Data-driven Reachability using Christoffel Functions and Conformal Prediction

An important mathematical tool in the analysis of dynamical systems is the approximation of the reach set, i.e., the set of states reachable after a given time from a given initial state. This set is difficult to compute for complex systems even if the system dynamics are known and given by a system of ordinary differential equations with known coefficients. In practice, parameters are often unknown and mathematical models difficult to obtain. Data-based approaches are promised to avoid these difficulties by estimating the reach set based on a sample of states. If a model is available, this training set can be obtained through numerical simulation. In the absence of a model, real-life observations can be used instead. A recently proposed approach for data-based reach set approximation uses Christoffel functions to approximate the reach set. Under certain assumptions, the approximation is guaranteed to converge to the true solution. In this paper, we improve upon these results by notably improving the sample efficiency and relaxing some of the assumptions by exploiting statistical guarantees from conformal prediction with training and calibration sets. In addition, we exploit an incremental way to compute the Christoffel function to avoid the calibration set while maintaining the statistical convergence guarantees. Furthermore, our approach is robust to outliers in the training and calibration set

Brain wave classification using long short - term memory based OPTICAL predictor

Brain-computer interface (BCI) systems having the ability to classify brain waves with greater accuracy are highly desirable. To this end, a number of techniques have been proposed aiming to be able to classify brain waves with high accuracy. However, the ability to classify brain waves and its implementation in real-time is still limited. In this study, we introduce a novel scheme for classifying motor imagery (MI) tasks using electroencephalography (EEG) signal that can be implemented in real-time having high classification accuracy between different MI tasks. We propose a new predictor, OPTICAL, that uses a combination of common spatial pattern (CSP) and long short-term memory (LSTM) network for obtaining improved MI EEG signal classification. A sliding window approach is proposed to obtain the time-series input from the spatially filtered data, which becomes input to the LSTM network. Moreover, instead of using LSTM directly for classification, we use regression based output of the LSTM network as one of the features for classification. On the other hand, linear discriminant analysis (LDA) is used to reduce the dimensionality of the CSP variance based features. The features in the reduced dimensional plane after performing LDA are used as input to the support vector machine (SVM) classifier together with the regression based feature obtained from the LSTM network. The regression based feature further boosts the performance of the proposed OPTICAL predictor. OPTICAL showed significant improvement in the ability to accurately classify left and right-hand MI tasks on two publically available datasets. The improvements in the average misclassification rates are 3.09% and 2.07% for BCI Competition IV Dataset I and GigaDB dataset, respectively. The Matlab code is available at https://github.com/ShiuKumar/OPTICAL