Tracking biosignals is crucial for monitoring wellness and preempting the development of severe medical conditions. Today, wearable devices can conveniently record various biosignals, creating the opportunity to monitor health status without disruption to one's daily routine. Despite widespread use of wearable devices and existing digital biomarkers, the absence of curated data with annotated medical labels hinders the development of new biomarkers to measure common health conditions. In fact, medical datasets are usually small in comparison to other domains, which is an obstacle for developing neural network models for biosignals. To address this challenge, we have employed self-supervised learning using the unlabeled sensor data collected under informed consent from the large longitudinal Apple Heart and Movement Study (AHMS) to train foundation models for two common biosignals: photoplethysmography (PPG) and electrocardiogram (ECG) recorded on Apple Watch. We curated PPG and ECG datasets from AHMS that include data from ${\sim} 141$K participants spanning ${\sim} 3$ years. Our self-supervised learning framework includes participant level positive pair selection, stochastic augmentation module and a regularized contrastive loss optimized with momentum training, and generalizes well to both PPG and ECG modalities. We show that the pre-trained foundation models readily encode information regarding participants' demographics and health conditions. To the best of our knowledge, this is the first study that builds foundation models using large-scale PPG and ECG data collected via wearable consumer devices $\textendash$ prior works have commonly used smaller-size datasets collected in clinical and experimental settings. We believe PPG and ECG foundation models can enhance future wearable devices by reducing the reliance on labeled data and hold the potential to help the users improve their health.

Robust principal component analysis (PCA) is a key technique for dynamical high-dimensional data analysis, including background subtraction for surveillance video. Typically, robust PCA requires all observations to be stored in memory before processing. The batch manner makes robust PCA inefficient for big data. In this paper, we develop an efficient online robust PCA method, namely, online moving window robust principal component analysis (OMWRPCA). Unlike the existing algorithms, OMWRPCA can successfully track not only slowly changing subspaces but also abruptly changing subspaces. Embedding hypothesis testing into the algorithm enables OMWRPCA to detect change points of the underlying subspaces. Extensive numerical experiments, including real-time background subtraction, demonstrate the superior performance of OMWRPCA compared with other state-of-the-art approaches.

We further delay embedding framework application to multiple time series for extraction of their potential causal interactions. We introduce a novel geometric model-free causality measure that can efficiently detect linear and nonlinear causal interactions between time series with no prior information or parameter estimation. Using multivariate delay embedding, we construct a point cloud from a set of time domain signals and propose the inverse of its fractal dimension as a causal interaction measure between the corresponding time series. Correlation dimension estimation is fully exploited as a fractal-based method for uncovering the dimensions of generated point clouds. Extensive simulation results are presented to substantiate the capabilities of the proposed approach.

Parkinson's disease (PD) is a debilitating neurodegenerative disease excessively affecting millions of patients. Early diagnosis of PD is critical as manifestation of symptoms occur many years after the onset of neurodegenration, when more than 60\% of dopaminergic neurons are lost. Since there is no definite diagnosis of PD, the early management of disease is a significant challenge in the field of PD therapeutics. Therefore, identifying valid biomarkers that can characterize the progression of PD has lately received growing attentions in PD research community. In this paper, we employ a multi-task learning regression framework for prediction of Parkinson's disease progression, where each task is the prediction of PD rating scales at one future time point. We then use the model to identify the important biomarkers predictive of disease progression. We adopt a graph regularization approach to capture the relationship between different tasks and penalize large variations of the model at consecutive future time points. We have carried out comprehensive experiments using different categories of measurements at baseline from Parkinson's Progression Markers Initiative (PPMI) database to predict the severity of PD, measured by unified PD rating scale. We use the learned model to identify the biomarkers with significant contribution in prediction of PD progression. Our results confirm some of the important biomarkers identified in existing medical studies, validate some of the biomarkers that have been observed as a potential marker of PD and discover new biomarkers that have not yet been investigated.

Four characteristic points of pulse pressure waves-the systolic peak, the anacrotic notch, the dicrotic notch, and the diastolic foot-are used to estimate various aspects of cardiovascular function, such as heart rate and augmentation index. We propose a novel approach to extracting these characteristic points using a topological signal processing framework. We characterize the topology of the signals using a collection of persistence intervals, which are encapsulated in a persistence diagram. The characteristic points are identified based on their time of occurrence and their distance from the identity line in the persistence diagram. We validate this approach by collecting radial pulse pressure data from twenty-eight participants using a wearable tonometer, and computing the peripheral augmentation index using a traditional derivative-based method and our novel persistence-based method. The augmentation index values computed using the two methods are statistically indistinguishable, suggesting that this representation merits further exploration as a tool for analyzing pulse pressure waves.

The goal of this study is to find evidence of cyclicity or periodicity in data with low computational complexity and high accuracy. Using delay embeddings, we transform the timedomain signal into a point cloud, whose topology reflects the periodic behavior of the signal. Persistent homology is employed to determine the underlying manifold of the point cloud, and the Euler characteristic provides for a fast computation of topology of the resulting manifold. We apply the introduced approach to breathing sound signals for wheeze detection. Our experiments substantiate the capabilities of the proposed method.

We propose a new approach to detect and quantify the periodic structure of dynamical systems using topological methods. We propose to use delay-coordinate embedding as a tool to detect the presence of harmonic structures by using persistent homology for robust analysis of point clouds of delay-coordinate embeddings. To discover the proper delay, we propose an autocorrelation like (ACL) function of the signals, and apply the introduced topological approach to analyze breathing sound signals for wheeze detection. Experiments have been carried out to substantiate the capabilities of the proposed method.