What are the differences among Proper Orthogonal Decomposition (POD), Singular value decomposition (SVD) and principal component analysis (PCA)?

Proper Orthogonal Decomposition (POD), singular value decomposition (SVD), and principal component analysis (PCA) are three eigenvalue methods used to reduce a high-dimensional data set into fewer dimensions while retaining important information. Online articles,e.g., this Wikipedia article, say that these methods are 'related' but does not specify the exact relation.

What is the intuitive relationship between POD, and SVD, PCA? How to decide which one to choose?

• This is not a proper answer, but I believe the 3 are the same mathematically. In particular, SVD is the operation of decomposing a matrix into orthogonal modes. If the data is a time series, then it is referred to as POD; if the data is sampled from some statistical measure, then it is referred to as PCA. Apr 18, 2022 at 21:36

1. Singular Value Decomposition (SVD): SVD is a mathematical technique that decomposes a matrix into three matrices: U, Σ, and V*. It is a general matrix factorization method that can be applied to any matrix, not just data matrices. In the context of data analysis, SVD can be applied to the data covariance matrix to extract its eigenvectors (principal components) and eigenvalues (variance explained). PCA is actually a specific application of SVD to perform dimensionality reduction on data.

2. Principal Component Analysis (PCA): PCA is a statistical method that uses SVD as its core technique. The principal components obtained through PCA are orthogonal to each other and capture the most significant directions of variation in the data.

3. Proper Orthogonal Decomposition (POD): POD is closely related to SVD and is often used in fluid dynamics, structural mechanics, and data analysis. It is a specific application of SVD to analyze and extract dominant modes of variation in data. In POD, SVD is applied to snapshot data (time-series data) to identify the most important spatial modes, which represent the primary patterns of variation in the data. POD is widely used in the analysis of fluid flow simulations, modal analysis of structures, and other fields where data can be represented as snapshots in time.

How to Decide Which Method to Choose: The choice between PCA, POD, and SVD depends on the specific problem and the nature of the data:

1. PCA is generally used for standard data analysis tasks when the goal is to reduce dimensionality, visualize data, or identify important features.

2. POD is more specific to problems where data is represented as snapshots in time, like fluid flow simulations or structural dynamics. It is well-suited for analyzing time-dependent datasets.

3. SVD, being a general matrix factorization method, is applicable to various problems, including data analysis, image processing, and natural language processing.