Research in CMS Project Listings 2018

These days, it is very common to monitor simultaneously many attributes that may vary over time. Times at which the underlying data generating mechanism changes, known as changepoints, are of particular interest in many applications. For instance, in internet security monitoring, changepoints may indicate a distributed denial of service attack, while in functional Magnetic Resonance Imaging studies, a changepoint may indicate a significant neurological activity. Frequently, changes occur only in a relatively sparse subset of the attributes. For instance, in the case of stock price data, it may well be the case that market shocks affect only a particular industry sector. A recent method, called `inspect', has been proposed to estimate sparse changes in the mean vector of a high-dimensional data stream (Wang and Samworth, 2018). The aim of this project is to generalise the `inspect' methodology to take advantage of known or hypothesised structures in the mean changes. Examples include spatial clustering of changes in fMRI or other images and sector grouping in financial time series. This project would be particularly suitable for a statistician who has just finished Part II (though it may be accessible to a very strong Part IB student too). It has the potential to involve a mixture of methodological development, generalisation of existing theoretical results, implementation of algorithms in R and applications to real-world datasets, though the weights of this mixture can be tailored to skills and interests of the student involved.

Wang, T., Samworth, R. J. (2018) High-dimensional change point estimation via sparse projection. J. Roy. Statist. Soc., Ser. B, 80, 57–83.

Nearest neighbour methods are a family of techniques whose wide-ranging influence can be felt in many areas of data science. Their use in statistics dates back at least as far as Fix and Hodges (1951) in which the search for a fully nonparametric classification rule led the authors to propose a $k$-nearest neighbour classifier and density estimator. A large part of the popularity of nearest neighbour methods is undoubtedly due to their simplicity and analytic tractability, which make efficient practical implementation possible and open up the challenge of providing a thorough theoretical understanding. For their use in recent work on entropy estimation, classification and independence testing see Berrett, Samworth and Yuan (2018), Cannings, Berrett and Samworth (2017) and Berrett and Samworth (2017). As with many nonparametric techniques there is a tuning parameter, in this case $k$, whose value may affect the performance of the procedures significantly. Heuristically speaking, in many applications $k$ can be seen as controlling a bias--variance trade-off where larger values of $k$ result in larger bias and smaller values of $k$ result in larger variance. Theoretical results provide some insight into the optimal choice of $k$ (e.g. Cannings, Berrett and Samworth, 2017), but in practice $k$ is usually chosen heuristically or empirically, often by cross-validation (where possible). For this project the student would investigate the practical considerations of nearest neighbour methods in one or more of entropy estimation, classification and independence testing. This will principally involve the choice of $k$, but could also include choice of weights in weighted nearest neighbour methods or a study of whether standardising the data (prewhitening) reduces error. One further possibility is to explore the use of approximate nearest neighbours and the inherent trade-off between computational cost and statistical accuracy (e.g. Muja and Lowe, 2014).

References: Berrett, T. B. and Samworth, R. J. (2017) Nonparametric independence testing via mutual information. Available at arXiv:1711.06642.

Berrett, T. B., Samworth, R. J. and Yuan, M. (2018) Efficient multivariate entropy estimation via $k$-nearest neighbour distances. Ann. Statist., to appear.

Cannings, T. I., Berrett, T. B. and Samworth, R. J. (2017) Local nearest neighbour classification with applications to semi-supervised learning. Available at arXiv:1704.00642.

Fix, E. and Hodges, J. L. (1951) Discriminatory analysis -- nonparametric discrimination: Consistency properties. Technical Report number 4, USAF School of Aviation Medicine, Randolph Field, Texas.

Muja, M. and Lowe, D.~G. (2014) Scalable nearest neighbour algorithms for high dimensional data. IEEE Trans. Pattern Anal. Mach. Intell., 36, 2227-2240.