1Blurb:: 2Construct a Taylor Series expansion around a point 3Description:: 4The Taylor series model is purely a local approximation method. That 5is, it provides local trends in the vicinity of a single point in 6parameter space. 7 8The order of the 9Taylor series may be either first-order or second-order, which is 10automatically determined from the gradient and Hessian specifications 11in the responses specification (see \ref responses for info on 12how to specify gradients and Hessians) 13for the truth model 14 15<em>Known Issue: When using discrete variables, there have been 16sometimes significant differences in surrogate behavior observed 17across computing platforms in some cases. The cause has not yet been 18fully diagnosed and is currently under investigation. In addition, 19guidance on appropriate construction and use of surrogates with 20discrete variables is under development. In the meantime, users 21should therefore be aware that there is a risk of inaccurate results 22when using surrogates with discrete variables.</em> 23 24 25Topics:: 26Examples:: 27Theory:: 28 29The first-order Taylor series expansion is: 30\anchor eq-taylor1 31\f{equation} 32\hat{f}({\bf x}) \approx f({\bf x}_0) + \nabla_{\bf x} f({\bf x}_0)^T 33({\bf x} - {\bf x}_0) 34\f} 35and the second-order expansion is: 36\anchor eq-taylor2 37\f{equation} 38\hat{f}({\bf x}) \approx f({\bf x}_0) + \nabla_{\bf x} f({\bf x}_0)^T 39({\bf x} - {\bf x}_0) + \frac{1}{2} ({\bf x} - {\bf x}_0)^T 40\nabla^2_{\bf x} f({\bf x}_0) ({\bf x} - {\bf x}_0) 41\f} 42 43where \f${\bf x}_0\f$ is the expansion point in \f$n\f$-dimensional parameter 44space and \f$f({\bf x}_0)\f$, \f$\nabla_{\bf x} f({\bf x}_0)\f$, and 45\f$\nabla^2_{\bf x} f({\bf x}_0)\f$ are the computed response value, 46gradient, and Hessian at the expansion point, respectively. 47 48As 49dictated by the responses specification used in building the local 50surrogate, the gradient may be analytic or numerical and the Hessian 51may be analytic, numerical, or based on quasi-Newton secant updates. 52 53In general, the Taylor series model is accurate only in the region of 54parameter space that is close to \f${\bf x}_0\f$ . While the accuracy is 55limited, the first-order Taylor series model reproduces the correct 56value and gradient at the point \f$\mathbf{x}_{0}\f$, and the second-order 57Taylor series model reproduces the correct value, gradient, and 58Hessian. This consistency is useful in provably-convergent 59surrogate-based optimization. The other surface fitting methods do not 60use gradient information directly in their models, and these methods 61rely on an external correction procedure in order to satisfy the 62consistency requirements of provably-convergent SBO. 63 64Faq:: 65See_Also:: 66