Parkinson’s Disease (PD) is a progressive neurodegenerative disease with multiple motor and non-motor characteristics. PD patients commonly face vocal impairments during the early stages of the disease. In this article, the aim is to explain the Unified Parkinson’s Disease Rating Scale (UPDRS) as a measure of the progression of Parkinson’s disease using a set of covariates obtained from voice signals. In particular, a Support Vector Regression (SVR) model based on a combination of kernel functions is introduced. Theoretically, this proposal, that relies on a mixed kernel (global and local) produces an admissible kernel function. The optimal fitting was obtained for the combination given by the product of radial and polynomial basis. Important results are the non-linear relationships inferred from the features to the response, as well as a considerable improvement in prediction performance metrics, when compared to other learning approaches. Furthermore, with knowledge on factors such as age and gender, it is possible to describe the dynamics of patients’ UPDRS from the data collected during their monitoring. In summary, these advances could expand learning processes and intelligent systems to assist in monitoring the evolution of Parkinson’s disease.
Herein, theoretical results are presented to provide insights into the effectiveness of subsampling methods in reducing the amount of instances required in the training stage when applying support vector machines (SVMs) for classification in big data scenarios. Our main theorem states that under some conditions, there exists, with high probability, a feasible solution to the SVM problem for a randomly chosen training subsample, with the corresponding classifier as close as desired (in terms of classification error) to the classifier obtained from training with the complete dataset. The main theorem also reflects the curse of dimensionalityin that the assumptions made for the results are much more restrictive in large dimensions; thus, subsampling methods will perform better in lower dimensions. Additionally, we propose an importance sampling and bagging subsampling method that expands the nearest-neighbors ideas presented in previous work. Using different benchmark examples, the method proposed herein presents a faster solution to the SVM problem (without significant loss in accuracy) compared with the available state-of-the-art techniques.
The use of quadratic forms of the empirical process for the two-sample problem in the context of functional data is considered. The convergence of the family of statistics proposed to a Gaussian limit is established under metric entropy conditions for smooth functional data. The applicability of the proposed methodology is evaluated in examples.
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