CN106295043A - A kind of parameter determination method of gaussian radial basis function agent model - Google Patents

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

A kind of parameter determination method of gaussian radial basis function agent model

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；x_iIt is the position vector of i-th sample point；N is sample point number；r_i=| | x- x_i| | for Euclidean distance；w_iWeight coefficient for i-th basic function.

The core width cs that parameter to be asked is Gaussian bases in gaussian radial basis function agent model_iWith weight coefficient w_i.Core Width cs_iDetermination 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:

$x_k=\frac{X_k-X_k^L}{X_k^U-X_k^L}k=\mathrm{1,2},...,n---\left(3\right)$

In formula,It is respectively the bound of kth dimension design variable；X_k、x_kAfter 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:

$\mathrm{\ρ}\left(x_i\right)=\underset{j=1}{\overset{N}{\mathrm{\Σ}}}{e}^{-\frac{{\left|\right|x_i-x_j\left|\right|}^2}{c^2}}=\underset{j=1}{\overset{N}{\mathrm{\Σ}}}{e}^{-\frac{{(x_i-x_j)}^T(x_i-x_j)}{c^2}}---\left(4\right)$

In formula, take:

$\mathrm{\ρ}\left(x_i\right)=\underset{j=1}{\overset{N}{\mathrm{\Σ}}}{e}^{-\frac{{\left|\right|x_i-x_j\left|\right|}^2}{c^2}}=\underset{j=1}{\overset{N}{\mathrm{\Σ}}}{e}^{-\frac{{(x_i-x_j)}^T(x_i-x_j)}{c^2}}=\underset{j=1}{\overset{N}{\mathrm{\Σ}}}{e}^{-N^{2/n}{(x_i-x_j)}^T(x_i-x_j)}---\left(5\right)$

3rd step: calculate the sample point x that local density is minimum_sMinimum range d to other sample point_{S, min}；

The sample point x that local density is minimum_sMinimum range d to other sample point_{S, min}For:

$d_{s,\min }=\underset{j=1}{\overset{N}{\min }}\left[{(x_s-x_j)}^T(x_s-x_j)\right];---\left(6\right)$

4th step: determine the core width of each sample point；

Following formula is used to determine the core width of each sample point:

${\mathrm{\σ}}_i=\sqrt[n]{\mathrm{\ρ}\left(x_s\right)/\mathrm{\ρ}\left(x_i\right)}{d}_{s,\min };---\left(7\right)$

5th step: determine the weight coefficient w of each sample point correspondence basic function_i(i=1,2 ..., N)；

By sample S:[x_i, y_i] (i=1,2 ..., N) substitution formula:

It is able to basis function weights coefficient w_i(i=1,2 ..., N) be the N-dimensional system of linear equations of unknown number:

In formula,Solve equation group and obtain w_i.So far, formulaIn unknown parameter σ_iWith w_iThe 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 bases_iCharacterize the size of i-th sample point influence area, σ_iN Power should be with density function ρ (x_i) be inversely proportional to, i.e.

$\frac{{\mathrm{\σ}}_i^n}{{\mathrm{\σ}}_j^n}=\frac{1/\mathrm{\ρ}\left(x_i\right)}{1/\mathrm{\ρ}\left(x_j\right)}=\frac{\mathrm{\ρ}\left(x_j\right)}{\mathrm{\ρ}\left(x_i\right)}---\left(10\right)$

Formula (10) comprises N-1 independent equation, core width cs to be determined_iFor 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 minimum_sIf its core width is σ_s, its basic function is away from x_sNearest sample point Numerical value is

If σ_sNumerical value is too small, the core width cs of remaining sample point_iAlso by too small, RBF agent model will be caused not only Sliding；If otherwise σ_sNumerical value is excessive, will cause dragon lattice phenomenon.Reasonably σ_sValue should ensure that sample point x_sThe 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 σ_s/d_{S, min}Numerical value close System is as shown in Figure 1.Work as σ_s/d_{S, min}During ＜ 0.6592,Numerical value is less than normal, i.e. sample point x_sImpact on the weak side；When σ_s/d_{S, min}During ＞ 1,The present invention takes σ_s/d_{S, min}=1 to ensure sample point x_sThe domain of influence arrive from it Near sample point, it is ensured that agent model is smooth, i.e.

σ_s=d_{S, 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-x_i)^T(x-x_i), but (x-x_i)^T(x-x_i) determine that weight coefficient w equally_iThe 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 σ_s/d_{S, min}Numerical 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 minimum_sMinimum range d to other sample point_{S, min}:

Following formula is used to calculate d_{S, min}:

$d_{s,\min }=\underset{j=1}{\overset{N}{\min }}\left[{(x_s-x_j)}^T(x_s-x_j)\right]---\left(13\right)$

Step 4: determine the core width cs of each sample point_i:

Following formula is used to determine σ_i:

${\mathrm{\σ}}_i=\sqrt[n]{\mathrm{\ρ}\left(x_s\right)/\mathrm{\ρ}\left(x_i\right)}{d}_{s,\min }---\left(14\right)$

Step 5: determine the weight coefficient w of each sample point correspondence basic function_i(i=1,2 ..., N):

By sample S:[x_i, y_i] (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 w_i.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 d_{I, max}For 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 d_maxFor 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:

$R^2=1-\frac{\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{(y_i-{\hat{y}}_i)}^2}{\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{(y_i-\stackrel{\‾}{y})}^2}---\left(16\right)$

In formula, n is checking sample size；y_iFor archetype actual value；For approximate model numerical value；For y_iAverage 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 |x_i|≤5 (18)

Function III (low-dimensional higher-order function):

$f\left(x\right)=(10+x_1\cos \left(x_1\right))(3+\exp (-x_2^2))\left|x_i\right|\≤5---\left(19\right)$

Function IV (higher-dimension lowfunction):

$\begin{array}{c}f\left(x\right)=2{(x_1-1)}^2+x_2^2-x_1{x}_2+{(4-x_3)}^2+3{(x_4-6)}^2+\\ 0.5{(x_5-2)}^2+{(x_6-10)}^2+4{(x_7-9)}^2+\\ 2{(x_8-5)}^2+33\left|x_i\right|\≤20\end{array}---\left(20\right)$

Function V (higher-dimension higher-order function):

$f\left(x\right)=\underset{i=1}{\overset{8}{\mathrm{\Σ}}}\exp \left(x_i\right)(x_i-\ln \left(\underset{k=1}{\overset{8}{\mathrm{\Σ}}}\exp \left(x_k\right)\right))\left|x_i\right|\≤4---\left(21\right)$

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:

$\begin{array}{c}\mathrm{Min}:f\left(x\right)=1.10471x_1^2{x}_2+0.04811x_3{x}_4(14.0+x_2)\\ s.t.g_1\left(x\right)=\mathrm{\τ}\left(x\right)-{\mathrm{\τ}}_{\max }\≤0\\ g_2\left(x\right)=\mathrm{\σ}\left(x\right)-{\mathrm{\σ}}_{\max }\≤0\\ g_3\left(x\right)=x_1-x_4\≤0\\ g_4\left(x\right)=0.10471x_1^2+0.04811x_3{x}_4(14.0+x_2)-5.0\≤0\\ g_5\left(x\right)=0.125-x_1\≤0\\ g_6\left(x\right)=\mathrm{\δ}\left(x\right)-{\mathrm{\δ}}_{\max }\≤0\\ g_7\left(x\right)=P-P_c\left(x\right)\≤0\end{array}---\left(22\right)$

In formula:

$\mathrm{\τ}\left(x\right)=\sqrt{{\left({\mathrm{\τ}}^{\′}\right)}^2+2{\mathrm{\τ}}^{\′}{\mathrm{\τ}}^{\′\′}\frac{x_2}{2R}+{\left({\mathrm{\τ}}^{\′\′}\right)}^2}$

${\mathrm{\τ}}^{\′}=\frac{P}{\sqrt{2}{x}_1{x}_2},{\mathrm{\τ}}^{\′\′}=\frac{\mathrm{MR}}{J},M=P(L+\frac{x_2}{2})$

$R=\sqrt{\frac{x_2^2}{4}+{\left(\frac{x_1+x_3}{2}\right)}^2}$

$J=2\left\{\sqrt{2}{x}_1{x}_2\right[\frac{x_2^2}{12}+{\left(\frac{x_1+x_3}{2}\right)}^2\left]\right\}$

$\mathrm{\σ}\left(x\right)=\frac{6\mathrm{PL}}{x_4{x}_3^2},\mathrm{\δ}\left(x\right)=\frac{{4\mathrm{PL}}^3}{{\mathrm{Ex}}_3^3{x}_4}$

$P_c\left(x\right)=\frac{4.013E\sqrt{x_3^2{x}_4^6/36}}{L^2}(1-\frac{x_3}{2L}\sqrt{\frac{E}{4G}})$

P=6000lb, L=14in, E=30 × 10⁶Psi, G=12 × 10⁶psi

τ_max=13600psi, σ_max=30000psi, δ_max=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:

$x_k=\frac{X_k-X_k^L}{X_k^U-X_k^L},k=1,2,...,n---\left(3\right)$

In formula,It is respectively the bound of kth dimension design variable；X_k、x_kUnit 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:

$\mathrm{\ρ}\left(x_i\right)=\underset{j=1}{\overset{N}{\Σ}}{e}^{-\frac{\left|\right|x_i-x_j|{|}^2}{c^2}}=\underset{j=1}{\overset{N}{\Σ}}{e}^{-\frac{{(x_i-x_j)}^T(x_i-x_j)}{c^2}}---\left(4\right)$

In formula, take:

$\mathrm{\ρ}\left(x_i\right)=\underset{j=1}{\overset{N}{\Σ}}{e}^{-\frac{\left|\right|x_i-x_j|{|}^2}{c^2}}=\underset{j=1}{\overset{N}{\Σ}}{e}^{-\frac{{(x_i-x_j)}^T(x_i-x_j)}{c^2}}=\underset{j=1}{\overset{N}{\Σ}}{e}^{-N^{2/n}{(x_i-x_j)}^T(x_i-x_j)};---\left(5\right)$

3rd step: calculate the sample point x that local density is minimum_sMinimum range d to other sample point_{S, min}；

The sample point x that local density is minimum_sMinimum range d to other sample point_{S, min}For:

$d_{s,\min }=\underset{j=1}{\overset{N}{\min }}\[{(x_s-x_j)}^T(x_s-x_j)\];---\left(6\right)$

4th step: determine the core width of each sample point；

Following formula is used to determine the core width of each sample point:

${\mathrm{\σ}}_i=\sqrt[n]{\mathrm{\ρ}\left(x_s\right)/\mathrm{\ρ}\left(x_i\right)}{d}_{s,\min };---\left(7\right)$

5th step: by sample S:[x_i, y_i] (i=1,2 ..., N) substitution formula:

It is able to basis function weights coefficient w_i(i=1,2 ..., N) be the N-dimensional system of linear equations of unknown number:

In formula,Solve equation group and obtain w_i；So far, formula

In unknown parameter σ_iWith w_iComplete Entirely determine, i.e. complete the parameter determination process of gaussian radial basis function agent model.