[论文解读] Longitudinal data analysis using matrix completion.

Łukasz Kidziński, Trevor Hastie|arXiv (Cornell University)|Sep 24, 2018

Sparse and Compressive Sensing Techniques参考文献 18被引用 11

一句话总结

本文提出了一种基于迭代SVD的矩阵补全框架，用于从稀疏、不规则的生物医学数据中估计纵向进展曲线。该方法成功建模了脑瘫患儿的运动损伤趋势，解释了30%的变异，并通过低秩表示揭示了不同亚型间的显著进展模式。

ABSTRACT

In clinical practice and biomedical research, measurements are often collected sparsely and irregularly in time while the data acquisition is expensive and inconvenient. Examples include measurements of spine bone mineral density, cancer growth through mammography or biopsy, a progression of defect of vision, or assessment of gait in patients with neurological disorders. Since the data collection is often costly and inconvenient, estimation of progression from sparse observations is of great interest for practitioners. From the statistical standpoint, such data is often analyzed in the context of a mixed-effect model where time is treated as both random and fixed effect. Alternatively, researchers analyze Gaussian processes or functional data where observations are assumed to be drawn from a certain distribution of processes. These models are flexible but rely on probabilistic assumptions and require very careful implementation. In this study, we propose an alternative elementary framework for analyzing longitudinal data, relying on matrix completion. Our method yields point estimates of progression curves by iterative application of the SVD. Our framework covers multivariate longitudinal data, regression and can be easily extended to other settings. We apply our methods to understand trends of progression of motor impairment in children with Cerebral Palsy. Our model approximates individual progression curves and explains 30% of the variability. Low-rank representation of progression trends enables discovering that subtypes of Cerebral Palsy exhibit different progression trends.

研究动机与目标

• 解决从稀疏且时间点不规则的临床测量中估计疾病进展的挑战。
• 为纵向数据提供一种灵活的、非概率性的混合效应模型和高斯过程的替代方法。
• 在多变量和回归设置中实现个体进展曲线的估计。
• 通过利用进展数据中的低秩结构，发现疾病亚型的潜在趋势。

提出的方法

• 对稀疏纵向观测数据矩阵应用迭代奇异值分解（SVD）以完成矩阵填充。
• 使用低秩逼近将个体进展曲线建模为数据中的潜在模式。
• 将问题建模为矩阵补全问题，其中缺失条目对应于未观测到的时间点。
• 通过迭代使用SVD更新低秩估计，直至收敛，以改善曲线估计。
• 通过将协变量整合到矩阵结构中，将该框架扩展至多变量纵向数据和回归问题。
• 通过秩约束优化，在保留个体轨迹的同时捕捉群体水平趋势。

实验结果

研究问题

• RQ1 通过SVD进行矩阵补全是否能有效估计来自稀疏纵向生物医学数据的个体进展曲线？
• RQ2 低秩表示在多大程度上捕捉了脑瘫患儿运动损伤进展中的有意义趋势？
• RQ3 当使用该框架建模时，脑瘫的不同亚型是否表现出不同的进展模式？
• RQ4 与传统统计模型相比，该方法在解释纵向进展变异方面能达到多大程度？

主要发现

• 矩阵补全方法成功地从稀疏临床测量中近似了个体进展曲线。
• 该模型解释了脑瘫患儿运动损伤进展中30%的变异。
• 低秩表示揭示了脑瘫不同亚型表现出截然不同的进展趋势。
• 该方法为混合效应模型和高斯过程提供了一种稳健的、非概率性的替代方案。
• 该框架可轻松扩展至多变量纵向数据和回归设置，且修改极少。
• 基于迭代SVD的矩阵补全产生了稳定且可解释的个体轨迹估计。