The Graded of Membership (GoM) model is a powerful tool for inferring latent classes in categorical data, which enables subjects to belong to multiple latent classes. However, its application is limited to categorical data with nonnegative integer responses, making it inappropriate for datasets with continuous or negative responses. To address this limitation, this paper proposes a novel model named the Weighted Grade of Membership (WGoM) model. Compared with GoM, our WGoM relaxes GoM's distribution constraint on the generation of a response matrix and it is more general than GoM. We then propose an algorithm to estimate the latent mixed memberships and the other WGoM parameters. We derive the error bounds of the estimated parameters and show that the algorithm is statistically consistent. The algorithmic performance is validated in both synthetic and real-world datasets. The results demonstrate that our algorithm is accurate and efficient, indicating its high potential for practical applications. This paper makes a valuable contribution to the literature by introducing a novel model that extends the applicability of the GoM model and provides a more flexible framework for analyzing categorical data with weighted responses.

翻译：隶属度等级（GoM）模型是推断分类数据中潜在类别的有力工具，允许个体同时属于多个潜在类别。然而，其应用仅限于响应为非负整数的分类数据，因此不适用于包含连续值或负值响应的数据集。为克服这一局限，本文提出了一种名为加权隶属度等级（WGoM）模型的新型模型。与GoM相比，我们的WGoM模型放宽了GoM对响应矩阵生成的分布约束，且更具通用性。随后，我们提出了一种算法来估计潜在混合隶属度及其他WGoM参数。我们推导了估计参数的误差界，并证明该算法具有统计一致性。在合成数据集和真实数据集上的实验验证了算法的性能。结果表明，我们的算法准确且高效，显示出在实际应用中的巨大潜力。本文通过引入这一新型模型，扩展了GoM模型的适用性，并为分析加权响应分类数据提供了更灵活的框架，从而对相关文献做出了有价值的贡献。