题 目：CONCEPT OF DENSITY FOR FUNCTIONAL DATA

汇报人：Professor Peter G. Hall（The University of Melbourne）

时 间：2010年7月21日（周三）上午10:00

地 点：拉斯维加斯9888治理学院新楼217教室

提要：The data in a sample of time series, for example graphs of average temperature or average rainfall at different weather stations, can be considered to be different realisations of the series. As for any dataset we can ask which realisations are more extreme; that is, what realisations lie in the tails, and what realisations lie towards the centre (for example, near the mode) of the distribution. Questions such as these raise the notion of probability density for time-series realisations, or for functional data. While it is possible to rank points in a function space in terms of their density within a ball of given nonzero radius, the conventional concept of a probability density function, constructed with respect to a ball of infinitesimal radius, is not well defined, not least because there is no natural analogue of Lebesgue measure in a function space. We suggest instead a transparent and meaningful surrogate for density, defined as the average value of the logarithms of the densities of the distributions of principal component scores, for a given dimension. This `density approximation' is readily estimable from data, and leads directly to estimators of the mode of a distribution of functions. In particular, the mode of a distribution of random functions is well defined, even if the density is not. Methodology for estimating densities of principal component scores is of independent interest; it reveals shape differences that have not