In this work, the concept of Braced Fourier Continuation and Regression
(BFCR) is introduced. BFCR is a novel and computationally efficient means of
finding nonlinear regressions or trend lines in arbitrary one-dimensional data
sets. The Braced Fourier Continuation (BFC) and BFCR algorithms are first
outlined, followed by a discussion of the properties of BFCR as well as
demonstrations of how BFCR trend lines may be used effectively for anomaly
detection both within and at the edges of arbitrary one-dimensional data sets.
Finally, potential issues which may arise while using BFCR for anomaly
detection as well as possible mitigation techniques are outlined and discussed.
All source code and example data sets are either referenced or available via
GitHub, and all associated code is written entirely in Python.

本篇论文介绍了 Braced Fourier Continuation and Regression（BFCR）的概念，它是一种在任意一维数据集中寻找非线性回归或趋势线的新颖且计算高效的方法。BFCR 算法和 BFCR 趋势线的性质被讨论，并展示了如何有效使用 BFCR 趋势线在任意一维数据集的内部和边缘进行异常检测。最后，介绍和讨论了使用 BFCR 进行异常检测可能出现的潜在问题和可能的缓解技术。所有源代码和示例数据集都可在 GitHub 上找到或引用，并且所有相关代码完全由 Python 编写。

加强型傅里叶延续与回归用于异常检测

Braced Fourier Continuation and Regression for Anomaly Detection

For minimization problems without 2nd derivative information, methods that
estimate Hessian ma- trices can be very effective. However, conventional
techniques generate dense matrices that are prohibitive for large problems.
Limited-memory compact representations express the dense arrays in terms of a
low rank representation and have become the state-of-the-art for software
implementations on large deterministic problems. We develop new compact
representations that are parameterized by a choice of vectors and that reduce
to existing well known formulas for special choices. We demonstrate
effectiveness of the compact representations for large eigenvalue computations,
tensor factorizations and nonlinear regressions.

无需二阶导数信息的最小化问题，通过估计 Hessian 矩阵的方法可以非常有效。然而，传统技术产生的密集矩阵对于大型问题来说是不可行的。有限内存紧凑表示以低秩表示方式表达密集数组，并成为大型确定性问题软件实现的最新技术。我们开发了一种新的紧凑表示方法，通过选择向量进行参数化，并对特定选择的现有公式进行减少。我们展示了紧凑表示在大量特征值计算、张量分解和非线性回归中的有效性。