Introduction

We want to illustrate the RPointCloud package (Version 0.6.2) with a AML10.node287 data set. Not surprisingly, we start by loading the package.

library(RPointCloud)

We also load several other useful packages (some of which may eventually get incorporated into the package requirements).

suppressMessages( library(Mercator) )
library(ClassDiscovery)
library(Polychrome)
data(Dark24)
data(Light24)
suppressMessages( library(igraph) )
suppressMessages( library("ape") )
suppressPackageStartupMessages( library(circlize) )

Now we fetch the sample AML10.node287 data set that is included with the package.

data(cytof)
ls()

##  [1] "AML10.node287"      "AML10.node287.rips" "Arip"               "Dark24"            
##  [5] "G"                  "G2"                 "H"                  "K"                 
##  [9] "L"                  "Light24"            "Lt"                 "P"                 
## [13] "U"                  "V"                  "W"                  "X"                 
## [17] "Y"                  "angle.df"           "annote"             "clinical"          
## [21] "colorScheme"        "colset"             "cyc"                "cyc1"              
## [25] "cyc2"               "cyc3"               "cycles"             "d0"                
## [29] "d1"                 "d2"                 "daisydist"          "diag"              
## [33] "e"                  "eb"                 "edges"              "ef"                
## [37] "featMU"             "featRai"            "keg"                "mds"               
## [41] "mercury"            "mu"                 "nu"                 "nzero"             
## [45] "ob"                 "oopt"               "pal"                "persistence"       
## [49] "poof"               "rate"               "ripdiag"            "shape"             
## [53] "sigma"              "vd"                 "xx"                 "yy"

dim(AML10.node287)

## [1] 214  18

colnames(AML10.node287)

##  [1] "CD45RA" "CD133"  "CD47"   "p21"    "CD90"   "CycA"   "CycB"   "PCNA"   "Ki-67"  "pRb"   
## [11] "CD99"   "CD13"   "pCDK1"  "H2AX"   "cPARP"  "pS6"    "pH3"    "DNA-2"

amldist <- dist(AML10.node287)

The AML10.node187 object is a numeric matrix from a patient identified as “AML10”. The complete data set [Behbehani et al.] was processed by the SPADE algorithm [Qiu et al.], with individual cells clustered into distinct nodes based on the expression patterns of cell surface markers. All cells in “node187” were identified as early monocytes. The columns in this data matrix are additional protein/antibody markers that were measured in the experiment, which focused on the cell cycle. The AML10.node187.rips object is a “Rips diagram” produced by applying the TDA algorithm to these data.

TDA Built-in Visualizations of the Rips Diagram

Here are some plots of the TDA results using tools from the original package. (I am not sure what any of these are really good for.)

diag <- AML10.node287.rips[["diagram"]]
opar <- par(mfrow = c(1,2))
plot(diag, barcode = TRUE, main = "Barcode")
plot(diag, main = "Rips Diagram")

Figure 1 : The Rips barcode diagram from TDA.

par(opar)
rm(opar)

Mercator Visualizations of the Underlying Data and Distance Matrix

Now we use our Mercator package to view the underlying data.

mercury <- Mercator(amldist, metric = "euclidean", method = "hclust", K = 8)
mercury <- addVisualization(mercury, "mds")
mercury <- addVisualization(mercury, "tsne")
mercury <- addVisualization(mercury, "umap")
mercury <- addVisualization(mercury, "som")

opar <- par(mfrow = c(3,2), cex = 1.1)
plot(mercury, view = "hclust")
plot(mercury, view = "mds", main = "Mult-Dimensional Scaling")
plot(mercury, view = "tsne", main = "t-SNE")
plot(mercury, view = "umap", main = "UMAP")
barplot(mercury, main = "Silhouette Width")
plot(mercury, view = "som", main = "Self-Organizing Maps")

Figure 3 : Mercator Visualizations of the distance matrix.

par(opar)
rm(opar)

Dimension Zero

Here is a picture of the “zero-cycle” data, which can also be used ultimately to cluster the points (where each point is a patient). The connected lines are similar to a single-linkage clustering structure, showing when individual points are merged together as the TDA parameter increases.

nzero <- sum(diag[, "dimension"] == 0)
cycles <- AML10.node287.rips[["cycleLocation"]][1:nzero]
L <- sapply(cycles, length)
cycles <- cycles[L > 0]
W <- mercury@view[["umap"]]$layout
plot(W, main = "Connected Zero Cycles")
for (cyc in cycles) {
  points(W[cyc[1], , drop = FALSE], pch = 16,col = "red")
  X <- c(W[cyc[1], 1], W[cyc[2],1])
  Y <- c(W[cyc[1], 2], W[cyc[2],2])
  lines(X, Y)
}

Figure 4 : Hierarchical connections between zero cycles.

Using iGraph

We can convert the 0-dimensional cycle structure into a dendrogram, by first passing them through the igraph package. We start by putting all the zero-cycle data together, which can be viewed as an “edge-list” from the igraph perspective.

edges <- t(do.call(cbind, cycles)) # this creates an "edgelist"
G <- graph_from_edgelist(edges)
G <- set_vertex_attr(G, "label", value = attr(amldist, "Labels"))

Note that we attached the sample names to the graph, obtaining them from the daisy distance matrix. Now we use two different algorithms to decide how to layout the graph.

set.seed(2734)
Lt <- layout_as_tree(G)
L <- layout_with_fr(G)

opar <- par(mfrow = c(1,2), mai = c(0.01, 0.01, 1.02, 0.01))
plot(G, layout = Lt, main = "As Tree")
plot(G, layout = L, main = "Fruchterman-Reingold")

Figure 5 : Two igraph depictions of the zero cycle structure.

par(opar)

Note that the Fruchterman-Reingold layout gives the most informative structure.

Community Structure

There are a variety of community-finding algorithms that we can apply. (Communities in graph theory are similar to clusters in other machine learning areas of study.) “Edge-betweenness” seems to work best.

keg <- cluster_edge_betweenness(G)
table(membership(keg))

## 
##  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 
##  7 25  6  4  6 11  9 22 25  8  1  6  6  9  7  6 12  9  9  7  6  6  7

pal <- Dark24[membership(keg)]

The first line in the next code chunk shows that we did actually produce a tree. We explore three different ways ro visualize it

is.hierarchical(keg)

## [1] TRUE

H <- as.hclust(keg)
H$labels <- attr(amldist, "Labels")
K <-  10
colset <- Light24[cutree(H, k=K)]
G2 <- set_vertex_attr(G, "color", value = colset)
e <- 0.01
opar <- par(mai = c(e, e, e, e))
plot(G2, layout = L)

Figure 6 : Community structure, simplified.

par(opar)

P <- as.phylo(H)
opar <- par(mai = c(0.01, 0.01, 1.0, 0.01))
plot(P, type = "u", tip.color = colset, cex = 1.2, main = "Ape/Cladogram")

Figure 8 : Cladogram realization, from the ape package.

par(opar)
rm(opar)

Visualizing Features

In any of the “scatter plot views” (e.g., MDS, UMAP, t-SNE) from Mercator, we may want to overlay different colors to represent different AML10.node287 features.

U <- mercury@view[["mds"]]
V <- mercury@view[["tsne"]]$Y
W <- mercury@view[["umap"]]$layout

featKi67 <- Feature(AML10.node287[,"Ki-67"], "Ki-67", c("cyan", "red"), c("Low", "High"))
featCD99 <- Feature(AML10.node287[,"CD99"], "CD99", c("green", "magenta"), c("Low", "High"))
opar <- par(mfrow = c(1,2))
plot(W, main = "UMAP; Ki-67", xlab = "U1", ylab = "U2")
points(featKi67, W, pch = 16, cex = 1.4)
plot(W, main = "UMAP; CD99", xlab = "U1", ylab = "U2")
points(featCD99, W, pch = 16, cex = 1.4)

Figure 10 : UMAP visualizations with AML10.node287 features.

par(opar)
rm(opar)

Significance

We have a statistical approach to deciding which of the detected cycles are statistically significant. Empirically, the persistence of 0-dimensional cycles looks like a gamma distribution, while the persistence of higher dimensional cycles looks like an exponential distribution. In both cases, we use an empirical Bayes approach, treating the observed distribution as a mixture of either gamma or exponential (as appropriate) with an unknown distribution contributing to heavier tails.

persistence <- diag[, "Death"] - diag[, "Birth"]

Zero-Cycles (Connected Components)

d0 <- persistence[diag[, "dimension"] == 0]
d0 <- d0[d0 < 5]
summary(d0)

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##  0.8442  1.4682  1.6475  1.7155  1.9176  3.6600

mu <- mean(d0)
nu <- median(d0)
sigma <- sd(d0)
shape <- mu^2/sigma^2
rate <- mu/sigma^2
xx <- seq(0, 4, length = 100)
yy <- dgamma(xx, shape = shape, rate = rate)
hist(d0, breaks = 123, freq = FALSE)
lines(xx, yy, col = "purple", lwd = 2)

One-Cycles (Loops)

Now we want to determine if there are significant “loops” in the data, and, if so, how many?

d1 <- persistence[diag[, "dimension"] == 1]
ef <- ExpoFit(d1) # should be close to log(2)/median? 
plot(ef)

eb <- EBexpo(d1, 200)
hist(eb)

plot(eb, prior = 0.56)

sum(d1 > cutoff(0.8, 0.56, eb)) # posterior 80%, prior 0.56

## [1] 9

cutoff(0.8, 0.56, eb)

## [1] 0.2373232

sum(d1 > 0.237) # post 80%

## [1] 9

which(d1 > 0.237)

## [1]  35  59  60  75  87  96 103 116 133

which.max(d1)

## [1] 60

Let’s pick out the most persistent 1-cycle.

cyc1 <- Cycle(AML10.node287.rips, 1, 103, "forestgreen")
cyc1@index

##      [,1] [,2]
## [1,]    4  116
## [2,]  116  211
## [3,]   12   36
## [4,]    4   12
## [5,]  120  211
## [6,]   36  120

Each row represents an edge, by listing the IDs of the points at either end of the line segment. In this case, there are nine edges that link together to form a connect loop (or topological circle).

cyc2 <- Cycle(AML10.node287.rips, 1, 96, "red")
cyc3 <- Cycle(AML10.node287.rips, 1, 87, "purple")

opar <- par(mfrow = c(1, 3))
plot(cyc1, W, lwd = 2, pch = 16, col = "gray", xlab = "U1", ylab = "U2", main = "UMAP")
lines(cyc2, W, lwd=2)
lines(cyc3, W, lwd=2)

plot(U, pch = 16, col = "gray", main = "MDS")
lines(cyc1, U, lwd = 2)
lines(cyc2, U, lwd = 2)
lines(cyc3, U, lwd = 2)

plot(V, pch = 16, col = "gray", main = "t-SNE")
lines(cyc1, V, lwd = 2)
lines(cyc2, V, lwd = 2)
lines(cyc3, V, lwd = 2)

par(opar)
rm(opar)

poof <- angleMeans(W, AML10.node287.rips, cyc3, AML10.node287)
poof[is.na(poof)] <- 0
angle.df <- poof[, c("Ki-67", "CD99", "pRb", "PCNA",
                     "CycA", "CycB")]
colorScheme <- list(c(M = "green", U = "magenta"),
                    c(Hi = "cyan", Lo ="red"),
                    c(Hi = "blue", Lo = "yellow"),
                    c(Hi = "#dddddd", Lo = "#111111"),
                    c(No = "#dddddd", Yes = "brown"),
                    c(No = "#dddddd", Yes = "purple"))
annote <- LoopCircos(cyc1, angle.df, colorScheme)
image(annote)

Two-Cycles (Voids)

Now we want to determine if there are significant “voids” (empty interiors of spheres) in the data, and, if so, how many?

d2 <- persistence[diag[, "dimension"] == 2]
ef <- ExpoFit(d2) # should be close to log(2)/median? 
plot(ef)

eb <- EBexpo(d2, 200)
hist(eb)

plot(eb, prior = 0.75)

sum(d2 > cutoff(0.8, 0.75, eb)) # posterior 80%, prior 0.56

## [1] 8

sum(d2 > cutoff(0.95, 0.75, eb)) # posterior 90%, prior 0.56

## [1] 3

cutoff(0.95, 0.75, eb)

## [1] 0.1260304

sum(d2 > 0.032) # post 90%

## [1] 15

which(d2 > 0.032)

##  [1]  1  2  7 13 14 15 20 22 24 26 27 29 32 33 34

vd <- getCycle(AML10.node287.rips, 2)
mds <- cmdscale(amldist, k = 3)
support <- cycleSupport(vd, mds)

options(oopt)
#rm(list = ls())

RPointCloud: A Mass Cytometry Example

Kevin R. Coombes and Jake Reed