We want to illustrate the `RPointCloud`

package (Version 0.8.0) with a clinical data set. Not surprisingly, we start by loading the package.

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 clinical data set that is included with the package.

`## [1] "Dark24" "Light24" "clinical" "daisydist" "oopt" "ripdiag"`

`## [1] 266 29`

```
## [1] "AgeAtDx" "Sex" "Race"
## [4] "mutation.status" "Light.chain.subtype" "Zap70Protein"
## [7] "Rai.Stage" "CatRAI" "Serum.beta.2.microglobulin"
## [10] "LogB2M" "CatB2M" "Serum.LDH"
## [13] "LogLDH" "White.blood.count" "LogWBC"
## [16] "CatWBC" "CatCD38" "Hypogammaglobulinemia"
## [19] "Massive.Splenomegaly" "Matutes" "Hemoglobin"
## [22] "Platelets" "Prolymphocytes" "stat13"
## [25] "stat11" "stat12" "stat17"
## [28] "Dohner" "Purity"
```

The `clinical`

object is a numeric matrix containing the clinical data (binary values have been converted to 0-1; categorical values to integers). The `daisydist`

is a distance matrix (stored as a `dist`

object). Coombes and colleagues [J Biomed Inform. 2021 Jun;118:103788] showed that this is the best way to measure distances between mixed-type clinical data. The `ripdiag`

object is a “Rips diagram” produced by applying the `TDA`

algorithm to the daisy distances.

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

Now we use our `Mercator`

package to view the underlying data.

```
mercury <- Mercator(daisydist, metric = "daisy", 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")
```

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 <- ripdiag[["cycleLocation"]][2:nzero]
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)
}
```

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(daisydist, "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.

```
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")
```

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

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

```
##
## 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
## 10 21 22 11 21 23 7 8 9 9 20 8 11 26 14 11 8 15 7 5
```

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

```
## Warning: `is.hierarchical()` was deprecated in igraph 2.0.0.
## ℹ Please use `is_hierarchical()` instead.
## This warning is displayed once every 8 hours.
## Call `lifecycle::last_lifecycle_warnings()` to see where this warning was generated.
```

`## [1] TRUE`

```
H <- as.hclust(keg)
H$labels <- attr(daisydist, "Labels")
K <- 7
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)
```

```
P <- as.phylo(H)
opar <- par(mai = c(0.01, 0.01, 1.0, 0.01))
plot(P, type = "u", tip.color = colset, cex = 0.8, main = "Ape/Cladogram")
```

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 clinical features.

```
featMU <- Feature(clinical[,"mutation.status"], "Mutation Status", c("cyan", "red"),
c("Mutated", "Unmutated"))
featRai <- Feature(clinical[,"CatRAI"], "Rai Stage", c("green", "magenta"), c("High", "Low"))
opar <- par(mfrow = c(1,2))
plot(W, main = "UMAP; Mutation Status", xlab = "U1", ylab = "U2")
points(featMU, W, pch = 16, cex = 1.4)
plot(W, main = "UMAP; Rai Stage", xlab = "U1", ylab = "U2")
points(featRai, W, pch = 16, cex = 1.4)
```

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.

```
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.003652 0.054455 0.076682 0.084373 0.108333 0.227561
```

```
mu <- mean(d0)
nu <- median(d0)
sigma <- sd(d0)
shape <- mu^2/sigma^2
rate <- mu/sigma^2
xx <- seq(0, 0.23, length = 100)
yy <- dgamma(xx, shape = shape, rate = rate)
```

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?
eb <- EBexpo(d1, 200)
opar <- par(mfrow = c(1,3))
plot(ef)
hist(eb)
plot(eb, prior = 0.56)
```

```
## Warning in d1 > cutoffSignificant(eb, 0.8, 0.56): longer object length is not a multiple of
## shorter object length
```

`## [1] 145`

`## [1] 1`

`## [1] 50 91 93 123 214 236 249`

`## [1] 236`

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

```
## [,1] [,2]
## [1,] 99 212
## [2,] 4 7
## [3,] 2 99
## [4,] 110 212
## [5,] 44 59
## [6,] 4 75
## [7,] 7 59
## [8,] 75 110
## [9,] 2 44
```

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 connected loop (or topological circle).

```
cyc2 <- Cycle(ripdiag, 1, 123, "red")
cyc3 <- Cycle(ripdiag, 1, 214, "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)
```

```
poof <- angleMeans(W, ripdiag, cyc1, clinical)
poof[is.na(poof)] <- 0
angle.df <- poof[, c("mutation.status", "CatB2M", "CatRAI", "CatCD38",
"Massive.Splenomegaly", "Hypogammaglobulinemia")]
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)
```

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?
eb <- EBexpo(d2, 200)
opar <- par(mfrow = c(1, 3))
plot(ef)
hist(eb)
plot(eb, prior = 0.75)
```

```
## Warning in d2 > cutoff(0.8, 0.75, eb): longer object length is not a multiple of shorter object
## length
```

`## [1] 87`

```
## Warning in d2 > cutoff(0.95, 0.75, eb): longer object length is not a multiple of shorter
## object length
```

`## [1] 82`

```
## $low
## [1] 8.703499e-05
##
## $high
## [1] 0.03263812
```

`## [1] 2`

`## [1] 95`

```
vd <- Cycle(ripdiag, 2, 95, "purple")
mds <- cmdscale(daisydist, k = 3)
voidPlot(vd, mds)
voidFeature(featMU, mds, radius = 0.011) # need to increase radius when you overlay one sphere on another
rgl::rglwidget()
```