Abstract The classic model-based paradigm in time series analysis is rooted in the Wold decomposition of the data-generating process into an uncorrelated white noise process. By design, this universal decomposition is indifferent to particular features of a specific prediction problem (e. g., forecasting or signal extraction) – or features driven by the priorities of the data-users. A single optimization principle (one-step ahead forecast error minimization) is proposed by this classical paradigm to address a plethora of prediction problems. In contrast, this paper proposes to reconcile prediction problem structures, user priorities, and optimization principles into a general framework whose scope encompasses the classic approach. We introduce the linear prediction problem (LPP), which in turn yields an LPP objective function. Then one can fit models via LPP minimization, or one can directly optimize the linear filter corresponding to the LPP, yielding the Direct Filter Approach. We provide theoretical results and practical algorithms for both applications of the LPP, and discuss the merits and limitations of each. Our empirical illustrations focus on trend estimation (low-pass filtering) and seasonal adjustment in real-time, i. e., constructing filters that depend only on present and past data.

中文翻译：

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线性预测问题的最佳实时滤波器

摘要时间序列分析中经典的基于模型的范式植根于将数据生成过程的Wold分解为不相关的白噪声过程。通过设计，这种通用分解对于特定的预测问题的特定特征（例如，预测或信号提取）或由数据用户的优先级驱动的特征无动于衷。这种经典范式提出了一种单一的优化原理（提前一步，将预测误差最小化）来解决过多的预测问题。相反，本文提出将预测问题的结构，用户优先级和优化原则协调为一个通用框架，该框架的范围涵盖了经典方法。我们介绍了线性预测问题（LPP），它又产生了LPP目标函数。然后，可以通过LPP最小化来拟合模型，或者可以直接优化与LPP相对应的线性滤波器，从而产生直接滤波器方法。我们为LPP的两种应用提供了理论结果和实用算法，并讨论了每种方法的优缺点。我们的经验说明着重于趋势估计（低通滤波）和实时季节性调整，即 例如，构造仅依赖于当前和过去数据的过滤器。我们的经验说明着重于趋势估计（低通滤波）和实时季节性调整，即 例如，构造仅依赖于当前和过去数据的过滤器。我们的经验说明着重于趋势估计（低通滤波）和实时季节性调整，即 例如，构造仅依赖于当前和过去数据的过滤器。