Proper Orthogonal Decomposition

POD（Proper Orthogonal Decomposition，正交分解）是一种在工程领域广泛应用的有效且精妙的数据分析方法。在高维过程中，POD能够提供数据的低维近似描述，特别适用于实验或数值模拟数据集的模态分解需求。该方法关键在于获取一组正交基函数，以捕捉数据的主要动态特性，这些基函数通常称为经验模态。正交分解在数据压缩、噪声去除、系统识别和流体动力学等领域有广泛应用。文中详述了POD方法的三种主要形式：Karhunen-Loève分解（KLD）、主成分分析（PCA）和奇异值分解（SVD），这些方法在处理POD问题时理论上等效。KLD通过最优线性正交展开提取连续时间随机过程的特征函数集，而PCA则转换可能相关联的变量为主成分，以减少数据维度并保留信息。SVD则是一种在信号和图像处理中广泛应用的线性代数分解方法。文中强调了这些方法在处理离散随机向量的POD问题时的数学一致性，不论从理论还是实际应用角度均具有重要意义。作者还突出了POD方法在工程应用中的重要性和趋于普及，特别强调了方法间的联系描述对工程实践的重要性。

使用带行置换的QR分解计算稀疏矩阵的NULL空间和ORTHOGONAL基的两个简单函数。对于FULL矩阵，Matlab库存函数NULL和ORTH使用SVD分解，这不适用于SPARSE矩阵。从Matlab 2009B开始，QR分解可用于稀疏矩阵，能够有效估计正交基，而无需将矩阵转换为FULL形式。

对字符串中，存在各种特殊符号的，可同时按多种符号（或特殊符号），分解字符串，按字符位置顺序返回。

很不错的Matlab代码，可以很好的解决奇异值分解问题。

矩阵分解的推荐算法MATLAB实现，直接运行main.m

(4) 三角分解: [L,U]=lu(A) 将 A 做对角线分解，使得 A=LU，其中 L 为 下三角矩阵，U 为 上三角矩阵。注意：L 实际上是一个“心理上”的 下三角矩阵*，它事实上是一个置换矩阵 P 的逆矩阵与一个真正下三角矩阵 L1（其对角线元素为 1）的乘积。 例： a=[1 2 3;4 5 6;7 8 9] 比较： [l1,u1,p]=lu(a) 与 [l,u]=lu(a)

1D Signal Two-Level Wavelet Decomposition Overview Two-Level Approximation Decomposition: The original signal is averaged every 4 values to capture the approximate components at this level. Two-Level Detail Decomposition: The difference between every 2 consecutive values in the original signal provides the detailed components at this level. One-Level Detail Decomposition: The difference between the odd and even-indexed values of the original signal is calculated to extract finer details. Signal Recovery: After the decomposition, the signal can be reconstructed by combining the approximations and details from each level. In MATLAB, you can implement these wavelet decompositions to analyze various signals effectively, applying discrete wavelet transform (DWT) functions for both decomposition and reconstruction steps.

讨论了在Matlab环境下进行下上分解的实现方法及其应用。下上分解是解决线性系统的有效工具。

The Conservative Value Method MATLAB code Distribution_Planning_Lot_sizing_Decomposition.m is used for the Lagrangian Relaxation Method and decomposition algorithms applied to high-speed railway (CSHR) catering services. These programs are coded based on the following works: the time-varying demand and pedestrian congestion-based high-speed railway catering distribution planning problem and the batch-based model and decomposition algorithm developed by the Beijing Jiaotong University Research Team under the guidance of Professor Nie Lei. All these codes were written by Dr. Wu Xin. For any inquiries, please contact him. Your feedback is important to us, and the code will continue to be updated and improved in the future. The code includes three main parts: Main Program: main.m is the key component that initiates all related algorithms. The CPLEX solver used in the file can decompose the mixed-integer programming model into submodels. Therefore, the program will work only if the CPLEX interface is correctly installed in the MATLAB environment. Convex Program: A program that solves a series of single-variable convex maximization submodels. The submodels can be solved using the fmincon function in MATLAB's optimization toolbox. HCEA Functions: All files prefixed with HCEA_ embed the Convex Group Method (Frank Wolfe algorithm) as part of the Hybrid Cross-Entropy Algorithm (HCEA). The implementation of HCEA can be used to compare with the proposed decomposition method. Default settings are provided for various configurations.