Sparsity and entropy are pillar notions of modern theories in signal processing and information t... more Sparsity and entropy are pillar notions of modern theories in signal processing and information theory. However, there is no clear consensus among scientists on the characterization of these notions. Previous efforts have contributed to understand individually sparsity or entropy from specific research interests. This paper proposes a mathematical formalism, a joint axiomatic characterization, which contributes to comprehend (the beauty of) sparsity and entropy. The paper gathers and introduces inherent and first principles criteria as axioms and attributes that jointly characterize sparsity and entropy. The proposed set of axioms is constructive and allows to derive simple or \emph{core functions} and further generalizations. Core sparsity generalizes the Hoyer measure, Gini index and $pq$-means. Core entropy generalizes the R\'{e}nyi entropy and Tsallis entropy, both of which generalize Shannon entropy. Finally, core functions are successfully applied to compressed sensing and...
The analysis of the complexity of biological systems — a proved parameter indicative of the prope... more The analysis of the complexity of biological systems — a proved parameter indicative of the proper functioning of the human body — traditionally involves highly complex algorithms. In this work we use a well-known measure of similarity, the Nor-malized Compression Distance (NCD), to compute the variation of complexity of the human gait. We define the incremental NCD (iNCD) and analyze the duration of the gait cycle time series. To validate iNCD as a metric for this type of analysis, we perform experiments using a four-nodes Wireless Sensor Network (WSN), with one trained volunteer running on a treadmill during one hour, at a comfortable velocity. We show that the joint use of a WSN with iNCD analysis as a useful tool for detecting human gait anomalies at controlled computational load.
Given the high uncertainty in the dynamics of Wireless Sensor Networks (WSN), it is essential to ... more Given the high uncertainty in the dynamics of Wireless Sensor Networks (WSN), it is essential to monitor the Quality-of-Service metrics in order to find the links with poor characteristics. This would allow to implement fault-tolerant protocols to reroute the control packets and avoid zones with untrusted links. However, network monitoring is not used as a permanent analysis tool. This is due both to the conventional use of highly-complex inference algorithms and active approaches such as injection of probing packets. In this paper, we introduce a tree-graph representation for WSN with a general topology, and also suitable for link parameters inference problems. Factor Graphs and the Sum-Product algorithm are considered to develop low complexity schemes for permanent and passive monitoring of multiple link parameters such as Link Loss Rate, Link Error Rate and Link Delay.
In this work we present a Wireless Sensor Network (WSN) system designed for the on-board determin... more In this work we present a Wireless Sensor Network (WSN) system designed for the on-board determination of human gait entropy. The usage of nonlinear entropy-based metrics has proven to be a useful tool for analyzing the complexity of biological systems. The final goal of entropy calculation in this type of biological system is to identify possible causes of future injuries (in order to improve aging) and the early injury detection (ideal for elite athletes). Existing systems for human gait analysis are limited to traditional data gathering, e.g. continuous measurement and wireless transmission to a Data Fusion Center (DFC), due to the computational burden of entropy calculation. In addition, actual systems are likely to interfere the natural movement due to their cumbersome nature. The WSN presented here uses four sensor nodes, located in both ankles and hip sides, and are equipped with triaxial accelerometers. We propose the use of low-complexity algorithms in order to perform on-b...
Sparsity and entropy are pillar notions of modern theories in signal processing and information t... more Sparsity and entropy are pillar notions of modern theories in signal processing and information theory. However, there is no clear consensus among scientists on the characterization of these notions. Previous efforts have contributed to understand individually sparsity or entropy from specific research interests. This paper proposes a mathematical formalism, a joint axiomatic characterization, which contributes to comprehend (the beauty of) sparsity and entropy. The paper gathers and introduces inherent and first principles criteria as axioms and attributes that jointly characterize sparsity and entropy. The proposed set of axioms is constructive and allows to derive simple or \emph{core functions} and further generalizations. Core sparsity generalizes the Hoyer measure, Gini index and $pq$-means. Core entropy generalizes the R\'{e}nyi entropy and Tsallis entropy, both of which generalize Shannon entropy. Finally, core functions are successfully applied to compressed sensing and...
The analysis of the complexity of biological systems — a proved parameter indicative of the prope... more The analysis of the complexity of biological systems — a proved parameter indicative of the proper functioning of the human body — traditionally involves highly complex algorithms. In this work we use a well-known measure of similarity, the Nor-malized Compression Distance (NCD), to compute the variation of complexity of the human gait. We define the incremental NCD (iNCD) and analyze the duration of the gait cycle time series. To validate iNCD as a metric for this type of analysis, we perform experiments using a four-nodes Wireless Sensor Network (WSN), with one trained volunteer running on a treadmill during one hour, at a comfortable velocity. We show that the joint use of a WSN with iNCD analysis as a useful tool for detecting human gait anomalies at controlled computational load.
Given the high uncertainty in the dynamics of Wireless Sensor Networks (WSN), it is essential to ... more Given the high uncertainty in the dynamics of Wireless Sensor Networks (WSN), it is essential to monitor the Quality-of-Service metrics in order to find the links with poor characteristics. This would allow to implement fault-tolerant protocols to reroute the control packets and avoid zones with untrusted links. However, network monitoring is not used as a permanent analysis tool. This is due both to the conventional use of highly-complex inference algorithms and active approaches such as injection of probing packets. In this paper, we introduce a tree-graph representation for WSN with a general topology, and also suitable for link parameters inference problems. Factor Graphs and the Sum-Product algorithm are considered to develop low complexity schemes for permanent and passive monitoring of multiple link parameters such as Link Loss Rate, Link Error Rate and Link Delay.
In this work we present a Wireless Sensor Network (WSN) system designed for the on-board determin... more In this work we present a Wireless Sensor Network (WSN) system designed for the on-board determination of human gait entropy. The usage of nonlinear entropy-based metrics has proven to be a useful tool for analyzing the complexity of biological systems. The final goal of entropy calculation in this type of biological system is to identify possible causes of future injuries (in order to improve aging) and the early injury detection (ideal for elite athletes). Existing systems for human gait analysis are limited to traditional data gathering, e.g. continuous measurement and wireless transmission to a Data Fusion Center (DFC), due to the computational burden of entropy calculation. In addition, actual systems are likely to interfere the natural movement due to their cumbersome nature. The WSN presented here uses four sensor nodes, located in both ankles and hip sides, and are equipped with triaxial accelerometers. We propose the use of low-complexity algorithms in order to perform on-b...
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