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Flexible Modeling of non-Gaussian Longitudinal Data: Some Approaches using Copula
(Indian Statistical Institute, Kolkata, 2026-03-16) Chattopadhyay, Subhajit
Longitudinal data are common in medical and biological sciences, where measurements are gathered
from subjects over time to explore relationships with explanatory variables (covariates) and to uncover
the underlying mechanisms of dependence among these measurements. The responses observed at each instance can be either discrete or continuous. One of the primary challenges in longitudinal data analysis lies in the non-Gaussian nature of the response variables. As a result, there are relatively few multivariate models in the literature that effectively address the specific characteristics observed in such datasets. In this dissertation, we address four problems concerning longitudinal data analysis by developing new statistical models. These models specifically address the time-related relationships found in various types of non-Gaussian longitudinal data by employing suitable classes of parametric copulas. In the third chapter of this dissertation, we examine a motivating dataset from a recent HIV-AIDS study conducted in Livingstone district, Zambia. The histogram plots of the repeated measurements at each time point reveal asymmetry in the marginal distributions, and pairwise scatter plots uncover nonelliptical dependence patterns. Traditional linear mixed models, typically used for longitudinal data, struggle to capture these complexities effectively. We introduced skew-elliptical copula based mixed models to analyze this continuous data, where we use generalized linear mixed models (GLMM) for the marginals (e.g., Gamma mixed model), and address the temporal dependence of repeated measurements by utilizing copulas associated with skew-elliptical distributions (such as skew-normal/skew-t). The proposed class of copula-based mixed models addresses asymmetry, between-subject variability, and non-standard temporal dependence simultaneously, thereby extending beyond the limitations of standard linear mixed models based on multivariate normality. We estimate the model parameters using the IFM (inference function of margins) method, and outline the procedure for obtaining standard errors of the parameter estimates. To evaluate the performance of this approach under finite sample conditions, rigorous simulation studies are conducted, encompassing skewed and symmetric marginal distributions along with various copula selections. Finally, we apply these models to the HIV dataset and present the insight gained from the analysis. In the fourth chapter of this dissertation, we introduce factor copula models tailored for unbalanced non-Gaussian longitudinal data. Modeling the joint distribution of such data, where subjects may have varying numbers of repeated measurements and responses can be continuous or discrete, poses practical challenges, especially with numerous measurements per subject. Factor copula models, which are canonical vine copulas, leverage latent variables to elucidate the underlying dependence structure of multivariate data. This approach aids in interpretation and implementation for unbalanced longitudinal datasets, enhancing our ability to model complex dependencies effectively. We develop regression models for continuous, binary and ordinal longitudinal data, incorporating covariates, using factor copula constructions with subject-specific latent variables. With consideration for homogeneous within-subject dependence, the proposed models enable feasible parametric inference in moderate to high dimensional scenarios, employing a two-stage (IFM) estimation method. We also present a method for evaluating the residuals of factor copula models to visually assess the goodness of fit. The performance of the proposed models in finite samples is assessed through extensive simulation studies. In empirical analyses, we apply these models to analyze various longitudinal responses from two real-world datasets. Furthermore, we compare the performance of these models with widely used random effects models using standard selection techniques, revealing significant improvements. Our findings suggest that factor copula models can serve as viable alternatives to random effect models, offering deeper insights into the temporal dependence of longitudinal data across diverse contexts.
In the fifth chapter of this dissertation, we address the issue of modeling complex and hidden temporal
dependence of count longitudinal data. Multivariate elliptical copulas are typically preferred in statistical literature to analyze dependence between repeated measurements of longitudinal data since they allow for different choices of the correlation structure. But these copulas lack in flexibility to model dependence and inference is only feasible under parametric restrictions. In this chapter, we propose the
use of finite mixtures of elliptical copulas to enhance the modeling of temporal dependence in discrete
longitudinal data. This approach enables the utilization of distinct correlation matrices within each component of the mixture copula. We theoretically explore the dependence properties of finite mixtures of copulas before employing them to construct regression models for count longitudinal data. Inference for this proposed class of models is based on a composite likelihood approach, and we evaluate the finite sample performance of parameter estimates through extensive simulation studies. To validate the fitting of the proposed models, we extend traditional techniques and introduce the t-plot method to accommodate finite mixtures of elliptical copulas. Finally we apply the proposed models to analyze the temporal dependence within two real-world count longitudinal datasets and demonstrate their superiority over standard elliptical copulas. In the final contributing chapter of this dissertation, we introduce a novel multivariate copula based on the multivariate geometric skew-normal (GSN) distribution. This asymmetric copula serves as an alternative to the skew-normal copula proposed by Azzalini. Unlike the standard skew-normal copula, the multivariate GSN copula retains closure properties under marginalization, which offers computational advantages for modeling multivariate discrete data. In this chapter, we outline the construction of the geometric skew-normal copula and its application in modeling the temporal dependence observed in non-Gaussian longitudinal data. We begin by exploring the theoretical properties of the proposed multivariate copula. Subsequently, we develop regression models tailored for both continuous and discrete longitudinal data using this innovative framework. Notably, the quantile function of this copula remains independent of the correlation matrix of its respective multivariate distribution, offering computational advantages in likelihood inference compared to copulas derived from skew-elliptical distributions proposed by Azzalini. Furthermore, composite likelihood inference becomes feasible for this multivariate copula, allowing for parameter estimation from ordered probit models with the same dependence structure as the geometric skew-normal distribution. We conduct extensive simulation studies to validate the geometric skew-normal copula based models and apply them to analyze the longitudinal dependence of two real-world data sets. Finally, We present our findings in terms of the improvements over regression models based on multivariate Gaussian copulas.
Development of Some Scalable Pattern Recognition Algorithms for Real Life Data Analysis
(2017-11-20) Garai, Partha
A huge amount of data is being generated continuously as a result of recent advancement
and wide use of high-throughput technologies. With the rapid increase in size of data
distributed worldwide, understanding the data has become critical. In this regard, dimensionality
reduction and clustering have become the necessary preprocessing steps of
multiple research areas and applications. One of the important problems of real life large
data sets is uncertainty. Some of the sources of this uncertainty include imprecision in
computation and vagueness in class denitions. The uncertainty may also be present in
the denition of class membership function.
In this background, the thesis addresses the problem of dimensionality reduction and
clustering of real life data sets, in the presence of noise and uncertainty. The thesis rst
presents the problem of feature selection using both type-1 and interval type-2 fuzzyrough
sets, which are eective for dimensionality reduction of real life data sets when
uncertainty is present in the data set. The properties of fuzzy-rough sets allow greater
exibility in handling noisy and real valued data. While the concept of lower approximation
and boundary region of rough sets deals with uncertainty, incompleteness, and vagueness
in class denition, the use of either type-1 or interval type-2 fuzzy sets enables ecient
handling of overlapping classes in uncertain environment. Moreover, a new concept of
\simultaneous attribute selection and feature extraction" is introduced for dimensionality
reduction, integrating judiciously the merits of both feature selection and extraction.
A scalable rough-fuzzy clustering algorithm is introduced for large real life data sets,
where the theory of rough hypercuboid approach, interval type-2 fuzzy sets, and c-means
algorithm are integrated judiciously to handle the uncertainty present in a data set. While
the concept of rough hypercuboid approach deals with uncertainty, incompleteness, and
vagueness in cluster denition, the use of fuzzy membership of interval type-2 fuzzy sets
in the boundary region of a cluster enables ecient handling of overlapping partitions in
uncertain environment. Finally, the application of both clustering and feature selection
algorithms is demonstrated by grouping functionally similar microRNAs from microarray
data. The proposed approach can automatically select the optimum set of features while
clustering the microRNAs, making the complexity of the algorithm lower.
Implementing a Health Recommendation System from Wearable Data
(Indian Statistical Institute, Kolkata, 2025-07-22) Pramanick Priti
The rising interest in personalized health monitoring has created a demand for intelligent systems that not only evaluate an individual’s health status but also offer actionable recommendations. This dissertation presents a data-driven approach to assess overall health by calculating a weekly health score using multi-dimensional data sources such as sleep patterns, nutrition, cardiovascular activity, fitness levels, and metabolic parameters. The system integrates and processes data stored in MongoDB using Python, applies scoring logic tailored to each health domain, and aggregates them into a unified health score. Addi- tionally, the system generates a detailed summary and leverages a language model to extract personalized recommendations aimed at improving user well-being. A comprehensive PDF health report is produced, featuring score visualizations and advice tailored to the individual. The implementation was tested across multiple profiles, and evaluation metrics indicate that the approach is both adaptive and insightful. This work not only demonstrates a scalable pipeline for health analysis but also opens up opportunities for future integration of machine learning and deeper behavioral insights.
A Study of the SHA-2 Cryptographic Hash Family
(Indian Statistical Institute, Kolkata, 2009-02-01) Sanadhya Somitra Kumar
Dynamic Sparsification in Secure Gradient Aggregation for Federated Learning
(Indian Statistical Institute, Kolkata, 2025-07-23) Samanta, Bikash
Secure aggregation is a critical component of privacy-preserving federated learning.
However, existing fixed-sparsity approaches often incur unnecessary communication
overhead. We present DynamicSecAgg, a novel framework that introduces dynamic
sparsity while preserving coordinate-level privacy. Our method achieves significant
improvements in communication efficiency while maintaining — and in some cases
improving — model accuracy across both IID and non-IID user distributions. The
framework maintains information-theoretic privacy guarantees via adaptive gradient
thresholding and polynomial-based aggregation, proving particularly effective under
heterogeneous data settings. These results establish dynamic sparsity as a key optimization
for efficient and privacy-preserving federated learning.