A recently discovered universal rank-based matrix method to extract trends from noisy time series is described in [1] but the formula for the output matrix elements, implemented there as an open-access supplement MATLAB computer code, is ${\cal O}(N^4)$, with $N$ the matrix dimension. This can become prohibitively large for time series with hundreds of sample points or more. Based on recurrence relations, here we derive a much faster ${\cal O}(N^2)$ algorithm and provide code implementations in MATLAB and in open-source JULIA. In some cases one has the output matrix and needs to solve an inverse problem to obtain the input matrix. A fast algorithm and code for this companion problem, also based on the above recurrence relations, are given. Finally, in the narrower, but common, domains of (i) trend detection and (ii) parameter estimation of a linear trend, users require, not the individual matrix elements, but simply their accumulated mean value. For this latter case we provide a yet faster ${\cal O}(N)$ heuristic approximation that relies on a series of rank one matrices. These algorithms are illustrated on a time series of high energy cosmic rays with $N > 4 \times 10^4$. [1] Universal Rank-Order Transform to Extract Signals from Noisy Data, Glenn Ierley and Alex Kostinski, Phys. Rev. X 9 031039 (2019).

中文翻译：

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一种用于计算矩阵变换的快速算法，用于检测噪声数据中的趋势

最近发现的一种从嘈杂时间序列中提取趋势的通用基于秩的矩阵方法在 [1] 中有描述，但输出矩阵元素的公式在那里作为开放访问补充 MATLAB 计算机代码实现，是 ${\cal O} (N^4)$，其中 $N$ 是矩阵维度。对于具有数百个或更多样本点的时间序列，这可能会变得非常大。基于递推关系，我们在这里推导出更快的 ${\cal O}(N^2)$ 算法，并在 MATLAB 和开源 JULIA 中提供代码实现。在某些情况下，有输出矩阵，需要求解逆问题才能获得输入矩阵。给出了这个伴随问题的快速算法和代码，也基于上述递推关系。最后，在更窄但很常见的地方，(i) 趋势检测和 (ii) 线性趋势的参数估计领域，用户需要的不是单个矩阵元素，而是它们的累积平均值。对于后一种情况，我们提供了一个更快的 ${\cal O}(N)$ 启发式近似，它依赖于一系列的 rank 1 矩阵。这些算法在 $N > 4 \times 10^4$ 的高能宇宙射线时间序列上进行了说明。[1] 从噪声数据中提取信号的通用秩阶变换，Glenn Ierley 和 Alex Kostinski，物理学。修订版 X 9 031039（2019 年）。4 \times 10^4$。[1] 从噪声数据中提取信号的通用秩阶变换，Glenn Ierley 和 Alex Kostinski，物理学。修订版 X 9 031039（2019 年）。4 \times 10^4$。[1] 从噪声数据中提取信号的通用秩阶变换，Glenn Ierley 和 Alex Kostinski，物理学。修订版 X 9 031039（2019 年）。