Cluster Analysis Classifying the Exoplanets

Cluster Analysis Simple idea, difficult execution Used for indexing large amounts of data in databases. (very hot skill to have 70/hour) “The best form of cluster analysis is ordination, because ordination is not a form of cluster analysis.” –Morgan Byron –No formal def. of a cluster –Results are descriptive and subjective.

R Commands library("scatterplot3d") scatterplot3d(log(planets$mass), log(planets$period), log(planets$eccen), type = "h", angle = 55, scale.y = 0.7, pch = 16, y.ticklabs = seq(0, 10, by = 2), y.margin.add = 0.1) –Taking the log of the each data point –Setting the angle and the physical scale so it looks like a box –Pch is the symbol used for the data point –Seq() function sets the numeric scales –Y.margin.add adds a bit to the vertical margins

Interpretation No real insight after our first view of the data, but it looks neat.

R Commands rge <- apply(planets, 2, max) - apply(planets, 2, min) –Stores the range of the data 2 indicates the column margin of the data matrix planet.dat <- sweep(planets, 2, rge, FUN = "/") –Divides each element in the matrix by the range of the column margin n <- nrow(planet.dat) wss <- rep(0, 10) –Creates a 10 dimensional vector of all 0’s wss[1] <- (n-1)*sum(apply(planet.dat, 2, var)) –This is the sum of squares of all the points – if we partition the data in 1 group. for (i in 2:10) wss[i] <- sum(kmeans(planet.dat, centers = i)$withinss) –Using the kmeans method, as the number of partitions increases, calculates the sum of squares of the members of each group.

The K-Means Method This method uses different ways of minimizing a numerical value - often a notion of distance- by partitioning the data. The method used in this analysis is minimizing the sums of squares of data within a group, and finding a number of groups that has the lowest SS This method can be impractical with the number of partitions increasing very quickly as the number of groups and data points increases.

The “Elbow” In choosing a good number of partitions, the “elbow” or the sharpest angle in the graph is an easy approach. –The steepest angles look to be at 3 and 5 number of groups.

Number of planets in the groups planet_kmeans3 <- kmeans(planet.dat, centers = 3) –We chose to try 3 groups table(planet_kmeans3$cluster) – – ccent <- function(cl) { –f <- function(i) colMeans(planets[cl == i, ]) Finds the mean for each cluster –x <- sapply(sort(unique(cl)), f) Sorts –colnames(x) <- sort(unique(cl)) –return(x) }

The results > ccent(planet_kmeans3$cluster) Cluster mass period eccen Number

Model-Based Clustering in brief –The subjective decision or assumption is the number of clusters. –After that, it becomes a problem of maximizing the likelihood that a partition is the best.

Mclust function Mclust find an appropriate model AND the optimal number of groups. –Not Free?!! Need a liscence agreement from University of Washington. R Commands: –Library(“mclust”) –Planet_mclust <- Mclust(planet.dat) –Plot(planet_mclust, planet.dat) –Print(planet_mclust) The best model is of diagonal clusters of varying volume and shape with 3 groups

Homework Spend 30 minutes attempting exercise 15.1 and send me what you get done. Stick it to the Man! Then practice your air guitar