Univariate Clustering {Ckmeans.1d.dp} | R Documentation |
Perform optimal univariate k-means or k-median clustering in linear (fastest), loglinear, or quadratic (slowest) time.
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"))
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 |
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). 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. 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)); the "loglinear"
option, with a runtime of O(kn lg n), is slightly slower but uses the least memory; the slowest "quadratic"
option, 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 x n dynamic programming matrices in memory, we recommend the heuristic solutions implemented in the kmeans
function in package stats.
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
.
Joe Song and Haizhou Wang
Wang, H. and Song, M. (2011) Ckmeans.1d.dp: optimal k-means clustering in one dimension by dynamic programming. The R Journal 3(2), 29–33. Retrieved from https://journal.r-project.org/archive/2011-2/RJournal_2011-2_Wang+Song.pdf
ahist
, plot.Ckmeans.1d.dp
, print.Ckmeans.1d.dp
in this package.
kmeans
in package stats that implements several heuristic k-means algorithms.
# 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"))