For smoothing covariance functions, we propose two fast algorithms that scale linearly with the number of observations per function. Most available methods and software cannot smooth covariance matrices of dimension $J \times J$ with $J>500$; the recently introduced sandwich smoother is an exception, but it is not adapted to smooth covariance matrices of large dimensions such as $J \ge 10,000$. Covariance matrices of order $J=10,000$, and even $J=100,000$, are becoming increasingly common, e.g., in 2- and 3-dimensional medical imaging and high-density wearable sensor data. We introduce two new algorithms that can handle very large covariance matrices: 1) FACE: a fast implementation of the sandwich smoother and 2) SVDS: a two-step procedure that first applies singular value decomposition to the data matrix and then smoothes the eigenvectors. Compared to existing techniques, these new algorithms are at least an order of magnitude faster in high dimensions and drastically reduce memory requirements. The new algorithms provide instantaneous (few seconds) smoothing for matrices of dimension $J=10,000$ and very fast ($<$ 10 minutes) smoothing for $J=100,000$. Although SVDS is simpler than FACE, we provide ready to use, scalable R software for FACE. When incorporated into R package {\it refund}, FACE improves the speed of penalized functional regression by an order of magnitude, even for data of normal size ($J <500$). We recommend that FACE be used in practice for the analysis of noisy and high-dimensional functional data.
In recent years, samples of time-varying object data such as time-varying networks that are not in a vector space have been increasingly collected. These data can be viewed as elements of a general metric space that lacks local or global linear structure and therefore common approaches that have been used with great success for the analysis of functional data, such as functional principal component analysis , cannot be applied. In this paper we propose metric covariance, a novel association measure for paired object data lying in a metric space $(Ω,d)$ that we use to define a metric auto- cova…
An Evaluation of Sparse Inverse Covariance Models for Group Functional Connectivity in Molecular Imaging
通过神经成像技术测量大脑区域之间的功能关系，可以深入了解大脑如何在宏观尺度上共享信息。许多功能性连接方法已被开发用于动态成像模式，如功能性MRI（fMRI），而较少的工作集中在静态分子成像技术，如FDG-PET和Tc-99 m HMPAO SPECT跨人群的模型。在这项工作中，我们提供了一个定量评估如何以及三个功能connectivity模型的基础上稀疏逆协方差估计可以准确地恢复黄金标准连接模式跨多个队列和数据集大小。我们比较了学习正则化逆协方差阵的准确性，这些学习独立于使用两个不同的基于组的正则化模型。通过使用大量的SPECT扫描，我们能够对模型在恢复金标准功能连接方面的准确性进行定量评估。
Unified Principal Component Analysis for Sparse and Dense Functional Data under Spatial Dependency
We consider spatially dependent functional data collected under a geostatistics setting, where locations are sampled from a spatial point process. The functional response is the sum of a spatially dependent functional effect and a spatially independent functional nugget effect. Observations on each function are made on discrete time points and contaminated with measurement errors . Under the assumption of spatial stationarity and isotropy, we propose a tensor product spline estimator for the spatio-temporal covariance function . When a coregionalization covariance structure is further assumed,…
Fast Multichannel Source Separation Based on Jointly Diagonalizable Spatial Covariance Matrices
Center for Advanced Intelligence Project (AIP), RIKEN（理化学研究所先进智能项目中心）；Graduate School of Informatics, Kyoto University（京都大学信息学研究生院）；National Institute of Advanced Industrial Science and Technology (AIST)（产业技术综合研究所）
Smoothing and mean-covariance estimation of functional data with a Bayesian hierarchical model
Functional data, with basic observational units being functions (e.g., curves, surfaces) varying over a continuum, are frequently encountered in various applications. While many statistical tools have been developed for functional data analysis, the issue of smoothing all functional observations simultaneously is less studied. Existing methods often focus on smoothing each individual function separately, at the risk of removing important systematic patterns common across functions. We propose a nonparametric Bayesian approach to smooth all functional observations simultaneously and nonparametr…
许多实验记录了连续的轨迹，其中每个轨迹由零附近的振荡和波动组成。这样的轨迹可以被视为零均值函数数据。当高阶矩中存在结构性突变（在轨迹序列上）时，仅通过目视检查并不总是容易发现这些突变。基于脑信号分析中的这一具有挑战性的问题，我们提出了一种检测和测试程序来找到函数协方差的变化点。检测过程是基于累积和统计量（Cumulative sum statistics，CSUM）。函数数据的经典检验过程依赖于零分布，该零分布依赖于无穷多个未知参数，尽管实际上只有有限数量的未知参数可以用于变点存在性的假设检验。本文提供了一些理论见解的影响参数的数量。同时，给出了估计变点的渐近性质。所提出的方法的有效性进行了数值验证，在模拟研究和应用程序，以调查在大鼠脑信号的变化后，实验诱导中风。
Adaptive Functional Thresholding for Sparse Covariance Function Estimation in High Dimensions
Covariance function estimation is a fundamental task in multivariate functional data analysis and arises in many applications. In this paper, we consider estimating sparse covariance functions for high-dimensional functional data, where the number of random functions p is comparable to, or even larger than the sample size n. Aided by the Hilbert--Schmidt norm of functions, we introduce a new class of functional thresholding operators that combine functional versions of thresholding and shrinkage, and propose the adaptive functional thresholding estimator by incorporating the variance effects o…
