Minimal volume ellipsoids

In bounded-error estimation one is interested in characterizing the set § of all values of the parameters of a model which are consistent with the data in the sense that the corresponding errors fall between known prior bounds. the problem treated here is the computation of a minimal volume ellipsoid guaranteed to contain §. Although this ellipsoidal approach to parameter bounding was initially applied to models linear in their parameters, where the only error to be accounted for was an output error, it can be extended to deal with errors-in-variables problems. This makes it possible to consider models nonlinear in their parameters or dynamical models where both inputs and outputs are subject to bounded errors. Recursive algorithms are considered first. the basic algorithm of Fogel and Huang (1982), developed from the seminal work of Schweppe (1973) and modified as suggested by Belforte and co-workers (1985, 1990), is derived in detail, which makes it possible to clarify the nature of the tests needed. This algorithm is proved to be mathematically equivalent to a recursively optimal algorithm independently developed by Todd (1980) and Konig and Pallaschke (1981) in the context of linear programming after the celebrated work of Khachiyan (1979). A recursive approach, also developed in this context, is suggested for errors-in-variables problems. Recursively optimal ellipsoids, however, are not globally optimal because of the approximations committed at each step. Using a methodology borrowed from experiment design, we obtain optimality conditions which can be used to derive a number of algorithms guaranteed to converge to the global optimum.