题' '&' ' 目：CONCEPT OF DENSITY FOR FUNCTIONAL DATA

报告人：Professor Peter G. Hall（The University of Melbourne）

时' '&' ' 间：2010年7月21日（周三）上午10:00

地' '&' ' 点：伟易博治理学院新楼217课堂

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摘要：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