Structural and Multidisciplinary Optimization

, Volume 42, Issue 6, pp 811–821 | Cite as

Estimating training data boundaries in surrogate-based modeling

  • Luis E. Pineda
  • Benjamin J. Fregly
  • Raphael T. Haftka
  • Nestor V. QueipoEmail author
Research Paper


Using surrogate models outside training data boundaries can be risky and subject to significant errors. This paper presents a computationally efficient approach to estimate the boundaries of training data inputs in surrogate modeling using the Mahalanobis distance (MD). This distance can then be used as a threshold for deciding whether or not a particular prediction site is within the boundaries of the training data inputs, and has the potential of a likelihood/probabilistic interpretation. The approach is evaluated using two and four dimensional analytical restricted input spaces and a complex biomechanical six dimensional problem. The proposed approach: i) gives good approximations for the boundaries of the restricted input spaces, ii) exhibits reasonable error rates when classifying prediction sites as inside or outside known restricted input spaces and iii) reflects expected error trends for increasing values of the MDs similar to those obtained using a computationally expensive convex hull approach.


Surrogate modeling Training data boundaries Mahalanobis distance 



Balanced error rate


Covariance matrix




Latin hypercube sampling


Number of training data


Mahalanobis distance


Number of input variables


Probability of a prediction site being within the training data boundaries


Set of real numbers of dimension p


Surrogate model


Training data


Input variables


Response variables


Statistical significance level

\(\chi_p^{2} \)

Chi-square distribution—p degrees of freedom




Relative error





boundary estimation


median of top 20% largest Mahalanobis distances


largest Mahalanobis distance


training data



This work was supported in part by the National Science Foundation CBET Division under Grant No. 0602996 to B. J. Fregly and R. T. Haftka.


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Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  • Luis E. Pineda
    • 1
  • Benjamin J. Fregly
    • 2
  • Raphael T. Haftka
    • 2
  • Nestor V. Queipo
    • 1
    Email author
  1. 1.Applied Computing InstituteUniversity of ZuliaMaracaiboVenezuela
  2. 2.Department of Mechanical and Aerospace EngineeringUniversity of FloridaGainesvilleUSA

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