Tests for separability in nonparametric covariance operators of random surfaces
Tests for separability in nonparametric covariance operators of random surfaces
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DOI：
10.1214/16-aos1495
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发表时间：
2015-05
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影响因子：
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通讯作者：
J. Aston;D. Pigoli;Shahin Tavakoli
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文献类型：
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作者：
J. Aston;D. Pigoli;Shahin Tavakoli
随机图像或超曲面的协方差算子的可分性假设在应用中可以有很大的用处，特别是在由于计算原因或由于小样本大小而无法准确估计完整协方差结构的情况下。然而，推理工具，以验证这一假设是有点缺乏在高维或功能{数据分析}设置，这一假设是最相关的。我们建议在这里测试可分性，专注于$K$维投影之间的差异的协方差算子和非参数可分离的近似。我们投影到的子空间是由在可分性假设下估计的协方差算子的本征函数生成的，否定了估计完整的不可分协方差的需要。我们证明了样本协方差算子的重标差及其可分离近似是渐近高斯的。作为这一结果的副产品，我们在高斯假设下推导出渐近关键检验，并提出了近似检验统计量分布的自举方法。我们探讨有限样本性能，通过模拟研究，并提出了一个应用程序，从语音语言学数据集的对数声谱图图像。
The assumption of separability of the covariance operator for a random image or hypersurface can be of substantial use in applications, especially in situations where the accurate estimation of the full covariance structure is unfeasible, either for computational reasons, or due to a small sample size. However, inferential tools to verify this assumption are somewhat lacking in high-dimensional or functional {data analysis} settings, where this assumption is most relevant. We propose here to test separability by focusing on $K$-dimensional projections of the difference between the covariance operator and a nonparametric separable approximation. The subspace we project onto is one generated by the eigenfunctions of the covariance operator estimated under the separability hypothesis, negating the need to ever estimate the full non-separable covariance. We show that the rescaled difference of the sample covariance operator with its separable approximation is asymptotically Gaussian. As a by-product of this result, we derive asymptotically pivotal tests under Gaussian assumptions, and propose bootstrap methods for approximating the distribution of the test statistics. We probe the finite sample performance through simulations studies, and present an application to log-spectrogram images from a phonetic linguistics dataset.