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.Index Terms-Algebraic topology algorithms, audio analysis, biomedical signal processing, topological signal analysis.

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Robust PCA methods are typically batch algorithms which requires loading all observations into memory before processing. This makes them inefficient to process big data. In this paper, we develop an efficient online robust principal component methods, namely online moving window robust principal component analysis (OMWRPCA). Unlike existing algorithms, OMWRPCA can successfully track not only slowly changing subspace but also abruptly changed subspace. By embedding hypothesis testing into the algorithm, OMWRPCA can detect change points of the underlying subspaces. Extensive simulation studies demonstrate the superior performance of OMWRPCA compared with other state-of-art approaches. We also apply the algorithm for real-time background subtraction of surveillance video. * Corresponding author.data. For example, data collected through sensors, cameras and websites are often very noisy and contains error entries or outliers.Various versions of robust PCA have been developed in the past few decades, including [9,6,10,11,12]. Among them, the Robust PCA based on Principal Component Pursuit (RPCA-PCP) [6,13] is the most promising one, as it has both good practical performance and strong theoretical performance guarantees. A comprehensive review of the application of Robust PCA based on Principal Component Pursuit in surveillance video can be found in [8]. RPCA-PCP decomposes the observed matrix into a low-rank matrix and a sparse matrix by solving Principal Component Pursuit. Under mild condition, both the low-rank matrix and the sparse matrix can be recovered exactly.Most robust PCA methods including RPCA-PCP are implemented in the batch manner.

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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.Index Terms-Algebraic topology algorithms, pulse pressure waves, topological signal analysis, wearable sensors.

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