Functional Object Data Analysis and its Applications

当语言学家试图确定不同的语言之间是如何联系的，或者神经科学家希望知道大脑的一部分是如何与另一部分联系的，如何分析既复杂又大量的数据是一个基本问题。然而，统计的一个领域，即功能数据分析，其中数据被描述为数学函数而不是数字或向量，最近被证明在这些情况下非常强大。该奖学金旨在采取功能数据分析并推进它，以便可以调查更复杂的数据。这将需要建立一个仔细的统计框架来分析这些函数，即使在这些函数之间有严格关系的情况下也是如此。通过考虑函数所在的底层数学空间，可以构建有效的统计程序，从而保持这些关系，例如函数需要是正定的，或者函数需要通过图或网络联系起来。例如，不同语言之间的比较（例如，法语在数量上与意大利语有何不同）可以在功能数据的框架内进行，但不能没有具体考虑如何分析数据以考虑其特定属性。例如，在试图找到从一种语言到另一种语言的路径时，尝试只通过其他可行的声音是明智的。这在数学上与形状分析有关，一个简单的例子可能是如何描述从伦敦到悉尼。最短的路径是穿过地球的中心，但这是不明智的，所以你必须绕地球一圈。在形状分析和功能数据之间建立联系是本研究的主要目标。此外，大多数大脑分析目前将大脑分成许多称为体素的元素，然后逐个分析这些体素。然而，大脑实际上是一个物体（或复杂的三维物体），应该一起分析。这是功能数据的另一个例子，在这个研究中开发的方法将使大脑作为一个单一的对象进行分析。这将通过检查脑成像数据观察之间的依赖类型来完成，并利用这些来构建这样一个对象。特别令人感兴趣的将是对特定任务产生的大脑连接的分析，这将需要功能数据分析和图形或网络分析的混合。然而，在此之前，可以做到这一点，并由此对大脑的见解发现，需要发展所需的统计方法。

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.