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A set of curves or images of similar shape is an increasingly common functional data set collected in the sciences. Principal Component Analysis (PCA) is the most widely used technique to decompose variation in functional data. However, the…
In climate studies, detecting spatial patterns that largely deviate from the sample mean still remains a statistical challenge. Although a Principal Component Analysis (PCA), or equivalently a Empirical Orthogonal Functions (EOF)…
Motivated by modern data forms such as images and multi-view data, the multi-attribute graphical model aims to explore the conditional independence structure among vectors. Under the Gaussian assumption, the conditional independence between…
Covariance operators are fundamental in functional data analysis, providing the canonical means to analyse functional variation via the celebrated Karhunen--Lo\`eve expansion. These operators may themselves be subject to variation, for…
Testing for the significance of a subset of regression coefficients in a linear model, a staple of statistical analysis, goes back at least to the work of Fisher who introduced the analysis of variance (ANOVA). We study this problem under…
Principal component analysis (PCA) is a dimensionality reduction method in data analysis that involves diagonalizing the covariance matrix of the dataset. Recently, quantum algorithms have been formulated for PCA based on diagonalizing a…
The functional ANOVA expansion of a multivariate mapping plays a fundamental role in statistics. The expansion is unique once a unique distribution is assigned to the covariates. Recent investigations in the environmental and climate…
In this brief note, we formulate Principal Component Analysis (PCA) over datasets consisting not of points but of distributions, characterized by their location and covariance. Just like the usual PCA on points can be equivalently derived…
Optimal transport has found widespread applications in signal processing and machine learning. Among its many equivalent formulations, optimal transport seeks to reconstruct a random variable/vector with a prescribed distribution at the…
Principal component analysis (PCA) has achieved great success in unsupervised learning by identifying covariance correlations among features. If the data collection fails to capture the covariance information, PCA will not be able to…
Recent findings for optimal transport maps between distribution functions sharing the same copula show that componentwise the solution is the optimal map between marginal distributions. This is an important discovery since in the…
For multivariate nonparametric regression, functional analysis-of-variance (ANOVA) modeling aims to capture the relationship between a response and covariates by decomposing the unknown function into various components, representing main…
Principal component analysis (PCA) is arguably the most widely used approach for large-dimensional factor analysis. While it is effective when the factors are sufficiently strong, it can be inconsistent when the factors are weak and/or the…
Principal Component Analysis (PCA) is a cornerstone of dimensionality reduction, yet its classical formulation relies critically on second-order moments and is therefore fragile in the presence of heavy-tailed data and impulsive noise.…
We propose a new high dimensional semiparametric principal component analysis (PCA) method, named Copula Component Analysis (COCA). The semiparametric model assumes that, after unspecified marginally monotone transformations, the…
Principal component analysis (PCA) is fundamental to statistical machine learning. It extracts latent principal factors that contribute to the most variation of the data. When data are stored across multiple machines, however, communication…
Machine learning and data processing techniques relying on covariance information are widespread as they identify meaningful patterns in unsupervised and unlabeled settings. As a prominent example, Principal Component Analysis (PCA)…
We introduce a variant of (sparse) PCA in which the set of feasible support sets is determined by a graph. In particular, we consider the following setting: given a directed acyclic graph $G$ on $p$ vertices corresponding to variables, the…
How can we discern whether the covariance operator of a stochastic process is of reduced rank, and if so, what its precise rank is? And how can we do so at a given level of confidence? This question is central to a great deal of methods for…
Permutation tests enable testing statistical hypotheses in situations when the distribution of the test statistic is complicated or not available. In some situations, the test statistic under investigation is multivariate, with the multiple…