n=1}{\overset{N}{\&Sigma;}}(\frac{{\widehat{\mathrm{\&beta;}}}_{k-1}^2}{{\widehat{\mathrm{\&beta;}}}_{k-1}+a_n\left(k\right)}+(a_n\left(k\right)-1\left)\right).\end{array}---<8>$

Wherein,Represent kth time iteration intermediate variable,Represent scale parameter Kth time iterative value,Represent the kth time iterative value of form parameter,Represent-1 iterative value of kth of scale parameter, Represent-1 iterative value of kth of form parameter, r_nRepresent the n-th sea clutter model amplitude.

Step 4, utilizes the moments estimation value of the scale parameter of sea clutter data RMoments estimation value with form parameterIterative initial value respectively as maximal possibility estimation mesoscale parameterIterative initial value with form parameter

$\begin{array}{c}{\widehat{\mathrm{\&mu;}}}_0={\widehat{\mathrm{\&mu;}}}_{MoM}\\ {\widehat{\mathrm{\&beta;}}}_0={\widehat{\mathrm{\&beta;}}}_{MoM}\end{array}---<9>$

Step 5, according to the two-parameter iterative formula of the maximal possibility estimation in step (3.2), to scale parameter μ and shape Parameter beta is iterated, and tries to achieve the maximum likelihood estimator of the scale parameter of sea clutter data RWith the maximum of form parameter seemingly So estimated value

(5.1) as k=n, n represents nth iteration, utilizes N number of clutter amplitude data: x in step (1)₁,x₂,..., x_n,....,x_N, according to the two-parameter iterative formula of maximal possibility estimation, obtain the kth time iterative value of scale parameterShape The kth of parameter time iterative value

$\begin{array}{c}{\widehat{\mathrm{\&mu;}}}_k=\frac{1}{N}\underset{n=1}{\overset{N}{\&Sigma;}}\frac{a_n^2\left(k\right)+3{\widehat{\mathrm{\&beta;}}}_{k-1}({\widehat{\mathrm{\&beta;}}}_{k-1}+a_n(k\left)\right)}{a_n^2\left(k\right)({\widehat{\mathrm{\&beta;}}}_{k-1}+a_n(k\left)\right)}{x}_n^2,\\ {\widehat{\mathrm{\&beta;}}}_k=\frac{1}{N}\underset{n=1}{\overset{N}{\&Sigma;}}(\frac{{\widehat{\mathrm{\&beta;}}}_{k-1}^2}{{\widehat{\mathrm{\&beta;}}}_{k-1}+a_n\left(k\right)}+(a_n\left(k\right)-1\left)\right).\end{array}---<10>$

Wherein,Represent kth time iteration intermediate variable value,Represent scale parameter -1 iterative value of kth,Represent-1 iterative value of kth of form parameter, x_nRepresent the n-th clutter amplitude data, n=1, 2 ..., N, N represent sea clutter data amount check；

(5.2) k=n+1 is made, repeated execution of steps (5.1), until k gets maximum iteration time L=100, stop iteration, Obtain the maximum likelihood estimator of the scale parameter of sea clutter data RMaximum likelihood estimator with form parameter

Below in conjunction with emulation experiment, the effect of the present invention is described further.

1. simulation parameter

The sea clutter data used in emulation experiment are inverse Gauss data texturings that in MATLAB, random simulation produces and equal Value is 0, and variance is the multiple Gauss speckle data of 1, and sample data amount is N=10000.

2. emulation experiment content

Emulation experiment is respectively adopted the inventive method and moment estimation method obtains sea clutter width based on inverse Gauss texture The estimation of degree distributed model parameter, compares the effect of two kinds of methods of estimation by root-mean-square error checking R MSE methods analyst.All Square error checking R MSE value is the least, and specification error is the least, and parameter estimation is the most accurate.

Emulation experiment 1

Taking scale parameter is μ=1, form parameter be β from 0.1 to 20, be spaced apart 0.1 inverse Gauss grain distribution produce Sea clutter data, are respectively adopted the present invention on this basis and moments estimation method tries to achieve form parameter estimated value, iteration of the present invention time Several 100 times, experiment number 10⁴Secondary, draw the RMSE root-mean-square error curve of its form parameter estimated value, as in figure 2 it is shown, Fig. 2 Transverse axis represents form parameter true value, and the longitudinal axis represents RMSE root-mean-square error.

From figure 2 it can be seen that compared with moment estimation method, the estimation of form parameter is had less by the inventive method Error.

Emulation experiment 2

Taking scale parameter is μ=10, form parameter be β from 0.1 to 20, be spaced apart 0.1 inverse Gauss grain distribution produce Sea clutter data, be respectively adopted the present invention on this basis and moments estimation method try to achieve form parameter estimated value, iteration of the present invention Number of times 100 times, experiment number 10⁴Secondary, draw the RMSE root-mean-square error curve of its form parameter estimated value, as it is shown on figure 3, figure 3 transverse axis represent form parameter true value, and the longitudinal axis represents its RMSE root-mean-square error.

From figure 3, it can be seen that compared with moment estimation method, the estimation of form parameter is had less by the inventive method Error.Contrast simulation experiment 1, it is also possible to find, the change of the scale parameter of clutter amplitude has little influence on form parameter and estimates Error.

Emulation experiment 3

Taking form parameter is β=0.5, scale parameter μ from 0.1 to 20, be spaced apart 0.1 inverse Gauss grain distribution produce Sea clutter data, are respectively adopted the present invention on this basis and moments estimation method tries to achieve scale parameter estimated value, iteration of the present invention time Several 100 times, experiment number 10⁴Secondary, draw the RMSE root-mean-square error curve of its scale parameter estimated value, as shown in Figure 4, Fig. 4 Transverse axis represents scale parameter true value, and the longitudinal axis represents its RMSE root-mean-square error.

Figure 4, it is seen that the inventive method is the most consistent with moments estimation to the estimation of form parameter, all have the least Error.The RMSE root-mean-square error curve of scale parameter estimated value is almost level simultaneously, illustrates that its error is joined with yardstick The change of number is unrelated.

Emulation experiment 4

Taking form parameter is β=10, scale parameter be μ from 0.1 to 20, be spaced apart 0.1 inverse Gauss grain distribution produce Sea clutter data, be respectively adopted the present invention on this basis and moments estimation method try to achieve scale parameter estimated value, iteration of the present invention Number of times 100 times, experiment number 10⁴Secondary, draw the RMSE root-mean-square error curve of its scale parameter estimated value, as it is shown in figure 5, figure 5 transverse axis represent scale parameter true value, and the longitudinal axis represents its RMSE root-mean-square error.

From figure 5 it can be seen that the inventive method is more better than moments estimation to the estimation of scale parameter, have the least Error.

In sum, the maximum likelihood of the sea clutter Amplitude Distributed Model parameter based on inverse Gauss texture that the present invention proposes Method of estimation, when data volume deficiency, it is possible to make full use of sample information, improves and estimates performance, to sea clutter complex Gaussian width Degree distributed model parameter carries out more effectively, estimating more accurately.

Claims (5)

1. the maximum Likelihood of the sea clutter amplitude model parameter of an inverse Gauss texture, it is characterised in that including:

(1) produce sea clutter data R based on inverse Gauss texture by emulation, from R, choose the distance only comprising clutter data Unit, chooses N number of clutter amplitude data: x altogether₁, x₂..., x_n...., x_N, utilize this clutter amplitude data, estimate according to square Meter method obtains the moments estimation value of the scale parameter of sea clutter data RMoments estimation value with form parameter

(2) utilize the probability density function f (r, μ, β) of sea clutter Amplitude Distributed Model based on inverse Gauss texture, calculate N number of sea Clutter Model amplitude r₁, r₂..., r_n...., r_NLog-likelihood function Ψ (r₁, r₂..., r_n...., r_N| μ, β):

$\begin{array}{c}\mathrm{\&Psi;}\left(r_1,r_2,...,r_n,\mathrm{....},r_N|\mathrm{\&mu;},\mathrm{\&beta;}\right)=N\left(\ln 2+\ln \mathrm{\&beta;}+{\mathrm{\&beta;}}^{-1}-\ln \mathrm{\&mu;}\right)\\ -3\underset{n=1}{\overset{N}{\mathrm{\&Sigma;}}}\ln \left(a_n\right)+\underset{n=1}{\overset{N}{\mathrm{\&Sigma;}}}\ln \left(r_n\right)+\underset{n=1}{\overset{N}{\mathrm{\&Sigma;}}}\ln \left(1+{\mathrm{\&beta;}}^{-1}{a}_n\right)-{\mathrm{\&beta;}}^{-1}\underset{n=1}{\overset{N}{\mathrm{\&Sigma;}}}{a}_n,\end{array}$

Wherein,Representing iteration intermediate variable, μ represents the scale parameter of this Amplitude Distributed Model, and β represents The form parameter of this Amplitude Distributed Model, r represents sea clutter amplitude, r_nRepresent the n-th sea clutter model amplitude, n=1, 2 ..., N, N represent sea clutter data amount check；

(3) N number of sea clutter model amplitude r is calculated₁, r₂..., r_n...., r_NLog-likelihood function Ψ (r₁, r₂..., r_n...., r_N| μ, β) about form parameter μ and the partial derivative of scale parameter β, and to make it be 0:

$\left\{\begin{array}{c}\frac{\&part;\mathrm{\&Psi;}(r_1,r_2,...,r_n,\mathrm{....},r_N|\mathrm{\&mu;},\mathrm{\&beta;})}{\&part;\mathrm{\&mu;}}=0\\ \frac{\&part;\mathrm{\&Psi;}(r_1,r_2,...,r_n,\mathrm{....},r_N|\mathrm{\&mu;},\mathrm{\&beta;})}{\&part;\mathrm{\&beta;}}=0\end{array}\right.$

Obtain maximal possibility estimation about scale parameter μ and the two-parameter iterative formula of form parameter β:

${\widehat{\mathrm{\&mu;}}}_k=\frac{1}{N}\underset{n=1}{\overset{N}{\&Sigma;}}\frac{a_n^2\left(k\right)+3{\widehat{\mathrm{\&beta;}}}_{k-1}({\widehat{\mathrm{\&beta;}}}_{k-1}+a_n(k\left)\right)}{a_n^2\left(k\right)({\widehat{\mathrm{\&beta;}}}_{k-1}+a_n(k\left)\right)}{r}_n^2,$

${\widehat{\mathrm{\&beta;}}}_k=\frac{1}{N}\underset{n=1}{\overset{N}{\&Sigma;}}(\frac{{\widehat{\mathrm{\&beta;}}}_{k-1}^2}{{\widehat{\mathrm{\&beta;}}}_{k-1}+a_n\left(k\right)}+(a_n\left(k\right)-1\left)\right).$

Wherein,Represent kth time iteration intermediate variable,Represent scale parameter Kth time iterative value,Represent the kth time iterative value of form parameter,Represent-1 iterative value of kth of scale parameter, Represent-1 iterative value of kth of form parameter, r_nRepresent the n-th sea clutter model amplitude；

(4) the moments estimation value of the scale parameter of sea clutter data R is utilizedMoments estimation value with form parameterMake respectively Iterative initial value for maximal possibility estimation mesoscale parameterIterative initial value with form parameter

$\begin{array}{c}{\widehat{\mathrm{\&mu;}}}_0={\widehat{\mathrm{\&mu;}}}_{MoM}\\ {\widehat{\mathrm{\&beta;}}}_0={\widehat{\mathrm{\&beta;}}}_{MoM}\end{array};$

(5) according to the two-parameter iterative formula of the maximal possibility estimation in step (3), scale parameter μ and form parameter β are carried out Iteration, until iterations arrives maximum iteration time L=100, stops iteration, tries to achieve the scale parameter of sea clutter data R Maximum likelihood estimatorMaximum likelihood estimator with form parameter

2. the method for claim 1, wherein obtains the yardstick of sea clutter data R according to moment estimation method in step (1) The moments estimation value of parameterMoments estimation value with form parameter

${\widehat{\mathrm{\&mu;}}}_{MoM}={\hat{m}}_2$

${\widehat{\mathrm{\&beta;}}}_{MoM}=\frac{{\hat{m}}_4}{2{\hat{m}}_2^2}-1$

Wherein,Represent that the second moment of sea clutter data R is estimated,Represent sea clutter data R Fourth-order moment estimate, x_nRepresent the n-th clutter amplitude data, n=1,2 ..., N, N represent sea clutter data amount check.

3. the method for claim 1, wherein utilizes sea clutter amplitude distribution mould based on inverse Gauss texture in step (2) The probability density function f (r, μ, β) of type calculates N number of sea clutter model amplitude r₁, r₂..., r_n...., r_NLog-likelihood letter Number Ψ (r₁, r₂..., r_n...., r_N| μ, β), carry out as follows:

2a) calculate N number of sea clutter model amplitude r respectively₁, r₂..., r_n...., r_NProbability density function f (r_n, μ, β):

$f(r_n,\mathrm{\&mu;},\mathrm{\&beta;})=\frac{2{\mathrm{\&beta;e}}^{1/\mathrm{\&beta;}}{r}_n}{\mathrm{\&mu;}}{(1+\frac{2{{\mathrm{\&beta;r}}_n}^2}{\mathrm{\&mu;}})}^{-3/2}(1+\frac{1}{\mathrm{\&beta;}}\sqrt{1+\frac{2{{\mathrm{\&beta;r}}_n}^2}{\mathrm{\&mu;}}})\exp (-\frac{1}{\mathrm{\&beta;}}\sqrt{1+\frac{2{{\mathrm{\&beta;r}}_n}^2}{\mathrm{\&mu;}}})$

Wherein, μ represents the scale parameter of this Amplitude Distributed Model, and β represents the form parameter of this Amplitude Distributed Model, r_nRepresent n-th Individual sea clutter model amplitude, n=1,2 ..., N；

2b) calculate N number of sea clutter model amplitude r₁, r₂..., r_n...., r_NProbability density function f (r_n, μ, β) product, Obtain likelihood function Ω (r₁, r₂..., r_n...., r_N| μ, β):

$\begin{array}{c}\mathrm{\&Omega;}\left(r_1,r_2,...,r_n,\mathrm{....},r_N|\mathrm{\&mu;},\mathrm{\&beta;}\right)=\underset{n=1}{\overset{N}{\mathrm{\&Pi;}}}f\left(r_n,\mathrm{\&mu;},\mathrm{\&beta;}\right)\\ ={\left(\frac{2{\mathrm{\&beta;e}}^{1/\mathrm{\&beta;}}}{\mathrm{\&mu;}}\right)}^N\underset{n=1}{\overset{N}{\mathrm{\&Pi;}}}{r}_n{\left(1+\frac{2{{\mathrm{\&beta;r}}_n}^2}{\mathrm{\&mu;}}\right)}^{-3/2}\left(1+\frac{1}{\mathrm{\&beta;}}\sqrt{1+\frac{2{{\mathrm{\&beta;r}}_n}^2}{\mathrm{\&mu;}}}\right)\exp \left(-\frac{1}{\mathrm{\&beta;}}\sqrt{1+\frac{2{{\mathrm{\&beta;r}}_n}^2}{\mathrm{\&mu;}}}\right),\end{array}$

2c) by likelihood function Ω (r₁, r₂..., r_n...., r_N| μ, β) take the logarithm, obtain log-likelihood function Ψ (r₁, r₂..., r_n...., r_N| μ, β):

$\begin{array}{c}\mathrm{\&Psi;}\left(r_1,r_2,...,r_n,\mathrm{....},r_N|\mathrm{\&mu;},\mathrm{\&beta;}\right)=\ln \&lsqb;\mathrm{\&Omega;}\left(r_1,r_2,...,r_n,\mathrm{....},r_N|\mathrm{\&mu;},\mathrm{\&beta;}\right)\&rsqb;\\ =N\left(\ln 2+\ln \mathrm{\&beta;}+{\mathrm{\&beta;}}^{-1}-\ln \mathrm{\&mu;}\right)\\ -3\underset{n=1}{\overset{N}{\mathrm{\&Sigma;}}}\ln \left(a_n\right)+\underset{n=1}{\overset{N}{\mathrm{\&Sigma;}}}\ln \left(r_n\right)+\underset{n=1}{\overset{N}{\mathrm{\&Sigma;}}}\ln \left(1+{\mathrm{\&beta;}}^{-1}{a}_n\right)-{\mathrm{\&beta;}}^{-1}\underset{n=1}{\overset{N}{\mathrm{\&Sigma;}}}{a}_n\end{array}$

Wherein,Represent iteration intermediate variable.

4. the method for claim 1, wherein utilizes N number of sea clutter model amplitude r in step (3)₁, r₂..., r_n...., r_NLog-likelihood function Ψ (r₁, r₂..., r_n...., r_N| μ, β), obtain maximal possibility estimation and join about yardstick Number μ and the two-parameter iterative formula of form parameter β, carried out as follows:

3a) calculate log-likelihood function Ψ (r respectively₁, r₂..., r_n...., r_N| μ, β) about form parameter μ and scale parameter β Partial derivative, and to make it be 0, obtains the two-parameter formula of value formula:

$\mathrm{\&mu;}=\frac{1}{N}\underset{n=1}{\overset{N}{\mathrm{\&Sigma;}}}\frac{a_n^2+3\mathrm{\&beta;}(\mathrm{\&beta;}+a_n)}{a_n^2(\mathrm{\&beta;}+a_n)}{r}_n^2$

$\mathrm{\&beta;}=\frac{1}{N}\underset{n=1}{\overset{N}{\&Sigma;}}(\frac{\mathrm{\&beta;}}{\mathrm{\&beta;}+a_n}+(a_n-1\left)\right)$

Wherein,Representing iteration intermediate variable, μ represents the scale parameter of this Amplitude Distributed Model, β table Show the form parameter of this Amplitude Distributed Model, r_nRepresenting the n-th sea clutter model amplitude, N represents sea clutter data amount check；

3b) utilize step 3a) in two-parameter be most worth formula, obtain the two-parameter iterative formula of maximal possibility estimation:

${\widehat{\mathrm{\&mu;}}}_k=\frac{1}{N}\underset{n=1}{\overset{N}{\&Sigma;}}\frac{a_n^2\left(k\right)+3{\widehat{\mathrm{\&beta;}}}_{k-1}({\widehat{\mathrm{\&beta;}}}_{k-1}+a_n(k\left)\right)}{a_n^2\left(k\right)({\widehat{\mathrm{\&beta;}}}_{k-1}+a_n(k\left)\right)}{r}_n^2$

${\widehat{\mathrm{\&beta;}}}_k=\frac{1}{N}\underset{n=1}{\overset{N}{\&Sigma;}}(\frac{{\widehat{\mathrm{\&beta;}}}_{k-1}^2}{{\widehat{\mathrm{\&beta;}}}_{k-1}+a_n\left(k\right)}+(a_n\left(k\right)-1\left)\right)$

Wherein,Represent kth time iteration intermediate variable,Represent the kth time of scale parameter Iterative value,Represent the kth time iterative value of form parameter,Represent-1 iterative value of kth of scale parameter,Represent shape -1 iterative value of the kth of shape parameter.

5. the method for claim 1, wherein two-parameter according to the maximal possibility estimation in step (3) in step (5) Iterative formula, is iterated scale parameter μ and form parameter β, and the maximum likelihood of the scale parameter trying to achieve sea clutter data R is estimated EvaluationMaximum likelihood estimator with form parameterCarry out as follows:

5a) as k=n, n represents nth iteration, utilizes N number of clutter amplitude data: x in step (1)₁,x₂,...,x_n,...., x_N, according to the two-parameter iterative formula of maximal possibility estimation, obtain the kth time iterative value of scale parameterThe kth of form parameter Secondary iterative value

${\widehat{\mathrm{\&mu;}}}_k=\frac{1}{N}\underset{n=1}{\overset{N}{\&Sigma;}}\frac{a_n^2\left(k\right)+3{\widehat{\mathrm{\&beta;}}}_{k-1}({\widehat{\mathrm{\&beta;}}}_{k-1}+a_n(k\left)\right)}{a_n^2\left(k\right)({\widehat{\mathrm{\&beta;}}}_{k-1}+a_n(k\left)\right)}{x}_n^2$

${\widehat{\mathrm{\&beta;}}}_k=\frac{1}{N}\underset{n=1}{\overset{N}{\&Sigma;}}(\frac{{\widehat{\mathrm{\&beta;}}}_{k-1}^2}{{\widehat{\mathrm{\&beta;}}}_{k-1}+a_n\left(k\right)}+(a_n\left(k\right)-1\left)\right)$

Wherein,Represent kth time iteration intermediate variable value,Represent the of scale parameter K-1 iterative value,Represent-1 iterative value of kth of form parameter, x_nRepresent the n-th clutter amplitude data, n=1, 2 ..., N, N represent sea clutter data amount check；

5b) make k=n+1, repeated execution of steps 5a), until k gets maximum iteration time L=100, stop iteration, obtain sea miscellaneous The maximum likelihood estimator of the scale parameter of wave datum RMaximum likelihood estimator with form parameter