ABSTRACT The boxplot as a summary display of a one‐dimensional dataset is typically credited to John Tukey (1915–2000). His ideas for these displays became widespread when Exploratory Data Analysis (EDA) (Tukey 1977) was published; there he defined first the “box‐and‐whisker plot” and then the “schematic plot,” later called “boxplot”. Some displays with a similar flavor appeared 2–3 decades earlier. As a testament to its popularity, enhancements to the boxplot and extensions to weighted and multidimensional data have been proposed in the 50 years since EDA . In this article, I describe the boxplot's predecessors, define the standard boxplot, and review several other versions and recent enhancements that have appeared in the past 50 years. References to R implementations are cited where available. The goal is to clarify the boxplot's definition and origins, provide a review of boxplot‐inspired displays and extensions, including some for bivariate data (but excluding circular and functional data), and illustrate its widespread use for displaying data across many fields. This article is categorized under: Statistical and Graphical Methods of Data Analysis > Statistical Graphics and Visualization Statistical Learning and Exploratory Methods of the Data Sciences > Exploratory Data Analysis Statistical and Graphical Methods of Data Analysis > Robust Methods
ABSTRACT Network‐valued random vectors (NVRVs) provide a statistical framework for settings in which each observational unit is a network rather than a scalar, vector, image, or functional observation. Such data occur in social networks, omics and gene‐regulatory systems, functional brain connectivity, policy and intervention networks, and other domains where relational structure is itself the object of inference. NVRVs incorporate dependence through structured interactions among nodes and edges, thereby presenting significant challenges for statistical modeling, regression, and inference. This article provides an advanced review of inference techniques for NVRVs, with technical expositions focusing on the problem of measuring and testing network change in dynamic and heterogeneous settings. We review approaches for modeling networks as both responses and covariates, emphasizing key strategies such as edge‐wise models, summary‐based regression, latent variable methods, and Bayesian hierarchical formulations. We discuss how heterogeneity and temporal dynamics complicate inference, particularly in the context of network change detection, where current approaches frequently target isolated network features instead of the full network distribution. We consider the perspectives from both frequentist and Bayesian paradigms to identify fundamental gaps in current methodology including limitations in global network testing and the lack of theoretical guarantees in heterogeneous network data. Using the MRN‐114 dataset from the Mind Research Network database as empirical test cases, we perform integrative experiments demonstrating use of the techniques, discussing the advantages and pitfalls of each. Overall, our reviews compare techniques covering the representation of network‐valued observations, the measurement of dynamic change, and the inferential consequence of heterogeneity.
ABSTRACT Functional graphical models (FGMs) extend classical graphical models from multivariate vectors to multivariate random functions, enabling inference on conditional dependence and, in directed settings, directional or causal relationships across functional domains. This review provides a unified overview of recent developments in both undirected and directed FGMs. We summarize key theoretical foundations, estimation methods, and computational challenges arising from infinite dimensionality, operator non‐invertibility, dimension reduction, and non‐Gaussian behavior. Applications in brain connectivity, protein signaling, transportation systems, and longitudinal microbiome studies illustrate the broad potential of FGMs. We also highlight open research directions, including dynamic networks, adaptive truncation, and methodological extensions beyond Gaussian assumptions.
ABSTRACT Signal‐plus‐noise type matrices provide a natural extension of classical sample covariance models and arise widely in high‐dimensional statistics, signal processing, wireless communications, and machine learning. This review surveys recent advances in the spectral theory of such matrices from both global and local perspectives. We first introduce the basic signal‐plus‐noise and generalized signal‐plus‐noise models, emphasizing the role of deterministic signal components and heterogeneous noise structures. We then review the limiting spectral distribution and its characterization through Stieltjes transform equations, together with related analytic properties such as density regularity and support determination. Building on these first‐order results, we summarize spectral separation and exact separation phenomena, which describe how gaps in the limiting spectrum determine the location and number of sample eigenvalues. We further discuss edge behavior of extreme eigenvalues, including Tracy–Widom law, as well as the asymptotic theory of spiked eigenvalues and eigenvectors under general noise environments. In addition, the review covers recent progress on linear spectral statistics and their central limit theorems, highlighting the impact of deterministic signals on second‐order fluctuations. Overall, this survey provides a unified overview of the main spectral phenomena of signal‐plus‐noise type matrices and illustrates their theoretical significance and practical value in high‐dimensional inference, wireless communications, and modern machine learning problems such as noisy manifold learning and kernel‐based sensor fusion.
ABSTRACT Sufficient dimension reduction (SDR) refers to supervised methods of dimension reduction that apply in the context of regression, interpreted broadly. SDR started in the early 1990's with methodology to reduce linearly the predictor dimension without loss of information about the conditional distribution of the response given the predictors. The field grew quickly and today it is vast. The early ideas and methods have been formalized, extended, specialized and adapted to many problems in statistics. A comprehensive synopsis of everything covered by SDR would be truly substantial. Instead, the focus of this overview is on the ideas and philosophy of SDR. While the field is vast, there are a few foundational ideas that define the area generally and have been adapted to different problems. In this overview, we elucidate these ideas by explaining the historical ambience, exploring the genesis of the ideas and discussing how they are adapted for various problems. There is also literature on ‘dimensionality reduction’ that is beyond the scope of this overview. Examples include uniform manifold approximation and projection, neural PCA, kernel PCA, and locally linear embedding. Dimension reduction of data that are not meaningfully stochastic is also outside the scope of this article. The worlds of dimensionality reduction and sufficient dimension reduction were largely developed independently, even when in retrospect the developments involve overlap.
