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Further applications of the Cauchon algorithm to rank determination and bidiagonal factorization
(2018)
For a class of matrices connected with Cauchon diagrams, Cauchon matrices, and the Cauchon algorithm, a method for determining the rank, and for checking a set of consecutive row (or column) vectors for linear independence is presented. Cauchon diagrams are also linked to the elementary bidiagonal factorization of a matrix and to certain types of rank conditions associated with submatrices called descending rank conditions.
The class of square matrices of order n having a negative determinant and all their minors up to order n-1 nonnegative is considered. A characterization of these matrices is presented which provides an easy test based on the Cauchon algorithm for their recognition. Furthermore, the maximum allowable perturbation of the entry in position (2,2) such that the perturbed matrix remains in this class is given. Finally, it is shown that all matrices lying between two matrices of this class with respect to the checkerboard ordering are contained in this class, too.
In 1970, B.A. Asner, Jr., proved that for a real quasi-stable polynomial, i.e., a polynomial whose zeros lie in the closed left half-plane of the complex plane, its finite Hurwitz matrix is totally nonnegative, i.e., all its minors are nonnegative, and that the converse statement is not true. In this work, we explain this phenomenon in detail, and provide necessary and sufficient conditions for a real polynomial to have a totally nonnegative finite Hurwitz matrix.
In this paper, rectangular matrices whose minors of a given order have the same strict sign are considered and sufficient conditions for their recognition are presented. The results are extended to matrices whose minors of a given order have the same sign or are allowed to vanish. A matrix A is called oscillatory if all its minors are nonnegative and there exists a positive integer k such that A^k has all its minors positive. As a generalization, a new type of matrices, called oscillatory of a specific order, is introduced and some of their properties are investigated.
Positive systems play an important role in systems and control theory and have found applications in multiagent systems, neural networks, systems biology, and more. Positive systems map the nonnegative orthant to itself (and also the non-positive orthant to itself). In other words, they map the set of vectors with zero sign variation to itself. In this article, discrete-time linear systems that map the set of vectors with up to k-1 sign variations to itself are introduced. For the special case k = 1 these reduce to discrete-time positive linear systems. Properties of these systems are analyzed using tools from the theory of sign-regular matrices. In particular, it is shown that almost every solution of such systems converges to the set of vectors with up to k-1 sign variations. It is also shown that these systems induce a positive dynamics of k-dimensional parallelotopes.
Probabilistic Short-Term Low-Voltage Load Forecasting using Bernstein-Polynomial Normalizing Flows
(2021)
The transition to a fully renewable energy grid requires better forecasting of demand at the low-voltage level. However, high fluctuations and increasing electrification cause huge forecast errors with traditional point estimates. Probabilistic load forecasts take future uncertainties into account and thus enables various applications in low-carbon energy systems. We propose an approach for flexible conditional density forecasting of short-term load based on Bernstein-Polynomial Normalizing Flows where a neural network controls the parameters of the flow. In an empirical study with 363 smart meter customers, our density predictions compare favorably against Gaussian and Gaussian mixture densities and also outperform a non-parametric approach based on the pinball loss for 24h-ahead load forecasting for two different neural network architectures.
Nowadays, the importance of early active patient mobilization in the recovery and rehabilitation phase has increased significantly. One way to involve patients in the treatment is a gamification-like approach, which is one of the methods of motivation in various life processes. This article shows a system prototype for patients who require physical activity because of active early mobilization after medical interventions or during illness. Bedridden patients and people with a sedentary lifestyle (predominantly lying in bed) are also potential users. The main idea for the concept was non-contact system implementation for the patients making them feel effortless during its usage. The system consists of three related parts: hardware, software, and game application. To test the relevance and coherence of the system, it was used by 35 people. The participants were asked to play a video game requiring them to make body movements while lying down. Then they were asked to take part in a small survey to evaluate the system's usability. As a result, we offer a prototype consisting of hardware and software parts that can increase and diversify physical activity during active early mobilization of patients and prevent the occurrence of possible health problems due to predominantly low activity. The proposed design can be possibly implemented in hospitals, rehabilitation centers, and even at home.
This study investigates the application of Force Sensing Resistor (FSR) sensors and machine learning algorithms for non-invasive body position monitoring during sleep. Although reliable, traditional methods like Polysomnography (PSG) are invasive and unsuited for extended home-based monitoring. Our approach utilizes FSR sensors placed beneath the mattress to detect body positions effectively. We employed machine learning techniques, specifically Random Forest (RF), K-Nearest Neighbors (KNN), and XGBoost algorithms, to analyze the sensor data. The models were trained and tested using data from a controlled study with 15 subjects assuming various sleep positions. The performance of these models was evaluated based on accuracy and confusion matrices. The results indicate XGBoost as the most effective model for this application, followed by RF and KNN, offering promising avenues for home-based sleep monitoring systems.