CN105824994A - Zoning-based combined approximate model building method - Google Patents

Abstract

The invention relates to a zoning-based combined approximate model building method which has high adaptability to unknown linear and nonlinear problems. The zoning-based combined approximate model building method is characterized in that a combined approximate model is built under the condition that the input values and the output values of a certain number groups are given to approximate the input and output analytic relation of a system, and the output of the system is forecasted under the condition that the input of the system is given. The zoning-based combined approximate model building method is mainly applied to approximation of the functional relation between optimization objectives and variables during optimal design. As for the problem that the strong linear or nonlinear features of local areas cannot be well captured because traditional approximate models adopt modeling on the whole test area, the concept of modeling by dividing the whole test area is adopted. On one hand, the precision forecasting advantage of the combined approximate model to a single approximate model is utilized, and on the other hand, the purpose of improving the forecasting precision to the local area is achieved, so that the overall forecasting precision of the model is improved. Therefore, the computing time and resources, consumed by iterative optimal solution solving, can be greatly shortened and reduced in the optimization process.

Description

A kind of combination approximation model building method based on zoning

Technical field

The present invention relates to the method for building up of a kind of combination approximation model that linearity and non-linearity unknown problem is had high-adaptability, the method is to come input and the output analytical relation of approximation system in the case of the input of given some groups with output valve by setting up combination approximation model, and in the case of known system inputs, the output to system provides predictive value.It is mainly used in optimizing when designing the approximation of functional relationship between optimization aim and variable.

Background technology

Optimize in the every field that design has been widely used in industry.In optimization problem, parsing relation engineering staff often between optimization aim and optimized variable cannot directly obtain during optimizing, therefore, by arranging a number of sample point in given pilot region, gather variable (input) and target (output) value of some groups, and then use the method for difference or matching to set up approximate model to the approximate relation providing between optimization aim and optimized variable.So, solve optimal value by pairing approximation relational expression, provide optimizing design scheme.Visible, the precision of prediction (difference between predictive value and actual value) of this approximate model can produce great impact to optimum results.In simple terms, approximate model precision of prediction is the highest, and the iterative steps finding optimal solution will be the fewest, thus can save substantial amounts of calculating time and computational costs.

Relation between optimization aim and optimized variable is approximated only with single approximate model in whole pilot region by many scholars when being optimized design.But, single approximate model is not to obtain higher precision of prediction when process has linear and nonlinear characteristic, higher-dimension and low-dimensional problem.This is accomplished by us and the several approximate models having corresponding advantage for different types of problem is combined, and precision of prediction is higher, stability (can obtain the ability of higher forecasting precision for different types of problem) preferably combination approximation model to obtain to use average weighted mode.

But, this combination approximation model set up for whole pilot region yet suffers from certain defect.Such as, in whole pilot region, the relation between optimization aim and optimized variable is based on linear character, but some regional area present the strongest non-linear or in whole pilot region relation between optimization aim and optimized variable based on stronger nonlinear characteristic, but present linear character at borderline region or some regional area.So, if still setting up combination approximation model for whole pilot region, it will make the model set up be substantially reduced at the precision of prediction of these regional areas.For such situation, the present invention proposes the combination approximation model building method of a kind of zoning, on the one hand the combination approximation model precision of prediction advantage to single approximate model has been used for reference, on the other hand it is intended to the precision of prediction improving combination approximation model for regional area, and then improves the global prediction precision of model.

Summary of the invention

The construction method of combination approximation model:

(1) basic submodel is set up

First, the present invention have chosen two submodels, is multinomial model and RBF model respectively.It is intended to utilize multinomial model to have higher forecasting precision for low order linear and nonlinear problem and RBF model has the advantage of higher forecasting precision for high-order nonlinear problem.

A. multinomial model

For simplicity while ensureing approximation quality, the present invention selects quadratic polynomial model.Y (x) is made to represent real receptance function,Being its approximate function, quadratic polynomial can be expressed as:

$\hat{y}={\mathrm{\β}}_0+\underset{i=1}{\overset{m}{\Σ}}{\mathrm{\β}}_i{x}_i+\underset{i=1}{\overset{m}{\Σ}}{\mathrm{\β}}_{ii}{x}_i^2+\underset{i=1}{\overset{m-1}{\Σ}}\underset{j=i+1}{\overset{m}{\Σ}}{\mathrm{\β}}_{ij}{x}_i{x}_j$

Wherein x_iAnd x_jRepresenting m variable, β represents unknowm coefficient.Arranging n sample point in pilot region, correspondence will produce n system response value therewith.

B. RBF model

RBF model is initially used to dispersion multivariate data is carried out interpolation.This method uses a series of almost symmetry Radial Equations to carry out approximate target receptance function.Y (x) is made to represent real receptance function,Being its approximate function, RBF model representation is as follows:

$\hat{y}\left(x\right)=\underset{i=1}{\overset{n}{\Σ}}{a}_i\mathrm{\φ}(||x-x_i||)$

Wherein n represents sample point sum, and x represents independent variable vector, x_iRepresent the i-th sample point in independent variable vector, | | x-x_i| | represent the Euclidean distance between two vectors.φ represents basic function, a_iRepresent unknowm coefficient.Basic function equation selection Gaussian bases in the present invention:

$\mathrm{\&phi;}(||x-x_i||)=e^{-c{||x-x_i||}^2},0<c\&le;1$

(2) n-quadrant crosscheck technology

Whole pilot region is divided into N number of subregion, and the most all of sample point is also correspondingly assigned in every sub regions.Sample point in each subregion is not involved in the approximate model in this region and sets up and be served only for weighing the precision of prediction of the approximate model that one's respective area is set up.Precision of prediction uses residual error to measure and be defined as the deviation between predictive value and actual value:

$R_i={\hat{y}}_i-y_i$

Residual error characterizes the deviation between predictive value and the true response value of system.Residual error is the least, and model prediction accuracy is the highest.

Such as, such as Fig. 1, whole pilot region is divided into 4 sub regions.Only with the sample point in region 2,3 and 4 when setting up approximate model 1, and the sample point in region 1 is served only for checking the precision of prediction of approximate model 1.

(3) L parameter is set up

In every sub regions, we cannot learn that the functional relationship between variable and target presents linear or nonlinear characteristic.In order to make full use of the advantage of submodel in step 1, in every sub regions, it is modeled with multinomial model and RBF model respectively.Then the higher submodel of precision of prediction is chosen to set up the approximate model of this subregion.To this end, developed L parameter and judged the precision of prediction of approximate model with it.L parameter is defined as follows:

$PESS=\underset{i=1}{\overset{q}{\&Sigma;}}{(y_i-{\hat{y}}_i)}^2$

$PSST=\underset{i=1}{\overset{q}{\&Sigma;}}{(y_i-\stackrel{\&OverBar;}{y})}^2$

$L=\frac{\sqrt{\frac{1}{q}\underset{i=1}{\overset{q}{\&Sigma;}}{(y_i-{\hat{y}}_i)}^2}}{\sqrt{\frac{1}{q}\underset{i=1}{\overset{q}{\&Sigma;}}{(y_i-\stackrel{\&OverBar;}{y})}^2}}$

Wherein q represents the sum of check point.The value of L parameter is the least, and the global prediction precision of approximate model is the highest.

(4) combination approximation model is set up

After respectively each subregion is set up approximate model, the approximate model that every sub regions is set up is needed to constitute a combination approximation model by average weighted mode.Weight coefficient can determine according to the precision of prediction height of approximate model in each subregion.In this case, it was predicted that the model that precision is high can have bigger weight coefficient, otherwise, it was predicted that the weight coefficient of the model that precision is low can be less.Final combination approximation model can be expressed as:

${\hat{y}}_{avg}^{^}=\underset{i=1}{\overset{N}{\&Sigma;}}{w}_i{y}_i$

The sum of the subregion divided during wherein N represents whole pilot region；w_iRepresent weight coefficient；Represent combination approximation model,Represent the approximate model that i-th subregion is set up.Weight coefficient meets following relation:

$\underset{i=1}{\overset{N}{\&Sigma;}}{w}_i=1$

Weight coefficient is that the L parameter value of the approximate model set up according to each subregion determines:

$w_i=\frac{1-(L_i/\underset{i=1}{\overset{N}{\&Sigma;}}{L}_i)}{N-1}$

Wherein L_iRepresent the L parameter value of the approximate model of i-th subregion foundation.

Accompanying drawing explanation

Fig. 1 is to the division of whole pilot region and schematic diagram how to set up approximate model in subregion 1 in the present invention；

The image of tri-kinds of typical test functions of Fig. 2；

Detailed description of the invention

Table 1 lists three kinds of typical test function analytical relations.They mainly show linearly, lower order nonlinear and high-order nonlinear feature (such as Fig. 2).Three kinds of test functions are for checking method proposed by the invention.

1 three kinds of typical test functions of table

For each test function, in pilot region, uniform experiment design method is used to arrange 24 sample points.Meanwhile, it is divided into 4 parts, every part to comprise 6 sample points whole pilot region.In order to check the precision of prediction of combination approximate model, in pilot region, arrange 100 check points.Sample point quantity and pilot region are as shown in table 2.

Table 2 sample point quantity and pilot region

According to the construction method of combination approximation model in summary of the invention, whole pilot region is divided into 4 sub regions, sets up combination approximation model with multinomial model and RBF model for submodel.Meanwhile, for relative analysis, setting up again two approximate models, one of them approximate model carries out subregion modeling only with multinomial model, and another approximate model carries out subregion modeling only with RBF model.Meanwhile, having research display, the precision of prediction using single submodel to carry out the approximate model obtained by two-zone model is higher than the precision of prediction of the approximate model using single submodel obtained by whole pilot region models.

(1) linear function

For linear function problem, the precision of prediction comparing result of three kinds of approximate models is as shown in table 3.Result shows: for linear function problem, and the precision of prediction of the combination approximation model constructed by the present invention is higher than the precision of prediction carrying out approximate model obtained by two-zone model only with single submodel.Therefore can learn further, the combination approximation model constructed by the present invention can improve the precision of prediction for linear function problem further.

Prediction effect contrast (linear function) of table 3 approximate model

The wherein definition of NMAE:

$MAE=\underset{i=1\&RightArrow;q}{max}\left|y_i-{\hat{y}}_i\right|$

$NMAE=\frac{\underset{i=1\&RightArrow;q}{max}\left|y_i-{\hat{y}}_i\right|}{\sqrt{\frac{1}{q}\underset{i=1}{\overset{q}{\&Sigma;}}{(y_i-\stackrel{\&OverBar;}{y})}^2}}$

Wherein NMAE value is the least, and the local prediction precision of approximate model is the highest.

(2) lower order nonlinear function

For lower order nonlinear function problem, the precision of prediction comparing result of three kinds of approximate models is as shown in table 4.Result shows: for lower order nonlinear function problem, and the precision of prediction of the combination approximation model constructed by the present invention is higher than the precision of prediction carrying out approximate model obtained by two-zone model only with single submodel.

Prediction effect contrast (lower order nonlinear function) of table 4 approximate model

(3) high-order nonlinear function

For high-order nonlinear function problem, the precision of prediction comparing result of three kinds of approximate models is as shown in table 5.Result shows: for high-order nonlinear function problem, and the precision of prediction of the combination approximation model constructed by the present invention is higher than (being at least not less than) and carries out the precision of prediction of approximate model obtained by two-zone model only with single submodel.

Prediction effect contrast (high-order nonlinear function) of table 5 approximate model

Sum up:

For linear problem, lower order nonlinear problem and high-order nonlinear problem, the precision of prediction of the combination approximation model constructed by the present invention is higher than the precision of prediction carrying out approximate model obtained by two-zone model only with single submodel.Further, the precision of prediction of the combination approximation model constructed by the present invention is higher than the precision of prediction of traditional approximate model only with single submodel obtained by whole pilot region models.Meanwhile, the combination approximation model constructed by the present invention all has higher precision of prediction for linear problem, lower order nonlinear problem and high-order nonlinear problem and illustrates that this more traditional approximate model of combination approximation model has more preferable stability.

Claims (1)

1. the combination approximation model building method that linearity and non-linearity unknown problem is had high-adaptability based on zoning, it is characterised in that:

(1) propose one and linear character and nonlinear characteristic are all had preferable adaptive combination approximation method for establishing model；

(2) this method overcomes interpolation model and must be introduced into the shortcoming that extra sample point (cannot use the sample point used when setting up model) carrys out testing model precision of prediction；

(3) in whole trial zone, the input of system only may present linear or nonlinear characteristic at regional area with output (response), thus whole pilot region is divided into N number of subregion, this method can make full use of and constitute the basic approximate model of combination approximation model for subregion internal linear feature or the advantage of the higher forecasting precision of nonlinear characteristic；

(4) this combination approximation model is for the system of unknown function analytic expression, the most uncertain analytic expression be linear or nonlinear characteristic account for leading or in whole pilot region regional area nonlinear characteristic account for the situations such as leading, there is well adapting to property and higher precision of prediction；

(5) the basic approximate model constituting combination approximation model can be not limited to multinomial model and the RBF mould that this patent uses, and can be extended to any other basic approximate model (artificial nerve network model and Kriging model etc.) according to concrete requirement of engineering.And, the quantity of the basic approximate model constituting combination approximation model freely can also be selected by engineering staff according to concrete engineering problem.