今天我们继续深入分析NMF，看看NMF到底能带给我们什么。参考文献是Single-Cell Transcriptomic Analysis of Primary and Metastatic Tumor Ecosystems in Head and Neck Cancer,发表于cell，顶级期刊，这里我们只关心NMF分析的部分。当然，文章的其他部分也很重要，我们后续分享。

首先来看作者进行NMF分析的目的。

Intra-tumoral Expression Heterogeneity of the Malignant Compartment。（研究肿瘤细胞的异质性）。

We next explored how expression states varied among different malignant cells within the same tumor, focusing on ten tumors from which the largest numbers of malignant cell transcriptomes were acquired.（不同病人相同肿瘤）。We used non-negative matrix factorization to uncover coherent sets of genes that were preferentially co-expressed by subsets of malignant cells（这个地方就采用了NMF的方法，来识别恶性细胞亚群优先共表达的相关基因集，类似于WGCNA，但是原理目的都不一样，注意区分 ）。For example, we defined six gene signatures that vary among malignant cells of MEEI25（例子，在癌变细胞亚群中NMF找到的特异基因模块）。

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Applying the approach to each of the ten tumors defined a total of 60 gene signatures that coherently vary across individual cells in at least one tumor。Next, we used hierarchical clustering to distill these 60 signatures into meta-signatures that reflect common expression programs that vary within multiple tumors（识别的基因模块进行层次聚类）。

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The high concordance between signatures from different tumors suggests that they reflect common patterns of intra-tumoral expression heterogeneity.（这是聚类的意义）。

接下来就是对识别到的模块基因的特征进行分析了，Seven expression programs were preferentially expressed by subsets of malignant cells in at least two tumors.来看NMF的主要作用就是用来识别肿瘤内部不同亚群之间的特异基因模块。非常重要，一定要认真学习并学以致用。

接下来我们看一看具体的方法。

For each of the 10 tumors, non-negative matrix factorization (as implemented by the MATLAB nnmf function, with the number of factors set to 10（注意这里的因子数，跟样本一致，这个地方需要格外注意，视情况可以进行人为的划分）) was used to identify variable expression programs。NNMF was applied to the relative expression values (Er), by transforming all negative values to zero.（这个地方大家要对比NMF和我们常用的PCA的区别）。Notably, undetected genes include many drop-out events (genes that are expressed but are not detected in particular cells due to the incomplete transcriptome coverage), which introduce challenges for normalization of single-cell RNA-seq; since NNMF avoids the exact normalized values of undetected genes (as they are all zero), it may be beneficial in analysis of single-cell RNA-seq (data not shown).(这个地方也是非常重要的，NMF相对于PCA的优势)。We retained only programs for which the standard deviation in cell scores within the respective tumor was larger than 0.8, which resulted in a total of 60 programs across the 10 tumors.（特征的提取也是一门学问）。The 60 programs were compared by hierarchical clustering (data not shown), using one minus the Pearson correlation coefficient over all gene scores as a distance metric.（如上图），Six clusters of programs were identified manually and used to define meta-signatures. For each cluster, NNMF gene scores were log2-transformed and then averaged across the programs in the cluster, and genes were ranked by their average scores。The top 30 genes for each cluster were defined as the meta signature that was used to define cell scores(这个地方是精髓，有的文章是提取前50个基因)。each of those genes had average scores above 1 and a t test p value below 0.05, based ontheir scores across the individual programs in the cluster. Since the number of programs in a cluster was small this analysis was not powered to correct for multiple testing and thus we refer to an uncorrected p value and selected the top ranked genes。However, while confidence is difficult to establish for individual genes in each meta-program, each gene-set defined as a meta-program is highly significant in its co-variation in tumors. For each of the meta-programs, and within each of the tumors included in those meta-programs (2-8 tumors for each meta-program), the average Pearson correlation between all pairs of genes included in the gene-set (calculated across single malignant cells from the respective tumor) was higher than that obtained for 10,000 control gene-sets, which were selected to reproduce the overall distribution of expression levels of the meta-program genes（分数的求解过程我们下一篇详细分享）。

To show the robustness of the NNMF-derived programs with regards to the number of NNMF factors, we repeated the NNMF analysis with the number of factors between 5 and 15（这个迭代过程也很重要）。Each of the seven meta-signatures was robustly identified with each of the NNMF parameters.

想深入学习生物信息的同学，基础的数学知识还是要掌握一些的。

生活很好，等你超越