This paper studies a class of Spatial Dynamic Panel Data (SDPD) models in which spatial units are grouped into clusters that have the same parameters within clusters but different across clusters. This specific formulation of the model is called the Clustered SDPD model. When exogenous components are included, efficient estimation of their associated parameters becomes crucial for accurate cluster identification. To address this, we propose a two-step estimation procedure that enhances the efficiency of parameter estimation in a Heterogeneous SDPD framework, where each location has its own parameters, and provides a direct method for estimating fixed effects. This leads to increased precision in cluster detection. Theoretical results establish the consistency and asymptotic properties of the proposed estimators. Monte Carlo simulations and a real data application demonstrate substantial efficiency gains and superior clustering performance compared to standard methods.
This work discusses the article Independent component analysis by robust distance correlation appeared within this same issue of this journal. Particularly, it focuses into the novel bowl transform, a proposal embedded into an Independent Component Analysis strategy to obtain solutions which are resistant to the presence of contamination in the data. The proposal is discussed, and an alternative solution is suggested. This latter exploits data depth contours to embed a multivariate winsorization step within the same proposed strategy.
We are grateful to the editors of Advances in Data Analysis and Classification for bringing together this excellent collection of comments from many esteemed colleagues. We also sincerely thank the discussants for the considerable effort they invested in their thoughtful and stimulating remarks. Their contributions span a wide range of perspectives, from broader reflections on the philosophy of robust ICA to concrete suggestions regarding methodological building blocks and promising directions for future research. In particular, they propose alternative robust whitening procedures, different robustness transformations, and further theoretical developments that enrich the scope of our work. In what follows, we address the discussants’ comments by grouping them according to the main topics that were raised.
This discussion addresses PICARD, a robust approach to Independent Component Analysis (ICA) recently introduced by Leyder et al. My comments center on the distinction between two levels of robustness: that of the local dependence criterion — in particular the newly introduced bowl transform — and that of the full unmixing functional, the composite pipeline mapping the data distribution to the estimated sources. To clarify the mechanisms and trade-offs behind this robustness, I propose targeted empirical diagnostics, notably ablation studies and q-stability paths, where q is the scale parameter of the bowl transform. I further argue for a unified framework in which the dependence contrast, whitening functional, optimization scheme, and perturbation model are treated jointly.
In the correspondence analysis literature, advocates over the past 45 years (or so) have discussed the advantages of studying the square root of the cell counts of a contingency table, rather than the cell counts themselves. Doing so means that the Freeman-Tukey statistic is used as an alternative measure of association to Pearson’s chi-squared statistic. While there are advantages to using the classic Freeman-Tukey statistic, it is not always beneficial since it does not always behave like a chi-squared random variable. To address this issue, various improvements to the classic Freeman-Tukey statistic have been proposed, including a class of Freeman-Tukey statistics introduced in the early 1990s. This paper shows how correspondence analysis can be performed using two special cases of this class. By using these two special cases, we discuss the construction and properties of a low-dimensional space that visually represents the structure of a statistically significant association. We also quantify and interpret the distance between two row (or column) points in this low-dimensional space in terms of the link between their Hellinger and chi-squared differences.
The rapid expansion of digital data has intensified the need for computational methods capable of analyzing complex latent structures across a variety of domains, including textual data. Latent topic models, particularly latent Dirichlet allocation (LDA), are widely used to uncover latent structures in large text corpora. However, the Dirichlet prior on topic proportions imposes structural limitations that reduce the model’s ability to capture complex dependencies among topics. In this paper, we introduce the extended flexible latent Dirichlet allocation (EFLDA), a probabilistic model that extends LDA by allowing richer patterns of dependence among topics. The enriched parametrization of EFLDA improves the model’s ability to represent complex thematic structures, leading to great interpretability in real-world settings. Furthermore, we introduce the concept of sub-topics, defined as specific combinations of topics that provide a deeper understanding of corpora. We develop a collapsed Gibbs sampler for efficient inference and conduct an extensive evaluation on both synthetic data and multiple real-world applications, including mental health discourse, news articles, and microbiome data. Empirical results show that EFLDA outperforms classical LDA and recent alternative approaches in terms of topic coherence, sub-topic detection, and interpretability, while remaining robust across heterogeneous data settings characterized by complex and overlapping latent structures.
Leyder et al. (2026) make an important contribution to robust independent component analysis by introducing PICARD, a sequential procedure that estimates independent sources through a robust dependence criterion based on distance correlation. A central role is played by the bowl transform, which maps far outlying observations close to the origin while remaining continuous and injective, thereby preserving the characterization of independence through vanishing distance correlation. In this discussion, we focus on the statistical role of this transformation within the overall robustness of PICARD. We argue that the bowl transform is not only a device for robustifying a dependence measure, but also induces a geometry that can be interpreted in terms of observation weighting. This leads to two related weighting schemes: one based on the norm of the bowl-transformed observations, and one based on the radial component of the transform. To isolate the effect of such weights, we examine them in a simple projection pursuit setting, which provides a natural projection-based counterpart to ICA. The resulting exploratory study suggests that bowl-induced weights can transfer part of the robustness mechanism of PICARD to projection-based procedures. More broadly, the discussion highlights the importance of disentangling the contributions of robust whitening, bowl-transform geometry, and robust distance correlation in explaining the empirical success of PICARD.
This study explores the PICARD framework for Independent Component Analysis (ICA), focusing on robust implementation methods. We utilize the "Cocktail party" and handwritten digits datasets to discuss features of the framework's application and performance.
In this work, new efficient and robust clustering algorithms for large datasets of mixed-type data are proposed and implemented in a new Python package called db_robust_clust. Their performance is analyzed in rather complex mixed-type datasets, where a wide variety of scenarios is considered regarding size, dimensionality, number of clusters, cluster separation, proportion and type of outlying units, cluster sphericity and imbalance, as well as several interesting interactions. The simulation study comprises extensive computational experiments starting from a baseline configuration (control scenario) in order to evaluate the robustness and efficiency of the algorithms to the progressive degradation of the ideal baseline conditions. Their performance is compared to that of state-of-the-art clustering alternatives. Finally, the goodness and computing time of the methods under evaluation are tested on real datasets of varying sizes and patterns. Mathematical Subject Classifiction: 62H30, 68T09
In this discussion, we comment on the paper by Leyder et al., which introduces a new framework for robust independent component analysis based on robust distance correlation. Our discussion focuses on the role of whitening, the choice of scatter matrices, and the implications of the independence property in robust ICA. We further argue that the minimum distance index is preferable to the Amari index as a performance measure, as it admits a direct connection to the limiting distribution of ICA estimators. Overall, the paper by Leyder et al. constitutes an important contribution to robust blind source separation and opens several promising avenues for future research.
In many scientific studies in recent years, data have been collected at such a high frequency that they can be considered as functional data. In the case when both the response variable to be predicted and the covariates are functions, we provide a novel and easy-to-implement method addressing function-on-function linear modelling and obtain interpretable parameters. Two main types of models are considered: (i) the concurrent model which explains the response curve Y_i(t) at time t from the values at same time t of the covariates X_i^l(t) ; (ii) the (feed-forward) integral model which explains Y_i(t) based on the values of covariate curves X_i^l(s) observed at any times s≤ t . A regularized inference approach is proposed, which accurately selects an appropriate set of basis functions that can be used for functional data reconstruction and at the same time provides smooth and interpretable functional parameters. A functional confidence interval procedure is also proposed which uses the conformalization framework. Numerical studies on simulated data with different scenarios illustrate the good performance of our method to capture the relationship between covariates and response. The method is finally applied to well-known data and compared to a baseline: on Canadian weather data (predicting precipitations from temperature measurements) and on Hawaii ocean data (predicting ocean salinity from temperature, oxygen, chloropigments and density measurements). Our method shows significant improvements on prediction error.
We discuss the robust independent component analysis (ICA) method of Leyder et al. (2026), called PICARD, with particular emphasis on the role of whitening and sphering transformations. We show that robust sphering alters the geometry of the ICA problem, so that the classical reduction to an orthogonal mixing model does not apply in general. Nevertheless, PICARD appears empirically stable despite deviations from exact decorrelation and unit variance, suggesting that these conditions may not be strictly necessary in practice. We further characterize the induced perturbation asymptotically and discuss alternative formulations that incorporate the resulting geometry into the estimation procedure.
Decision trees are powerful predictive models widely used in machine learning. However, simple trees like CART (Classification and Regression Trees) struggle to capture linear relationships. Linear model trees address this by fitting linear models in the nodes, combining tree structure with linear regression for better interpretability. Recently, a new approach called PILOT (Piecewise Linear Organic Tree) was introduced, offering the speed of traditional decision trees with the flexibility of linear model trees. This paper proposes a weighted extension of the PILOT tree algorithm, allowing its full potential to be exploited in a broader range of applications. During training, each observation can be assigned a weight that reflects its importance in the loss function. Crucially, incorporating these weights does not change the computational complexity of the PILOT algorithm. Three applications demonstrate the benefits of weighted PILOT trees. First, they are used in a one-step boosting model that fits two complementary PILOT trees. This approach strikes a balance between the simplicity of a single PILOT tree and the performance of an entire (random) forest of PILOT trees. Second, weighted PILOT is applied to imbalanced regression tasks. By increasing the weights of underrepresented points, prediction accuracy improves in critical regions. Finally, in the third application, we apply weighted PILOT to the covariance shift problem. Applying weights effectively mitigates covariate shift, leading to improved performance compared to the unweighted version.
While the mixture of linear experts models are widely used for capturing heterogeneity in regression and clustering, their traditional Gaussian based formulations often fail with skewed, heavy tailed, or outlier contaminated data. This paper develops a robust mixture of linear experts (MoLE) framework based on normal mean–variance mixture (NMVM) distributions, explicitly modeling both asymmetric probability structures and heterogeneous tail behavior across expert components. The proposed estimation framework employs an optimized ECME algorithm that simultaneously enhances computational efficiency and estimation accuracy. Through extensive simulation studies, we demonstrate the model’s accuracy and robustness under challenging data conditions. Applications to real datasets, including musical tone perception and U.S. county level poverty analysis, highlight the model’s practical advantages in both predictive performance and interpretability. These results suggest that the NMVM–MoLE framework is a powerful and adaptable tool for analyzing real world data with nonstandard distributional features.
This is a discussion of PICARD, the robust independent component analysis (ICA) estimator built on a transformed distance correlation and on the bowl transform.I develop two threads.The first reads the bowl transform as a multivariate redescending ψ-function, makes its exponential decay explicit, and describes how the transformed norm concentrates as the dimension grows.The second examines the reach of the strong-consistency result, relating its population target to classical ICA identifiability and to the behavior of robust whitening under non-elliptical sources.
The objective of this study is to address the challenge of imbalanced data in data mining. To solve the problem, different methods are used to increase the minority class or reduce the majority class. However, reducing the number of samples eliminates important information, and increasing the number of samples can grow, increasing the computational cost and producing synthetic noise samples. In order to address the problem, we propose a technique that utilizes Gas Neural Network to uncover the topology of the distribution of the minority class. This approach aims to decrease the likelihood of generating synthetic samples, which results in additional overhead. We evaluated our method by comparing it with seven other widely recognized oversampling methods, not only on multiple online datasets from the Keel repository (consisting of 13 datasets) but also on two locally collected datasets to ensure the proposed method's independence from data collection methods. To evaluate the performance of the proposed method, we employ five different classifiers, including MLP, SVM, LVQ, RBFN, and C4.5. Furthermore, we compared the proposed method with seven different approaches for dealing with imbalanced datasets, including ADASYN, ADOMS, AHC, Borderline, SMOTE, ROD, and SPIDER. The performance is measured using the Area Under the Curve (AUC) metric. Experimental results demonstrate that our method outperforms conventional methods in classifying imbalanced datasets. Moreover, the process is robust enough to perform better on datasets that have been locally collected by ourselves. The proposed method provides a comprehensive solution to the challenge of imbalanced data in data mining. By utilizing the Neural Gas Network and uncovering the topology of the minority class distribution, it efficiently diminishes the necessity of generating synthetic samples. The experimental results emphasize the superiority of our method in classifying imbalanced datasets compared to conventional approaches.
Random cut forest (RCF) algorithms have been developed for anomaly detection, particularly in time series data. The RCF algorithm is an improved version of the isolation forest (IF) algorithm. Unlike the IF algorithm, the RCF algorithm can determine whether real-time input contains an anomaly by inserting the input into the constructed tree network. Various RCF algorithms, including Robust RCF (RRCF), have been developed, where the cutting procedure is adaptively chosen probabilistically. The RRCF algorithm demonstrates better performance than the IF algorithm, as dimension cuts are decided based on the geometric range of the data, whereas the IF algorithm randomly chooses dimension cuts. However, the overall data structure is not considered in both IF and RRCF, given that split values are chosen randomly. In this paper, we propose new IF and RCF algorithms, referred to as the weighted IF (WIF) and weighted RCF (WRCF) algorithms, respectively. Their split values are determined by considering the density of the given data. To introduce the WIF and WRCF, we first present a new geometric measure, a density measure, which is crucial for constructing the WIF and WRCF. We provide various mathematical properties of the density measure, accompanied by theorems that support and validate our claims through numerical examples. As the tree algorithms rely on geometric analysis, we first provide numerical examples for geometric data sets. Then we apply the WRCF to time series data sets and demonstrate that the proposed method performs better than the existing RRCF.
PICARD combines MCD whitening, a bowl transform, and distance-correlation minimization for robust independent component analysis (ICA). This discussion uses PICARD as a point of departure for reviewing several meanings of robustness in ICA, including resistance to high-leverage observations, robust preprocessing, bounded and slow-growing contrasts, divergence-based weighting, marginal transformations, missingness, and model misspecification. Within this broader landscape, PICARD is best understood as a geometric robustification of dependence-based ICA: it changes the Euclidean geometry in which empirical dependence is computed, thereby reducing the leverage of atypical observations that would otherwise dominate whitening or pairwise distances. Among all the benchmarks in the literature, probability integral transform (PIT) and copula-based mechanisms represent a different yet comparable branch, aimed more directly at marginal scale, distributional shape, and monotone invariance. We further discuss how PICARD’s geometric viewpoint relates to alternative distance-dependence criteria, temporal structure, group ICA, and non-Gaussian component analysis.
The multinomial probit model is a typical statistical model for multiple-choice data applied in many research areas. If we are interested in some quantiles of relative utilities for understanding the distribution of these utilities, the multinomial probit model is unsuitable because it can only interpret the expectation of relative utilities. We thus propose quantile regression analysis methods for multinomial choice data based on joint quantile regression and a multinomial probit model to compare relative utilities with quantiles. Using a joint quantile regression model allows us to consider the conditional quantile points of the relative utilities and explicitly describe the correlation structure in the latent variables. We derive the full conditional distribution under several prior distributions and estimate the model’s parameters from the posterior distribution by Gibbs sampling. The calculation by Gibbs sampling is not only computationally less expensive than the Metropolis–Hastings method, but also easier to implement. We also apply the proposed model to several datasets. Consequently, we obtain interpretable results about different parameters by quantile.
