Recently, Liu and Zhang studied the rather challenging problem of time series forecasting from the perspective of compressed sensing. They proposed a no-learning method, named Convolution Nuclear Norm Minimization (CNNM), and proved that CNNM can exactly recover the future part of a series from its observed part, provided that the series is convolutionally low-rank. While impressive, the convolutional low-rankness condition may not be satisfied whenever the series is far from being seasonal, and is in fact brittle to the presence of trends and dynamics. This paper tries to approach the issues by integrating a learnable, orthonormal transformation into CNNM, with the purpose for converting the series of involute structures into regular signals of convolutionally low-rank. We prove that the resultant model, termed Learning-Based CNNM (LbCNNM), strictly succeeds in identifying the future part of a series, as long as the transform of the series is convolutionally low-rank. To learn proper transformations that may meet the required success conditions, we devise an interpretable method based on Principal Component Pursuit (PCP). Equipped with this learning method and some elaborate data argumentation skills, LbCNNM not only can handle well the major components of time series (including trends, seasonality and dynamics), but also can make use of the forecasts provided by some other forecasting methods; this means LbCNNM can be used as a general tool for model combination. Extensive experiments on 100,452 real-world time series from Time Series Data Library (TSDL) and M4 Competition (M4) demonstrate the superior performance of LbCNNM.
翻译:最近,刘刘和张从压缩遥感的角度研究了相当具有挑战性的时间序列预测问题。他们提出了一种不学习的方法,名为“革命核规范最小化 ” ( CNNM ), 并证明CNNM能够完全从观察到的系列中的今后部分恢复其观察部分, 但前提是该系列是进化式低级的。 虽然令人印象深刻, 进化式低级条件可能无法满足, 只要该系列的变换远不是季节性的, 并且事实上与趋势和动态存在相去甚远。 本文试图通过将可学习的、 异常的转化转化为CNM CN, 目的是将演动性结构系列转换成正常的低级信号。 我们证明, 成型的模型,即以学习为基础的CN4 CN4 (LbCNNM ), 只能通过学习模型和精细化的周期性变换模型, 来展示这个周期性变异性模型(LNMM 4), 并且只能用这种高级数据序列的精细度预测方法,, 也可以用这种高级工具 和精细的周期性预测方法, 。