Classical second-order spectral analysis, which is based on the Fourier transform of the autocovariance functions, focuses on summarizing the oscillatory behaviors of a time series. However, this type of analysis is subject to two major limitations: first, being covariance-based, it cannot captures oscillatory information beyond the second moment, such as time-irreversibility and kurtosis, and cannot accommodate heavy-tail dependence and infinite variance; second, focusing on a single time series, it is unable to quantify the association between multiple time series and other covariates of interests. In this article, we propose a novel nonparametric approach to the spectral analysis of multiple time series and the associated covariates. The procedure is based on the copula spectral density kernel, which inherits the robustness properties of quantile regression and does not require any distributional assumptions such as the existence of finite moments. Copula spectral density kernels of different pairs are modeled jointly as a matrix to allow flexible smoothing. Through a tensor-product spline model of Cholesky components of the conditional copula spectral density matrix, the approach provides flexible nonparametric estimates of the copula spectral density matrix as nonparametric functions of frequency and covariate while preserving geometric constraints. Empirical performance is evaluated in simulation studies and illustrated through an analysis of stride interval time series.

中文翻译：

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重复时间序列的稳健条件谱分析

基于自协方差函数的傅里叶变换的经典二阶谱分析侧重于总结时间序列的振荡行为。然而，这种类型的分析有两个主要限制：第一，基于协方差，它不能捕获超过二阶矩的振荡信息，例如时间不可逆性和峰态，并且不能适应重尾依赖和无限方差；其次，专注于单个时间序列，无法量化多个时间序列与其他兴趣协变量之间的关联。在本文中，我们提出了一种新的非参数方法来对多个时间序列和相关协变量进行谱分析。该过程基于 copula 谱密度内核，它继承了分位数回归的稳健性属性，并且不需要任何分布假设，例如有限矩的存在。不同对的 Co​​pula 谱密度内核被联合建模为矩阵以允许灵活平滑。通过条件 copula 谱密度矩阵的 Cholesky 分量的张量积样条模型，该方法提供了 copula 谱密度矩阵的灵活非参数估计，作为频率和协变量的非参数函数，同时保留几何约束。在模拟研究中评估经验性能，并通过分析步幅间隔时间序列来说明。不同对的 Co​​pula 谱密度内核被联合建模为矩阵以允许灵活平滑。通过条件 copula 谱密度矩阵的 Cholesky 分量的张量积样条模型，该方法提供了 copula 谱密度矩阵的灵活非参数估计，作为频率和协变量的非参数函数，同时保留几何约束。在模拟研究中评估经验性能，并通过分析步幅间隔时间序列来说明。不同对的 Co​​pula 谱密度内核被联合建模为矩阵以允许灵活平滑。通过条件 copula 谱密度矩阵的 Cholesky 分量的张量积样条模型，该方法提供了 copula 谱密度矩阵的灵活非参数估计，作为频率和协变量的非参数函数，同时保留几何约束。在模拟研究中评估经验性能，并通过分析步幅间隔时间序列来说明。该方法提供 copula 谱密度矩阵的灵活非参数估计，作为频率和协变量的非参数函数，同时保留几何约束。在模拟研究中评估经验性能，并通过分析步幅间隔时间序列来说明。该方法提供 copula 谱密度矩阵的灵活非参数估计，作为频率和协变量的非参数函数，同时保留几何约束。在模拟研究中评估经验性能，并通过分析步幅间隔时间序列来说明。