A mathematical comparison of non‐negative matrix factorization related methods with practical implications for the analysis of mass spectrometry imaging data

RATIONALE: Non‐negative matrix factorization (NMF) has been used extensively for the analysis of mass spectrometry imaging (MSI) data, visualizing simultaneously the spatial and spectral distributions present in a slice of tissue. The statistical framework offers two related NMF methods: probabilistic latent semantic analysis (PLSA) and latent Dirichlet allocation (LDA), which is a generative model. This work offers a mathematical comparison between NMF, PLSA, and LDA, and includes a detailed evaluation of Kullback–Leibler NMF (KL‐NMF) for MSI for the first time. We will inspect the results for MSI data analysis as these different mathematical approaches impose different characteristics on the data and the resulting decomposition. METHODS: The four methods (NMF, KL‐NMF, PLSA, and LDA) are compared on seven different samples: three originated from mice pancreas and four from human‐lymph‐node tissues, all obtained using matrix‐assisted laser desorption/ionization time‐of‐flight mass spectrometry (MALDI‐TOF MS). RESULTS: Where matrix factorization methods are often used for the analysis of MSI data, we find that each method has different implications on the exactness and interpretability of the results. We have discovered promising results using KL‐NMF, which has only rarely been used for MSI so far, improving both NMF and PLSA, and have shown that the hitherto stated equivalent KL‐NMF and PLSA algorithms do differ in the case of MSI data analysis. LDA, assumed to be the better method in the field of text mining, is shown to be outperformed by PLSA in the setting of MALDI‐MSI. Additionally, the molecular results of the human‐lymph‐node data have been thoroughly analyzed for better assessment of the methods under investigation. CONCLUSIONS: We present an in‐depth comparison of multiple NMF‐related factorization methods for MSI. We aim to provide fellow researchers in the field of MSI a clear understanding of the mathematical implications using each of these analytical techniques, which might affect the exactness and interpretation of the results.