Functional Object Data Analysis and its Applications

When linguists are trying to determine how different languages are related or neuroscientists wish to know how one part of the brain is associated with another, how to analyse data which is both complex and massive is a fundamental question. However, an area of Statistics, namely Functional Data Analysis, where the data is described as mathematical functions rather than numbers or vectors, has recently been shown to be very powerful in these situations.

This fellowship aims to take functional data analysis and advance it so that much more complex data can be investigated. This will require establishing a careful statistical framework for the analysis of such functions even in situations where the functions have strict relationships. By considering the underlying mathematical spaces which the functions lie in, it is possible to construct valid statistical procedures, which preserve these relationships, such as the functions needing to be positive definite or the functions needing to be related by a graph or network.

As an example, comparison between different languages (for example, how is French quantitatively different from Italian) can be carried out in the framework of functional data but not without considering specifically how the data should be analysed to take into account its particular properties. For example in trying to find a path from one language to another, it would be sensible to try to only go via other feasible acoustic sounds. This turns out to be mathematically related to shape analysis, a simple example of which might be how to describe going from London to Sydney. The shortest path is through the centre of the Earth, but this is not sensible, so you have to go round the world. Establishing links between shape analysis and functional data is a major aim of this fellowship.

In addition, most brain analysis currently splits the brain up into lots of elements know as voxels, and then analyses these voxels one by one. However, the brain is really one object (or complex 3-D object) which should be analysed together. This is another example of functional data and the methods developed in this fellowship will enable the analysis of the brain as a single object. This will be done by examining the types of dependence between observations in brain imaging data, and using these to build such an object. Of particular interest will be the analysis of brain connections resulting from particular tasks which will require a mixture of functional data analysis and graphical or network analysis. However, before this can be done and the resulting insights into the brain found, the statistical methods required to do this need to be developed.

There are three main areas of impact from this fellowship.

Firstly there are the academic disciplines associated with the projects within the fellowship. The main academic beneficiary will be statistical science, not simply those working on Functional Data Analysis, but also those in the other areas of shape analysis and network or graphical modelling, as well as applied statisticians making use of methodology in these areas. However, those working in the application areas under investigation will also greatly benefit from the methodology developed. This is evidenced by the strong support shown by the three non-statisticians who have written letters to say that they feel the research will benefit their groups and disciplines more generally.

Secondly, there are the non academic counterparts of those mentioned above. Statistics is a discipline where statisticians working outside academia outnumber those within, and the methodology developed here will be of benefit to these statisticians in government and industry as well, particularly as software development is a critical part of the dissemination of this research. For clinical neuroscientists and speech technologists, the resulting application driven research that will be undertaken will also be of considerable use, and dissemination through mediums such as subject specific journals will allow them to make use of the results.

Finally, there is the wider public engagement that a project of this kind will engender. The ideas of using statistics to determine linguistic relations such that ancient languages can be reheard or so that the function of the brain in particular settings can be understood is something that will naturally be of wide interest, given the public fascination with science. Making full use of this engagement opportunity through webcasts and public lectures will help to engage the public in the work undertaken.