크리깅 메타모델의 검증을 위한 평균제곱오차 및 응답 적분법

Abstract

메타모델기법이란 제한된 정보를 이용하여 설계변수와 시뮬레이션 응답 간의 관계를 간단한 수학적 함수로 정의하고 예측응답을 구하는 방법이다. 이러한 메타모델기법의 적용은 시뮬레이션 모델 기반 구조최적설계 문제와 같이 반복되는 해석으로 인해 해석 시간 및 계산 비용이 급증하는 경우, 이를 해결하고 효율성을 증대시킬 수 있기 때문에 널리 연구가 진행되고 있다. 그러나 이러한 장점에도 불구하고 메타모델기법이 실질적인 공학 문제에 널리 적용되는 데에는 많은 문제점을 갖고 있다. 이 문제점 중 근사모델의 정확성을 엄밀하게 평가하여 근사모델의 신뢰도를 확보할 수 있는 근사모델의 검증기법의 개발이 절실히 요구된다. 본 연구에서는 크리깅모델의 정확성 및 수렴 여부를 판단하기 위한 기준으로 평균제곱오차 및 응답 적분법을 제안한다. 크리깅모델이란 시뮬레이션 모델의 응답과 같이 측정 오차나 무작위 오차가 발생하지 않는 전산실험의 결과를 근사화하는데 적합하며 비선형성이 강한 응답을 정확하게 표현하는 보간모델의 하나이다. 평균제곱오차 및 응답 적분법이란 크리깅모델이 제공하는 예측응답과 그 통계적 오차인 평균제곱오차를 적분하여 크리깅모델의 상대 오차 및 상대 변동량을 평가하기 위한 기준 값으로 정의하고 근사화된 메타모델의 정확도를 평가하기 위한 검증방법이다. 평균제곱오차 및 응답의 적분은 크리깅모델을 이용하여 해석적으로 적분을 수행하기 때문에 부가적 수치연산은 요구되지 않는다. 제안된 검증방법의 우수성을 확인하기 위해 다양한 수학 예제를 크리깅모델을 이용하여 순차적으로 근사화 하고 매 단계 갱신되는 크리깅모델의 정확도 및 변동량에 대한 수렴 여부를 판단하는 실험을 실시하였다. 또한 실제 공학 문제에 대한 적용 가능성을 확인하기 위해 소형 진동자의 1차 고유 진동수를 크리깅모델로 근사화하고 평가하였다. 제안된 방법은 교차검증법과 비교하여 우수성을 보여주고 있다.; Metamodels have been successfully substituted computationally expensive and time consuming simulation models in a variety of engineering fields. Especially, in design optimization, an implementation of metamodels has great benefits such as reduction of computational cost and analysis time. Thus, various researches have been performed in order to improve the efficiency and the accuracy of metamodels, for example, optimization algorithms that are proper for application of metamodel, strategies for selecting sample points and validation measures for validation of various approximate models. The research area relating to the metamodel techniques can be classified into three parts: sampling techniques, modeling techniques and validation techniques. Sampling techniques determine how to fill the design domain with sample points effectively. Thus, sampling strategies are an important factor for accuracy as well as efficiency of metamodels because additional sample points for accuracy is in need of extra analysis of systems or experiments. Modeling techniques define a relation between design variables and responses with mathematical expressions to predict responses at untried inputs accurately. Metamodels are divided into regression models and interpolation models depending on the type of fitting responses. Validation techniques have been developed in order to verify the accuracy of metamodels. Definitely, the verified metamodels guarantee predictions of responses. Consequently, validation technique is a last key and the most significant area for metamodel methodology in order to come into wide use in practical engineering fields. For response surface model, analysis of variance (ANOVA) is performed for quantifying the accuracy of a response surface model. On the other hand, for interpolation models such as kriging model and radial basis function, leave-k-out cross validation strategy is only available in actual application. Leave-k-out cross validation requires to build a number of metamodels up to k in order to validate an approximate model and computational cost for quantifying the accuracy increases remarkably. Thus, the efficiency of leave-k-out cross validation gets worse if the number of dimension becomes higher or large number of sample points is needed for metamodeling. Furthermore, functional requirements of validation techniques are extended. For example, validation techniques not only evaluate the accuracy of metamodels, but also uses as the stop criterion in sequential metamodeling. In these reasons, the useful validation technique is needed in order to find exact and reliable predictions or optimum solution in metamodel-based optimization. In this paper, integrated mean squared errors and responses (IMSER) method is proposed as a validation technique and also a stop criterion for kriging model. Kriging model which is the one of approximate model can describe highly nonlinear responses and predict unknown responses in real time. The mean squared error is provided by kriging model and it is the prediction error on untried sample points. Theoretically, the mean squared error becomes zero if kriging model fits the responses exactly. The summation of mean squared error is converged toward zero and the summation of responses within design domain is closed toward a certain value as kriging model updated precisely. Thus IMSE and IR measures can be applied as a stop criterion and a validation measure for kriging model. This paper is constituted as follows. At first, traditional validation measures are reviewed and their limitations as a robust validation and also a stop criterion are described. Then, a mathematical function which has a feature of highly nonlinear is fitted sequentially with kriging model. During sequential sampling, IMSE and IR method examines an approximate model to validate their reliability. Also, histories of IMSE and IR measures are compared with results of conventional validation measures in order to confirm a behavior of convergence. Additionally, the proposed method is adapted as a validation measure and a stop criterion for a kriging model of a practical simulation model, the lateral vibrating circular spring to verify accuracy of an approximated model and stop sequential sampling process. Finally, the results are reviewed and conclusion is presented.