Functional Object Data Analysis and its Applications

当语言学家试图确定不同语言之间的关联，或者神经学家希望知道大脑的一个部分如何与另一个部分关联时，如何分析既复杂又海量的数据是一个基本问题。然而，统计领域的一个领域，即函数数据分析，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。这个奖学金的目的是进行功能性数据分析，并推动它的发展，以便能够调查更复杂的数据。这将需要建立一个仔细的统计框架来分析这些职能，即使在这些职能有严格关系的情况下也是如此。例如，不同语言之间的比较(例如，法语和意大利语在数量上的不同)可以在函数数据的框架内进行，但不能在不考虑如何分析数据以考虑其特定属性的情况下进行。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学声音是明智的。事实证明，这与形状分析有关，其中一个简单的例子可能是如何描述从伦敦到悉尼的过程。最短的路径是穿过地球的中心，但这是不明智的，所以你必须绕世界一圈。建立形状分析和功能数据之间的联系是该研究会的主要目标。此外，目前大多数大脑分析将大脑分割成许多被称为体素的元素，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3-D物体)，应该一起分析。这是功能数据的另一个例子，在这项研究中开发的方法将使作为单一对象的大脑分析成为可能。这将通过检查大脑成像数据中观察数据之间的依赖类型来完成，并使用这些类型来建立这样的对象。特别令人感兴趣的是对特定任务所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，在能够做到这一点并由此发现对大脑的洞察之前，需要开发完成这一任务所需的统计方法。

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.