A range of procedures in both robustness and diagnostics require optimisation of a target functional over all subsamples of given size. Whereas such combinatorial problems are extremely difficult to solve exactly, something less than the global optimum can be ‘good enough’ for many practical purposes, as shown by example. Again, a relaxation strategy embeds these discrete, high-dimensional problems in continuous, low-dimensional ones. Overall, nonlinear optimisation methods can be exploited to provide a single, reasonably fast algorithm to handle a wide variety of problems of this kind, thereby providing a certain unity. Four running examples illustrate the approach. On the robustness side, algorithmic approximations to minimum covariance determinant (MCD) and least trimmed squares (LTS) estimation. And, on the diagnostic side, detection of multiple multivariate outliers and global diagnostic use of the likelihood displacement function. This last is developed here as a global complement to Cook’s (in J. R. Stat. Soc. 48:133–169, 1986) local analysis. Appropriate convergence of each branch of the algorithm is guaranteed for any target functional whose relaxed form is—in a natural generalisation of concavity, introduced here—‘gravitational’. Again, its descent strategy can downweight to zero contaminating cases in the starting position. A simulation study shows that, although not optimised for the LTS problem, our general algorithm holds its own with algorithms that are so optimised. An adapted algorithm relaxes the gravitational condition itself.

Systematic departures from a reference sampling scheme, such as multivariate normality, can affect all the observed ordered Mahalanobis distances (OMDs) – not just a few extreme values – causing distinctive changes to their overall pattern. Envelope OMD plots provide a useful framework against which to ‘read’ such changes, often suggestive of further, confirmatory analyses. Illustrative examples include heavy-tailed and skew distributions and the presence of (clusters of) outliers. The exact distributions involved being intractable, the above framework is found by simulation. Fast, accurate approximations to it are also developed for use when such simulation is computationally prohibitive.

The present paper focuses on the case sensitivity function approach to diagnostics and robustness that are combinatorial by definition and hard to solve exactly. Attention is also given to the visual displays.

Summary The case sensitivity function approach to influence analysis is introduced as a natural smooth extension of influence curve methodology in which both the insights of geometry and the power of (convex) analysis are available. In it, perturbation is defined as movement between probability vectors defining weighted empirical distributions. A Euclidean geometry is proposed giving such perturbations both size and direction. The notion of the salience of a perturbation is emphasized. This approach has several benefits. A general probability case weight analysis results. Answers to a number of outstanding questions follow directly. Rescaled versions of the three usual finite sample influence curve measures—seen now to be required for comparability across different-sized subsets of cases—are readily available. These new diagnostics directly measure the salience of the (infinitesimal) perturbations involved. Their essential unity, both within and between subsets, is evident geometrically. Finally it is shown how a relaxation strategy, in which a high dimensional (O(nCm)) discrete problem is replaced by a low dimensional (O(n)) continuous problem, can combine with (convex) optimization results to deliver better performance in challenging multiple-case influence problems. Further developments are briefly indicated.

Strip electrodes have been used to augment acute cortical electrocorticography (ECoG) in 100 consecutive surgeries for epilepsy. The strip electrodes have been especially helpful in defining the extent of mesial temporal and mesial frontal foci. When dealing with temporal lobe foci, strip electrodes can help identify epileptiform activity residing in posterior regions of hippocampus which should be removed. Thus, we have found that strip electrodes can significantly augment information gained during acute ECoG recordings as well as decrease the time needed for recordings.

Methohexital was used as an activating agent in 62 patients undergoing focal cortical resections of epileptogenic foci and in six patients undergoing chronic electroencephalogram (EEG)/video monitoring with intracranial strip electrodes. In 87% of cases, methohexital caused selective activation of the epileptogenic focus during acute electrocorticography (ECoG). This activation appeared to be specific for the epileptogenic focus and did not cause epileptiform spiking from adjacent nonepileptogenic cortex. This ECoG activation occurred whether patients received local or general anesthesia. Sixty-five percent of patients demonstrated adequate activation with as little as 25 mg of drug. Methohexital is a safe and reliable method of activating epileptogenic foci during acute ECoG recordings and can decrease the time for ECoG recordings during surgery.