Univariate Clustering {Ckmeans.1d.dp} | R Documentation |

## Optimal (Weighted) Univariate Clustering

### Description

Perform optimal univariate `k`

-means or `k`

-median clustering in linear (fastest), loglinear, or quadratic (slowest) time.

### Usage

```
Ckmeans.1d.dp(x, k=c(1,9), y=1,
method=c("linear", "loglinear", "quadratic"),
estimate.k=c("BIC", "BIC 3.4.12"))
Ckmedian.1d.dp(x, k=c(1,9), y=1,
method=c("linear", "loglinear", "quadratic"),
estimate.k=c("BIC", "BIC 3.4.12"))
```

### Arguments

`x` |
a numeric vector of data to be clustered. All |

`k` |
either an exact integer number of clusters, or a vector of length two specifying the minimum and maximum numbers of clusters to be examined. The default is |

`y` |
a value of 1 (default) to specify equal weights of 1 for each element in |

`method` |
a character string to specify the speedup method to the original cubic runtime dynamic programming. The default is |

`estimate.k` |
a character string to specify the method to estimate optimal |

### Details

`Ckmean.1d.dp`

minimizes unweighted or weighted within-cluster sum of squared distance (L2).

`Ckmedian.1d.dp`

minimizes within-cluster sum of distance (L1). Only unweighted solution is implemented and guarantees optimality.

In contrast to the heuristic `k`-means algorithms implemented in function `kmeans`

, this function optimally assigns elements in numeric vector `x`

into `k`

clusters by dynamic programming (Wang and Song 2011; Song and Zhong 2020). It minimizes the total of within-cluster sums of squared distances (`withinss`) between each element and its corresponding cluster mean. When a range is provided for `k`

, the exact number of clusters is determined by Bayesian information criterion (Song and Zhong 2020). Different from the heuristic `k`-means algorithms whose results may be non-optimal or change from run to run, the result of Ckmeans.1d.dp is guaranteed to be optimal and reproducible, and its advantage in efficiency and accuracy over heuristic `k`-means methods is most pronounced at large `k`.

The `estimate.k`

argument specifies the method to select optimal `k`

based on the Gaussian mixture model using the Bayesian information criterion (BIC). When `estimate.k="BIC"`

, it effectively deals with variance estimation for a cluster with identical values. When `estimate.k="BIC 3.4.12"`

, it uses the code in version 3.4.12 and earlier to estimate `k`

.

The `method`

argument specifies one of three options to speed up the original dynamic programming taking a runtime cubic in sample size `n`. The default `"linear"`

option, giving a total runtime of `O(n \lg n + kn)`

or `O(kn)`

(if `x`

is already sorted in ascending order) is the fastest option but uses the most memory (still `O(kn)`

) (Song and Zhong 2020); the `"loglinear"`

option, with a runtime of `O(kn \lg n)`

, is slightly slower but uses the least memory (Song and Zhong 2020); the slowest `"quadratic"`

option (Wang and Song 2011), with a runtime of `O(kn^2)`

, is provided for the purpose of testing on small data sets.

When the sample size `n` is too large to create two `k \times n`

dynamic programming matrices in memory, we recommend the heuristic solutions implemented in the `kmeans`

function in package stats.

### Value

An object of class "`Ckmeans.1d.dp`

" or "`Ckmedian.1d.dp`

". It is a list containing the following components:

`cluster` |
a vector of clusters assigned to each element in |

`centers` |
a numeric vector of the (weighted) means for each cluster. |

`withinss` |
a numeric vector of the (weighted) within-cluster sum of squares for each cluster. |

`size` |
a vector of the (weighted) number of elements in each cluster. |

`totss` |
total sum of (weighted) squared distances between each element and the sample mean. This statistic is not dependent on the clustering result. |

`tot.withinss` |
total sum of (weighted) within-cluster squared distances between each element and its cluster mean. This statistic is minimized given the number of clusters. |

`betweenss` |
sum of (weighted) squared distances between each cluster mean and sample mean. This statistic is maximized given the number of clusters. |

`xname` |
a character string. The actual name of the |

`yname` |
a character string. The actual name of the |

Each class has a print and a plot method, which are described along with `print.Ckmeans.1d.dp`

and `plot.Ckmeans.1d.dp`

.

### Author(s)

Joe Song and Haizhou Wang

### References

Song M, Zhong H (2020).
“Efficient weighted univariate clustering maps outstanding dysregulated genomic zones in human cancers.”
*Bioinformatics*, **36**(20), 5027–5036.
doi:10.1093/bioinformatics/btaa613.

Wang H, Song M (2011).
“Ckmeans.1d.dp: Optimal `k`

-means clustering in one dimension by dynamic programming.”
*The R Journal*, **3**(2), 29–33.
doi:10.32614/RJ-2011-015.

### See Also

`ahist`

, `plot.Ckmeans.1d.dp`

, `print.Ckmeans.1d.dp`

in this package.

`kmeans`

in package stats that implements several heuristic `k`

-means algorithms.

### Examples

```
# Ex. 1 The number of clusters is provided.
# Generate data from a Gaussian mixture model of three components
x <- c(rnorm(50, sd=0.2), rnorm(50, mean=1, sd=0.3), rnorm(100,
mean=-1, sd=0.25))
# Divide x into 3 clusters
k <- 3
result <- Ckmedian.1d.dp(x, k)
plot(result, main="Optimal univariate k-median given k")
result <- Ckmeans.1d.dp(x, k)
plot(result, main="Optimal univariate k-means given k")
plot(x, col=result$cluster, pch=result$cluster, cex=1.5,
main="Optimal univariate k-means clustering given k",
sub=paste("Number of clusters given:", k))
abline(h=result$centers, col=1:k, lty="dashed", lwd=2)
legend("bottomleft", paste("Cluster", 1:k), col=1:k, pch=1:k,
cex=1.5, bty="n")
# Ex. 2 The number of clusters is determined by Bayesian
# information criterion
# Generate data from a Gaussian mixture model of three components
x <- c(rnorm(50, mean=-3, sd=1), rnorm(50, mean=0, sd=.5),
rnorm(50, mean=3, sd=1))
# Divide x into k clusters, k automatically selected (default: 1~9)
result <- Ckmedian.1d.dp(x)
plot(result, main="Optimal univariate k-median with k estimated")
result <- Ckmeans.1d.dp(x)
plot(result, main="Optimal univariate k-means with k estimated")
k <- max(result$cluster)
plot(x, col=result$cluster, pch=result$cluster, cex=1.5,
main="Optimal univariate k-means clustering with k estimated",
sub=paste("Number of clusters is estimated to be", k))
abline(h=result$centers, col=1:k, lty="dashed", lwd=2)
legend("topleft", paste("Cluster", 1:k), col=1:k, pch=1:k,
cex=1.5, bty="n")
# Ex. 3 Segmenting a time course using optimal weighted
# univariate clustering
n <- 160
t <- seq(0, 2*pi*2, length=n)
n1 <- 1:(n/2)
n2 <- (max(n1)+1):n
y1 <- abs(sin(1.5*t[n1]) + 0.1*rnorm(length(n1)))
y2 <- abs(sin(0.5*t[n2]) + 0.1*rnorm(length(n2)))
y <- c(y1, y2)
w <- y^8 # stress the peaks
res <- Ckmeans.1d.dp(t, k=c(1:10), w)
plot(res)
plot(t, w, main = "Time course weighted k-means",
col=res$cluster, pch=res$cluster,
xlab="Time t", ylab="Transformed intensity w",
type="h")
abline(v=res$centers, col="chocolate", lty="dashed")
text(res$centers, max(w) * .95, cex=0.5, font=2,
paste(round(res$size / sum(res$size) * 100), "/ 100"))
```

*Ckmeans.1d.dp*version 4.3.5 Index]