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

Laplacian和PCA貌似是同一种性质的方法，坐标系变换。

 

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，通常的经过函数名字就能够看到R代码，但这里你输入函数名，如sc3_calc_dists，看到的只是如下的封装好的代码：

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，因此只能经过函数名去GitHub里面查了。

GitHub真的很优秀，能够直接查文件内部代码，能够很快定位到sc3_calc_dists。

再配合这个目录插件，效率提升了很多，https://www.octotree.io/?utm_source=lite&utm_medium=extension

如下是封装前的代码：

#' 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)　　

经过setMethod连接到一块儿的。

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是他们实验室本身用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），而后作完PCA，返回了rotation，不知道这样作有什么意义？

还真有用PCA来处理距离类似度矩阵的，MDS，目的就是降维，由于后面要用kmean聚类；

而后对每个降维了的矩阵用kmeans；

consensus用的是这个算法：Cluster-based Similarity Partitioning Algorithm (CSPA)，作这个的意义何在？输入是每一个细胞的多重聚类结果，而后作了一个一致性统一。

 

 

参考：

拉普拉斯矩阵（Laplacian matrix）