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Machine learning and artificial intelligence techniques have been applied to construct online portfolio selection strategies recently. A popular and state-of-the-art family of strategies is to explore the reversion phenomenon through online learning algorithms and statistical prediction models. Despite gaining promising results on some benchmark datasets, these strategies often adopt a single model based on a selection criterion (e.g., breakdown point) for predicting future price. However, such model selection is often unstable and may cause unnecessarily high variability in the final estimation, leading to poor prediction performance in real datasets and thus non-optimal portfolios. To overcome the drawbacks, in this article, we propose to exploit the reversion phenomenon by using combination forecasting estimators and design a novel online portfolio selection strategy, named Combination Forecasting Reversion  (CFR), which outputs optimal portfolios based on the improved reversion estimator. We further present two efficient CFR implementations based on online Newton step (ONS) and online gradient descent (OGD) algorithms, respectively, and theoretically analyze their regret bounds, which guarantee that the online CFR model performs as well as the best CFR model in hindsight. We evaluate the proposed algorithms on various real markets with extensive experiments. Empirical results show that CFR can effectively overcome the drawbacks of existing reversion strategies and achieve the state-of-the-art performance.

Abstract:In this paper we make a Lie symmetry analysis of a generalized nonlinear beam equation with both second-order and fourth-order wave terms, which is extended from the classical beam equation arising in the historical events of travelling wave behavior in the Golden Gate Bridge in San Francisco. We perform a complete Lie symmetry group classification by using the equivalence transformation group theory for the equation under consideration. Lie symmetry reductions of a nonlinear beam-like equation which are singled out from the classification results are investigated. Some classes of exact solutions, including solitary wave solutions, triangular periodic wave solutions and rational solutions of the nonlinear beam-like equations are constructed by means of the reductions and symbolic computation.

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