Univariate Clustering: 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"))

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.

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.

Author

Joe Song and Haizhou Wang

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 Wang2011Ckmeans,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 song2020wucCkmeans.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 \(O(kn)\)) song2020wucCkmeans.1d.dp; the "loglinear" option, with a runtime of \(O(kn \lg n)\), is slightly slower but uses the least memory song2020wucCkmeans.1d.dp; the slowest "quadratic" option Wang2011CkmeansCkmeans.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.

References

Examples

Run this code

# 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"))

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