Construct a Taylor Series expansion around a point

Specification

Alias: none

Argument(s): none

Description

The Taylor series model is purely a local approximation method. That is, it provides local trends in the vicinity of a single point in parameter space.

The order of the Taylor series may be either first-order or second-order, which is automatically determined from the gradient and Hessian specifications in the responses specification (see responses for info on how to specify gradients and Hessians) for the truth model.

Theory

The first-order Taylor series expansion is:

$\begin{equation} \hat{f}({\bf x}) \approx f({\bf x}_0) + \nabla_{\bf x} f({\bf x}_0)^T ({\bf x} - {\bf x}_0) \end{equation}$

and the second-order expansion is:

$\begin{equation} \hat{f}({\bf x}) \approx f({\bf x}_0) + \nabla_{\bf x} f({\bf x}_0)^T ({\bf x} - {\bf x}_0) + \frac{1}{2} ({\bf x} - {\bf x}_0)^T \nabla^2_{\bf x} f({\bf x}_0) ({\bf x} - {\bf x}_0) \end{equation}$

where ${\bf x}_0$ is the expansion point in $n$ -dimensional parameter space and $f({\bf x}_0)$ , $\nabla_{\bf x} f({\bf x}_0)$ , and $\nabla^2_{\bf x} f({\bf x}_0)$ are the computed response value, gradient, and Hessian at the expansion point, respectively.

As dictated by the responses specification used in building the local surrogate, the gradient may be analytic or numerical and the Hessian may be analytic, numerical, or based on quasi-Newton secant updates.

In general, the Taylor series model is accurate only in the region of parameter space that is close to ${\bf x}_0$ . While the accuracy is limited, the first-order Taylor series model reproduces the correct value and gradient at the point $\mathbf{x}_{0}$ , and the second-order Taylor series model reproduces the correct value, gradient, and Hessian. This consistency is useful in provably-convergent surrogate-based optimization. The other surface fitting methods do not use gradient information directly in their models, and these methods rely on an external correction procedure in order to satisfy the consistency requirements of provably-convergent SBO.