This paper investigates the asymptotic power of a Spearman's rank correlation-based test for assessing independence between the joint regressors and the error term in a generalized partially linear logistic regression model. The proposed test statistic is constructed using general order differences of the observed binary responses and their fitted probabilities, leveraging the property that independence between joint covariates and the error term implies asymptotic independence between these order differences under mild regularity conditions. We introduce a sequence of contiguous alternatives to derive the limiting distribution of the test and to compute its asymptotic power. Simulation studies demonstrate strong finite-sample performance and confirm the theoretical power results. A real-data application further highlights the practical usefulness of the proposed method.
Survival analysis involves modeling time-to-event data using multiple models, such as the Accelerated Failure Time (AFT) model. Choosing the most appropriate model is crucial. Various model selection criteria are available in the literature. This study examines the performance of new model selection criteria based on Kullback's Symmetric Divergence within the Parametric AFT model framework. Simulations and real data were used to compare the Kullback Information Criterion (KIC) and its corrected version (KICc) with traditional information criteria, such as the Akaike Information Criterion (AIC), corrected AIC (AIC(c)), improved AIC (AIC(sur)), and Bayesian Information Criterion. Results showed that KICc outperforms other criteria in selecting the true model order in small samples and maintains competitive predictive accuracy across conditions. KIC and KICc outperform the AIC family criteria for moderate and large sample sizes. The real data shows that AIC and AIC(sur) tend to select overfitted models, while KICc and AIC(c) favor underfitted models.
Partial Multi-Label Learning (PML) is a weakly supervised learning framework in which each instance is associated with a set of candidate labels, among which only a subset are correct. The presence of noisy labels in the candidate set poses significant challenges for accurate classification and effective feature selection. Existing methods often overlook either the noise in the feature space or the structural relationships within the label space, limiting their robustness and generalizability. In this paper, we propose a novel PML approach based on Shared Subspace Representation, which jointly learns a low-dimensional latent space for both feature and label information. Our method decomposes the noisy candidate label matrix into a low-rank ground-truth label component and a sparse noise component, leveraging the inherent structure and sparsity of label noise. Simultaneously, a feature correlation matrix maps original features into a compact subspace, enhancing discriminative power while mitigating the impact of feature noise. To preserve the geometric structure of the data, manifold regularization is incorporated via a graph Laplacian constructed from the feature space, enforcing local consistency between similar instances. We formulate the overall objective as a unified optimization problem, which is efficiently solved using block coordinate descent and gradient-based methods. Extensive experiments on benchmark datasets demonstrate that our approach outperforms state-of-the-art methods in both label recovery and feature selection, confirming the effectiveness of shared subspace modeling in partial multi-label learning.
We established the complete solution set for a specific exponential Diophantine equation involving polynomial bases with three unknown exponents (k, m, n). Our analysis demonstrated that the equation possesses a unique positive integer solution for the exponents (k, m, n), namely (1, 1, 2), which holds true for every positive integer value of the parameter 'w', subject to a set of arithmetic constraints. The proof is constructed through a rigorous synthesis of established arithmetic methods (such as modular congruences and the Quadratic Reciprocity Law) and advanced analytic tools derived from transcendental number theory, specifically leveraging a powerful bound on p-adic measurement of logarithmic sums developed by Bugeaud. The proof's numerical bounds and specific calculations were performed and confirmed using Python code.
Retinography is a non-invasive medical imaging technique for analyzing the human visual system. As one of the top three global causes of blindness, diabetic retinopathy's early identification is critical, with exudates often being the initial visible indicator. The presence of the optic disc can interfere with the detection of bright lesions, particularly exudates. For this reason, accurate detection of the optic disc is considered essential. The primary goal of this research is the detection and segmentation of the optic disc. This work presents a new model for the automatic localization of the optic disc in retinal images from a South African patient cohort.. The method utilized images sourced from the Eye Centre fundus photo database. An efficient method for segmenting the optic disc for retinal image analysis is described. This approach integrates wavelet and wavelet packet denoising techniques with morphological operations. The algorithm was tested on a set of 100 images, which included 60 normal images and 40 images presenting exudates. The roundness metric was measured, achieving values of 0.8 or higher in 90% of the normal images and in 87.5% of the images with exudates. The algorithm demonstrated a high degree of accuracy in detecting the optic disc in the majority of cases. In contrast to other methods found in the literature, this algorithm's effectiveness is not compromised by the presence of exudates in retinal images.
