pca-based testing

This study addresses the limited power of traditional goodness-of-fit tests in high-dimensional settings by proposing a novel approach based on the joint distribution of multiple samples. The method employs principal component analysis for dimensionality reduction and constructs confidence sets using k-nearest neighbor estimates of high-density regions, effectively integrating information from order statistics, empirical distribution function values, and moments. It is further extended to the two-sample case. To enhance test power, the approach incorporates non-uniform transformations and probability integral transforms, with inference carried out via permutation testing. Simulation studies demonstrate that the proposed method outperforms or matches existing classical and graphical tests across a range of alternative hypotheses, substantially improving both power and applicability in high-dimensional scenarios.

This paper addresses the goodness-of-fit testing problem for parametric conditional distribution models. We propose a novel method based on residual-marked empirical processes and Conditional Principal Component Analysis (CPCA). Our key contributions are threefold: (i) we introduce CPCA into the distributional testing framework for the first time, constructing three types of test statistics—global, single-directional, and smoothed-combination; (ii) we design an adaptive component selection mechanism that automatically identifies the most discriminative principal components, thereby enhancing directional sensitivity and statistical power; and (iii) Monte Carlo experiments demonstrate that the proposed method significantly outperforms classical tests (e.g., Kolmogorov–Smirnov and Cramér–von Mises types) in finite samples, especially under strong heterogeneity or tail deviations. The approach provides a flexible, interpretable, and powerful framework for conditional distribution validation.

The proportion of variance explained (PVE) by principal components—widely reported in PCA yet long lacking rigorous statistical inference—remains theoretically underdeveloped, particularly for sample-derived components. Method: We introduce the first population-level PVE definition conditioned on sample singular vectors, bridging a fundamental gap in inferential theory. Our approach integrates random matrix theory, high-dimensional asymptotics, and conditional inference frameworks, augmented by bootstrap calibration and scree-plot modeling. Contribution/Results: This enables valid, efficient inference—including confidence intervals, p-values, and point estimates—for arbitrary subsets of principal components, even those selected data-adaptively (e.g., via the elbow rule). Extensive simulations and real gene expression data analyses demonstrate nominal coverage for confidence intervals, high statistical power for hypothesis testing, and substantially improved interpretability and statistical reliability of PCA outputs.

This paper addresses the challenge of simultaneously covering an unknown fixed parameter (e.g., the population mean) and an unobserved random data point (e.g., a new test sample) within the frequentist framework. We propose joint coverage regions (JCRs)—the first method unifying confidence sets and prediction sets under frequentist inference. JCRs are constructed via conditional pivotal quantities, leveraging data splitting and set optimization to guarantee finite-sample joint coverage probability for both the parameter and the new observation. Unlike conventional separate inference procedures, JCRs achieve both statistical validity and computational tractability. Empirical evaluations on mean estimation and structured prediction tasks demonstrate substantial improvements in set compactness and practical utility. Crucially, JCRs establish the first theoretical unification of classical confidence inference with distribution-free prediction, bridging two foundational paradigms in statistical learning.

In principal component analysis (PCA), near-degenerate eigenvalues induce the “isotropy curse,” causing instability in principal direction estimation and diminished interpretability. Method: This paper proposes a novel modeling framework leveraging the eigenvalue multiplicity hierarchy of the covariance matrix. Generalizing probabilistic PCA (PPCA), it employs the ordered geometric structure of flag manifolds to characterize maximum-likelihood estimation under joint eigenvalue multiplicity constraints on signal and noise subspaces. Contribution/Results: We introduce, for the first time, a hierarchical partial-order model selection criterion enabling compact, interpretable low-dimensional modeling. Experiments demonstrate that our method significantly outperforms PPCA—particularly in small-sample regimes and when eigenvalue gaps are weak—achieving superior trade-offs between model complexity and fitting accuracy on both synthetic and real-world data.

This study investigates the capacity and stability of principal component analysis (PCA) to recover latent structures at observational scales ranging from billions to trillions. Through large-scale empirical experiments on datasets with 10 billion and 1 trillion samples—both random and containing embedded latent factors—this work provides the first validation of PCA’s convergence behavior at the trillion-sample scale. The results demonstrate that PCA achieves practical convergence well before reaching a trillion observations: outputs remain highly stable on random data, while in datasets with latent factors, the top three principal components account for 99.996% of the variance, accurately reconstructing the underlying ground-truth structure. These findings offer both theoretical grounding and empirical evidence supporting the application of PCA to ultra-large-scale data.

This work addresses the challenge of accurately attributing detected change points in multivariate time series to specific subsets of variables. The authors propose a post-hoc, nonparametric testing framework that, after an offline change point has been identified, determines whether the change occurs in one of two pre-specified coordinate blocks or in both. Built upon two-sample nonparametric hypothesis testing, the method offers rigorous theoretical guarantees for Type I error control. Empirical evaluations on both synthetic and real-world datasets demonstrate that the proposed approach achieves high attribution accuracy and strong robustness in identifying the components responsible for the change.

This work proposes a dimensionality reduction and structural decomposition approach for multivariate probability density functions characterized by relative and constrained properties, formulated within the Bayesian paradigm. By embedding density functions into a Hilbert space via the centered log-ratio (clr) transformation and applying functional principal component analysis (FPCA), the method achieves effective dimensionality reduction. An orthogonal decomposition is further introduced to disentangle independent and interactive components. The core contribution lies in establishing an optimal variance decomposition framework for multivariate densities in the Bayes space, which endows FPCA outcomes with clear geometric and statistical interpretability. Experimental results on housing and geological datasets demonstrate the method’s efficacy in producing interpretable low-dimensional representations and accurately identifying constituent components.

This study addresses the challenge of goodness-of-fit assessment for black-box models and the limitations of traditional parametric tests when applied to high-dimensional, flexible learners. To this end, we propose SPARK, a general testing framework that projects residuals onto near-orthogonal directions via kernel functions to extract residual signals, combined with a debiasing strategy. The method accommodates both continuous and binary responses alongside predictors of arbitrary dimensionality. We establish the asymptotic properties of the test statistic and prove bootstrap consistency. Simulation studies and real-data analyses validate the effectiveness and flexibility of the proposed approach, demonstrating its capacity to significantly enhance accuracy evaluation for complex machine learning models.

This study addresses the limitation of conventional global goodness-of-fit tests in multivariate settings, which often fail to pinpoint localized model misspecifications. To overcome this, the authors propose a local calibration test based on adaptive partitioning via Beta-trees. Departing from single-statistic global frameworks, the method evaluates whether predicted probabilities fall within finite-sample confidence intervals across data-driven subregions, enabling precise identification and visualization of model inadequacies. By leveraging k-means clustering to generate null distributions and constructing rigorous confidence intervals, the approach effectively detects local deviations in both simulated and real-world datasets, demonstrating superior performance in tasks such as selecting the number of components in mixture models.