Principal Moment Analysis is a method designed for dimension reduction, analysis and visualization of high dimensional multivariate data. It generalizes Principal Component Analysis and allows for significant statistical modeling flexibility, when approximating an unknown underlying probability distribution, by enabling direct analysis of general approximate measures. Through https://principalmomentanalysis.github.io/ we provide an implementation, together with a graphical user interface, of a simplex based version of Principal Moment Analysis.
In 2019, Yoshida et al. introduced a notion of tropical principal component analysis (PCA). The output is a tropical polytope with a fixed number of vertices that best fits the data. We here apply tropical PCA to dimension reduction and visualization of data sampled from the space of phylogenetic trees. Our main results are twofold: the existence of a tropical cell decomposition into regions of fixed tree topology and the development of a stochastic optimization method to estimate the tropical PCA using a Markov Chain Monte Carlo (MCMC) approach. This method performs well with simulation studi…
Principal component analysis (PCA) is a popular dimension reduction technique often used to visualize high-dimensional data structures. In genomics, this can involve millions of variables, but only tens to hundreds of observations. Theoretically, such extreme high-dimensionality will cause biased or inconsistent eigenvector estimates, but in practice the principal component scores are used for visualization with great success. In this paper, we explore when and why the classical principal component scores can be used to visualize structures in high-dimensional data, even when there are few obs…
Fan et al. [$\mathit{Annals}$ $\mathit{of}$ $\mathit{Statistics}$ $\textbf{47}$(6) (2019) 3009-3031] constructed a distributed principal component analysis (PCA) algorithm to reduce the communication cost between multiple servers significantly. However, their algorithm's guarantee is only for sub-Gaussian data. Spurred by this deficiency, this paper enhances the effectiveness of their distributed PCA algorithm by utilizing robust covariance matrix estimators of Minsker [$\mathit{Annals}$ $\mathit{of}$ $\mathit{Statistics}$ $\textbf{46}$(6A) (2018) 2871-2903] and Ke et al. [$\mathit{Statistical…
University of Florida（佛罗里达大学）；University of North Carolina Wilmington（北卡罗来纳大学威尔明顿分校）；University of Waterloo（滑铁卢大学）
Because of high dimensionality, correlation among covariates, and noise contained in data, dimension reduction (DR) techniques are often employed to the application of machine learning algorithms. Principal Component Analysis (PCA), Linear Discriminant Analysis (LDA), and their kernel variants (KPCA, KLDA) are among the most popular DR methods. Recently, Supervised Kernel Principal Component Analysis (SKPCA) has been shown as another successful alternative. In this paper, brief reviews of these popular techniques are presented first. We then conduct a comparative performance study based on thr…
University of Tsukuba（筑波大学）；The Institute of Statistical Mathematics（统计数理研究所）
Principal component analysis (PCA) is a widely used method for data processing, such as for dimension reduction and visualization. Standard PCA is known to be sensitive to outliers, and thus, various robust PCA methods have been proposed. It has been shown that the robustness of many statistical methods can be improved using mode estimation instead of mean estimation, because mode estimation is not significantly affected by the presence of outliers. Thus, this study proposes a modal principal component analysis (MPCA), which is a robust PCA method based on mode estimation. The proposed method …
Georgia Institute of Technology（佐治亚理工学院）；Northwestern University（西北大学）；Ansys Inc.（Ansys公司）；Temple University（天普大学）
The optimization of electric machines at multiple operating points is crucial for applications that require frequent changes on speeds and loads, such as the electric vehicles, to strive for the machine optimal performance across the entire driving cycle. However, the number of objectives that would need to be optimized would significantly increase with the number of operating points considered in the optimization, thus posting a potential problem in regards to the visualization techniques currently in use, such as in the scatter plots of Pareto fronts, the parallel coordinates, and in the pri…
University of Konstanz（康斯坦茨大学）；University of Stuttgart（斯图加特大学）
We present a technique to perform dimensionality reduction on data that is subject to uncertainty. Our method is a generalization of traditional principal component analysis (PCA) to multivariate probability distributions. In comparison to non-linear methods, linear dimensionality reduction techniques have the advantage that the characteristics of such probability distributions remain intact after projection. We derive a representation of the PCA sample covariance matrix that respects potential uncertainty in each of the inputs, building the mathematical foundation of our new method: uncertain…
Texas A&M University（德克萨斯农工大学）；Yale School of Medicine（耶鲁医学院）
Visual summarization of clinical data collected on patients contained within the electronic health record (EHR) may enable precise and rapid triage at the time of patient presentation to an emergency department (ED). The triage process is critical in the appropriate allocation of resources and in anticipating eventual patient disposition, typically admission to the hospital or discharge home. EHR data are high-dimensional and complex, but offer the opportunity to discover and characterize underlying data-driven patient phenotypes. These phenotypes will enable improved, personalized therapeutic…
VinAI Research（VinAI 研究院）；Hong Kong Polytechnic University（香港理工大学）
Principal component analysis is a simple yet useful dimensionality reduction technique in modern machine learning pipelines. In consequential domains such as college admission, healthcare and credit approval, it is imperative to take into account emerging criteria such as the fairness and the robustness of the learned projection. In this paper, we propose a distributionally robust optimization problem for principal component analysis which internalizes a fairness criterion in the objective function. The learned projection thus balances the trade-off between the total reconstruction error and t…
Principal Component Analysis (PCA) is a well known procedure to reduce intrinsic complexity of a dataset, essentially through simplifying the covariance structure or the correlation structure. We introduce a novel algebraic, model-based point of view and provide in particular an extension of the PCA to distributions without second moments by formulating the PCA as a best low rank approximation problem. In contrast to hitherto existing approaches, the approximation is based on a kind of spectral representation, and not on the real space. Nonetheless, the prominent role of the eigenvectors is he…
The paper is concerned with asymptotic properties of the principal components analysis of functional data. The currently available results assume the existence of the fourth moment. We develop analogous results in a setting which does not require this assumption. Instead, we assume that the observed functions are regularly varying. We derive the asymptotic distribution of the sample covariance operator and of the sample functional principal components. We obtain a number of results on the convergence of moments and almost sure convergence. We apply the new theory to establish the consistency o…
