CN108427846A - A kind of multiple response model validation measure based on probability box framework - Google Patents
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Abstract
A kind of multiple response model validation measure based on probability box framework establishes corresponding Simulation Calculation according to the correlation theory and input condition of particular problem；Corresponding Simulation Calculation input parameter will be established and be divided into stochastic uncertainty, cognition uncertainty and Hybrid parameter matrix, and parameters are assessed, determine specific distribution；The double stratified sample of M × N is carried out to model, obtains the multidimensional sample of M group models output；The conversion of mahalanobis distance is carried out to every group of sample, obtains the sample of M group mahalanobis distances；Mahalanobis distance sample value is counted；The area calculated between two probability boxes measures section.Using probability cassette method, the problem of random uncertain and cognition uncertainty exists simultaneously can effectively be described simultaneously, existing probabilistic model, interval number and evidence structure can be converted directly into the form of probability box, and it describes uncertainty and meets engineering custom, is easy to be received and used by engineering staff.
Description
Technical field
The present invention relates to measures, are based on more particularly, to the one kind measured suitable for multiple-input and multiple-output model validation
The multiple response model validation measure of probability box framework.
Background technology
In face of increasingly huge and complicated engineering system, it is contemplated that carry out the difficulty of Physical Experiment and required costliness to it
Expense.In engineering research, scientific research personnel attempts to increasingly mature computer technology and carries out modeling and simulation calculating, passes through
Complicated and expensive actual physical experiment is replaced to the calculating of institute's established model.But it would generally be because when establishing computation model
To be simplified to model by the prior art, computational efficiency, or by the cognition of modeling personnel is limited and cause to be modeled
Type and realistic model have certain deviation.The problem of at this time just will appear the accuracy and confidence level of computation model, to calculating
Model, which carries out model validation, to be particularly important[1-3].Model validation is exactly simply accurately quantitative description computation model
Difference between experimental observed data[4-8]。
There are mainly four types of existing model validation methods[9]：Classical assumption method of inspection, Bayesian Factor method, Frequency Index method
Knead dough product metric method.What the confirmation result of classical assumption method of inspection and Bayesian Factor method was accepts or rejects the model, not
There is the accuracy qualitative assessment provided to model.And Frequency Index rule gives quantifying for difference between model and experimental result
Quantization, but only considered sample average this characteristic quantity, do not account under reaction uncertain condition sample dispersion degree etc. its
Its characteristic quantity.Three of the above can not fully meet the requirement for confirming measurement, the iterated integral that Ferson etc. is exported with model emulation
Folded area is as measurement results between cloth function and the empirical distribution function of experimental observed data, put forward area measure with
And the u_pooling methods confirmed for " multiple spot "[10]。
Area measure can with objective quantitative describe the difference between simulation calculation and experimental observed data, but only
Suitable for single output model, the model validation problem of multi output model present in Practical Project system can not be directed to.Side herein
On the basis of method, Li Wei 2014 etc. is proposed converts (Probability Integral based on probability integral
Transformation, PIT) area measure and t_pooling methods；Zhao Liang 2015, which is proposed, gives probability distribution distance
Multiple response model validation measurement；Zhao Lufeng etc. 2017 proposes the multi output model validation method converted based on mahalanobis distance；
Hu Jiarui etc. 2017 proposes the multi output model validation method based on core principle component analysis.These methods expand area measure
The confirmation measurement field of the model containing multiple correlation output amounts is opened up.
Probability box (probability box, p-box) can effectively embody random simultaneously when handling uncertain problem
It is uncertain with cognition[1,11], and have the engineer application potentiality of bigger, it is concerned in recent years in terms of model validation measurement,
The area of probability box confirms that measurement has extended to interval value[12].However, that existing probability cassette method is directed to is Dan Xiang mostly
The model validation problem that should be measured.
Bibliography：
[1]W.L.Oberkampf and C.J.Roy,Verification and Validation in
Scientific Computing:Cambridge University Press,2010.
[2] Guo Qintao, more, and Fei Qing states, " development of Structural Dynamics FEM updating --- model is true for order
Recognize, " Proceedings of Mechanics, vol.36, pp.36-42,2006.
[3] Liu Cuicui, " the VV＆A technique studies of modeling and simulation, " Harbin Engineering University, 2012.
[4]W.L.Oberkampf,M.N.Sindir,and A.T.Conlisk,"AIAA Guide for the
Verification and Validation of Computational Fluid Dynamics Simulations,"1998.
[5]W.L.Oberkampf,T.G.Trucano,and C.Hirsch,"Verification,validation,
and predictive capability in computational engineering and physics,"Applied
Mechanics Reviews,vol.57,p.345,2004.
[6]L.E.Schwer,"An overview of the PTC 60/V&V 10:guide for
verification and validation in computational solid mechanics,"Engineering
with Computers,vol.23,pp.245-252,2007.
[7]D.Sornette,A.B.Davis,K.Ide,K.R.Vixie,V.Pisarenko,and J.R.Kamm,"
Algorithm for model validation:theory and applications,"Proceedings of the
National Academy of Sciences of the United States of America,vol.104,p.6562,2007.
[8] Hu Jiarui and Lv Zhen cosmos, " the multi output model validation method based on core principle component analysis, " Beijing Aviation boat
Its college journal, vol.43, pp.1470-1480,2017.
[9]Y.Liu,W.Chen,P.Arendt,and H.Huang,"Towards A Better Understanding of
Model Validation Metrics,"Journal of Mechanical Design,vol.133,p.071005,2011.
[10]S.Ferson,W.L.Oberkampf,and L.Ginzburg,"Model validation and
predictive capability for the thermal challenge problem,"Computer Methods in
Applied Mechanics&Engineering,vol.197,pp.2408-2430,2008.
[11] Xiao Zhao, " structural uncertainty based on probability box theory propagates analysis, " Hunan University, 2016.
[12] before Liu Xinen, He Qinshu, and Chen Xue, " interval value area metric algorithm research, " in mechanics in China conference-
2015 thesis summary sets, 2015.
Invention content
The purpose of the present invention is the above-mentioned deficiencies for the prior art, while considering at random and recognizing unascertained information,
The confirmation metric question of multiple response amount can be handled by providing, so as to test output multiple response data and simulation data multiple response data into
Row comparison, a kind of multiple response model validation based on probability box framework of difference objective quantification result between being tested and being emulated
Measure.
The present invention includes the following steps：
1) according to the correlation theory of particular problem and input condition, corresponding Simulation Calculation is established；
2) step 1) is established into corresponding Simulation Calculation input parameter and is divided into stochastic uncertainty, cognition uncertainty
And Hybrid parameter matrix, and parameters are assessed, determine specific distribution；
3) double stratified sample that M × N is carried out to model, obtains the multidimensional sample of M group models output；
In step 3), the specific method of the double stratified sample that M × N is carried out to model can be：Outer layer is recognized first
Know 1 sampling of uncertain progress；Secondly, n times sampling is carried out to the random not confirmatory of internal layer, and calculates corresponding sample
Value；Finally repeat M outer layer sampling, obtains the sample of M group models output.
4) conversion that mahalanobis distance is carried out to every group of sample, obtains the sample of M group mahalanobis distances；
In step 4), the specific method of the conversion that mahalanobis distance is carried out to every group of sample can be：Mahalanobis distance
(Mahalanobis distance) is proposed by India's statistician's Mahalanobis (P.C.Mahalanobis), is
A method of calculating two unknown sample collection similarities；Compared with Euclidean distance, mahalanobis distance has many good qualities：1, it not by
The influence of dimension, i.e., the mahalanobis distance between 2 points are unrelated with the units of measurement of initial data；2, by standardized data and center
The mahalanobis distance changed between calculated 2 points of data (i.e. the difference of initial data and mean value) is identical；Moreover, mahalanobis distance can be examined
Consider the contact between various characteristics, the interference of the correlation between variable can be excluded；
If Y=(Y1,Y2,...,Ym) be m dimensions random sample, mean vector and covariance matrix be respectively μ=
(μ1,μ2,...,μm) and Σ can with mahalanobis distance construct statistic MD (Y), according to the calculation formula of mahalanobis distance：
If the sample value of Y is y=(y1,y2,...,ym), then the sample value of statistic MD (Y) can be obtained by following formula：
5) mahalanobis distance sample value is counted, defines mahalanobis distance cumulative distribution function：
FMD(md)=P (MD≤md) (3)
The empirical distribution function of M mahalanobis distance is obtained, and extracts its boundary, emulation probability box is obtained, to experimental data
Identical mahalanobis distance conversion is carried out, the probability box of experimental data is obtained.
6) area calculated between two probability boxes measures section.
In step 6), the method for the area measurement is a kind of confirmation measure based on probability distribution distance, is led to
The area between computation model response and the empirical cumulative distribution function of experimental observation is crossed, the amount to model accuracy can be provided
Change as a result, its mathematical definition is：
Wherein, FmFor the cumulative distribution function of simulation result, SeFor the empirical distribution function of experimental data, responded in model
In the case of probability box and experimental observed data probability box compare, the area measurement of probability box is extended into interval value, is used
A kind of to calculate this interval value based on the algorithm of interval theory, specific calculating process is provided by following equation：
Wherein, d (F, S) andThe respectively minimum value (lower bound) and maximum value (upper bound) in area measurement section,WithRespectively point piPlace's emulation probability box and the probability interval for testing probability box；
The detailed process of algorithm is：Probability axis is evenly dividing as N equal portions first, it is general that each minizone takes midpoint to obtain corresponding prediction
Rate box sectionWith experiment probability box sectionThen every a pair of of interval value is become
Amount, which is subtracted each other, asks absolute value to obtain corresponding section, and all section bounds of acquisition are averagely finally obtained interval value face respectively
The bound of product metric
For the model validation problem for allowing probability cassette method to be suitable for out there are multiple relevant response amounts, carried according to Zhao Lufeng
The mahalanobis distance conversion concept gone out, the present invention will introduce mahalanobis distance on the basis of the interval value area of probability box is measured
(Mahalanobis Distance, MD), by seeking mahalanobis distance, converts the random probability distribution of multidimensional to one-dimensional horse
The probability distribution of family name's distance.According to this feature of MD, it is one-dimensional the probability box of the joint probability distribution of multidimensional can be converted dimension
Mahalanobis distance probability box.After the probability box for obtaining mahalanobis distance by computer sim- ulation probability box and experiment probability box it
Between area measurement obtain area measurement interval value.
Compared with the prior art, the invention has the advantages that：
1) probability cassette method is used, it is uncertain simultaneous can effectively to describe random uncertain and cognition simultaneously
Problem, existing probabilistic model, interval number and evidence structure can be converted directly into the form of probability box, and its description is not true
It is qualitative to meet engineering custom, it is easy to be received and used by engineering staff.
2) the confirmation metric question for handling multiple response model for amount of translation using mahalanobis distance, remains each response quautity
Between correlation information, can confirmation measurement effectively be carried out to multiple response model.This method is relative to probability integral simultaneously
(PIT) method of transformation, avoids the finding process of multivariate joint probability distribution function, higher on operation efficiency.
Description of the drawings
Fig. 1 is double stratified sample flow chart.In Fig. 1, solid box is that outer layer cognition is uncertain, and dotted line frame is that internal layer is random
It is uncertain.
Fig. 2 is that the area based on probability distribution distance measures method schematic diagram.
Fig. 3 is that probability box area measures schematic diagram.
Fig. 4 is the front view of storage tank.
Fig. 5 is the side view of storage tank.
Fig. 6 is the mahalanobis distance probability box result of the mahalanobis distance probability box and experiment of emulation.
Specific implementation mode
Following embodiment will the present invention is further illustrated in conjunction with attached drawing.
The present invention includes the following steps：
1) according to the correlation theory of particular problem and input condition, corresponding Simulation Calculation is established；
2) mode input parameter is divided into stochastic uncertainty, cognition uncertainty and Hybrid parameter matrix, and to each
Parameter is just assessed, and determines its specific distribution；
3) double stratified sample that M × N is carried out to model, obtains the multidimensional sample of M group models output；
Specific implementation process is as shown in Figure 1：It is 1 sampling of cognition uncertainty progress to outer layer first；Later, internally
The random not confirmatory of layer carries out n times sampling, and calculates corresponding sample value；M outer layer sampling of progress is repeated the above process,
Finally obtain the sample of M group models output.
4) conversion that mahalanobis distance is carried out to every group of sample, obtains the sample of M group mahalanobis distances；
Mahalanobis distance (Mahalanobis distance) is by India's statistician's Mahalanobis
(P.C.Mahalanobis) it proposes, is a kind of method calculating two unknown sample collection similarities.Compared with Euclidean distance,
Mahalanobis distance has many good qualities：1, it is not influenced by dimension, i.e., the measurement list of mahalanobis distance and initial data between 2 points
Position is unrelated；2, by the geneva between standardized data and calculated 2 points of centralization data (i.e. the difference of initial data and mean value)
Apart from identical.And mahalanobis distance can consider the contact between various characteristics, can exclude the dry of the correlation between variable
It disturbs.
If Y=(Y1,Y2,...,Ym) be m dimensions random sample, mean vector and covariance matrix be respectively μ=
(μ1,μ2,...,μm) and Σ can with mahalanobis distance construct statistic MD (Y), according to the calculation formula of mahalanobis distance：
If the sample value of Y is y=(y1,y2,...,ym), then the sample value of statistic MD (Y) can be obtained by following formula：
5) mahalanobis distance sample value is counted, defines mahalanobis distance cumulative distribution function：
FMD(md)=P (MD≤md) (10)
The empirical distribution function of M mahalanobis distance is obtained, and extracts its boundary, obtains emulation probability box.To experimental data
Identical mahalanobis distance conversion is carried out, the probability box of experimental data is obtained；
6) area calculated between two probability boxes measures section.
Area measure is a kind of confirmation measure based on probability distribution distance, as shown in Fig. 2, by calculating mould
Type responds the area (dash area in figure) between the empirical cumulative distribution function of experimental observation, and it is accurate to model to provide
The quantized result of property, mathematical definition are：
Wherein, FmFor the cumulative distribution function of simulation result, SeFor the empirical distribution function of experimental data.It is responded in model
In the case of probability box and experimental observed data probability box compare, the area measurement of probability box is extended into interval value.It uses
A kind of to calculate this interval value based on the algorithm of interval theory, specific calculating process is provided by following equation：
Wherein, d (F, S) andThe respectively minimum value (lower bound) and maximum value (upper bound) in area measurement section,WithRespectively point piPlace's emulation probability box and the probability interval for testing probability box.
The detailed process of algorithm is：Probability axis is evenly dividing as N equal portions first, it is general that each minizone takes midpoint to obtain corresponding prediction
Rate box sectionWith experiment probability box sectionThen to every a pair of of interval value variable
Subtract each other and absolute value is asked to obtain corresponding section, all section bounds of acquisition are averagely finally obtained into interval value area respectively
The bound of measurementFig. 3 intuitively gives the geometric meaning of the interval value bound, and upper dividing value is
The area of light dash area in figure, floor value are the area of dark-shaded part in figure.
From Fig. 3 it can be found that when calculating the metric upper bound, this algorithm will emulate probability box and experiment probability box
The area of lap also counts together.So, the definition of measure is confirmed according to area, the area meeting of this part
So that the upper dividing value in the measurement section calculated is overly conservative.In the calculating of the present invention, which has been carried out centainly
Modification eliminates the area of this part when the computation interval value upper bound.
This hair is illustrated using the model verification and validation challenge of U.S.'s Sandia National Labs proposition in 2014
The bright specific implementation step on engineering problem：
The problem is output object, diagnostic cast with the maximum (normal) stress of storage tank under specific environment, load and maximum normal strain
The critical issue solved is needed during type verification and validation.Figure 4 and 5 show storage tank dimension information.
Model for simulation calculation is a black box, can be described with following formula：
The definition of parameters is as shown in table 1.
The description of 1 model parameter of table
1, corresponding Simulation Calculation is established.Computation model is established using python language, is carried by the laboratories Sang Diya
For.
2, each parameter is divided into three classes：Stochastic uncertainty, cognition uncertainty and Hybrid parameter matrix, and pass through
Bootstrap methods estimate sample parameter, obtain the specific distribution (being shown in Table 2) of parameters.
The classification and its distribution of 2 uncertain parameters of table
3, the double stratified sample of M × N is carried out to computation model using the double-deck Latin Hypercube Sampling method of Dakota softwares,
Obtain the two dimensional sample of M group models output；It is 1 sampling of cognition uncertainty progress to outer layer first；Later, to internal layer
Confirmatory does not carry out n times sampling at random, and calculates corresponding sample value；M outer layer sampling of progress is repeated the above process, finally
Obtain the sample of M group models output.
4, the M group two dimensional samples in " * .out " file of matlab programs reading Dakota outputs are write, are calculated corresponding
Mahalanobis distance sample value, and draw the empirical distribution function of M mahalanobis distance in same figure, extract probability box side
Boundary.
5, identical mahalanobis distance conversion process is executed to experimental data, obtains the probability box boundary of experimental data.Emulation
Probability box and experiment probability box result are drawn on Fig. 6.
6, the area calculated between two probability boxes measures section.The program thread of Matlab is：Using mid-rectangle formula
Probability axis is evenly dividing as N equal portions by the method quadratured first, and each minizone takes midpoint to obtain corresponding prediction probability box
SectionWith experiment probability box sectionThen every a pair of of interval value variable is subtracted each other
It asks absolute value to obtain corresponding section, all section bounds of acquisition is finally averagely obtained to interval value area measurement respectively
Bound
7, the confirmation measurement section finally identified is：[0.0,0.5146].
Claims (4)
1. a kind of multiple response model validation measure based on probability box framework, it is characterised in that include the following steps：
1) according to the correlation theory of particular problem and input condition, corresponding Simulation Calculation is established；
2) step 1) is established into corresponding Simulation Calculation input parameter to be divided into stochastic uncertainty, cognition uncertain and mixed
Uncertainty is closed, and parameters are assessed, determines specific distribution；
3) double stratified sample that M × N is carried out to model, obtains the multidimensional sample of M group models output；
4) conversion that mahalanobis distance is carried out to every group of sample, obtains the sample of M group mahalanobis distances；
5) mahalanobis distance sample value is counted, defines mahalanobis distance cumulative distribution function：
FMD(md)=P (MD≤md)
The empirical distribution function of M mahalanobis distance is obtained, and extracts its boundary, emulation probability box is obtained, experimental data is carried out
Identical mahalanobis distance conversion, obtains the probability box of experimental data；
6) area calculated between two probability boxes measures section.
2. a kind of multiple response model validation measure based on probability box framework as described in claim 1, it is characterised in that
In step 3), the specific method of the double stratified sample that M × N is carried out to model is：First to the cognition uncertainty of outer layer into
1 sampling of row；Secondly, n times sampling is carried out to the random not confirmatory of internal layer, and calculates corresponding sample value；Finally repeat
M outer layer sampling is carried out, the sample of M group models output is obtained.
3. a kind of multiple response model validation measure based on probability box framework as described in claim 1, it is characterised in that
In step 4), the specific method of the conversion that mahalanobis distance is carried out to every group of sample is：Mahalanobis distance is by India's statistics
Family's Mahalanobis proposes, is a kind of method calculating two unknown sample collection similarities, is not influenced by dimension, i.e., two
Mahalanobis distance between point is unrelated with the units of measurement of initial data；By standardized data and centralization data, that is, initial data with
Mahalanobis distance between calculated 2 points of the difference of mean value is identical；And mahalanobis distance can consider the connection between various characteristics
System excludes the interference of the correlation between variable；
If Y=(Y1,Y2,...,Ym) it is the random sample that a m is tieed up, mean vector and covariance matrix are respectively μ=(μ1,
μ2,...,μm) and Σ can with mahalanobis distance construct statistic MD (Y), according to the calculation formula of mahalanobis distance：
If the sample value of Y is y=(y1,y2,...,ym), then the sample value of statistic MD (Y) is obtained by following formula：
4. a kind of multiple response model validation measure based on probability box framework as described in claim 1, it is characterised in that
In step 6), the method for the area measurement is a kind of confirmation measure based on probability distribution distance, passes through computation model
Area between response and the empirical cumulative distribution function of experimental observation, provides the quantized result to model accuracy, mathematics
Definition is：
Wherein, FmFor the cumulative distribution function of simulation result, SeFor the empirical distribution function of experimental data, in model response probability
In the case of box and experimental observed data probability box compare, the area measurement of probability box is extended into interval value, uses one kind
This interval value is calculated based on the algorithm of interval theory, specific calculating process is provided by following equation：
Wherein,d(F, S) andThe respectively minimum value and maximum value in area measurement section,WithRespectively point piPlace's emulation probability box and the probability interval for testing probability box；The detailed process of algorithm is：
Probability axis is evenly dividing as N equal portions first, each minizone takes midpoint to obtain corresponding prediction probability box sectionWith experiment probability box sectionThen every a pair of of interval value variable is subtracted each other and asks exhausted
Corresponding section is obtained to value, all section bounds of acquisition are finally averagely obtained to the upper of interval value area measurement respectively
Lower bound
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| CN112528417B (en) * | 2020-12-18 | 2024-05-10 | 北京机电工程研究所 | Assessment method for semi-physical simulation of aircraft |
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| CN113946953A (en) * | 2021-10-14 | 2022-01-18 | 厦门大学 | Method for calculating global sensitivity under probability box framework |
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Application publication date: 20180821 |