CN106295043A - A kind of parameter determination method of gaussian radial basis function agent model - Google Patents
A kind of parameter determination method of gaussian radial basis function agent model Download PDFInfo
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Abstract
The invention belongs to field of information processing, be specifically related to the parameter determination method of a kind of gaussian radial basis function agent model.The technical solution used in the present invention is: by sample space Linear Mapping to cubic unit body;Each sample point local density is calculated according to sample distribution situation;Calculate the sample point minimum range to other sample point of local density's minimum;Determine the core width of each sample point;Determine the weight coefficient of each sample point correspondence basic function.The invention has the beneficial effects as follows: the method clear logic, easy and simple to handle, easy to carry out that the present invention proposes;Basic function core width of the present invention determines that method is substantially without the increase bringing amount of calculation;The gaussian radial basis function agent model using the inventive method to obtain is common to uniformly/nonuniform sample, and has distinguishing feature reliable, efficient, high-precision.
Description
Technical field
The invention belongs to field of information processing, be specifically related to the parameter determination side of a kind of gaussian radial basis function agent model
Method.
Background technology
Agent model method is the method that project analysis, Design and optimization process frequently involve, and its basic thought is with one
Relatively simple analytical function approximation replaces original complicated calculations to analyze model, utilizes the response of known discrete data point (sample)
Information predicts unknown point response value.RBF agent model is the most reliable in terms of precision and robustness, is respectively
The agent model that engineering field is widely used.Gaussian function has good seriality and the property led, and is widely used as radially
The kernel function of basic function, its expression-form is:
In formula, the core width of σ gaussian kernel function.
The primitive form of gaussian radial basis function agent model is:
In formula, x is design variable vector;xiIt is the position vector of i-th sample point;N is sample point number;ri=| | x-
xi| | for Euclidean distance;wiWeight coefficient for i-th basic function.
The core width cs that parameter to be asked is Gaussian bases in gaussian radial basis function agent modeliWith weight coefficient wi.Core
Width csiDetermination agent model precision of prediction is had decisive influence, determined that method mainly has and directly determined method and base
In method two class optimized.Previous class method amount of calculation is little, and the approximate model obtained for being uniformly distributed sample can realize higher
Precision, but when sample distribution is uneven, approximation quality is without Reliable guarantee;The precision phase of latter class method correspondence agent model
To higher, but amount of calculation is the most relatively large.
Summary of the invention
For improving the reliability of gaussian radial basis function agent model, computational efficiency and approximation quality, the present invention further
The parameter determination method of a kind of gaussian radial basis function agent model is proposed.The technical scheme is that
The parameter determination method of a kind of gaussian radial basis function agent model, specifically includes following steps:
The first step: by sample space Linear Mapping to n dimension cubic unit body;
Use following formula by sample space Linear Mapping to n dimension cubic unit body:
In formula,It is respectively the bound of kth dimension design variable;Xk、xkAfter being respectively original design space and mapping
The value of kth dimension design variable in unit cube.
Second step: calculate each sample point local density according to sample distribution situation;
The each sample point local density of employing following formula calculating:
In formula, take:
3rd step: calculate the sample point x that local density is minimumsMinimum range d to other sample pointS, min;
The sample point x that local density is minimumsMinimum range d to other sample pointS, minFor:
4th step: determine the core width of each sample point;
Following formula is used to determine the core width of each sample point:
5th step: determine the weight coefficient w of each sample point correspondence basic functioni(i=1,2 ..., N);
By sample S:[xi, yi] (i=1,2 ..., N) substitution formula:
It is able to basis function weights coefficient wi(i=1,2 ..., N) be the N-dimensional system of linear equations of unknown number:
In formula,Solve equation group and obtain wi.So far, formulaIn unknown parameter σiWith wiThe most really
Fixed, i.e. complete the parameter determination process of gaussian radial basis function agent model.
For being best understood from the present invention, the origin of formula (7) is described as follows:
The core width cs of each sample point correspondence Gaussian basesiCharacterize the size of i-th sample point influence area, σiN
Power should be with density function ρ (xi) be inversely proportional to, i.e.
Formula (10) comprises N-1 independent equation, core width cs to be determinediFor N number of, try to achieve arbitrary sample point core width
After i.e. can determine that the core width of remaining sample point.
Consider the sample point x that local density is minimumsIf its core width is σs, its basic function is away from xsNearest sample point
Numerical value is
If σsNumerical value is too small, the core width cs of remaining sample pointiAlso by too small, RBF agent model will be caused not only
Sliding;If otherwise σsNumerical value is excessive, will cause dragon lattice phenomenon.Reasonably σsValue should ensure that sample point xsThe domain of influence of basic function
The position of the sample point of its nearest neighbours should be arrived, i.e.Numerical value be sufficiently large,With σs/dS, minNumerical value close
System is as shown in Figure 1.Work as σs/dS, minDuring < 0.6592,Numerical value is less than normal, i.e. sample point xsImpact on the weak side;When
σs/dS, minDuring > 1,The present invention takes σs/dS, min=1 to ensure sample point xsThe domain of influence arrive from it
Near sample point, it is ensured that agent model is smooth, i.e.
σs=dS, min (12)
The computing formula of each sample point core width as shown in formula (7) is i.e. obtained by formula (10) and formula (12).
The invention has the beneficial effects as follows:
(1) present invention propose gaussian radial basis function agent model parameter determination method clear logic, easy and simple to handle,
Easy to carry out;
(2) present invention determine that the main amount of calculation of method of Gaussian bases core width is to calculate the distance between sample point
(x-xi)T(x-xi), but (x-xi)T(x-xi) determine that weight coefficient w equallyiThe amount needed, in other words, basic function core width of the present invention
Degree determines that method is substantially without the increase bringing amount of calculation;
(3) the logical uniformly/nonuniform sample that is suitable to of gaussian radial basis function agent model using the inventive method to obtain, and
There is distinguishing feature reliable, efficient, high-precision.
Accompanying drawing explanation
Fig. 1 isWith σs/dS, minNumerical relation;
Fig. 2 is the flow process of the parameter determination method of the gaussian radial basis function agent model that the present invention proposes;
Fig. 3 is the MR of the gaussian radial basis function agent model that the inventive method obtains with other two kinds of methods2Value compares;
Fig. 4 is welded grider space structure schematic diagram.
Detailed description of the invention
The flow process of the parameter determination method of this gaussian radial basis function agent model proposed is as shown in Figure 2.Tie below
Close accompanying drawing, the detailed description of the invention of the present invention is further described.
Step one: by sample space Linear Mapping to n dimension cubic unit body:
According to formula (3) by sample space Linear Mapping to n dimension cubic unit body.
Step 2: according to each sample point local density of sample distribution situation calculating:
Each sample point local density ρ (x is calculated according to formula (5)i)。
Step 3: calculate the sample point x that local density is minimumsMinimum range d to other sample pointS, min:
Following formula is used to calculate dS, min:
Step 4: determine the core width cs of each sample pointi:
Following formula is used to determine σi:
Step 5: determine the weight coefficient w of each sample point correspondence basic functioni(i=1,2 ..., N):
By sample S:[xi, yi] (i=1,2 ..., N) substitute into formula (2), obtain N-dimensional system of linear equations:
In formula,Solve equation group and i.e. obtain weight coefficient wi.The most i.e. complete height
The parameter determination process of this RBF agent model.
For analyzing the performance characteristics of the present invention, the gaussian radial basis function agent model present invention obtained and following two
The model that method obtains compares analysis.
Existing method one: useDetermine each sample point basic function core width, wherein dI, maxFor i-th
Sample point is to the minimum range between other sample points, the determination method of the weight coefficient of each sample point correspondence basic function and class of the present invention
Seemingly.(list of references: S.Kitayama, M.Arakawa, K.Yamazaki.Sequential Approximate
Optimization using Radial Basis Function network for engineering optimization
[J] .Optimization and Engineering.2011,12:535-557.)
Existing method two: useDetermine each sample point basic function core width, wherein dmaxFor between all sample points
Ultimate range, the determination method of the weight coefficient of each sample point correspondence basic function is similar with the present invention.(list of references:
H.Nakayama, M.Arakawa, R.Sasaki.Simulation-based optimization using
Computational intelligence [J] .Optimization and Engineering.2002 (3): 201-214.)
Use the gaussian radial basis function agent model that following formula comparison between the standards the inventive method obtains with other two kinds of methods
Approximation quality:
In formula, n is checking sample size;yiFor archetype actual value;For approximate model numerical value;For yiAverage
Value.R2≤ 1, it is closer to 1 and shows that approximate model precision is the highest, work as R2Approximate model error at checking sample is shown when=1
It is 0.
Take 5 typical test functions as follows.
Function I (one-dimensional functions):
F (x)=x sin (1.5x) 0≤x≤10 (17)
Function II (low-dimensional lowfunction):
F (x)=sin (x1+x2)+(x1-x2)2-1.5x1+2.5x2+1 |xi|≤5 (18)
Function III (low-dimensional higher-order function):
Function IV (higher-dimension lowfunction):
Function V (higher-dimension higher-order function):
The method validation core in this paper width in document [24] is used to determine the performance of method,
Proof procedure is: function I takes 10 random sample points, and function II, III take 50 random sample points, function IV, V
Take 200 random sample points, determine agent model by the inventive method with existing method one, two respectively, for ensureing approximation quality
R2The accuracy calculated, takes fully enough checking sample points, and quantity is 1000, carries out 20 independent random number emulation real
Test, calculate R2With R2Meansigma methods MR2.Three kinds of methods obtain the MR of agent model2Value compares as shown in Figure 3.Result shows, this
The approximation quality of the gaussian radial basis function agent model that the parameter determination method that invention proposes obtains is significantly better than existing method
One, two.
The gaussian radial basis function agent model that the parameter determination method that the present invention proposes obtains can be widely used for engineering and divides
Analysis, Design and optimization field, and improve efficiency and performance.As a example by welded grider optimization problem, this problem sets for classical engineering optimization
Meter problem, as shown in Figure 4, the target optimizing design is to reduce manufacturing cost under conditions of satisfied constraint to its schematic diagram as far as possible,
Four design variables are x respectively1=h, x2=l, x3=t and x4=b, constraint has maximum stress to retrain, and maximum distortion retrains, shape
Shape constraint etc., between this optimization, the mathematic(al) representation of this optimization problem of topic is:
In formula:
P=6000lb, L=14in, E=30 × 106Psi, G=12 × 106psi
τmax=13600psi, σmax=30000psi, δmax=0.255in
Use and optimize Latin hypercube method 40 sample points of generation, calculate the object function of each sample point according to formula (22)
With binding occurrence, it is respectively adopted the inventive method and existing method one, two and sets up the gaussian radial basis function letter of object function and binding occurrence
Number agent model, is optimized on the basis of agent model, and optimum results is as shown in table 1.Obtained by table 1, the inventive method gained
Agent model correspondence optimal solution meets each constraints of master mould, for the feasible solution of master mould, design variable and target letter
Numerical value is closest to master mould optimal solution;Existing method one gained agent model correspondence optimal solution is also the feasible solution of master mould, but
Design variable is with target function value deviation master mould optimal solution farther out;Existing method two gained agent model correspondence optimal solution is former
The infeasible solution of model;The parameter determination method of the gaussian radial basis function agent model that the present invention proposes is when engineering optimization
Performance is better than existing method one, two.
The contrast of table 1 distinct methods gained agent model correspondence optimal solution
The present invention proposes clear logic, the parameter of gaussian radial basis function agent model easy and simple to handle, easy to carry out
Determining method, the method is logical is suitable to uniformly/nonuniform sample, and has distinguishing feature reliable, efficient, high-precision, is Gauss
The Perfected process of RBF agent model parameter determination.
In sum, although the present invention is disclosed above with preferred embodiment, so it is not limited to the present invention, any
Those of ordinary skill in the art, without departing from the spirit and scope of the present invention, when various change and retouching can be made, therefore this
Bright protection domain is when defining in the range of standard depending on claims.
Claims (1)
1. the parameter determination method of a gaussian radial basis function agent model, it is characterised in that the method specifically includes following
Step:
The first step: by sample space Linear Mapping to n dimension cubic unit body;
Use following formula by sample space Linear Mapping to n dimension cubic unit body:
In formula,It is respectively the bound of kth dimension design variable;Xk、xkUnit after being respectively original design space and mapping
The value of kth dimension design variable in cube;
Second step: calculate each sample point local density according to sample distribution situation;
The each sample point local density of employing following formula calculating:
In formula, take:
3rd step: calculate the sample point x that local density is minimumsMinimum range d to other sample pointS, min;
The sample point x that local density is minimumsMinimum range d to other sample pointS, minFor:
4th step: determine the core width of each sample point;
Following formula is used to determine the core width of each sample point:
5th step: by sample S:[xi, yi] (i=1,2 ..., N) substitution formula:
It is able to basis function weights coefficient wi(i=1,2 ..., N) be the N-dimensional system of linear equations of unknown number:
In formula,Solve equation group and obtain wi;So far, formula
In unknown parameter σiWith wiComplete
Entirely determine, i.e. complete the parameter determination process of gaussian radial basis function agent model.
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CN113759342A (en) * | 2021-08-31 | 2021-12-07 | 柳州柳工叉车有限公司 | Scanning method and device of laser radar, computer equipment and storage medium |
CN114218686A (en) * | 2022-02-21 | 2022-03-22 | 中国人民解放军国防科技大学 | Multi-precision data smooth scale approximate modeling method for aircraft |
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彭科 等: "《序列近似优化方法及其在火箭外形快速设计中的应用》", 《国防科技大学学报》 * |
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CN113759342A (en) * | 2021-08-31 | 2021-12-07 | 柳州柳工叉车有限公司 | Scanning method and device of laser radar, computer equipment and storage medium |
CN113759342B (en) * | 2021-08-31 | 2023-10-24 | 柳州柳工叉车有限公司 | Laser radar scanning method and device, computer equipment and storage medium |
CN114218686A (en) * | 2022-02-21 | 2022-03-22 | 中国人民解放军国防科技大学 | Multi-precision data smooth scale approximate modeling method for aircraft |
CN114218686B (en) * | 2022-02-21 | 2022-05-10 | 中国人民解放军国防科技大学 | Multi-precision data smooth scale approximate modeling method for aircraft |
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