4.5 Conclusions

Exploring data is an important part of successful statistical model building. General discussions of graphical tools may be found in [55,58,7,8,64] and [4], for example. Advanced exploratory software may be found in many commercial packages, but of special note is the XGobi ([46]) system and successors. Immersive environments are also of growing interest ([9]). A general visualization overview may be found in [66].

Especially when the data size grows, point-oriented methods should be supplemented by indirect visualization techniques based upon nonparametric density estimation or by parallel coordinates ([16,63]). Many density algorithms are available. The use of order-two algorithms is generally to be recommended. These should be calibrated by several techniques, starting with an oversmoothed bandwidth and the normal reference rule.

For data beyond three dimensions, density estimates may be computed and slices such as $\widehat{f}(x,y,z,t=t_0)$ visualized. If advanced hardware is available, the surfaces can be animated as $t$ varies continuously over an interval $(t_0,t_1)$; see Scott ([35,38]). Obviously, this is most useful for data in four and five dimensions. In any case, multivariate density estimation and visualization are important modern tools for EDA.

Acknowledgments. This research was supported in part by the National Science Foundation grants NSF EIA-9983459 (digital government) and DMS 02-04723 (non-parametric methodology).