Non-negative matrix factorization (NMF) has become a well-established class of methods for the analysis of non-negative data. In particular, a lot of effort has been devoted to probabilistic NMF, namely estimation or inference tasks in probabilistic models describing the data, based for example on Poisson or exponential likelihoods. When dealing with time series data, several works have proposed to model the evolution of the activation coefficients as a non-negative Markov chain, most of the time in relation with the Gamma distribution, giving rise to so-called temporal NMF models. In this paper, we review four Gamma Markov chains of the NMF literature, and show that they all share the same drawback: the absence of a well-defined stationary distribution. We then introduce a fifth process, an overlooked model of the time series literature named BGAR(1), which overcomes this limitation. These temporal NMF models are then compared in a MAP framework on a prediction task, in the context of the Poisson likelihood.
翻译:非负矩阵因子化(NMF)已成为分析非负矩阵数据的一个既定方法类别,特别是大量努力致力于概率性NMF,即根据Poisson或指数可能性等来估计或推断描述数据的概率模型中的概率性任务。在处理时间序列数据时,若干工作提议将活化系数作为非负马克夫链的演变模式,大部分时间与伽马分布有关,从而产生所谓的时间性NMF模型。在本文件中,我们审查了四个NMF文献的伽马·马尔科夫链,并表明它们都具有相同的缺点:没有明确界定的静止分布。然后,我们引入了第五个进程,即一个被忽视的时间序列文献模型,即BGAR(1), 克服了这一限制。这些时间性NMF模型随后在PAM框架中结合Poisson的可能性,比较了预测任务。