The RSFAS seminar for Thursday 11 March was a double act, by Daning Bi and Adam Nie of ANU. Around 20 people Zoomed in for the talk.
Their problem was motivated by data such as measures of particulate matter in multiple locations in the atmosphere, often taken half-hourly for long periods e.g. six months. Similar series arise in economics, with daily prices of multiple stocks over the course of a year; and also in population health, looking at mortality curves for every year of age in a given country over a period of time. Both of these examples are characterised by large numbers of series over a period not much longer than the number of series.
Interest in describing these series leads to interest in the spectral properties of the autocovariance matrices, which leads to the problem that Daning and Adam solved around a Central Limit Theorem for spiked eigenvalues from those matrices (the spike refers to the first few in order from largest to smallest).
The asymptotic theory is easy (!) if the number of eigenvalues, p, is fixed, but not so simple as p goes to infinity. Adam showed with some theory and some simulation studies that so long as a certain threshold is reached, asymptotically Normal distributions arise for the eigenvalues. This leads to z tests for comparisons e.g. the comparison of mortality curves for two countries.
Daning took the mortality curves example a little further, with a factor analysis that yielded groupings of countries requiring anywhere between one and five factors to describe the covariance matrix. Its my understanding that Daning and Adam carried out this work together while they were both PhD candidates, and I’d like to congratulate them on an elegant piece of mathematical statistics with a solid grounding in real-world applications.
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