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