CN105046346A - Method of using maximum relative error minimum multi-input multi-output support vector regression machine to carry out concrete multicomponent anti-prediction - Google Patents

Abstract

The invention relates to a method of using a maximum relative error minimum multi-input multi-output support vector regression machine to carry out simultaneous anti-prediction on various kinds of concrete components. In the method of the invention, a standard support vector machine is improved. A relative error is used to establish an error term to solve an optimization problem. Through introducing Lagrange, the optimization problem with m variables and k constrained conditions is converted into an extremum problem of an equation set with m+k variables. And the variables are not subject to any constraints. Through a Lagrangian multiplier method and dualistic transformation, the optimization problem originally possessing an equality constraint is converted into a new optimization problem. Then a KKT optimization condition is used to carry out solving. An RE-SVM as a fitness function. A PSO is used to carry out parameter optimization. Finally, the RE-SVM is used to establish an anti-prediction model so as to realize carrying out simultaneous anti-prediction on the various kinds of the concrete components. Anti-prediction time efficiency of partial components of the concrete is high, precision of a prediction result is high and an experiment result basically satisfies a precision requirement during engineering.

Description

A kind of maximum relative error minimum Multiinputoutput support vector regression that utilizes carries out the anti-method predicted of concrete polycomponent

Technical field

The present invention relates to a kind of maximum relative error minimum multiple-input and multiple-output support vector regression building method and a kind of method utilizing this support vector regression to carry out the anti-prediction simultaneously of multiple concrete component.

Background technology

Concrete is most important in civil engineering work and the material of high complexity.Usually, concrete is primarily of cement (x1), ground granulated blast furnace slag (x2), flyash (x3), water (x4), water reducer (x5), coarse aggregate (x6) and fine aggregate (x7), according to a certain ratio, uniform stirring, closely knit shaping, and form through certain length of time (x8) maintenance sclerosis.Concrete strength (y1), stream (y2), the slump (y3) are the core contents of concrete Quality Control, are also the important evidence of the civil infrastructure designs such as dam construction and construction thereof.In practical engineering application, due to extraneous factor interference such as different regions temperature, humidity, and physics, chemical reaction complicated between each component in concrete, make concrete strength (y1), stream (y2), the slump (y3) and concrete each component cement (x1), ground granulated blast furnace slag (x2), flyash (x3), water (x4), water reducer (x5), coarse aggregate (x6), between fine aggregate (x7) with the length of time (x8), become the nonlinear relationship of complexity.The method that actual proportioning is manually carried out in traditional utilization wastes a large amount of human and material resources and financial resources, therefore the artificial actual proportioning test result data of history collected and obtain is utilized, use the method for artificial intelligence, machine learning and data mining, learning machine (such as neural network, support vector machine etc.) is trained, thus prediction is carried out to the unknown situation of new input becomes inexorable trend.

At present, mainly concentrate in concrete strength or slump list output forward prediction concrete learning machine forecasting research, such as Wu De can use LS-SVM to predict concrete strength; The people such as He Xiaofeng and Li Gang in order to overcome artificial neural network speed of convergence slowly, be easily absorbed in local extremum problem use respectively PSO-BP and based on Regularized RBF Network prediction concrete strength; The people such as Yang Songsen apply Fuzzy System Method and set up forecast model; People such as Zhang Yan and forest rent is gloomy etc. that people uses Gaussian process machine learning to predict coagulation intensity; The people such as Yan Chunling set up the regeneration concrete slump adaptive nuero-fuzzy inference system model prediction slump.

Concrete proportioning method for designing conventional in current civil engineering work has Osadebe regression model, Ibearugbulem regression model, the mathematical model such as Optimized model based on the simplex theory of Scheffe.These methods are well suited for concrete proportioning optimization and can design effectively concrete constituents, but when input attributes and number of samples many time, model becomes very complicated, and training pattern will spend long time, and effect neither be very desirable.

Utilize that concrete strength, stream (FLOW), the slump (SLUMP) and constituent part are counter dopes the remaining various ingredients of concrete, belong to multiple-input and multiple-output Nonlinear inverse problem, and indirect problem is more complicated than direct problem, often there is ill-posedness and morbid state in indirect problem, is also difficult to provide strict effective proof to its existence and uniqueness and determinacy.The known concrete component of concrete strength, stream, the slump and part is not also used to carry out the pertinent literature report of anti-prediction to one or more unknown components at present.

Summary of the invention

The object of invention is the deficiency overcoming prior art, provides a kind of maximum relative error minimum Multiinputoutput support vector regression that utilizes to carry out the anti-method predicted of concrete polycomponent, first, uses relative error to set up error term to solve optimization problem.Then, the optimization problem of m variable and k constraint condition is had to be converted to the extreme-value problem that one has the system of equations of m+k variable by introducing Lagrange by one, its variable is under no restraint, by method of Lagrange multipliers and dualistic transformation thereof, the former optimization problem with equality constraint is converted to new optimization problem, uses KKT optimal condition to solve afterwards.Take RE-SVM as fitness function, use PSO to carry out parameter optimization.Finally, use RE-SVM to set up anti-forecast model, realize the anti-prediction simultaneously of multiple concrete component.The anti-predicted time efficiency of the present invention to concrete parts component is high, and the precision that predicts the outcome is high, and experimental result meets the accuracy requirement in engineering substantially.

The technical solution adopted for the present invention to solve the technical problems is: provide a kind of maximum relative error minimum Multiinputoutput support vector regression that utilizes to carry out the anti-method predicted of concrete polycomponent, it is characterized in that, based on the maximum relative error minimum multiple-input and multiple-output support vector regression building method of particle swarm optimization algorithm (PSO), comprise step:

A1, given training set for the input of M dimension, for N dimension exports;

A2, be optimized according to support vector machine maximize margin characteristic and maximum relative error minterm, optimization problem can be expressed as by formulism:

$\begin{array}{c}\underset{\mathrm{\ω},b,e}{\min }\frac{1}{2}{\mathrm{\ω}}^T\mathrm{\ω}+\frac{1}{2}C\underset{i}{\Σ}{e}_i^2\\ s.t.e_i=\frac{{\mathrm{\ω}}^T\mathrm{\φ}\left(x_i\right)+b-y_i}{y_i},\∀i\end{array}---\left(1\right)$

Wherein punish parameter C>0, φ (x _i) be a function being mapped to higher dimensional space, ω is weight vectors, b=[b ¹..., b ⁿ] be biased, ω ^tthe transposition of ω,

A3, to establish introduce method of Lagrange multipliers and (1) formula be converted to Lagrangian function form:

$L(\mathrm{\ω},b,e;\mathrm{\α})=J(\mathrm{\ω},e)-\underset{i}{\Σ}{\mathrm{\α}}_i\[{\mathrm{\ω}}^T\mathrm{\φ}\left(x_i\right)+b+y_i{e}_i-y_i\]---\left(2\right)$

Now optimization problem is converted to and has solved optimization problem, is converted to optimization problem again by antithesis and solves

A4, optimization problem for step 3, by introducing KKT condition, meeting under some conditions, and nonlinear programming problem has a necessary and sufficient condition of optimum solution to be:

$\{\begin{array}{c}\frac{\∂L}{\∂\mathrm{\ω}}=0\→\mathrm{\ω}=\underset{i}{\mathrm{\Σ}\Σ}{\mathrm{\α}}_i\mathrm{\φ}\left(x_i\right)\\ \frac{\∂L}{\∂b}=0-\→-\underset{i}{\Σ}{\mathrm{\α}}_i=0\\ \frac{\∂L}{\∂e_i}=0\→{\mathrm{Ce}}_i=y_i{\mathrm{\α}}_i\\ \frac{\∂L}{\∂{\mathrm{\α}}_i}=0\→y_i={\mathrm{\ω}}^T\mathrm{\φ}\left(x_i\right)+b+y_i{e}_i\end{array}---\left(3\right)$

By (3) formula cancellation ω and e _iafter can solve parameter with matrix form and be biased b and Lagrange multiplier α:

$\left(\begin{array}{cc}0& I^T\\ I& \mathrm{\Ω}+y^2{C}^{-1}I\end{array}\right)\left(\begin{array}{c}b\\ \mathrm{\α}\end{array}\right)=\left(\begin{array}{c}\stackrel{\→}{0}\\ y\end{array}\right)---\left(4\right)$

Wherein y=[y ₁..., y _n] ^t, α=[α ₁..., α _n] ^t, Ω=(Ω _ij) _{n × n}=K (x _i, x _j), K (x here _i, x _j) be Radial basis kernel function, I=[1 ..., 1] ^t, b=[b ¹..., b ⁿ], $\stackrel{\→}{0}=\[0^1,\mathrm{...},0^N\];$

A5, be symmetric positive semidefinite matrix by the known Ω of steps A 4, therefore Ω+y ²c ^-1i is symmetric positive definite matrix, so Matrix Solving has unique solution, namely $\left(\begin{array}{c}b\\ \mathrm{\α}\end{array}\right)={\left(\begin{array}{cc}0& I^T\\ I& \mathrm{\Ω}+y^2{C}^{-1}I\end{array}\right)}^{-1}\left(\begin{array}{c}\stackrel{\→}{0}\\ y\end{array}\right),$ Now can obtain parameter alpha and b.

Optimization problem can be converted into (5) formula by the Lagrangian function result of (3) formula be brought in (2) formula in addition:

$\begin{array}{c}\underset{\mathrm{\α}}{\max }\underset{i}{\Σ}{\mathrm{\α}}_i{y}_i-\frac{1}{2C}\underset{i}{\Σ}{\mathrm{\α}}_i^2{y}_i^2-\frac{1}{2}\underset{i,j}{\Σ}{\mathrm{\α}}_i{\mathrm{\α}}_{j}K(x_i,x_j)\\ s.t.\underset{i}{\Σ}{\mathrm{\α}}_i=0\end{array}---\left(5\right)$

A6, by steps A 5 Matrix Solving result and b ^*, the parameter of anti-forecast model can be obtained with

$\begin{array}{c}\stackrel{\‾}{\mathrm{\ω}}=\underset{i}{\Σ}{\mathrm{\α}}_i^{*}\mathrm{\φ}\left(x_i\right)\\ \stackrel{\‾}{b}=b^{*}\end{array}---\left(6\right)$

A7, by will (6) formula acquisition parameter with bring into in can obtain forecast model:

$g\left(x\right)=\underset{i}{\Σ}{\mathrm{\α}}_i^{*}K(x_i,x)+b^{*}---\left(7\right)$

(7) formula is the final anti-forecast model of optimum obtained.

The target of the variance sig2 in A8, particle swarm optimization algorithm optimization punishment parameter C and radial basis function minimizes relative error, and the fitness function of use is as shown in (8) formula:

$Fitness=max\left(\frac{1}{n}\underset{i=1}{\overset{n}{\Σ}}\right|\frac{g\left(x_i\right)-y_i}{y_i}\left|\right)---\left(8\right)$

The process of the minimum support vector machine method of the maximum relative error based on particle swarm optimization algorithm is:

B1, population initialization, comprise population scale, random initializtion position and speed;

B2, evaluate the fitness of each particle according to formula (8) fitness function;

B3, calculate the speed of each particle and position formula is as shown in (9), (10) and (11)

$V_i^d:={\mathrm{wV}}_i^d+c_1*r_1({\mathrm{pbest}}_i^d-X_i^d)+c_2*r_2({\mathrm{gbest}}^d-X_i^d)---\left(9\right)$

$V_i^d:=\{\begin{array}{cc}{V}_{max},& V_i^d>V_{\max }\\ -V_{max},& V_i^d<-V_{\max }\end{array}---\left(10\right)$

$X_i^d:=X_i^d+V_i^d---\left(11\right)$

Wherein w is Inertia Weight, c ₁and c ₂speedup factor, r ₁and r ₂be the random number belonged between 0 to 1, speed is limited in maximal rate V by (10) formula _maxwith the m-V of minimum speed _max, represent the history optimal value of i-th particle in d dimension, gbest ^drepresent the history optimal value of population in d dimension;

B4, upgrade each particle and colony's history optimal location;

B5, judge whether to reach maximum iteration time or reach the end condition of best adaptive value, then enter step B6 if the judgment is Yes, then enter step B2 if the judgment is No;

B6, output optimizing parametric optimal solution, the punishment parameter C namely in formula (4) and the variance sig2 in Radial basis kernel function;

B7, the optimizing parametric results drawn by step B6 are brought in steps A 5, solve α and b by the matrix operation of steps A 5, finally obtain the optimum prediction model of formula (7).

Preferably, the method utilizing this support vector regression to carry out the anti-prediction simultaneously of multiple concrete component comprises the steps:

C1, determine the concrete various ingredients that will predict, utilize the mixed earth historical data structure training sample data collection of actual measurement;

C2, pre-service is carried out to training sample data collection, namely delete training data and concentrate prediction group to be divided into the sample of zero;

C3, using the minimum multiple-input and multiple-output support vector regression of maximum relative error as fitness function, use particle swarm optimization algorithm parameter optimization is carried out to this support vector machine;

C4, the final optimizing parameter of use step C3 are to this multiple-input and multiple-output support vector regression modeling;

The anti-forecast model of multiple-input and multiple-output of C5, use step C4 is predicted the various ingredients that will predict is counter.

The invention has the beneficial effects as follows: 1. the maximum relative error minimum multiple-input and multiple-output support vector regression building method seeking ginseng optimization based on PSO of the present invention, be applicable to small sample, there will not be over-fitting, Generalization Ability is strong.2. the present invention uses the nonlinear relationship of the physical and chemical reaction process of the method matching concrete complexity of multiple-input and multiple-output support vector machine, can realize according to engineering demand that multiple component is counter to be predicted fast, thus greatly reduce the man power and material needed for artificial experiment.Experimental result meets engineering demand substantially.

Below in conjunction with drawings and Examples, the present invention is described in further detail; But of the present inventionly a kind ofly utilize maximum relative error minimum Multiinputoutput support vector regression to carry out the anti-method predicted of concrete polycomponent to be not limited to embodiment.

Accompanying drawing explanation

Fig. 1 is the multiple-input and multiple-output model schematic of the present invention's i-th sample, and each sample has M input, N number of output;

Fig. 2 is anti-forecast model in the present invention, and wherein C1, C2, C3, C4, C5, C6, C7 are seven concrete components (cement, flyash, water, ground granulated blast furnace slag, water reducer, coarse aggregate and fine aggregates, order can arbitrarily).

Fig. 3 is the minimum model construction of SVM process of maximum relative error of the present invention.

Fig. 4 is the minimum support vector machine method of maximum relative error (PSO-RE-SVM) schematic flow sheet that the present invention is based on PSO.

Fig. 5 is slump model of the present invention.

Fig. 6 to Figure 18 is counter the predict the outcome table of the present invention for concrete component.

Embodiment

This patent why adopt prediction export maximum relative error minimum instead of picture least square method using minimum for error sum of squares as support vector machine export optimization object function, because in multi-input multi-output system, the error sum of squares that usage forecastings exports is minimum as optimization aim, large especially by causing exporting less attribute error, especially when different output attribute value difference progression is larger, the judgment criteria of inapplicable engineer applied.And use maximum relative error minimum as optimization aim, effectively can avoid this phenomenon, more realistic engineer applied.

The minimum multiple-input and multiple-output support vector regression of maximum relative error is based on statistical theory and structural risk minimization theory, improves generalization ability, realize empiric risk and fiducial range minimizes by structuring least risk.Adopt Radial basis kernel function that sample is mapped to higher dimensional space, thus nonlinear situation is become linear solution, avoid dimension disaster.This machine learning method is applicable to small sample, there will not be over-fitting, and Generalization Ability is strong.

Utilize maximum relative error minimum multiple-input and multiple-output support vector regression to carry out the anti-Forecasting Methodology of concrete component, can realize fast simultaneously to the anti-prediction of various ingredients.This patent utilizes any 5 components in known concrete strength (y1), stream (y2), the slump (y3) 3 attributes and 7 component cement (x1), ground granulated blast furnace slag (x2), flyash (x3), water (x4), water reducer (x5), coarse aggregate (x6), fine aggregate (x7) as input, other 2 components are as output, and the data configuration of many group the type becomes many this support vector regression models of group multiple input multiple output data collection training.Utilize this data set to train this maximum relative error minimum multiple-input and multiple-output support vector regression model, utilize the support vector regression model trained, can be good at carrying out various ingredients anti-prediction simultaneously to a kind of new concrete, greatly reduce artificial experimental cost.

Owing to adopting maximum relative error minimum as constraint condition, therefore require that for predicted value can not be zero.When realization, first pre-service can be carried out to the data that predicted value is zero.In reality, predicted value is zero represent this mixed earth and there is not this component, and predict without the need to counter, therefore this pre-service does not affect the scope of application of this method.

Embodiment 1

Shown in Fig. 1 and Fig. 2, a kind of maximum relative error minimum Multiinputoutput support vector regression that utilizes of the present invention carries out the anti-method predicted of concrete polycomponent, comprising: given training set T={ (x ₁, y ₁) ..., (x _n, y _n), M inputs n exports in this patent, concrete data set has 103 samples, and 8 inputs 2 export, and namely concrete strength (Mpa), stream (cm), the slump (cm) add cement (kg/m ^{^3}), ground granulated blast furnace slag (kg/m ^{^3}), flyash (kg/m ^{^3}), water (kg/m ^{^3}), water reducer (kg/m ^{^3}), coarse aggregate (kg/m ^{^3}) and fine aggregate (kg/m ^{^3}) in 5, anti-prediction residue 2 components.Here it is bad because of carrying out anti-prediction effect to 3 kinds of components for not using 7 inputs 3 to export support vector regression, due to the ill-posedness of indirect problem, the applicability problem of method.

Maximum relative error minimum multiple-input and multiple-output support vector regression optimization problem can be described as:

$\begin{array}{c}\underset{\mathrm{\&omega;},b,e}{\min }\frac{1}{2}{\mathrm{\&omega;}}^T\mathrm{\&omega;}+\frac{1}{2}C\underset{i}{\&Sigma;}{e}_i^2\\ s.t.e_i=\frac{{\mathrm{\&omega;}}^T\mathrm{\&phi;}\left(x_i\right)+b-y_i}{y_i},\&ForAll;i\end{array}---\left(1\right)$

Wherein punish parameter C>0, φ (x _i) be a mapping function, lower dimensional space is mapped to higher dimensional space, and ω is weight vectors, b=[b ¹, b ¹..., b ⁿ] be biased, ω ^tit is the transposition of ω.

(1) formula uses support vector machine largest interval thought, and minimizes maximum relative error, makes forecast model more accurate.By using linear equality constraints instead of inequality constrain that calculated amount can be greatly reduced, accelerate training speed.Not adopting error sum of squares minimum in algorithm is because in multi-input multi-output system as optimization aim, the error of the property value that output valve is less can be caused minimum for error sum of squares large especially as optimization aim, when difference output progression differs greatly, very large error can be produced in practical application, and use that maximum relative error is minimum can avoid this phenomenon as optimization aim.

If then Lagrangian function is as shown in (2) formula.

$L(\mathrm{\&omega;},b,e;\mathrm{\&alpha;})=J(\mathrm{\&omega;},e)-\underset{i}{\&Sigma;}{\mathrm{\&alpha;}}_i\&lsqb;{\mathrm{\&omega;}}^T\mathrm{\&phi;}\left(x_i\right)+b+y_i{e}_i-y_i\&rsqb;---\left(2\right)$

Wherein α _ifor Lagrange multiplier, now optimization problem is converted to and has solved optimization problem, to be converted to optimization problem by Lagrange duality and to solve

Can be obtained by KKT optimal conditions:

$\left\{\begin{array}{c}\frac{\&part;L}{\&part;\mathrm{\&omega;}}=0\&RightArrow;\mathrm{\&omega;}=\underset{i}{\&Sigma;}{\mathrm{\&alpha;}}_i\mathrm{\&phi;}\left(x_i\right)\\ \frac{\&part;L}{\&part;b}=0\&RightArrow;-\underset{i}{\&Sigma;}{\mathrm{\&alpha;}}_i=0\\ \frac{\&part;L}{\&part;e_i}=0\&RightArrow;{\mathrm{Ce}}_i=y_i{\mathrm{\&alpha;}}_i\\ \frac{\&part;L}{\&part;{\mathrm{\&alpha;}}_i}=0\&RightArrow;y_i={\mathrm{\&omega;}}^T\mathrm{\&phi;}\left(x_i\right)+b+y_i{e}_i\end{array}\right.---\left(3\right)$

By (3) formula cancellation ω and e _ithe matrix form of dual problem can be obtained, that is:

$\left(\begin{array}{cc}0& I^T\\ I& \mathrm{\&Omega;}+y^2{C}^{-1}I\end{array}\right)\left(\begin{array}{c}b\\ \mathrm{\&alpha;}\end{array}\right)=\left(\begin{array}{c}\stackrel{\&RightArrow;}{0}\\ y\end{array}\right)---\left(4\right)$

Wherein y=[y ₁..., y _n] ^t, Ω=(Ω _ij) _{n × n}=K (x _i, x _j), K (x here _i, x _j) be Radial basis kernel function, I=[1 ..., 1] ^t, α=[α ₁..., α _n] ^t, b=[b ¹..., b ⁿ], $\stackrel{\&RightArrow;}{0}=\&lsqb;0^1,\mathrm{...},0^N\&rsqb;.$

By Matrix Solving, namely $\left(\begin{array}{c}b\\ \mathrm{\&alpha;}\end{array}\right)={\left(\begin{array}{cc}0& I^T\\ I& \mathrm{\&Omega;}+y^2{C}^{-1}I\end{array}\right)}^{-1}\left(\begin{array}{c}\stackrel{\&RightArrow;}{0}\\ y\end{array}\right),$ Parameter alpha and b can be obtained.By Ω=(Ω _ij) _{n × n}known Ω is symmetric positive semidefinite matrix, therefore Ω+y ²c ^-1i is symmetric positive definite matrix, effectively prevent the ill-conditioning problem of matrix inversion process.(5) formula (3) is brought into (2) formula abbreviation to obtain.

$\begin{array}{c}\underset{\mathrm{\&alpha;}}{\max }\underset{i}{\&Sigma;}{\mathrm{\&alpha;}}_i{y}_i-\frac{1}{2C}\underset{i}{\&Sigma;}{\mathrm{\&alpha;}}_i^2{y}_i^2-\frac{1}{2}\underset{i,j}{\&Sigma;}{\mathrm{\&alpha;}}_i{\mathrm{\&alpha;}}_{j}K(x_i,x_j)\\ s.t.\underset{i}{\&Sigma;}{\mathrm{\&alpha;}}_i=0\end{array}---\left(5\right)$

Be now the optimization problem that α is solved by former question variation, the parameter needed for the Matrix Solving directly can passing through (4) formula for this experiment obtains.Use step 5 Matrix Solving result and b ^*, the parameter of anti-forecast model with can be expressed as:

$\begin{array}{c}\stackrel{\&OverBar;}{\mathrm{\&omega;}}=\underset{i}{\&Sigma;}{\mathrm{\&alpha;}}_i^{*}\mathrm{\&phi;}\left(x_i\right)\\ \stackrel{\&OverBar;}{b}=b^{*}\end{array}---\left(6\right)$

Primal problem can be solved by (6) formula

$g\left(x\right)=\underset{i}{\&Sigma;}{\mathrm{\&alpha;}}_i^{*}K(x_i,x)+b^{*}---\left(7\right)$

The optimum prediction model of final acquisition (7) formula, algorithm terminates.

Embodiment 2

Concrete in engineer applied is generally made up of cement, ground granulated blast furnace slag, flyash, water, water reducer, coarse aggregate and fine aggregate component, and is formed through certain curing age.The slump is to the standard that concrete workability is passed judgment in engineer applied, comprising concrete water-retaining property, mobility and cohesiveness.By checking slump experimental result, if the larger explanation of the slump easily causing the segregation of mix, if the too little difficulty of construction that makes increases, now can change the consumption gathered materials when not changing water cement ratio, or adding grout to change.Therefore the slump can ensure normally carrying out of construction.The method of testing of the slump: with a 100mm suitable for reading, end opening 200mm, the trumpet-shaped slump bucket of high 300mm, penetration concrete divides three fillings, the rear beater of each filling hits 25 times, after tamping along the even ecto-entad of bucket wall, floating.Then pull up bucket, concrete produces slump phenomenon because of deadweight, deducts the height of concrete peak after slump, be called the slump, as shown in Figure 5 with bucket high (300mm).If difference is 100mm, then the slump is 100.At data-oriented collection, the minimum multiple-input and multiple-output of the maximum relative error based on the PSO support vector regression of this patent is adopted to predict concrete component is counter.The counter of concrete component is predicted the outcome as shown in Fig. 6 to Figure 18.

Embodiment 3

As shown in Figure 4, the minimum multiple-input and multiple-output support vector regression of the maximum relative error based on PSO method (PSO-RE-SVM) is applicable to small sample, there will not be over-fitting, and Generalization Ability is strong, and concrete steps are:

A1, given training set for the input of M dimension, for N dimension exports;

M=8, N=2 in this patent experiment, i.e. the confirmatory experiment of 8 input 2 outputs.

A2, be optimized according to support vector machine maximize margin characteristic and maximum relative error minterm, optimization problem can be expressed as by formulism:

$\begin{array}{c}\underset{\mathrm{\&omega;},b,e}{\min }\frac{1}{2}{\mathrm{\&omega;}}^T\mathrm{\&omega;}+\frac{1}{2}C\underset{i}{\&Sigma;}{e}_i^2\\ s.t.e_i=\frac{{\mathrm{\&omega;}}^T\mathrm{\&phi;}\left(x_i\right)+b-y_i}{y_i},\&ForAll;i\end{array}---\left(1\right)$

Wherein punish parameter C>0, φ (x _i) be a function being mapped to higher dimensional space, ω is weight vectors, b=[b ¹..., b ⁿ] be biased, ω ^tthe transposition of ω, it is multiple input single output model as N=1.

Conventional loss function is that error sum of squares is minimum, and use maximum relative error minimum as optimization aim here, reason is for multi-input multi-output system, when difference output difference progression is very large, the error sum of squares that usage forecastings exports is minimum as optimization aim, the error of the property value making output valve less is large especially, the judgment criteria of inapplicable engineer applied.And use maximum relative error minimum as optimization aim, this phenomenon can be avoided, more meet the application in engineering.

A3, to establish introduce method of Lagrange multipliers and (1) formula be converted to Lagrangian function form:

$L(\mathrm{\&omega;},b,e;\mathrm{\&alpha;})=J(\mathrm{\&omega;},e)-\underset{i}{\&Sigma;}{\mathrm{\&alpha;}}_i\&lsqb;{\mathrm{\&omega;}}^T\mathrm{\&phi;}\left(x_i\right)+b+y_i{e}_i-y_i\&rsqb;---\left(2\right)$

Now optimization problem is converted to and has solved optimization problem, is converted to optimization problem again by antithesis and solves

Introduce method of Lagrange multipliers, it is a kind of multivariate function bounding method found variable and limit by one or more condition, by the extreme-value problem having the optimization problem of m variable and k constraint condition to change into m+k variable, thus simplify former optimization problem.

A4, optimization problem for step 3, by introducing KKT condition, meeting under some conditions, and nonlinear programming problem has a necessary and sufficient condition of optimum solution to be:

$\left\{\begin{array}{c}\frac{\&part;L}{\&part;\mathrm{\&omega;}}=0\&RightArrow;\mathrm{\&omega;}=\underset{i}{\&Sigma;}{\mathrm{\&alpha;}}_i\mathrm{\&phi;}\left(x_i\right)\\ \frac{\&part;L}{\&part;b}=0\&RightArrow;-\underset{i}{\&Sigma;}{\mathrm{\&alpha;}}_i=0\\ \frac{\&part;L}{\&part;e_i}=0\&RightArrow;{\mathrm{Ce}}_i=y_i{\mathrm{\&alpha;}}_i\\ \frac{\&part;L}{\&part;{\mathrm{\&alpha;}}_i}=0\&RightArrow;y_i={\mathrm{\&omega;}}^T\mathrm{\&phi;}\left(x_i\right)+b+y_i{e}_i\end{array}\right.---\left(3\right)$

By (3) formula cancellation ω and e _iafter can solve parameter with matrix form and be biased b and Lagrange multiplier α:

$\left(\begin{array}{cc}0& I^T\\ I& \mathrm{\&Omega;}+y^2{C}^{-1}-I\end{array}\right)\left(\begin{array}{c}b\\ \mathrm{\&alpha;}\end{array}\right)=\left(\begin{array}{c}\stackrel{\&RightArrow;}{0}\\ y\end{array}\right)---\left(4\right)$

Wherein y=[y ₁..., y _n] ^t, α=[α ₁..., α _n] ^t, Ω=(Ω _ij) _{n × n}=K (x _i, x _j), K (x here _i, x _j) be Radial basis kernel function, I=[1 ..., 1] ^t, b=[b ¹..., b ⁿ], $\stackrel{\&RightArrow;}{0}=\&lsqb;0^1,\mathrm{...},0^N\&rsqb;.$

Meeting under some rule conditions, it is ensure that nonlinear programming problem has an adequate condition of optimum solution that KKT optimizes.When linearly inseparable, nonlinear problem is mapped to higher dimensional space, introduces Radial basis kernel function here and avoid calculating Nonlinear Mapping, effectively avoid dimension disaster problem, thus ensure that algorithm complex and sample characteristics Spatial Dimension have nothing to do.Conventional kernel function has Polynomial kernel function, Radial basis kernel function, Sigmoid kernel function.

A5, be symmetric positive semidefinite matrix by the known Ω of steps A 4, therefore Ω+y ²c ^-1i is symmetric positive definite matrix, so Matrix Solving has unique solution, namely $\left(\begin{array}{c}b\\ \mathrm{\&alpha;}\end{array}\right)={\left(\begin{array}{cc}0& I^T\\ I& \mathrm{\&Omega;}+y^2{C}^{-1}I\end{array}\right)}^{-1}\left(\begin{array}{c}\stackrel{\&RightArrow;}{0}\\ y\end{array}\right),$ Now can obtain parameter alpha and b.

Optimization problem can be converted into (5) formula by the Lagrangian function result of (3) formula be brought in (2) formula in addition:

$\begin{array}{c}\underset{\mathrm{\&alpha;}}{\max }\underset{i}{\&Sigma;}{\mathrm{\&alpha;}}_i{y}_i-\frac{1}{2C}\underset{i}{\&Sigma;}{\mathrm{\&alpha;}}_i^2{y}_i^2-\frac{1}{2}\underset{i,j}{\&Sigma;}{\mathrm{\&alpha;}}_i{\mathrm{\&alpha;}}_{j}K(x_i,x_j)\\ s.t.\underset{i}{\&Sigma;}{\mathrm{\&alpha;}}_i=0\end{array}---\left(5\right)$

A6, by steps A 5 Matrix Solving result and b ^*, the parameter of anti-forecast model can be obtained with

$\begin{array}{c}\stackrel{\&OverBar;}{\mathrm{\&omega;}}=\underset{i}{\&Sigma;}{\mathrm{\&alpha;}}_i^{*}\mathrm{\&phi;}\left(x_i\right)\\ \stackrel{\&OverBar;}{b}=b^{*}\end{array}---\left(6\right)$

A7, by will (6) formula acquisition parameter with bring into in can obtain forecast model:

$g\left(x\right)=\underset{i}{\&Sigma;}{\mathrm{\&alpha;}}_i^{*}K(x_i,x)+b^{*}---\left(7\right)$

(7) formula is the final anti-forecast model of optimum obtained.

The target of the variance sig2 in A8, particle swarm optimization algorithm optimization punishment parameter C and radial basis function minimizes relative error, and the fitness function of use is as shown in (8) formula:

$Fitness=max\left(\frac{1}{n}\underset{i=1}{\overset{n}{\&Sigma;}}\right|\frac{g\left(x_i\right)-y_i}{y_i}\left|\right)---\left(8\right)$

Embodiment 4

The process of the minimum support vector machine method of the maximum relative error based on particle swarm optimization algorithm is:

B1, population initialization, comprise population scale, random initializtion position and speed;

B2, evaluate the fitness of each particle according to formula (8) fitness function;

B3, calculate the speed of each particle and position formula is as shown in (9), (10) and (11)

$V_i^d:={\mathrm{wV}}_i^d+c_1*r_1({\mathrm{pbest}}_i^d-X_i^d)+c_2*r_2({\mathrm{gbest}}^d-X_i^d)---\left(9\right)$

$V_i^d:=\{\begin{array}{cc}{V}_{max},& V_i^d>V_{\max }\\ -V_{max},& V_i^d<-V_{\max }\end{array}---\left(10\right)$

$X_i^d:=X_i^d+V_i^d---\left(11\right)$

Wherein w is Inertia Weight, c ₁and c ₂speedup factor, r ₁and r ₂be the random number belonged between 0 to 1, speed is limited in maximal rate V by (10) formula _maxwith the m-V of minimum speed _max, represent the history optimal value of i-th particle in d dimension, gbest ^drepresent the history optimal value of population in d dimension;

B4, upgrade each particle and colony's history optimal location;

B5, judge whether to reach maximum iteration time or reach the end condition of best adaptive value, then enter step B6 if the judgment is Yes, then enter step B2 if the judgment is No;

B6, output optimizing parametric optimal solution, the punishment parameter C namely in formula (4) and the variance sig2 in Radial basis kernel function;

B7, the optimizing parametric results drawn by step B6 are brought in steps A 5, solve a and b by the matrix operation of steps A 5, finally obtain the optimum prediction model of formula (7).

Embodiment 5

The method utilizing this support vector regression to carry out the anti-prediction simultaneously of multiple concrete component comprises the steps:

C1, determine the concrete various ingredients that will predict, utilize the mixed earth historical data structure training sample data collection of actual measurement;

C2, pre-service is carried out to training sample data collection, namely delete training data and concentrate prediction group to be divided into the sample of zero;

C3, using the minimum multiple-input and multiple-output support vector regression of maximum relative error as fitness function, use particle swarm optimization algorithm parameter optimization is carried out to this support vector machine;

C4, the final optimizing parameter of use step C3 are to this multiple-input and multiple-output support vector regression modeling;

The anti-forecast model of multiple-input and multiple-output of C5, use step C4 is predicted the various ingredients that will predict is counter.

Above-described embodiment is only used for further illustrating a kind of maximum relative error minimum Multiinputoutput support vector regression that utilizes of the present invention and carries out the anti-method predicted of concrete polycomponent; but the present invention is not limited to embodiment; every above embodiment is done according to technical spirit of the present invention any simple modification, equivalent variations and modification, all fall in the protection domain of technical solution of the present invention.

Claims (3)

1. one kind utilizes maximum relative error minimum Multiinputoutput support vector regression to carry out the anti-method predicted of concrete polycomponent, it is characterized in that, maximum relative error based on particle swarm optimization algorithm minimum multiple-input and multiple-output support vector regression building method, comprises step:

A1, given training set for the input of M dimension, for N dimension exports;

A2, be optimized according to support vector machine maximize margin characteristic and maximum relative error minterm, optimization problem can be expressed as by formulism:

$\underset{\mathrm{\&omega;},b,e}{\min }\frac{1}{2}{\mathrm{\&omega;}}^T\mathrm{\&omega;}+\frac{1}{2}c\underset{i}{\&Sigma;}{e}_i^2$

(1)

$\begin{array}{cc}s.t.& e_i=\frac{{\mathrm{\&omega;}}^T\mathrm{\&phi;}\left(x_i\right)+b-y_i}{y_i},\&ForAll;i\end{array}$

Wherein punish parameter C>0, φ (x _i) be a function being mapped to higher dimensional space, ω is weight vectors, b=[b ¹..., b ⁿ] be biased, ω ^tthe transposition of ω,

A3, to establish introduce method of Lagrange multipliers and (1) formula be converted to Lagrangian function form:

$L(\mathrm{\&omega;},b,e;\mathrm{\&alpha;})=J(\mathrm{\&omega;},e)-\underset{i}{\&Sigma;}{\mathrm{\&alpha;}}_i\&lsqb;{\mathrm{\&omega;}}^T\mathrm{\&phi;}\left(x_i\right)+b+y_i{e}_i-y_i\&rsqb;---\left(2\right)$

Now optimization problem is converted to and has solved optimization problem, is converted to optimization problem again by antithesis and solves

A4, optimization problem for step 3, by introducing KKT condition, meeting under some conditions, and nonlinear programming problem has a necessary and sufficient condition of optimum solution to be:

$\{\begin{array}{c}\frac{\&part;L}{\&part;\mathrm{\&omega;}}=0\&RightArrow;\mathrm{\&omega;}=\underset{i}{\&Sigma;}{\mathrm{\&alpha;}}_i\mathrm{\&phi;}\left(x_i\right)\\ \frac{\&part;L}{\&part;b}=0\&RightArrow;-\underset{i}{\&Sigma;}{\mathrm{\&alpha;}}_i=0\\ \frac{\&part;L}{\&part;e_i}=0\&RightArrow;{\mathrm{Ce}}_i=y_i{\mathrm{\&alpha;}}_i\\ \frac{\&part;L}{\&part;{\mathrm{\&alpha;}}_i}=0\&RightArrow;y_i={\mathrm{\&omega;}}^T\mathrm{\&phi;}\left(x_i\right)+b+y_i{e}_i\end{array}---\left(3\right)$

By (3) formula cancellation ω and e _iafter can solve parameter with matrix form and be biased b and Lagrange multiplier α:

$\left(\begin{array}{cc}0& I^T\\ I& \mathrm{\&Omega;}+y^2{C}^{-1}I\end{array}\right)\left(\begin{array}{c}b\\ \mathrm{\&alpha;}\end{array}\right)=\left(\begin{array}{c}\stackrel{\&RightArrow;}{0}\\ y\end{array}\right)---\left(4\right)$

Wherein y=[y ₁..., y _n] ^t, α=[α ₁..., α _n] ^t, Ω=(Ω _ij) _{n × n}=K (x _i, x _j), K (x here _i, x _j) be Radial basis kernel function, I=[1 ..., 1] ^t, b=[b ¹..., b ⁿ], $\stackrel{\&RightArrow;}{0}=\&lsqb;0^1,\mathrm{...},0^N\&rsqb;;$

A5, be symmetric positive semidefinite matrix by the known Ω of steps A 4, therefore Ω+y ²c ^-1i is symmetric positive definite matrix, so Matrix Solving has unique solution, namely $\left(\begin{array}{c}b\\ \mathrm{\&alpha;}\end{array}\right)={\left(\begin{array}{cc}0& I^T\\ I& \mathrm{\&Omega;}+y^2{C}^{-1}I\end{array}\right)}^{-1}\left(\begin{array}{c}\stackrel{\&RightArrow;}{0}\\ y\end{array}\right),$ Now can obtain parameter alpha and b.

Optimization problem can be converted into (5) formula by the Lagrangian function result of (3) formula be brought in (2) formula in addition:

$\underset{\mathrm{\&alpha;}}{\max }\underset{i}{\&Sigma;}{\mathrm{\&alpha;}}_i{y}_i-\frac{1}{2C}\underset{i}{\&Sigma;}{\mathrm{\&alpha;}}_i^2{y}_i^2-\frac{1}{2}\underset{i,j}{\&Sigma;}{\mathrm{\&alpha;}}_i{\mathrm{\&alpha;}}_{j}K(x_i,x_j)$

$\begin{array}{cc}s.t.& \underset{i}{\mathrm{\&Sigma;}}{\mathrm{\&alpha;}}_i=0\end{array}$

A6, by steps A 5 Matrix Solving result and b ^*, the parameter of anti-forecast model can be obtained with

$\stackrel{\&OverBar;}{\mathrm{\&omega;}}=\underset{i}{\&Sigma;}{\mathrm{\&alpha;}}_{\text{i}}^{\text{*}}\mathrm{\&phi;}\left(x_i\right)\text{---}\left(6\right)$

$\stackrel{\&OverBar;}{b}=b^{\text{*}}$

A7, by will (6) formula acquisition parameter with bring into in can obtain forecast model:

$g\left(x\right)=\underset{i}{\&Sigma;}{\mathrm{\&alpha;}}_i^{*}K(x_i,x)+b^{*}---\left(7\right)$

(7) formula is the final anti-forecast model of optimum obtained.

The target of the variance sig2 in A8, particle swarm optimization algorithm optimization punishment parameter C and radial basis function minimizes maximum relative error, and the fitness function of use is as shown in (8) formula:

$Fitnss=max\left(\frac{1}{n}\underset{i=1}{\overset{n}{\&Sigma;}}\right|\frac{g\left(x_i\right)-y_i}{y_i}\left|\right)---\left(8\right)$

2. a kind of maximum relative error minimum Multiinputoutput support vector regression that utilizes according to claim 1 carries out the anti-method predicted of concrete polycomponent, it is characterized in that: the process of the minimum support vector machine method of the maximum relative error based on particle swarm optimization algorithm is:

B1, population initialization, comprise population scale, random initializtion position and speed;

B2, evaluate the fitness of each particle according to formula (8) fitness function;

B3, calculate the speed of each particle and position formula is as shown in (9), (10) and (11)

$V_i^d:={\mathrm{wV}}_i^d+c_1*-r_1({\mathrm{pbest}}_i^d-X_i^d)+c_2*-r_2({\mathrm{gbest}}^d-X_i^d)---\left(9\right)$

$V_i^d:=\left\{\begin{array}{cc}{V}_{max},& V_i^d>V_{max}\\ -V_{max},& V_i^d<-V_{max}\end{array}\right.\text{---}\left(\text{10}\right)$

$X_i^d:=X_i^d+V_i^d\text{---}\left(11\right)$

Wherein w is Inertia Weight, c ₁and c ₂speedup factor, r ₁and r ₂be the random number belonged between 0 to 1, speed is limited in maximal rate V by (10) formula _maxwith the m-V of minimum speed _max, represent the history optimal value of i-th particle in d dimension, gbest ^drepresent the history optimal value of population in d dimension;

B4, upgrade each particle and colony's history optimal location;

B5, judge whether to reach maximum iteration time or reach the end condition of best adaptive value, then enter step B6 if the judgment is Yes, then enter step B2 if the judgment is No;

B6, output optimizing parametric optimal solution, the punishment parameter C namely in formula (4) and the variance sig2 in Radial basis kernel function;

B7, the optimizing parametric results drawn by step B6 are brought in steps A 5, solve α and b by the matrix operation of steps A 5, finally obtain the optimum prediction model of formula (7).

3. a kind of maximum relative error minimum Multiinputoutput support vector regression that utilizes according to claim 1 carries out the anti-method predicted of concrete polycomponent, it is characterized in that: the method utilizing this support vector regression to carry out the anti-prediction simultaneously of multiple concrete component comprises the steps:

C1, determine the concrete various ingredients that will predict, utilize the mixed earth historical data structure training sample data collection of actual measurement;

C2, pre-service is carried out to training sample data collection, namely delete training data and concentrate prediction group to be divided into the sample of zero;

C3, using the minimum multiple-input and multiple-output support vector regression of maximum relative error as fitness function, use particle swarm optimization algorithm parameter optimization is carried out to this support vector machine;

C4, the final optimizing parameter of use step C3 are to this multiple-input and multiple-output support vector regression modeling;

The anti-forecast model of multiple-input and multiple-output of C5, use step C4 is predicted the various ingredients that will predict is counter.