CN109840069B - Improved self-adaptive fast iterative convergence solution method and system - Google Patents

Improved self-adaptive fast iterative convergence solution method and system Download PDF

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CN109840069B
CN109840069B CN201910183269.0A CN201910183269A CN109840069B CN 109840069 B CN109840069 B CN 109840069B CN 201910183269 A CN201910183269 A CN 201910183269A CN 109840069 B CN109840069 B CN 109840069B
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吴日恒
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Yantai Vocational College
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Abstract

The invention belongs to the technical field of signal detection, and discloses an improved adaptive fast iterative convergence solution method and system, wherein a parameter iterative process is approximated to Taylor second-order series expansion, so that the method is insensitive to the selection of an initial value, and compared with the traditional method, no calculation complexity is increased, and a second derivative and a third derivative of a log likelihood function are introduced into a correction term, namely, the slope and curvature characteristic mathematics of the log likelihood ratio function are increased, so that the fluctuation of the correction term can be adaptively adjusted, and an adaptive parameter estimation threshold value and an iterative convergence stop condition are developed by utilizing the correlation between the second derivative and the third derivative of the log likelihood function. The invention can not only ensure fast iteration, but also ensure convergence, and is insensitive to the selection of the initial value.

Description

Improved self-adaptive fast iterative convergence solution method and system
Technical Field
The invention belongs to the technical field of signal detection, and particularly relates to an improved self-adaptive fast iterative convergence solution method and system.
Background
Currently, the current state of the art commonly used in the industry is such that:
the prior art similar to the present solution is the one described in the following paragraphs.
In the technology related to the iterative solution, the traditional used technology comprises a Newton-Raphson iterative method, a scoring method, an EM method and the like, and the common characteristics of the methods are that an initial value is selected for a parameter to be estimated, then the parameter to be estimated is expanded according to a Taylor series primary term, and the parameter to be estimated is continuously approximated through iteration. However, this method may cause the iteration not to converge, for example, due to the large influence of data noise, when the second derivative of the log likelihood function is small, the iteration may not converge, and such a problem is common in the conventional iterative method because the fluctuation of the correction term is too large from one iteration term to another.
In summary, the problems of the prior art are as follows:
(1) in the prior art, iterative convergence speed in signal data detection is low, and a convergence threshold cannot be adaptively adjusted through actual conditions, so that a large number of parameter estimation is limited in related physical application.
(2) Because the conventional iterative method carries out iterative estimation based on the first-order Taylor series expansion principle, only the second derivative of the log-likelihood function is developed by the method, and the high-order derivative of the log-likelihood function is not developed, when the second derivative of the log-likelihood function is small, the parameter iteration is not converged due to large fluctuation of a correction term. The conventional method has no self-adaptive convergence threshold and lacks a mechanism for inhibiting fluctuation of a correction term, and the application of an iterative method is severely restricted by the problem.
(3) For example, in the application related to passive sensing, the receiving party is a sensor array, the transmitting party is a military radar of an enemy, the purpose of passive sensing is to estimate information such as azimuth angle and elevation angle of a local radar through the passive sensor array, in high-precision parameter estimation research, such problems are generally realized by an ML (maximum likelihood) method, which has the biggest defects of high computational complexity, incapability of meeting the real-time task computation requirements under engineering conditions, therefore, the numerical solution algorithm is often the first choice, the conventional numerical iteration method usually faces the problem that the iteration convergence can not be ensured, and the convergence value depends on the setting of the initial value to a great extent, sometimes even though convergence occurs, the convergence speed is slow, the task requirements are difficult to meet, the method provided by the invention not only ensures convergence, but also has high convergence speed and is insensitive to the setting of the initial value.
The significance of solving the technical problems is as follows: array signal processing is often used in high-dimensional parameter estimation, however, high-precision parameter estimation often requires closed-form solution with high computational complexity, and the invention provides a set of numerical solution scheme for high-precision parameter estimation and simplifying computational complexity, and has potential important application value in wider fields such as radar, sonar, electronic reconnaissance and wireless communication.
The invention provides a new method, which not only can ensure the convergence in the iterative process, but also can ensure the rapid convergence in the iterative process, and simultaneously develops the second derivative and the third derivative of the log-likelihood function, when the second derivative of the log-likelihood function is smaller, the fluctuation of a correction term is inhibited, and when the second derivative of the log-likelihood function is larger, the fluctuation of the correction term is strengthened, so the invention can adaptively select the iterative step length, and can adaptively adjust the convergence threshold through the actual condition, thereby fundamentally overcoming the problems in the traditional iterative method. It would be of great value for use in a number of physical applications involving parameter estimation.
Disclosure of Invention
Aiming at the problems in the prior art, the invention provides an improved self-adaptive fast iterative convergence solution method and system.
The invention is realized by an improved self-adaptive fast iterative convergence solution method, which comprises the following steps: in signal detection, a parameter iteration process is approximated to Taylor second-order series expansion, and an initial value is selected;
introducing a second derivative and a third derivative of a log-likelihood function into the correction term, increasing the slope and curvature characteristics of the log-likelihood ratio function, and adaptively adjusting the fluctuation of the correction term;
and acquiring an adaptive parameter estimation threshold and an iteration convergence stopping condition by utilizing the correlation between the second derivative and the third derivative of the log likelihood function.
Further, the improved adaptive fast iterative convergence solution method specifically includes:
step one, observing a data model,
x(n)=f(θ,n)+w(n),n=0,1,…,N-1;
step two, writing the parameter estimation method of the maximum likelihood estimation MLE into a form of a log likelihood ratio function, wherein the first derivative of the log likelihood function is
Figure RE-GDA0002024470120000031
Step three, setting an initial value theta0The above equation is approximated as a log likelihood function where θ is θ0Formulas for the second and third derivatives of time;
step four, making the first derivative g (theta) of the log likelihood function equal to 0;
step five, solving theta corresponding to the situation that g (theta) is 01
Figure RE-GDA0002024470120000041
Step six, performing operation by using an iterative method, and utilizing a previous predicted value thetakFind the next new predicted value thetak+1
Step seven, when theta is satisfiedk+1kWhen the value is less than or equal to gamma, the sequence of the predicted value is converged to an estimated value corresponding to a real zero value of g (theta) finally; completing an iterative convergence process;
and step eight, setting a correction term and inhibiting performance.
Further, in step one, the observation data model:
x(n)=f(θ,n)+w(n),n=0,1,…,N-1
where x (n) is the nth observed data sample, f (θ, n) is the nth function sample containing the unknown parameter θ to be estimated, and w (n) is the nth observed noise sample.
Further, the third step specifically comprises:
setting the first derivative of the solution number likelihood function
Figure RE-GDA0002024470120000042
Initial value of (a)0Handle bar
Figure RE-GDA0002024470120000043
Is approximately expressed as
Figure RE-GDA0002024470120000044
Wherein g' (θ)0) And g' (θ)0) Respectively representing log-likelihood functions at theta ═ theta0The second and third derivatives of time.
Further, the fifth step specifically comprises:
order to
Figure RE-GDA0002024470120000051
The formula is equal to zero, and theta corresponding to the case that g (theta) is equal to 0 is solved1To obtain
Figure RE-GDA0002024470120000052
Reuse this new predicted value theta1As a new predicted value θ0Performing quadratic term approximation on the function g (theta) again, and repeating the previous method to solve a new zero value; the sequence of predicted values will eventually converge to the estimated value corresponding to the true zero value of g (θ).
Further, the sixth step specifically comprises: based on the previous predicted value thetakThe next new predicted value θ is obtained by the following equationk+1And finishing the final iterative convergence process.
Figure RE-GDA0002024470120000053
When theta is satisfiedk+1kWhen gamma is less than or equal to gamma, the iterative convergence stops, gamma is a small positive number, thetakMonotonically decreasing; having a thetak+1≈θkAnd g (θ)k)≈0;g(θk) The log-likelihood function p (x; theta) derived function of
Figure RE-GDA0002024470120000054
Further, the method for inhibiting the performance of the correction term in the step eight comprises the following steps:
handle
Figure RE-GDA0002024470120000061
The correction term in (1) is defined as a function
Figure RE-GDA0002024470120000062
It is a further object of this invention to provide such an improved adaptive fast iterative convergence solution control system.
In summary, the advantages and positive effects of the invention are: in a high-dimensional parameter estimation occasion, the invention provides a numerical iteration solution with excellent performance, compared with the conventional numerical solution based on first-order Taylor series expansion, the invention has the advantages of higher convergence speed, insensitivity to an initial value and stronger noise resistance. Compared with closed solution methods such as ML and the like, the method has the advantages that the calculation complexity is greatly reduced, the method can be widely applied to occasions with high engineering real-time requirements, and the precision is not influenced.
In signal detection and parameter estimation, it is often necessary to estimate unknown parameters, such as in a Maximum Likelihood Estimation (MLE) algorithm, which estimates the unknown parameters in a data model from noisy data samples, which is a problem often encountered in statistical signals. Due to the fact that the statistical characteristics of noise are non-Gaussian or even Gaussian noise, the analytic solution of the parameter to be estimated is often difficult to calculate due to the nonlinear property of the parameter to be estimated in the data model, and the iterative solution is the only choice for statistical parameter estimation. In the estimation of statistical parameters such as MLE (maximum likelihood estimation), unbiased estimation of parameters to be estimated has important significance in a signal processing community, such as DOA (direction of arrival) estimation and the like, and has very important value and significance in both military fields and civil fields
The invention develops the second derivative and the third derivative of the log-likelihood function, the basic idea is to approximate the parameter iteration process to Taylor second-order series expansion, thus the invention is insensitive to the selection of the initial value, and compared with the traditional method, the invention does not increase any computational complexity.
Drawings
Fig. 1 is a flow chart of an improved adaptive fast iterative convergence solution provided by an embodiment of the present invention.
Fig. 2 is a flowchart for iteratively solving the parameter to be estimated according to the embodiment of the present invention.
Fig. 3 is a diagram of iterative convergence processes of two methods under the same initial value provided by the embodiment of the present invention.
Fig. 4 is a diagram of an iterative convergence process of two methods under different initial values according to an embodiment of the present invention.
Detailed Description
In order to make the objects, technical solutions and advantages of the present invention more apparent, the present invention is further described in detail with reference to the following embodiments. It should be understood that the specific embodiments described herein are merely illustrative of the invention and are not intended to limit the invention.
Because the conventional iterative method carries out iterative estimation based on the first-order Taylor series expansion principle, only the second derivative of the log-likelihood function is developed by the method, and the high-order derivative of the log-likelihood function is not developed, when the second derivative of the log-likelihood function is small, the parameter iteration is not converged due to large fluctuation of a correction term. The conventional method has no self-adaptive convergence threshold and lacks a mechanism for inhibiting fluctuation of a correction term, and the application of an iterative method is severely restricted by the problem.
To solve the above problems, the present invention will be described in detail with reference to specific embodiments.
The improved self-adaptive fast iterative convergence solution method provided by the embodiment of the invention comprises the following steps: in signal detection, the parameter iteration process is approximated to Taylor second-order series expansion, and an initial value is selected.
And introducing a second derivative and a third derivative of the log-likelihood function into the correction term, increasing the slope and curvature characteristics of the log-likelihood ratio function, and adaptively adjusting the fluctuation of the correction term.
And acquiring an adaptive parameter estimation threshold and an iteration convergence stopping condition by utilizing the correlation between the second derivative and the third derivative of the log likelihood function.
The invention is further described below with reference to the accompanying drawings.
As shown in fig. 1, the improved adaptive fast iterative convergence solution method provided by the embodiment of the present invention specifically includes:
s101, observation data model, where x (N) ═ f (θ, N) + w (N), and N ═ 0,1, …, N-1.
S102, writing the parameters into a log likelihood ratio function form by using a parameter estimation method of Maximum Likelihood Estimation (MLE), wherein the first derivative of the log likelihood function is,
Figure RE-GDA0002024470120000091
s103, setting an initial value theta0The above equation is approximated as a log likelihood function where θ is θ0The second and third derivatives of time.
And S104, making the first derivative g (theta) of the log likelihood function equal to 0.
S105, solving theta corresponding to the situation that g (theta) is 01
Figure RE-GDA0002024470120000092
S106, using an iterative method to operate and using the previous predicted value thetakFind the next new predicted value thetak+1
S107 when theta is satisfiedk+1kWhen the value is less than or equal to gamma, the sequence of predicted values finally converges to the estimated value corresponding to the real zero value of g (theta). The iterative convergence process is completed.
And S108, setting a correction term to inhibit the performance.
In an embodiment of the present invention, the signal model includes:
consider the observed data model:
x(n)=f(θ,n)+w(n),n=0,1,…,N-1 (1)
where x (n) is the nth observed data sample, f (θ, n) is the nth function sample containing the unknown parameter θ to be estimated, and w (n) is the nth observed noise sample. In MLE-based parameter estimation methods, the log-likelihood ratio function is usually written in the form of, say, instructions
Figure RE-GDA0002024470120000093
Where p (x; θ) is the Probability Density Function (PDF) of the data sample x ═ x (1), x (2), …, x (n), θ is the parameter to be estimated, and g (θ) represents the first derivative of the log-likelihood function.
In the embodiment of the present invention, it is assumed that an initial value θ for solving equation (2) is set0And suppose g (θ) is at θ0The vicinity is the approximation of quadratic nonlinearity, the assumption reduces parameter estimation errors, and the invention expresses the approximation of equation (2) as
Figure RE-GDA0002024470120000101
Wherein g' (θ)0) And g' (θ)0) Respectively representing log-likelihood functions at theta ═ theta0The second and third derivatives of time. In the present invention, (theta-theta) in (3)0) Regarding the whole as a whole, and making equation (3) equal to zero, solving for θ corresponding to when g (θ) is 01To obtain
Figure RE-GDA0002024470120000102
The present invention then reuses this new predicted value θ1As the new predicted value θ in (4)0The function g (θ) is again approximated by the quadratic term, and the previous method is repeated to solve for a new zero value, as shown in fig. 2, and this sequence of predicted values will eventually converge to the estimated value corresponding to the true zero value of g (θ). In summary, the iterative method of the present invention is based on the previous predicted value θkThe next new predicted value θ is obtained by the following equationk+1And finishing the final iterative convergence process.
Figure RE-GDA0002024470120000103
When theta is satisfiedk+1kWhen the value is less than or equal to gamma, the iteration convergence stops, and the gamma is a small positive number and is determined by a specific application scene. ThetakIs monotonically decreasing and therefore must converge. At this time, the present invention has a value of thetak+1≈θkAnd g (θ)k) 0. Because of g (theta)k) Is a log-likelihood function p (x; theta) is derived from the derivative function of the first order
Figure RE-GDA0002024470120000104
Figure RE-GDA0002024470120000111
In the embodiment of the present invention, the analysis of the suppression performance of the correction term includes:
the invention defines the correction term in (6) as a function
Figure RE-GDA0002024470120000112
Meanwhile, the invention also defines the correction term in the conventional iterative convergence equation
Figure RE-GDA0002024470120000113
As can be seen from FIG. 1, when k is small, g (θ)k)=O(g′(θk) O (-) represents a high order infinitesimal, so
Correction term v (x, theta)k) The fluctuation is relatively small and the convergence rate is slow. With increasing k, g' (θ)k) Gradually decrease, g (theta)k) Is also decreasing, but g (theta)k)≠O(g′(θk) So that the correction term v (x, θ) is madek) The fluctuations are large, so that the iterations may not converge. On the other hand, when k is small,
Figure RE-GDA0002024470120000114
so u (x, theta)k) With the gradual increase of k, the voltage is larger, and the iteration speed is higher.
Figure RE-GDA0002024470120000115
So u (x, theta)k) The fluctuation is small, the convergence speed becomes slow, and the iterative convergence can be ensured.
In summary, when the initial value of the correction term is far from the true value, the fluctuation is large, the convergence step length is long, and when the iteration value is close to the true value, the fluctuation of the correction term can be adaptively inhibited, and the convergence step length is small.
In the embodiment of the present invention, the convergence termination decision criterion includes: when the method of the invention is applied in consideration of actual conditions, an iterative convergence termination judgment threshold value is set, and when the threshold value is met
Figure RE-GDA0002024470120000121
When gamma is a very small constant threshold value, determined by practical application scenarios, the invention considers that iterative convergence is terminated, and theta is at the momentk+1≈θk
The invention is further described below in connection with specific performance analysis comparisons and simulations.
Performance analysis and comparison:
the performance of the method is superior to that of the traditional method, and an observation data sequence is assumed to exist
x(n)=rn+w(n),n=0,1,…,N-1 (9)
Where w (n) is the variance σ2White gaussian noise. The parameter r is an unknown parameter to be estimated, and r is greater than 0. The likelihood is estimated by finding the maximum likelihood of r, which is the value of r that maximizes the likelihood function,
Figure RE-GDA0002024470120000122
equivalently, the r value that minimizes the following equation is obtained
Figure RE-GDA0002024470120000123
Let dJ (r)/dr be 0, then
Figure RE-GDA0002024470120000124
The above formula is a nonlinear equation about r, so that an analytic solution cannot be directly obtained. Respectively solving the first derivative, the second derivative and the third derivative of the log likelihood function
Figure RE-GDA0002024470120000131
Figure RE-GDA0002024470120000132
Figure RE-GDA0002024470120000133
The traditional iteration method is based on Taylor first term expansion to obtain
Figure RE-GDA0002024470120000134
The iterative method of the invention is based on quadratic term expansion, substitutes (13) - (15) into (6), becomes
Figure RE-GDA0002024470120000135
Setting simulation parameters: 1) let r0The iteration number vs. the convergence process of the conventional method (16) and the inventive method (17) is shown in fig. 3, and it can be seen from the simulation that the iteration convergence of the present invention is fast, and the convergence can be ensured, and the convergence process can adaptively adjust the fluctuation of the correction term.
Setting simulation parameters: 2) let r0=1.1,r′01.2, γ is 0.002, and the conventional method (16) has an initial value of r0(closer to the true value), the initial value of the process (17) of the present invention is r'0(far from the true value), the iteration times vs. the convergence process of the two methods are shown in fig. 4, and it can be seen from the simulation that the iteration convergence process of the present invention is not sensitive to the initial value. This is also a performance advantage of the present invention over conventional approaches.
The above description is only for the purpose of illustrating the preferred embodiments of the present invention and is not to be construed as limiting the invention, and any modifications, equivalents and improvements made within the spirit and principle of the present invention are intended to be included within the scope of the present invention.

Claims (5)

1. An improved adaptive fast iterative convergence solution method, comprising: in signal detection, a parameter iteration process is approximated to Taylor second-order series expansion, and an initial value is selected;
introducing a second derivative and a third derivative of a log-likelihood function into the correction term, increasing the slope and curvature characteristics of the log-likelihood ratio function, and adaptively adjusting the fluctuation of the correction term;
obtaining an adaptive parameter estimation threshold and an iterative convergence stopping condition by utilizing the correlation between the second derivative and the third derivative of the log-likelihood function; the improved adaptive fast iterative convergence solution method specifically comprises the following steps:
step one, observing a data model, wherein x (N) ═ f (theta, N) + w (N), N ═ 0,1, …, and N-1;
wherein, x (n) is the nth observed data sample, f (theta, n) is the nth function sample containing the unknown parameter theta to be estimated, and w (n) is the nth observed noise sample;
step two, writing the parameter estimation method of the maximum likelihood estimation MLE into a form of a log likelihood ratio function, wherein the first derivative of the log likelihood function is
Figure FDA0002926841340000011
Wherein p (x; θ) is a log-likelihood function;
step three, setting an initial value theta0The above equation is approximated as a log likelihood function where θ is θ0Formulas for the second and third derivatives of time;
step four, making the first derivative g (theta) of the log likelihood function equal to 0;
step five, solving theta corresponding to the situation that g (theta) is 01
Figure FDA0002926841340000012
Step six, performing operation by using an iterative method, and utilizing a previous predicted value thetakFind the next new predicted value thetak+1
Step seven, when theta is satisfiedk+1kWhen the value is less than or equal to gamma, the sequence of the predicted value is converged to an estimated value corresponding to a real zero value of g (theta) finally; completing an iterative convergence process;
and step eight, setting a correction term and inhibiting performance.
2. The improved adaptive fast iterative convergence solution of claim 1, wherein step three specifically comprises:
setting the first derivative of the solution number likelihood function
Figure FDA0002926841340000021
Initial value of (a)0Handle bar
Figure FDA0002926841340000022
Is approximately expressed as
Figure FDA0002926841340000023
Wherein g' (θ)0) And g' (theta)0) Respectively representing log-likelihood functions at theta ═ theta0The second and third derivatives of time.
3. The improved adaptive fast iterative convergence solution of claim 1, wherein step five specifically comprises:
order to
Figure FDA0002926841340000024
The formula is equal to zero, and theta corresponding to the case that g (theta) is equal to 0 is solved1To obtain
Figure FDA0002926841340000025
Reuse this new predicted value theta1As a new predicted value θ0Performing quadratic term approximation on the function g (theta) again, and repeating the previous method to solve a new zero value; the sequence of predicted values will eventually converge to the estimated value corresponding to the true zero value of g (θ).
4. The improved adaptive fast iterative convergence solution of claim 1, wherein step six specifically comprises: based on the previous predicted value thetakThe next new predicted value θ is obtained by the following equationk+1And finishing the final iterative convergence process.
Figure FDA0002926841340000026
When theta is satisfiedk+1kWhen gamma is less than or equal to gamma, the iterative convergence stops, gamma is a small positive number, thetakIs monotonously decreasingSubtracting; having a thetak+1≈θkAnd g (θ)k)≈0;g(θk) Is a function of the log-likelihood function p (x; theta) derived function of
Figure FDA0002926841340000031
5. An improved adaptive fast iterative convergence solution as claimed in claim 1 wherein the step eight correction term performance suppressing method comprises:
handle
Figure FDA0002926841340000032
The correction term in (1) is defined as a function
Figure FDA0002926841340000033
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