Ckmeans.1d.dp: Optimal (Weighted) Univariate Clustering
In Ckmeans.1d.dp: Optimal, Fast, and Reproducible Univariate Clustering

Univariate Clustering

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 `NA` elements must be removed from `x` before calling this function. The function will run faster on sorted `x` (in non-decreasing order) than an unsorted input.
`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 `c(1,9)`. When `k` is a range, the actual number of clusters is determined by Bayesian information criterion.
`y`	a value of 1 (default) to specify equal weights of 1 for each element in `x`, or a numeric vector of unequal non-negative weights for each element in `x`. It is highly recommended to use positive (instead of zero) weights to account for the influence of every element. The weights have a strong impact on the clustering result. When the number of clusters `k` is given as a range, the weights should be linearly scaled to sum up to the observed sample size. Currently, `Ckmedian.1d.dp` only works with an equal weight of 1.
`method`	a character string to specify the speedup method to the original cubic runtime dynamic programming. The default is `"linear"`. All methods generate the same optimal results but differ in runtime or memory usage. See Details.
`estimate.k`	a character string to specify the method to estimate optimal `k`. This argument is effective only when a range for `k` is provided. The default is `"BIC"`. See Details.

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 \insertCiteWang2011Ckmeans,song2020wucCkmeans.1d.dp. 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 \insertCitesong2020wucCkmeans.1d.dp. 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)) \insertCitesong2020wucCkmeans.1d.dp; the "loglinear" option, with a runtime of O(kn \lg n), is slightly slower but uses the least memory \insertCitesong2020wucCkmeans.1d.dp; the slowest "quadratic" option \insertCiteWang2011CkmeansCkmeans.1d.dp, 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 `x`. Each cluster is indexed by an integer from 1 to `k`.
`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 `x` argument.
`yname`	a character string. The actual name of the `y` argument.

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

\insertAllCited

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 documentation built on Aug. 20, 2023, 1:08 a.m.