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Byline: Treena. Basu, Sebastian. Engel-Wolf, Olaf. Menzer Advances in Machine Learning and availability of state-of-the-art computational resources, along with digitized healthcare data, have set the stage for extensive application of [...]

Byline: Kanadpriya. Basu, Ritwik. Sinha, Aihui. Ong, Treena. Basu Artificially intelligent computer systems are used extensively in medical sciences. Common applications include diagnosing patients, end-to-end drug discovery and development, improving [...]

Fractional diffusion equations model phenomena exhibiting anomalous diffusion that cannot be modeled accurately by second-order diffusion equations. Because of the nonlocal property of fractional differential operators, numerical methods for space-fractional diffusion equations generate complicated dense or full coefficient matrices. Consequently, these numerical methods were traditionally solved by Gaussian elimination, which requires computational work of O(N³) per time step and O(N²) of memory, where N is the number of spatial grid points in the discretization. The significant computational work and memory requirement of the numerical methods impose a serious challenge for the numerical simulation of two- and especially three-dimensional space-fractional diffusion equations. We develop a fast and yet accurate solution method for the implicit finite difference discretization of space-fractional diffusion equations in two space dimensions by carefully analyzing the structure of the coefficient matrix of the finite difference method and delicately decomposing the coefficient matrix into a combination of sparse and structured dense matrices. The fast method has a computational work count of O(N log N) per iteration and a memory requirement of O(N), while retaining the same accuracy as the underlying finite difference method solved with Gaussian elimination. Numerical experiments show that the fast method has a significant reduction of CPU time, from two months and eight days as consumed by the traditional method to less than 40 minutes, with less than one ten-thousandth of the memory required by the traditional method, in the context of a two-dimensional space-fractional diffusion equation with 512 x 512 = 262,144 spatial nodes and 512 time steps. This demonstrates the utility of the method. [ABSTRACT FROM AUTHOR]

Fractional diffusion equations provide an adequate and accurate description of transport processes that exhibit anomalous diffusion that cannot be modeled accurately by classical second-order diffusion equations. However, numerical discretizations of fractional diffusion equations yield full coefficient matrices, which require a computational operation of O(N3) per time step and a memory of O(N2) for a problem of size N. In this paper we develop a fast second-order finite difference method for space-fractional diffusion equations, which only requires memory of O(N) and computational work of O(N log2 N). Numerical experiments show the utility of the method. [ABSTRACT FROM AUTHOR]

Stock trading has tremendous importance not just as a profession but also as an income source for individuals. Many investment account holders use the appreciation of their portfolio (as a combination of stocks or indexes) as income for their retirement years, mostly betting on stocks or indexes with low risk/low volatility. However, every stock-based investment portfolio has an inherent risk to lose money through negative progression and crash. This study presents a novel technique to predict such rare negative events in financial time series (e.g., a drop in the S&P 500 by a certain percent in a designated period of time). We use a time series of approximately seven years (2517 values) of the S&P 500 index stocks with publicly available features: the high, low and close price (HLC). We utilize a Siamese type neural network for pattern recognition in images followed by a bootstrapped image similarity distribution to predict rare events as they pertain to financial market analysis. Extending on literature about rare event classification and stochastic modeling in financial analytics, the proposed method uses a sliding window to store the input features as tabular data (HLC price), creates an image of the time series window, and then uses the feature vector of a pre-trained convolutional neural network (CNN) to leverage pre-event images and predict rare events. This research does not just indicate that our proposed method is capable of distinguishing event images from non-event images, but more importantly, the method is effective even when only limited and strongly imbalanced data is available. [ABSTRACT FROM AUTHOR]