Support Vector Regression for the simultaneous learning of a multivariate function and its derivatives

Incorporating prior knowledge in support vector regression

“…Many types of prior knowledge can be taken into account by the proposed method such as particular points with known values, prior knowledge on any derivative either provided by a prior model or available only at some points, bounds on the function or a derivative, etc. This extends and regroups the previous works of Mangasarian and Wild (2007) and Lázaro et al (2005b), which considered particular forms of prior knowledge. Moreover, a new method for the simultaneous approximation of multiple outputs linked by some prior knowledge has been proposed.…”

“…Knowledge on the derivatives In Lázaro et al (2005b), prior knowledge on the derivatives has been incorporated in the QP formulation of the SVR problem by considering a training set containing at every point the target values, not only for the function but also for the derivative. The optimization problem is extended to include the minimization of the error on the derivative and thus simultaneously approximate the function and its derivative.…”

“…This method allows to incorporate prior knowledge on any derivative of order k for any set of points possibly different for each k, while keeping the model in the form of (1). In comparison, the methods described in (Lázaro et al 2005a(Lázaro et al , 2005b require the prior derivative values on all the training points and these points only.…”

“…This problem is close to the setting of (Lázaro et al 2005a(Lázaro et al , 2005b as prior knowledge on the derivatives is given at every training point. However, here, the derivatives values are not directly available and are estimated by a prior model trained on another set of samples.…”

“…As the points of interest x p are different of the inputs x found in the training set, the methods of Lázaro et al (2005aLázaro et al ( , 2005b or cannot be used.…”