Dynamic covariance estimation via predictive Wishart process with an application on brain connectivity estimation
Dynamic covariance estimation via predictive Wishart process with an application on brain connectivity estimation
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DOI：
10.1016/j.csda.2023.107763
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发表时间：
2023-04
期刊：
影响因子：
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通讯作者：
Rui Meng;Fan Yang;Won Hwa Kim
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文献类型：
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作者：
Rui Meng;Fan Yang;Won Hwa Kim
多变量时间序列数据的复杂依赖关系建模是统计学和机器学习中的一个基本问题。传统上，该任务是通过多元自回归模型和多元广义自回归条件异方差模型等方法来解决的，而基于高斯过程的方法最近由于利用了非参数学习的灵活性而变得流行。然而，除了广义Wishart过程（GWP）之外，直接对协方差矩阵的动态建模的方法很少，而且由于多个高斯过程的极高计算量，广义Wishart过程在小数据集上的应用也受到限制。在这方面，提出了一种新的随机过程，称为预测Wishart过程（PWP），它提供了一个由输入变量索引的正半确定随机矩阵的集合。PWP将全球变暖潜能值的实现过程映射到一个较低维的子空间，以有效地估计每个全球变暖潜能值。考察了它的理论性质，探讨了与之相关的贝叶斯推理和有效的变分期望最大化。此外，在综合生成的时间序列数据上对PWP进行了实证测试，以验证具有竞争力的重建性能和有效的预测性能，并将其应用于人类连接组计划（HCP）的大规模真实功能磁共振成像（fMRI）数据集，以证明其实用性。对脑连通性进行了全面的可视化统计分析，并提出了基于pwp的多任务学习框架，从单个fmri中提取有意义的特征。
Modelling the complex dependence in multivariate time series data is a fundamental problem in statistics and machine learning. Traditionally, the task has been approached with methods such as multivariate autoregressive models and multivariate generalized autoregressive conditional heteroskedasticity models, and Gaussian process based methods are recently becoming popular by leveraging the flexibility of non-parametric learning. However, few methods exist that directly model the dynamics of the covariance matrices except generalized Wishart process (GWP), and even the generalized Wishart process is limited with applications on small dataset due to the extremely high computational capacity induced by multiple Gaussian processes. In this regard, a novel stochastic process named as Predictive Wishart Process (PWP) is proposed, which provides a collection of positive semi-definite random matrices indexed by input variables. The PWP projects process realizations of GWP to a lower dimensional subspace to efficiently estimate every GWP. The theoretical properties of it are examined, and both Bayesian inference and efficient variational expectation maximization are explored in relation to it. Moreover, the PWP is empirically tested on synthetically generated time-series data to validate competitive reconstructive performance and efficient predictive performance, and applied on a large-scale real functional magnetic resonance imaging (fMRI) dataset from Human Connectome Project (HCP) to demonstrate its practicality. A thorough statistical analysis with visualizations is conducted on the brain connectivity, and also a PWP-based multi-task learning framework is proposed to extract meaningful features from individual fMRIs.