SC3聚类 | 拉普拉斯矩阵 | Laplacian matrix | 图论 | R代码

Laplacian和PCA貌似是同一种性质的方法，坐标系变换。只是拉普拉斯属于图论的范畴，术语更加专业了。

要看就把一篇文章看完整，再看其中有什么值得借鉴的，总结归纳理解后的东西才是属于你的。

问题：

1. 这篇文章有哪些亮点决定他能发NM？单细胞，consensus，较好的表现，包装了一些专业的术语，显得自己很专业，其实真正做的东西很少；

2. consensus方法的本质是什么？

3. 工具的评估准则？ARI，silhouette index

4. SC3的最大缺点是什么？速度太慢，超过1000个细胞就非常耗费计算和存储资源

5. 能看懂SC3这个R包的逻辑吗？核心的就4步，多种距离度量，转换，kmeans聚类，consensus；

The main sc3 method explained above is a wrapper that calls several other SC3 methods in the following order:

• sc3_prepare
• sc3_estimate_k - Tracy-Widom theory - random matrix theory (RMT)
• sc3_calc_dists
• sc3_calc_transfs
• sc3_kmeans
• sc3_calc_consens
• sc3_calc_biology

6. 有很多问题没有回答，这篇文章偏技工！核心就是kmeans，打了个复杂的包而已。

• 不同距离的度量有什么差异？
• 为什么要做两种转换PCA和laplacian？
• 为什么选择了kmeans？不知道它有天然的劣势吗
• 做consensus的理论依据是什么？凭什么说做了一致性后结果就更好？

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最近在看SC3聚类这篇文章，SC3使用了这个工具。

SC3: consensus clustering of single-cell RNA-seq data

All distance matrices are then transformed using either principal component analysis (PCA) or by calculating the eigenvectors of the associated graph Laplacian (L = I – D–1/2AD–1/2, where I is the identity matrix, A is a similarity matrix (A = e–A′/max(A′)), where A′ is a distance matrix) and D is the degree matrix of A, a diagonal matrix that contains the row-sums of A on the diagonal (Dii = ΣjAij). The columns of the resulting matrices are then sorted in ascending order by their corresponding eigenvalues.

先看下该工具的功能：SC3 package manual

跑一下常规代码：

library(SingleCellExperiment)
library(SC3)
library(scater)

head(ann)
yan[1:3, 1:3]

# create a SingleCellExperiment object
sce <- SingleCellExperiment(
  assays = list(
    counts = as.matrix(yan),
    logcounts = log2(as.matrix(yan) + 1)
  ),
  colData = ann
)

# define feature names in feature_symbol column
rowData(sce)$feature_symbol <- rownames(sce)
# remove features with duplicated names
sce <- sce[!duplicated(rowData(sce)$feature_symbol), ]

# define spike-ins
isSpike(sce, "ERCC") <- grepl("ERCC", rowData(sce)$feature_symbol)

plotPCA(sce, colour_by = "cell_type1")

sce <- sc3(sce, ks = 2:4, biology = TRUE)
# sc3_interactive(sce)
# sc3_export_results_xls(sce)

######################################
sce <- sc3_prepare(sce)

sce <- sc3_estimate_k(sce)

sce <- sc3_calc_dists(sce)
names(metadata(sce)$sc3$distances)

sce <- sc3_calc_transfs(sce)
names(metadata(sce)$sc3$transformations)
metadata(sce)$sc3$distances

sce <- sc3_kmeans(sce, ks = 2:4)
names(metadata(sce)$sc3$kmeans)

col_data <- colData(sce)
head(col_data[ , grep("sc3_", colnames(col_data))])
sce <- sc3_calc_consens(sce)
names(metadata(sce)$sc3$consensus)
names(metadata(sce)$sc3$consensus$`3`)

col_data <- colData(sce)
head(col_data[ , grep("sc3_", colnames(col_data))])

sce <- sc3_calc_biology(sce, ks = 2:4)

sce <- sc3_run_svm(sce, ks = 2:4)
col_data <- colData(sce)
head(col_data[ , grep("sc3_", colnames(col_data))])

　　

接下来会尝试拆一下该工具。

怎么拆这个工具？

这种封装的很好的R包其实比较难拆，一般的通过函数名字就可以看到R代码，但这里你输入函数名，如sc3_calc_dists，看到的只是以下的封装好的代码：

new("nonstandardGenericFunction", .Data = function (object)
{
    standardGeneric("sc3_calc_dists")
}, generic = structure("sc3_calc_dists", package = "SC3"), package = "SC3",
    group = list(), valueClass = character(0), signature = "object",
    default = NULL, skeleton = (function (object)
    stop("invalid call in method dispatch to 'sc3_calc_dists' (no default method)",
        domain = NA))(object))　　

暂时还不熟悉这种形式，所以只能通过函数名去GitHub里面查了。

GitHub真的很优秀，可以直接查文件内部代码，可以很快定位到sc3_calc_dists。

再配合这个目录插件，效率提高了不少，https://www.octotree.io/?utm_source=lite&utm_medium=extension

以下是封装前的代码：

#' Calculate distances between the cells.
#'
#' This function calculates distances between the cells. It
#' creates and populates the following items of the \code{sc3} slot of the \code{metadata(object)}:
#' \itemize{
#'   \item \code{distances} - contains a list of distance matrices corresponding to
#'   Euclidean, Pearson and Spearman distances.
#' }
#'
#' @name sc3_calc_dists
#' @aliases sc3_calc_dists, sc3_calc_dists,SingleCellExperiment-method
#'
#' @param object an object of \code{SingleCellExperiment} class
#'
#' @return an object of \code{SingleCellExperiment} class
#'
#' @importFrom doRNG %dorng%
#' @importFrom foreach foreach %dopar%
#' @importFrom parallel makeCluster stopCluster
#' @importFrom doParallel registerDoParallel
sc3_calc_dists.SingleCellExperiment <- function(object) {
    dataset <- get_processed_dataset(object)

    # check whether in the SVM regime
    if (!is.null(metadata(object)$sc3$svm_train_inds)) {
        dataset <- dataset[, metadata(object)$sc3$svm_train_inds]
    }

    # NULLing the variables to avoid notes in R CMD CHECK
    i <- NULL

    distances <- c("euclidean", "pearson", "spearman")

    message("Calculating distances between the cells...")

    if (metadata(object)$sc3$n_cores > length(distances)) {
        n_cores <- length(distances)
    } else {
        n_cores <- metadata(object)$sc3$n_cores
    }

    cl <- parallel::makeCluster(n_cores, outfile = "")
    doParallel::registerDoParallel(cl, cores = n_cores)

    # calculate distances in parallel
    dists <- foreach::foreach(i = distances) %dorng% {
        try({
            calculate_distance(dataset, i)
        })
    }

    # stop local cluster
    parallel::stopCluster(cl)

    names(dists) <- distances

    metadata(object)$sc3$distances <- dists
    return(object)
}

#' @rdname sc3_calc_dists
#' @aliases sc3_calc_dists
setMethod("sc3_calc_dists", signature(object = "SingleCellExperiment"), sc3_calc_dists.SingleCellExperiment)　　

通过setMethod链接到一起的。

顺路找到了原函数：

#' Calculate a distance matrix
#'
#' Distance between the cells, i.e. columns, in the input expression matrix are
#' calculated using the Euclidean, Pearson and Spearman metrics to construct
#' distance matrices.
#'
#' @param data expression matrix
#' @param method one of the distance metrics: 'spearman', 'pearson', 'euclidean'
#' @return distance matrix
#'
#' @importFrom stats cor dist
#'
#' @useDynLib SC3
#' @importFrom Rcpp sourceCpp
#'
calculate_distance <- function(data, method) {
    return(if (method == "spearman") {
        as.matrix(1 - cor(data, method = "spearman"))
    } else if (method == "pearson") {
        as.matrix(1 - cor(data, method = "pearson"))
    } else {
        ED2(data)
    })
}

#' Distance matrix transformation
#'
#' All distance matrices are transformed using either principal component
#' analysis (PCA) or by calculating the
#' eigenvectors of the graph Laplacian (Spectral).
#' The columns of the resulting matrices are then sorted in
#' descending order by their corresponding eigenvalues.
#'
#' @param dists distance matrix
#' @param method transformation method: either 'pca' or
#' 'laplacian'
#' @return transformed distance matrix
#'
#' @importFrom stats prcomp cmdscale
#'
transformation <- function(dists, method) {
    if (method == "pca") {
        t <- prcomp(dists, center = TRUE, scale. = TRUE)
        return(t$rotation)
    } else if (method == "laplacian") {
        L <- norm_laplacian(dists)
        l <- eigen(L)
        # sort eigenvectors by their eigenvalues
        return(l$vectors[, order(l$values)])
    }
}

#' Calculate consensus matrix
#'
#' Consensus matrix is calculated using the Cluster-based Similarity
#' Partitioning Algorithm (CSPA). For each clustering solution a binary
#' similarity matrix is constructed from the corresponding cell labels:
#' if two cells belong to the same cluster, their similarity is 1, otherwise
#' the similarity is 0. A consensus matrix is calculated by averaging all
#' similarity matrices.
#'
#' @param clusts a matrix containing clustering solutions in columns
#' @return consensus matrix
#'
#' @useDynLib SC3
#' @importFrom Rcpp sourceCpp
#' @export
consensus_matrix <- function(clusts) {
    res <- consmx(clusts)
    colnames(res) <- as.character(c(1:nrow(clusts)))
    rownames(res) <- as.character(c(1:nrow(clusts)))
    return(res)
}

• 距离计算
• 转换
• consensus

都在这里。。。　　

ED2是他们实验室自己用Rcpp写的一个计算欧氏距离的工具。

transformation输入的是对称的距离矩阵（行列都是样本细胞），然后做完PCA，返回了rotation，不知道这样做有什么意义？

还真有用PCA来处理距离相似度矩阵的，MDS，目的就是降维，因为后面要用kmean聚类；

然后对每一个降维了的矩阵用kmeans；

consensus用的是这个算法：Cluster-based Similarity Partitioning Algorithm (CSPA)，做这个的意义何在？输入是每个细胞的多重聚类结果，然后做了一个一致性统一。

参考：

拉普拉斯矩阵（Laplacian matrix）