CN104615810A - Simulation model verification method based on functional data analysis - Google Patents

Abstract

The invention provides a simulation model verification method based on functional data analysis, and belongs to the field of simulation model verification. According to the method, output of a simulation model and output of a real system are regarded as two random processes, weighting functions are added to construct aggregative indicators, and the correlation coefficient of the two aggregative indicators is used for describing the output correlation of the simulation model and the real system. The method includes the steps that simulation data and real data are recorded first, data fitting is conducted, then a set of primary functions are selected, the fitting function and the weighting functions are expanded through the same base, and an equation is constructed to solve the correlation coefficient. According to the simulation model verification method, the condition limits in a traditional method can be avoided, the number of depending assumed conditions is small, the depending structural constraints are weak, the model estimability is improved, data analysis support is provided for simulation model verification, and it is guaranteed that the simulation model verification work can be conducted smoothly.

Description

A kind of Methods of Validating Simulation Model based on functional data analysis

Technical field

The invention belongs to Validation of Simulation Models field, be specifically related to a kind of Methods of Validating Simulation Model based on functional data analysis.

Background technology

Validation of Simulation Models, namely to certain emulation object, confirms that model behavior characteristic and system action Character Comparison precision meet the demands.Namely input-the output transform of input-output transform with enough accuracy representative system of model is confirmed.Validation of Simulation Models principle comprises two contents: detection model contrast precision and prove that this precision meets the requirement emulating object and specify with the behavioral trait of system.The checking of realistic model ensures the believable important method of analogue system, wherein export checking mainly through investigating under identical initial conditions, realistic model Output rusults and the degree of consistency whether consistent with real system Output rusults, this is the key areas of Validation of Simulation Models.

Just meeting the demands with to emulate object relevant with how high the contrast precision between the behavioral trait of system is of model, and be difficult to determine.Because during emulation, it is generally the overall characteristic of not knowing system.Although whether principle can meet after system cloud gray model the requirement of emulation object with the former simulation result of its product test.If meet requirement, then precision is contrasted to the behavioral trait of model and system and do actual observation record.This can be recorded as foundation when verifying the realistic model of homogeneous system by same emulation object later.But due to the type of system a lot, emulation object is also diversified, need the actual observation record done should be almost countless, and this is actually does not accomplish.Model is the abstract of system: retain main, ignores secondary.Which is main, and which is secondary is relative, depends on emulation object.So-called emulation object refer to want by emulation make about system which kind of decision-making or draw which kind of conclusion.The same with done by system experimentation result of this decision-making or conclusion.Such as: the capacity having the buffering of a cigarette automatic production line to deposit cigarette device has impact to output.In order to determine the impact damper optimum capacity that output can be made the highest, test with realistic model.The decision of cigarette device amount of capacity is deposited in the transformation made according to simulation result should with to make to test the conclusion drawn on automatic assembly line consistent.

Conventional data analysis method mainly comprises hypothesis test, time-domain analysis, frequency-domain analysis etc.Though verification method is a lot, there is certain limitation.The character of studied system should be considered and can be obtained it export the method that the factors such as the how many and emulation object of data select one or more suitable.If can obtain the sample that system exports, then more rational verification method is interval estimation method.Unfortunate this method only considered the randomness of problem, to interval limit value low to which kind of degree could meet emulation object this insoluble problem of requirement do not discuss.And null hypothesis method of inspection is opposing with emulation ultimate principle in general.Its assumed condition is higher than real needs in other words conj.or perhaps.Application conventional data analysis method can run into a lot of restriction, as: (1) emulated data must be identical with the leading zero's of True Data, otherwise invalid; (2) emulated data must be identical with the sampling interval of True Data, otherwise invalid; (3) emulated data must be identical with the data length of True Data, otherwise invalid; (4) a large amount of multivariate data efficiency of conventional data analysis method to the generation of process Complex simulation systems are not high.In addition, require that when carrying out data analysis data analyst has abundant knowledge to judge and will select what analytical approach to process, and proposes very high requirement to data analyst.Therefore, need new Methods of Validating Simulation Model to overcome available data analytical approach Problems existing.

Functional data analysis method is the common method of data analysis, is all widely used and wide prospect in many fields such as meteorology, biomechanics, economics, medical science.

Summary of the invention

For conventional data analysis method Problems existing, the present invention by functional data analysis method, provides data analysis to support to Validation of Simulation Models, guarantees Validation of Simulation Models everything goes well with your work to carry out.

The method that Validation of Simulation Models is the most basic, investigates exactly under identical initial conditions, and realistic model exports and real system exports whether consistent and conforming degree, namely carries out consistency check or consistency check to both.When applying conventional data analysis method, often require that emulated data and True Data meet the conforming requirement of time series, the data collected in practice often do not meet this type of condition.Data fitting is function by functional data analysis method, processes from the angle of function data, avoids similar condition restriction.Therefore the present invention proposes utility function type data correlation analysis method and carries out modelling verification.

The present invention proposes a kind of Methods of Validating Simulation Model based on functional data analysis, specifically comprises the following steps:

Step one: regard the output of realistic model and the output of real system as two stochastic processes, the N group recording realistic model and real system respectively exports data, if X _irepresent emulated data, Y _irepresent True Data, i=1,2 ..., N.Two weighting function ξ and η are set, in order to construct overall target z and w, describe the output correlativity of realistic model and real system by the related coefficient of two overall targets.

Step 2: emulated data and True Data are fitted to functional form respectively.

Step 3: choose one group of basis function φ ₁, φ ₂..., φ _m, X _i, Y _ifitting function and weighting function ξ, η all use this base function expansion.

Step 4: establish C, D is respectively X _i, Y _ithe expansion coefficient matrix of fitting function, a, b represent the expansion coefficient vector of weighting function respectively.Structural matrix K, wherein K (i, j)=< D ²φ _i, D ²φ _j>.Structural matrix J, wherein J (i, j)=< φ _i, φ _j>, if basis function is orthogonal basis, then J is unit battle array.Definition V ₁₁, V ₁₂, V ₂₂be respectively sample variance and covariance matrix, wherein

$V_{11}(v,p)=N^{-1}\underset{i}{\mathrm{\&Sigma;}}{c}_{\mathrm{iv}}{c}_{\mathrm{ip}},V_{12}=N^{-1}\underset{i}{\mathrm{\&Sigma;}}{c}_{\mathrm{iv}}{d}_{\mathrm{ip}},V_{22}=N^{-1}\underset{i}{\mathrm{\&Sigma;}}{d}_{\mathrm{iv}}{,V}_{22}=N^{-1}\underset{i}{\mathrm{\&Sigma;}}{d}_{\mathrm{iv}}{d}_{\mathrm{ip}}.$

Step 5: be constructed as follows equation:

$\left[\begin{array}{cc}0& {\mathrm{JV}}_{12}J\\ {\mathrm{JV}}_{21}J& 0\end{array}\right]\left[\begin{array}{c}a\\ b\end{array}\right]=\mathrm{\&rho;}\left[\begin{array}{cc}{\mathrm{JV}}_{11}J+{\mathrm{\&lambda;}}_{1}K& 0\\ 0& {\mathrm{JV}}_{22}J+{\mathrm{\&lambda;}}_{2}K\end{array}\right]\left[\begin{array}{c}a\\ b\end{array}\right]$

Solve this equation, obtain the solution of correlation coefficient ρ.

Step 6: wherein the maximal value of ρ is ρ _max, ρ _maxfor the related coefficient that realistic model exports and real system exports, according to ρ _maxclose to the degree of 1, draw two groups of similarities exporting data, thus realize the confidence level quantitatively judging this realistic model according to related coefficient.

Methods of Validating Simulation Model based on functional data analysis provided by the invention, has the following advantages and good effect:

(1) Methods of Validating Simulation Model of the present invention, can process the data of unlimited dimensional space (i.e. function type data), improve the packet content to modelling verification;

(2) Methods of Validating Simulation Model of the present invention, carries out the data analysing method of modelling verification compared to tradition, the present invention's functional data analysis used method relies on less assumed condition and more weak structural constraint, improves the estimability of model;

(3) Methods of Validating Simulation Model of the present invention, does not require that the data observation of different object of observation point is identical with observation frequency;

(4) Methods of Validating Simulation Model of the present invention, by the more important information of data can be excavated to the analysis of derivative curve or differential curve, such as by the difference between the investigative analysis curve to single order or higher derivative curve and curvilinear inner dynamic pattern etc., thus can the relation of more precise verification realistic model and true model.

Accompanying drawing explanation

Fig. 1 is the process flow diagram of the Methods of Validating Simulation Model based on functional data analysis of the present invention;

Fig. 2 is emulated data broken line graph in the embodiment of the present invention;

Fig. 3 is True Data broken line graph in the embodiment of the present invention;

Fig. 4 is emulated data fitting result schematic diagram in the embodiment of the present invention;

Fig. 5 is True Data fitting result schematic diagram in the embodiment of the present invention.

Embodiment

Below in conjunction with accompanying drawing and example, the present invention is described in further detail.

N observation is carried out in certain finite time interval T to two functions X, Y, obtains N to observation curve (X _i, Y _i), i=1,2 ... N.X, Y can be regarded as two vectors with unlimited dimension, the correlativity of X, Y is described, two vectors are needed comprehensively to become two overall targets, by the related coefficient of two overall targets, the correlativity between these two groups of variablees is described, find the weighting function ξ of two functions, η, constructs two overall targets:

Z=< X, ξ >=∫ X (t) ξ (t) dt and w=< Y, η >=∫ Y (t) η (t) dt.

The sample variance of X, Y and Semivariogram curve are as shown in formula (1), (2) and (3):

$v_{11}(x,t)=N^{-1}\mathrm{\&Sigma;}(X_i\left(s\right)-\stackrel{\&OverBar;}{X})(X_i\left(t\right)-\stackrel{\&OverBar;}{X})---\left(1\right)$

$v_{22}(s,t)=N^{-1}\mathrm{\&Sigma;}(Y_i\left(s\right)-\stackrel{\&OverBar;}{Y})(Y_i\left(t\right)-\stackrel{\&OverBar;}{Y})---\left(2\right)$

$v_{12}(s,t)=N^{-1}\mathrm{\&Sigma;}(X_i\left(s\right)-\stackrel{\&OverBar;}{X})(Y_i\left(t\right)-\stackrel{\&OverBar;}{Y})---\left(3\right)$

V ₁₁the sample variance that (s, t) is X, for the sample average of X, X _i(s) and X _it () is the different samples in X; v ₂₂the sample variance that (s, t) is Y, for the sample average of Y, Y _i(s) and Y _it () is the different samples in Y; v ₁₂the covariance that (s, t) is X and Y.s、t∈T。

The variance Var (z) of index z is as shown in formula (4):

Var(z)＝∫[ξ(s)∫v ₁₁(s,t)ξ(t)dt]ds (4)

The variance of index w is as shown in formula (5):

Var(w)＝∫[η(s)∫v ₂₂(s,t)η(t)dt]ds (5)

The covariance Cov (z, w) of index z and index w is as shown in formula (6):

Cov(z,w)＝∫[ξ(s)∫v ₁₂(s,t)η(t)dt]ds (6)

For the ease of representing, define corresponding operator V ₁₁, V ₂₂and V ₁₂if f represents the function of a time t, V ₁₁f is expressed as: correspondingly define V ₂₂, V ₁₂.The related coefficient of variable z, w square with ρ ²(z, w) represents, as shown in formula (7).

${\mathrm{\&rho;}}^2(z,w)=\frac{{\&lang;\mathrm{\&xi;}{v}_{12}\mathrm{\&eta;}\&rang;}^2}{\&lang;\mathrm{\&xi;},V_{11}\mathrm{\&xi;}\&rang;\&lang;\mathrm{\&eta;},V_{22}\mathrm{\&eta;}\&rang;}---\left(7\right)$

Above formula is equal to the following maximization problems solved under constraint condition, as shown in formula (8):

max＜ξ,V ₁₂η＞

(8)

s.t.＜ξ,V ₁₁＞＝＜η,V ₂₂η＞＝1

Can add coarse penalty term in constraint condition, then formula (8) can be converted to formula (9):

max＜ξ,V ₁₂η＞

(9)

s.t.＜ξ,V ₁₁＞+λ ₁||D ²ξ||＝＜η,V ₂₂η＞+λ ₂||D ²η||＝1

Wherein, λ ₁, λ ₂represent auxiliary undetermined coefficient, the expansion coefficient matrix of D representative function Y.

Further derivation, can obtain formula (10) as follows:

max＜ξ,V ¹²η＞

(10)

s.t.＜ξ,(V ₁₁+λ ₁D ⁴)ξ＞＝＜η,(V ₂₂+λ ₂D ⁴)η＞＝1

G is as follows for structure Lagrangian function:

G＝<ξ,V ₁₂η>-α(<ξ,(V ₁₁+λ ₁D ⁴)ξ>-1)-β(<η,(V ₂₂+λ ₂D ⁴)η>-1) (11)

Wherein, auxiliary coefficient when α and β is structure Lagrangian function.

Respectively to ξ, η differentiate, and make partial derivative equal zero, after arrangement, obtain the system of equations such as formula (12):

V ₁₂η＝ρ(V ₁₁+λ ₁D ₄)ξ

(12)

V ₂₁ξ＝ρ(V ₂₂+λ ₂D ₄)η

Correlation coefficient ρ has been there is in system of equations.

The system of equations of formula (12) is converted to matrix equation:

$\left(\begin{array}{cc}0& V_{12}\\ V_{21}& 0\end{array}\right)\left(\begin{array}{c}\mathrm{\&xi;}\\ \mathrm{\&eta;}\end{array}\right)=\mathrm{\&rho;}\left(\begin{array}{cc}{V}_{11}+{\mathrm{\&lambda;}}_1{D}^4& 0\\ 0& V_{22}+{\mathrm{\&lambda;}}_2{D}_4\end{array}\right)\left(\begin{array}{c}\mathrm{\&xi;}\\ \mathrm{\&eta;}\end{array}\right)---\left(13\right)$

The related coefficient of z, w wants the correlationship that can fully reflect between X, Y, so should get correlation coefficient ρ maximum two weighting functions ξ, η, makes ρ ²(z, w) maximizes, and asks for the maximal value of ρ, obtain the related coefficient of two groups of function curves, the span of correlation coefficient ρ is [-1,1], when related coefficient span is [0,1] time, two function positive correlations are described, value is larger, and the degree of correlation is higher, when value is 1, show that two groups of SYSTEM OF LINEAR VECTOR are correlated with.In like manner, when related coefficient value is [-1,0], two groups of vector negative correlation.When value is-1, two groups of vector reverse linears are correlated with.

The Methods of Validating Simulation Model based on functional data analysis that the present invention proposes, regard the output of realistic model and the output of real system as two stochastic processes, add weighting function structure overall target, describe the output correlativity of realistic model and real system by the related coefficient of two overall targets.As shown in Figure 1, specifically comprise the following steps:

Step one: register system exports data.

Regard the output of realistic model and the output of real system as two stochastic processes, the many groups of output data recording realistic model and real system are respectively analyzed.

The output data in N number of moment of realistic model and real system are designated as X _i, Y _i, i=1,2 ..., N, X _irepresent the output data in realistic model i moment, be also called emulated data, Y _irepresent the output data in real system i moment, be also called True Data, note matrix P=(X _i, Y _i), note weighting function is ξ, η.

Step 2: emulated data and True Data are fitted to functional form respectively.

Function Fitting has multiple approximating method, searches out the minimum approximating method of a kind of fitness bias by analyzing.

Step 3: choose one group of basis function, if φ ₁, φ ₂..., φ _mbe one group of suitable basis function, m represents basis function number.To X _i, Y _ifitting function and the same base function expansion of weighting function ξ, η.

If X _ithe simulation curve function of matching is according to obtaining a result after this group base function expansion as f ₁(x), f ₂(x) ..., f _n(x), Y _ithe real curve function of matching is according to obtaining a result after this group base function expansion as g ₁(x), g ₂(x) ..., g _n(x).

Step 4: structural matrix K, wherein K (i, j)=< D ²φ _i, D ²φ _j>.(the i-th row jth column element in (i, j) representing matrix) structural matrix J, wherein J (i, j)=< φ _i, φ _j>, if basis function is orthogonal basis, then J is unit battle array.In addition, define C, D and be respectively X _i, Y _iexpansion coefficient matrix, have with a, b difference representative function ξ, the expansion coefficient vector of η.Matrix C and D are the capable N column matrix of N.

Definition V ₁₁the sample variance matrix of representing matrix C, V ₂₂the sample variance matrix of representing matrix D, V ₁₂the covariance matrix of representing matrix C and D; Specific as follows:

$V_{11}(v,p)=N^{-1}\underset{i}{\mathrm{\&Sigma;}}{c}_{\mathrm{iv}}{c}_{\mathrm{ip}},V_{12}(v,p)=N^{-1}\underset{i}{\mathrm{\&Sigma;}}{c}_{\mathrm{iv}}{d}_{\mathrm{ip}},V_{22}(v,p)=N^{-1}\underset{i}{\mathrm{\&Sigma;}}{d}_{\mathrm{iv}}{d}_{\mathrm{ip}}$

Wherein, (v, p) represents that being positioned at the capable p of matrix v arranges; c _iv, c _ipv and p element of the i-th row in difference representing matrix C, d _iv, d _ipi-th row v and p element in representing matrix D.

Step 5: be constructed as follows equation:

$\left[\begin{array}{cc}0& {\mathrm{JV}}_{12}J\\ {\mathrm{JV}}_{21}J& 0\end{array}\right]\left[\begin{array}{c}a\\ b\end{array}\right]=\mathrm{\&rho;}\left[\begin{array}{cc}{\mathrm{JV}}_{11}J+{\mathrm{\&lambda;}}_{1}K& 0\\ 0& {\mathrm{JV}}_{22}J+{\mathrm{\&lambda;}}_{2}K\end{array}\right]\left[\begin{array}{c}a\\ b\end{array}\right]---\left(14\right)$

Solve this equation, the solution of the correlation coefficient ρ obtained.

With a, b difference representative function ξ, the expansion coefficient vector of η, obtains equation (14) to equation (13) through linear transformation.

Step 6: the maximal value obtaining ρ is ρ _max, be the related coefficient that realistic model exports and real system exports, according to ρ _maxclose to the degree of 1, draw the similarity of two groups of data, and then the confidence level of this realistic model can be quantitatively judged according to related coefficient.

Embodiment: the present invention have chosen certain flight simulator model as identifying object, according to the analysis process of Validation of Simulation Models automated analysis instrument, verifies the crucial output parameter displacement of certain flight simulation and speed respectively.Have employed corresponding data preprocessing method, model verification method, automatic partition analysis method, instance analysis is carried out to the implementation procedure of Validation of Simulation Models automated analysis instrument and validity thereof.

Step one: be function by data fitting with Basis Function Method.

Be function with Basis Function Method by data fitting, regard the output of realistic model and the output of real system as two stochastic processes, can analyze realistic model and real system record multi-group data respectively.Respectively the output of realistic model output and real system is designated as: X _i, Y _ii=1 ... N, X _irepresent realistic model to export, Y _irepresent the output of real system, weighting function is ξ, η.Three observation is carried out to X and obtains three groups of emulated datas as shown in Figure 2, three observation is carried out to Y and obtains three groups of True Datas as shown in Figure 3.In Fig. 2 ~ Fig. 5, abscissa representing time, unit is s, and ordinate represents acceleration, and unit is m/s ².

Step 2: by emulated data X _iwith True Data Y _ifit to functional form respectively.Function Fitting has multiple approximating method, needs to search out the minimum approximating method of a kind of fitness bias by analyzing.The deviation of matching is relevant with the features of shape of data and selected approximating method, here by constantly test, choose nine order polynomial fitting processs and carry out matching, error is in tolerance interval, therefore choose fitting of a polynomial to process, the function curve of matching as shown in Figure 4 and Figure 5.

Step 3: choose one group of basis function, if φ ₁, φ ₂..., φ _m, be a suitable base.To function X _i, Y _ilaunch with the same base of weighting function ξ, η.Choose as next group basis function:

φ ₁＝cos(5/23π×(x-27/5))φ ₂＝sin(5/23π×(x-27/5))

φ ₃＝cos(10/23π×(x-27/5))φ ₄＝sin(10/23π×(x-27/5))

φ ₅＝cos(15/23π×(x-27/5))φ ₆＝sin(15/23π×(x-27/5))

φ ₇＝cos(20/23π×(x-27/5))φ ₈＝sin(20/23π×(x-27/5))

Function after emulated data matching is:

f ₁(x)＝1.878×10 ^-6x ⁹-9.884×10 ^-5x ⁸+0.002206x ⁶-0.02721x ⁵

+0.2021x ⁴-0.9203x ³+2.532x ²-4.189x-1.147

f ₂(x)＝5.077×10 ^-6x ⁹-0.00025x ⁸+0.00523x ⁷-0.06063x ⁶

+0.4253x ⁵-1.846x ⁴+4.884x ³-7.679x ²+6.918x-2.088

f ₃(x)＝4.912×10 ^-6x ⁹-0.000256x ⁸+0.005674x ⁷-0.06978x ⁶

+0.5201x ⁵-2.406x ⁴+6.801x ³-11.3x ²+10.21x-3.119

Function after True Data matching is:

g ₁(x)＝3.145×10 ^-6x ⁹-0.0001476x ⁸+0.002918x ⁷-0.03163x ⁶

+0.2046x ⁵-0.8045x ⁴+1.909x ³-2.845x ²+2.989x-0.8644

g ₂(x)＝3.805×10 ^-6x ⁹-0.0001875x ⁸+0.003943x ⁷-0.04621x ⁶

+0.3294x ⁵-1.46x ⁴+3.968x ³-6.498x ²+6.275x-2.043

g ₃(x)＝-2.62×10 ^-6x ⁹+0.000145x ⁸-0.003394x ⁷+0.04402x ⁶

-0.3475x ⁵+1.729x ⁴-5.371x ³+9.736x ²-8.803x+3.635

Simulation curve function and real curve function according to obtaining result after this group base function expansion are:

f ₁(x)＝-0.2598φ ₁-1.079φ ₂+0.001594φ ₃+0.1173φ ₄

+0.01714φ ₅-0.004596φ ₆-0.008957φ ₇+0.04237φ ₈

f ₂(x)＝-0.2431φ ₁-1.010φ ₂-0.005960φ ₃+0.06863φ ₄

+0.003421φ ₅-0.03917φ ₆-0.004178φ ₇+0.03976φ ₈

f ₃(x)＝-0.2412φ ₁-0.8988φ ₂+0.0001208φ ₃+0.1028φ ₄

+0.0008439φ ₅-0.06252φ ₆+0.01559φ ₇+0.03881φ ₈

g ₁(x)＝-0.2319φ ₁-0.9843φ ₂+0.003080φ ₃+0.1094φ ₄

-0.008781φ ₅-0.05501φ ₆+0.4868×10 ^-4φ ₇+0.04684φ ₈

g ₂(x)＝-0.2422φ ₁-0.9782φ ₂+0.006256φ ₃+0.07485φ ₄

+0.00042φ ₅-0.03138φ ₆-0.003737φ ₇+0.04074φ ₈

g ₃(x)＝-0.2103φ ₁-1.027φ ₂+0.02230φ ₃+0.07588φ ₄

-0.01763φ ₅-0.04543φ ₆-0.003794φ ₇+0.05441φ ₈

Step 4: structural matrix K, wherein K (i, j)=< D ²φ _i, D ²φ _j>.Structural matrix J, wherein J (i, j)=< φ _i, φ _j>.

If basis function is orthogonal basis, then J is unit battle array.

In addition, define C, D and be respectively X _i, Y _iexpansion coefficient matrix, have with a, b difference representative function ξ, the expansion coefficient vector of η.Definition V ₁₁, V ₁₂, V ₂₂be respectively sample variance and covariance matrix.

Step 5: equationof structure (14), solves this equation, and the solution of the correlation coefficient ρ obtained is listed in Table 1.

The solution of table 1 matrix equation eigenwert

Sequence number	Eigenwert	Sequence number	Eigenwert
				1	-0.9314	9	-0.3303×10-5+0.1603×10-5i
2	0.9314	10	-0.3303×10-5-0.1603×10-5i
				3	-0.00214	11	0.3303×10-5+0.1603×10-5i
4	0.00214	12	0.3303×10-5-0.1603×10-5i
				5	-0.000125	13	0.1323×10-22+0.6056×10-6i
6	0.000125	14	0.1323×10-22-0.6056×10-6i

7	0.3282×10-2+0.6712×10-5	15	-0.1653×10-6
				8	0.3282×10-2-0.6712×10-5	16	0.1653×10-6

Step 6: the maximal value obtaining ρ is ρ _max=0.9314, be the related coefficient that realistic model exports and real system exports, due to 0.9314 closely 1, therefore think that the similarities of these two groups output data are very high, and then determine that the similarity of flight simulator displacement parameter is very high.

Claims (4)

1. based on a Methods of Validating Simulation Model for functional data analysis, it is characterized in that, comprise the following steps:

Step one: regard the output of realistic model and the output of real system as two stochastic processes, two weighting function ξ and η are set, in order to construct overall target z and w, the output correlativity of realistic model and real system is described by the related coefficient of two overall targets; The N group recording realistic model and real system respectively exports data; If X _irepresent emulated data, Y _irepresent True Data, i=1,2 ..., N;

Step 2: emulated data and True Data are fitted to functional form respectively;

Step 3: choose one group of basis function φ ₁, φ ₂..., φ _m, to X _i, Y _ifitting function and weighting function ξ, η all use this base function expansion;

Step 4: establish X _i, Y _ithe expansion coefficient matrix of fitting function be respectively C and D, the expansion coefficient vector of weighting function ξ, η is respectively a and b, structural matrix K, K (i, j)=< D ²φ _i, D ²φ _j>; Structural matrix J, J (i, j)=< φ _i, φ _j>;

If V ₁₁the sample variance matrix of representing matrix C, V ₂₂the sample variance matrix of representing matrix D, V ₁₂the covariance matrix of representing matrix C and D;

Step 5: be constructed as follows equation:

$\left[\begin{array}{cc}0& J{V}_{12}J\\ {\mathrm{JV}}_{21}J& 0\end{array}\right]\left[\begin{array}{c}a\\ b\end{array}\right]=\mathrm{\&rho;}\left[\begin{array}{cc}{\mathrm{JV}}_{11}J+{\mathrm{\&lambda;}}_{1}K& 0\\ 0& {\mathrm{JV}}_{22}J+{\mathrm{\&lambda;}}_{2}K\end{array}\right]\left[\begin{array}{c}a\\ b\end{array}\right]$

Solve this equation, obtain the solution of correlation coefficient ρ;

Step 6: the maximal value ρ obtaining ρ _max, this value is the related coefficient of the output of realistic model and real system, according to ρ _maxclose to the degree of 1, draw two groups of similarities exporting data.

2. a kind of Methods of Validating Simulation Model based on functional data analysis according to claim 1, is characterized in that, in described step 4, and matrix V ₁₁, V ₁₂and V ₂₂in element determine according to formula below respectively:

$V_{11}(v,p)=N^{-1}\underset{i}{\mathrm{\&Sigma;}}{c}_{\mathrm{iv}}{c}_{\mathrm{ip}},V_{12}(v,p)=N^{-1}\underset{i}{\mathrm{\&Sigma;}}{c}_{\mathrm{iv}}{d}_{\mathrm{ip}},V_{22}(v,p)=N^{-1}\underset{i}{\mathrm{\&Sigma;}}{d}_{\mathrm{iv}}{d}_{\mathrm{ip}}$

Wherein, (v, p) represents that being positioned at the capable p of matrix v arranges; c _iv, c _ipv in the i-th row respectively in representing matrix C and p element; d _iv, d _ipv in the i-th row respectively in representing matrix D and p element.

3. a kind of Methods of Validating Simulation Model based on functional data analysis according to claim 1, it is characterized in that, the equation described in step 5 is obtained by following process:

First, square ρ of the related coefficient of overall target z and w ²(z, w) is expressed as:

${\mathrm{\&rho;}}^2(z,w)=\frac{{<\mathrm{\&xi;},V_{12}\mathrm{\&eta;}>}^2}{<\mathrm{\&xi;},V_{11}\mathrm{\&xi;}><\mathrm{\&eta;},V_{22}\mathrm{\&eta;}>}$

This formula is equal to the following maximization problems solved under constraint condition:

max＜ξ,V ₁₂η＞

s.t.＜ξ,V ₁₁＞＝＜η,V ₂₂η＞＝1

Coarse penalty term is added, so be converted in constraint condition:

max＜ξ,V ₁₂η＞

s.t.＜ξ,V ₁₁＞+λ ₁||D ²ξ||＝＜η,V ₂₂η＞+λ ₂||D ²η||＝1

Wherein, λ ₁and λ ₂represent auxiliary undetermined coefficient;

Further derivation, obtains as follows:

max＜ξ,V ₁₂η＞

s.t.＜ξ,(V ₁₁+λ ₁D ⁴)ξ＞＝＜η,(V ₂₂+λ ₂D ⁴)η＞＝1

G is as follows for structure Lagrangian function:

G＝<ξ,V ₁₂η>-α(<ξ,(V ₁₁+λ ₁D ⁴)ξ>-1)-β(<η,(V ₂₂+λ ₂D ⁴)η>-1)

Wherein, auxiliary coefficient when α and β is structure Lagrangian function;

Respectively to ξ, η differentiate, and make partial derivative equal zero, obtain:

V ₁₂η＝ρ(V ₁₁+λ ₁D ₄)ξ

V ₂₁ξ＝ρ(V ₂₂+λ ₂D ₄)η

Finally, above formula system of equations is converted to matrix equation:

$\left(\begin{array}{cc}0& V_{12}\\ V_{21}& 0\end{array}\right)\left(\begin{array}{c}\mathrm{\&xi;}\\ \mathrm{\&eta;}\end{array}\right)=\mathrm{\&rho;}\left(\begin{array}{cc}{V}_{11}+{\mathrm{\&lambda;}}_1{D}^4& 0\\ 0& V_{22}+{\mathrm{\&lambda;}}_2{D}_4\end{array}\right)\left(\begin{array}{c}\mathrm{\&xi;}\\ \mathrm{\&eta;}\end{array}\right)$

With a, b difference representative function ξ, the expansion coefficient vector of η, obtains the equation in step 5 through linear transformation.

4. a kind of Methods of Validating Simulation Model based on functional data analysis according to claim 1, it is characterized in that, in described step 6, the span of correlation coefficient ρ is [-1,1], when correlation coefficient ρ span is [0,1], illustrate that realistic model and real system export two fitting function positive correlations of data, ρ value is larger, the degree of correlation is higher, when value is 1, shows that two groups of output data lines are correlated with; In like manner, when correlation coefficient ρ value is [-1,0], two groups of vector negative correlation, when value is-1 formula, two groups export data back linear correlation.