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An Intrinsic Characterization of Shannon's and Rényi's Entropy.
Entropy (Basel)
December 2024
Institute of Mathematics, University of Mannheim, 68131 Mannheim, Germany.
All characterizations of the Shannon entropy include the so-called chain rule, a formula on a hierarchically structured probability distribution, which is based on at least two elementary distributions. We show that the chain rule can be split into two natural components, the well-known additivity of the entropy in case of cross-products and a variant of the chain rule that involves only a single elementary distribution. The latter is given as a proportionality relation and, hence, allows a vague interpretation as self-similarity, hence intrinsic property of the Shannon entropy.
View Article and Find Full Text PDFReliable measures of rest-activity rhythm fragmentation: how many days are needed?
Eur Rev Aging Phys Act
October 2024
Epidemiology of Ageing and Neurodegenerative Diseases, Université Paris Cité, INSERM, U1153, CRESS, 10 Avenue de Verdun, Paris, 75010, France.
Background: A more fragmented, less stable rest-activity rhythm (RAR) is emerging as a risk factor for health. Accelerometer devices are increasingly used to measure RAR fragmentation using metrics such as inter-daily stability (IS), intradaily variability (IV), transition probabilities (TP), self-similarity parameter (α), and activity balance index (ABI). These metrics were proposed in the context of long period of wear but, in real life, non-wear might introduce measurement bias.
View Article and Find Full Text PDFMorphological Prediction of CPAP Associated Acute Respiratory Instability.
Ann Am Thorac Soc
September 2024
Beth Israel Deaconess Medical Center, Medicine, Boston, Massachusetts, United States.
Rationale: Multiple mechanisms are involved in the pathogenesis of obstructive sleep apnea (OSA). Elevated loop gain is a key target for precision OSA care and may be associated with treatment intolerance when the upper airway is the sole therapeutic target. Morphological or computational estimation of LG is not yet widely available or fully validated - there is a need for improved phenotyping/endotyping of apnea to advance its therapy and prognosis.
View Article and Find Full Text PDFAccurate identification of community-dwelling older adults at high fall risk can facilitate timely intervention and significantly reduce fall incidents. Analyzing gait and balance capabilities via feature extraction and modeling through sensor-based motion data has emerged as a viable approach for fall risk assessment. However, the existing approaches for extracting key features related to fall risk lack inclusiveness, with limited consideration of the non-linear characteristics of sensor signals, such as signal complexity, self-similarity, and local stability.
View Article and Find Full Text PDFMeasures of fragmentation of rest activity patterns: mathematical properties and interpretability based on accelerometer real life data.
BMC Med Res Methodol
June 2024
Université Paris Cité, INSERM, U1153, CRESS, Epidemiology of Ageing and Neurodegenerative Diseases, 10 Av de Verdun, 75010, Paris, France.
Accelerometers, devices that measure body movements, have become valuable tools for studying the fragmentation of rest-activity patterns, a core circadian rhythm dimension, using metrics such as inter-daily stability (IS), intradaily variability (IV), transition probability (TP), and self-similarity parameter (named ). However, their use remains mainly empirical. Therefore, we investigated the mathematical properties and interpretability of rest-activity fragmentation metrics by providing mathematical proofs for the ranges of IS and IV, proposing maximum likelihood and Bayesian estimators for TP, introducing the activity balance index (ABI) metric, a transformation of , and describing distributions of these metrics in real-life setting.
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