MNAR-$k$-means: A $k$-means Clustering for Data Missing Not at Random with Magnitude-Decaying Probability

2026-06-30

AI Summary

This work addresses the bias introduced by mean imputation in traditional $k$-means clustering when applied to data with missing not at random (MNAR) mechanisms—particularly in magnitude-dependent scenarios where smaller absolute values are more likely to be missing. To mitigate this issue, the authors propose the first statistically consistent variant of $k$-means tailored for such amplitude-attenuated MNAR settings. The method introduces a constrained loss function that incorporates bounds on imputed values and optimizes it via an alternating minimization algorithm. Theoretical analysis establishes the statistical consistency of the proposed estimator, while empirical results demonstrate a significant reduction in estimation bias and a marked improvement in clustering accuracy compared to conventional approaches.

Abstract

The classical $k$-means clustering, based on distances computed from all data features, cannot be directly applied to incomplete data with missing values. A natural extension of $k$-means to missing data is to involve only the observed positions in clustering, which is equivalent to imputing missing values by corresponding cluster means. However, for data missing not at random (MNAR), since missingness is related to data values, such a mean-imputation-based method may lead to the distortion of estimated cluster centers, resulting in a poor clustering result. Since MNAR mechanisms are very common in reality, it is necessary to improve the performance of $k$-means-based clustering methods for such data. In this paper, we focus on a magnitude-decaying MNAR scenario where data is more likely to be missing at positions with smaller absolute values, and we propose a novel $k$-means clustering method based on the constraint of the size of imputation values, which enjoys a good mathematical interpretation. Moreover, we establish the statistical consistency of the estimated cluster centers of the proposed method to the true cluster centers of fully observed data, and solve the optimization of the proposed loss function via an alternative minimization algorithm. Simulation experiments verify the effect of the proposed method in improving clustering results and reducing the bias of estimated cluster centers. Applications to real-world missing data further show the utility of the proposed method.

Problem

• MNAR
• k-means clustering
• missing data
• magnitude-decaying probability
• cluster center bias

Innovation

• MNAR
• k-means clustering
• magnitude-decaying missingness
• imputation constraint
• statistical consistency