Intensive longitudinal data (ILD) collected in mobile health (mHealth) studies contain rich information on multiple outcomes measured frequently over time that have the potential to capture short-term and long-term dynamics. Motivated by an mHealth study of smoking cessation in which participants self-report the intensity of many emotions multiple times per day, we propose a dynamic factor model that summarizes the ILD as a low-dimensional, interpretable latent process. This model consists of two submodels: (i) a measurement submodel -- a factor model -- that summarizes the multivariate longitudinal outcome as lower-dimensional latent variables and (ii) a structural submodel -- an Ornstein-Uhlenbeck (OU) stochastic process -- that captures the temporal dynamics of the multivariate latent process in continuous time. We derive a closed-form likelihood for the marginal distribution of the outcome and the computationally-simpler sparse precision matrix for the OU process. We propose a block coordinate descent algorithm for estimation. Finally, we apply our method to the mHealth data to summarize the dynamics of 18 different emotions as two latent processes. These latent processes are interpreted by behavioral scientists as the psychological constructs of positive and negative affect and are key in understanding vulnerability to lapsing back to tobacco use among smokers attempting to quit.

翻译：移动健康（mHealth）研究中收集的密集纵向数据（ILD）包含随时间频繁测量的多个结果变量的丰富信息，有望捕捉短期和长期动态。受一项戒烟mHealth研究的启发（该研究中参与者每日多次自我报告多种情绪强度），我们提出了一个动态因子模型，将ILD归纳为低维、可解释的潜在过程。该模型包含两个子模型：（i）测量子模型——因子模型——将多元纵向结果归纳为低维潜在变量；（ii）结构子模型——奥恩斯坦-乌伦贝克（OU）随机过程——捕捉连续时间下多元潜在过程的时域动态。我们推导了结果边际分布的闭式似然函数，以及用于OU过程的计算简化的稀疏精度矩阵。我们提出了一种块坐标下降算法进行估计。最后，我们将该方法应用于mHealth数据，将18种不同情绪的时域动态归纳为两个潜在过程。行为科学家将这些潜在过程解释为积极情感和消极情感的心理构念，这对于理解戒烟尝试者恢复吸烟行为的关键脆弱性具有重要意义。