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1. An Evolutionary View of the U.S. Supreme Court

   https://doi.org/10.3390/mca26020037  

   Giansiracusa, Noah (June 2021, Mathematical and Computational Applications)

   null (Ed.)

   The voting patterns of the nine justices on the United States Supreme Court continue to fascinate and perplex observers of the Court. While it is commonly understood that the division of the justices into a liberal branch and a conservative branch inevitably drives many case outcomes, there are finer, less transparent divisions within these two main branches that have proven difficult to extract empirically. This study imports methods from evolutionary biology to help illuminate the intricate and often overlooked branching structure of the justices’ voting behavior. Specifically, phylogenetic tree estimation based on voting disagreement rates is used to extend ideal point estimation to the non-Euclidean setting of hyperbolic metrics. After introducing this framework, comparing it to one- and two-dimensional multidimensional scaling, and arguing that it flexibly captures important higher-dimensional voting behavior, a handful of potential ways to apply this tool are presented. The emphasis throughout is on interpreting these judicial trees and extracting qualitative insights from them. 

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2. The dynamics of ideology drift among U.S. Supreme Court justices: A functional data analysis

   https://doi.org/10.1371/journal.pone.0269598  

   Zhou, Xiner; Müller, Hans-Georg (July 2022, PLOS ONE)

   Guimerà, Roger (Ed.)

   We study the U.S. Supreme Court dynamics by analyzing the temporal evolution of the underlying policy positions of the Supreme Court Justices as reflected by their actual voting data, using functional data analysis methods. The proposed fully flexible nonparametric method makes it possible to dissect the time-dynamics of policy positions at the level of individual Justices, as well as providing a comprehensive view of the ideology evolution over the history of Supreme Court since its establishment. In addition to quantifying individual Justice’s policy positions, we uncover average changes over time and also the major patterns of change over time. Additionally, our approach allows for representing highly complex dynamic trajectories by a few principal components which complements other models of analyzing and predicting court behavior. 

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3. Dynamic Inspection of Latent Variables in State-Space Systems

   https://doi.org/10.1109/TASE.2018.2884149  

   Feng, Tianshu; Qian, Xiaoning; Liu, Kaibo; Huang, Shuai (January 2019, IEEE Transactions on Automation Science and Engineering)

   The state-space models (SSMs) are widely used in a variety of areas where a set of observable variables are used to track some latent variables. While most existing works focus on the statistical modeling of the relationship between the latent variables and observable variables or statistical inferences of the latent variables based on the observable variables, it comes to our awareness that an important problem has been largely neglected. In many applications, although the latent variables cannot be routinely acquired, they can be occasionally acquired to enhance the monitoring of the state-space system. Therefore, in this paper, novel dynamic inspection (DI) methods under a general framework of SSMs are developed to identify and inspect the latent variables that are most uncertain. Extensive numeric studies are conducted to demonstrate the effectiveness of the proposed methods. 

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4. Individualized dynamic latent factor model for multi-resolutional data with application to mobile health

   https://doi.org/10.1093/biomet/asae015  

   Zhang, J; Xue, F; Xu, Q; Lee, J; Qu, A (April 2024, Biometrika)

   Summary Mobile health has emerged as a major success for tracking individual health status, due to the popularity and power of smartphones and wearable devices. This has also brought great challenges in handling heterogeneous, multi-resolution data that arise ubiquitously in mobile health due to irregular multivariate measurements collected from individuals. In this paper, we propose an individualized dynamic latent factor model for irregular multi-resolution time series data to interpolate unsampled measurements of time series with low resolution. One major advantage of the proposed method is the capability to integrate multiple irregular time series and multiple subjects by mapping the multi-resolution data to the latent space. In addition, the proposed individualized dynamic latent factor model is applicable to capturing heterogeneous longitudinal information through individualized dynamic latent factors. Our theory provides a bound on the integrated interpolation error and the convergence rate for B-spline approximation methods. Both the simulation studies and the application to smartwatch data demonstrate the superior performance of the proposed method compared to existing methods. 

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5. A spike-and-slab prior for dimension selection in generalized linear network eigenmodels

   https://doi.org/10.1093/biomet/asaf014  

   Loyal, Joshua D; Chen, Yuguo (March 2025, Biometrika)

   Abstract Latent space models are often used to model network data by embedding a network’s nodes into a low-dimensional latent space; however, choosing the dimension of this space remains a challenge. To this end, we begin by formalizing a class of latent space models we call generalized linear network eigenmodels that can model various edge types (binary, ordinal, nonnegative continuous) found in scientific applications. This model class subsumes the traditional eigenmodel by embedding it in a generalized linear model with an exponential dispersion family random component and fixes identifiability issues that hindered interpretability. We propose a Bayesian approach to dimension selection for generalized linear network eigenmodels based on an ordered spike-and-slab prior that provides improved dimension estimation and satisfies several appealing theoretical properties. We show that the model’s posterior is consistent and concentrates on low-dimensional models near the truth. We demonstrate our approach’s consistent dimension selection on simulated networks, and we use generalized linear network eigenmodels to study the effect of covariates on the formation of networks from biology, ecology, and economics and the existence of residual latent structure. 

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