Here, we illustrate optimal univariate clustering function
`Ckmeans.1d.dp`

. It uses a given number of clusters \(k\), or estimates \(k\) if a range is provided. If \(k\) is unspecified, a default range from 1
to 9 is used. It can also perform optimal weighted clustering when a
non-negative weight vector is specified as input. Weighted clustering
can be used to analyze signals such as time series data, spectral data,
genetic or epigenetic events along a chromosome.

Cluster data generated from a Gaussian mixture model of three components.

The number of clusters is provided.

Cluster data generated from a Gaussian mixture model of three components. The number of clusters is determined by Bayesian information criterion:

```
x <- c(rnorm(50, mean=-1, sd=0.3), rnorm(50, mean=1, sd=1), rnorm(50, mean=2, sd=0.4))
# Divide x into k clusters, k automatically selected (default: 1~9)
result <- Ckmeans.1d.dp(x)
k <- max(result$cluster)
colors <- brewer.pal(k, "Dark2")
```

`## Warning in brewer.pal(k, "Dark2"): minimal value for n is 3, returning requested palette with 3 different levels`

```
plot(x, col=colors[result$cluster], pch=result$cluster, cex=1.5,
main="Optimal univariate clustering with k estimated",
sub=paste("Number of clusters is estimated to be", k))
abline(h=result$centers, col=colors, lty="dashed", lwd=2)
legend("topleft", paste("Cluster", 1:k), col=colors, pch=1:k, cex=1, bty="n")
```

We segment a time course to identify peaks using weighted clustering. The input data is the time stamp of obtaining each intensity measurement; the weight is the signal intensity.

```
n <- 160
t <- seq(0, 2*pi*2, length=n)
n1 <- 1:(n/2)
n2 <- (max(n1)+1):n
y1 <- abs(sin(1.5*t[n1]) + 0.1*rnorm(length(n1)))
y2 <- abs(sin(0.5*t[n2]) + 0.1*rnorm(length(n2)))
y <- c(y1, y2)
w <- y^8 # stress the peaks
res <- Ckmeans.1d.dp(t, k=c(1:10), w)
k <- max(res$cluster)
colors <- brewer.pal(k, "Set1")
plot(res, col.clusters = colors)
grid()
```

```
plot(t, w, main = "Time course clustering / peak calling",
col=colors[res$cluster], pch=res$cluster, type="h",
xlab="Time t", ylab="Transformed intensity w")
grid()
abline(v=res$centers, col="gray", lty="dashed")
text(res$centers, max(w) * .95, cex=0.75, font=2,
paste(round(res$size / sum(res$size) * 100), "/ 100"))
```

It is often desirable to visualize boundaries between consecutive
clusters. The `ahist()`

function offers several ways to
estimate cluster boundaries. The simplest is to use the midpoint between
the two closest points in two consecutive clusters, as illustrated in
the code below.

```
x <- c(7, 4, 1, 8, 15, 22, -1)
k <- 3
ckm <- Ckmeans.1d.dp(x, k=k)
midpoints <- ahist(ckm, style="midpoints", data=x, plot=FALSE)$breaks[2:k]
colors <- brewer.pal(k, "Set2")
plot(ckm, col.clusters = colors, lwd=5,
main="Midpoints as cluster boundaries")
abline(v=midpoints, col="RoyalBlue", lwd=3, lty=2)
legend("topright", "Midpoints", lwd=3, lty=2, col="RoyalBlue")
```