Abstract
Chapter 5 explores the idea of using regression problems to estimate sensitivities. Section 5.1 explains how one might approximate the gradient of the QoI at a nominal point using a least-squares (regression) formulation. This naive approach requires more QoI evaluations than one-sided finite differences as described in the previous chapter. Section 5.2 introduces a regularization term into the least-squares minimization problem, allowing for useful solutions also for the case where fewer QoI evaluations than parameters are available; sparsity-promoting regularization (1-norm, LASSO) and a combination of 1-norm and 2-norm (elastic net) are considered. Section 5.3 adds cross-validation techniques for selecting the regularization parameters.
Wo! Nemo, toss a lasso to me now!
—Dona Smith
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Notes
- 1.
We have switched the notation for number of parameters here so that when we form matrices the indices will be the common i and j for row and column, respectively.
- 2.
The extra solve comes from needing to compute \(Q(\bar {\mathbf {x}})\).
- 3.
The constraint form can be changed into the penalty form by considering λ as a Lagrange multiplier. There is a one-to-one relationship between λ and s.
- 4.
Latin hypercube designs are covered in Sect. 7.2.2.
References
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Zou H, Hastie T (2005) Regularization and variable selection via the elastic net. J R Stat Soc Ser B (Stat Methodol) 67(2):301–320
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McClarren, R.G. (2018). Regression Approximations to Estimate Sensitivities. In: Uncertainty Quantification and Predictive Computational Science. Springer, Cham. https://doi.org/10.1007/978-3-319-99525-0_5
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DOI: https://doi.org/10.1007/978-3-319-99525-0_5
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