1. Grouping Data Methods of cluster analysis

2. Goals 1 1.We want to identify groups of similar artifacts or features or sites or graves, etc that represent cultural, functional, or chronological differences 2.We want to create groups as a measurement technique to see how they vary with external variables

3. Goals 2 3.We want to cluster artifacts or sites based on their location to identify spatial clusters

4. Real vs. Created Types Differences in goals –Real types are the aim of Goal 1 –Created types are the aim of Goal 2 Debate over whether Real types can be discovered with any degree of certainty Cluster analysis guarantees groups – you must confirm their utility

5. Initial Decisions 1 What variables to use? –All possible –Constructed variables (from principal components, correspondence analysis, or multi-dimensional scaling) –Restricted set of variables that support the goal(s) of creating groups (e.g. functional groups, cultural or stylistic groups)

6. Initial Decisions 2 How to transform the variables? –Log transforms –Conversion to percentages (to weight rows equally) –Size standardization (dividing by geometric mean) –Z – scores (to weight columns equally) –Conversion of categorical variables

7. Initial Decisions 3 How to measure distance? –Types of variables –Goals of the analysis –If uncertain, try multiple methods

8. Methods of Grouping Partitioning Methods – divide the data into groups Hierarchical Methods –Agglomerating – from n clusters to 1 cluster –Divisive – from 1 cluster to k clusters

9. Partitioning K – Means, K – Medoids, Fuzzy Measure of distance – but do not need to compute full distance matrix Specify number of groups in advance Minimizing within group variability Finds spherical clusters

10. Procedure Start with centers for k groups (user- supplied or random) Repeat up to iter.max times (default 10) –Allocate rows to their closest center –Recalculate the center positions Stop Different criteria for allocation Use multiple starts (e.g. 5 – 15)

11. Evaluation 1 Compute groups for a range of cluster sizes and plot within group sums of squares to look for sharp increases Cluster randomized versions of the data and compare the results Examine table of statistics by group

12. Evaluation 2 Plot groups in two dimensions with PCA, CA, or MDS Compare the groups using data or information not included in the analysis

13. Partitioning Using R Base R includes kmeans() for forming groups by partitioning Rcmdr includes KMeans() to iterate kmeans() for best solution Package cluster() includes pam() which uses medoids for more robust grouping and fanny() which forms fuzzy clusters

14. Example DarlPoints (not DartPoints) has 4 measurements for 23 Darl points Create Z-scores to weight variables equally with Data | Manage variables in active data set | Standardize variables … (or could use PCA and PC Scores)

15. Example (cont) Use Rcmdr to partition the data into 5, 4, 3, and 2 groups Statistics | Dimensional analysis | Cluster analysis | k-means cluster analysis … TWSS = 15.42, 19.78, 25.83, 34.24 Select group number and have Rcmdr add group to data set

18. Evaluation Evaluate groups against randomized data –Randomly permute each variable –Run k-means –Compare random and non-random results Evaluate groups against external criteria (location, material, age, etc)

19. KMPlotWSS <- function(data, ming, maxg) { WSS <- sapply(ming:maxg, function(x) kmeans(data, centers = x, iter.max = 10, nstart = 10)$tot.withinss) plot(ming:maxg, WSS, las=1, type="b", xlab="Number of Groups", ylab="Total Within Sum of Squares", pch=16) print(WSS) } KMRandWSS <- function(data, samples, min, max) { KRand <- function(data, min, max){ Rnd <- apply(data, 2, sample) sapply(min:max, function(y) kmeans(Rnd, y, iter.max= 10, nstart=5)$tot.withinss) } Sim <- sapply(1:samples, function(x) KRand(data, min, max)) t(apply(Sim, 1, quantile, c(0,.005,.01,.025,.5,.975,.99,.995, 1))) }

20. # Compare data to randomized sets KMPlotWSS(DarlPoints[,6:9], 1, 10) Qtiles <- KMRandWSS(DarlPoints[,6:9], 2000, 1, 10) matlines(1:10, Qtiles[,c(1, 5, 9)], lty=c(3, 2, 3), lwd=2, col="dark gray") legend("topright", c("Observed", "Median (Random)", "Max/Min Random"), col=c("black", "dark gray", "dark gray"), lwd=c(1, 2, 2), lty=c(1, 2, 3))

22. Hierarchical Methods Agglomerative – successive merging Divisive - successive splitting –Monothetic – binary data –Polythetic – interval/ratio

23. Agglomerative At the start all rows are in separate groups (n groups or clusters) At each stage two rows are merged, a row and a group are merged, or two groups are merged The process stops when all rows are in a single cluster

24. Agglomeration Methods How should clusters be formed? –Single Linkage, irregular shape groups –Average Linkage – spherical groups –Complete Linkage – spherical groups –Ward’s Method – spherical groups –Median – dendrogram inversions –Centroid – dendrogram inversions –McQuitty – similarity by reciprocal pairs

25. Agglomerating with R Base R includes hclus() for forming groups by partitioning Package cluster() includes agnes() Rcmdr uses hclus() via Statistics | Dimensional analysis | Cluster analysis | Hierarchical cluster analysis …

26. HClust Rcmdr menus provide –Cluster analysis and plot –Summary statistics by group –Adding cluster to data set To get traditional dendrogram: –plot(HClust.1, hang=-1, main= "Darl Points", xlab= "Catalog Number", sub="Method=Ward; Distance=Euclidian") –rect.hclust(HClust.1, 3)

29. summary(as.factor(cutree(HClust.1, k = 3))) # Cluster Sizes 1 2 3 11 6 6 by(model.matrix(~-1 + Z.Length + Z.Thickness + Z.Weight + Z.Width, DarlPoints), as.factor(cutree(HClust.1, k = 3)), mean) # Cluster Centroids INDICES: 1 Z.Length Z.Thickness Z.Weight Z.Width -0.1345150 -0.1585615 -0.2523805 -0.1241642 ------------------------------------------------------------ INDICES: 2 Z.Length Z.Thickness Z.Weight Z.Width -1.1085541 -0.9209550 -0.9400026 -0.8200594 ------------------------------------------------------------ INDICES: 3 Z.Length Z.Thickness Z.Weight Z.Width 1.355165 1.211651 1.402700 1.047694 > biplot(princomp(model.matrix(~-1 + Z.Length + Z.Thickness + Z.Weight + Z.Width, DarlPoints)), xlabs = as.character(cutree(HClust.1, k = 3)))

31. > cbind(HClust.1$merge, HClust.1$height) [,1] [,2] [,3] [1,] -12 -13 0.3983821 [2,] -2 -3 0.5112670 [3,] -9 -14 0.5247650 [4,] -10 -17 0.5572146 [5,] -15 3 0.7362171 [6,] -1 -11 0.7471874 [7,] -6 -18 0.8120594 [8,] -7 -8 0.8491895 [9,] 4 5 0.9841552 [10,] 2 6 1.2150606 [11,] -19 -21 1.2300507 [12,] 1 10 1.4059158 [13,] -22 11 1.4963400 [14,] -16 -20 1.5800167 [15,] -4 9 1.6195709 [16,] -5 12 2.1556543 [17,] -23 13 2.4007863 [18,] 7 14 2.4252670 [19,] 8 17 3.2632812 [20,] 16 18 4.9021149 [21,] 15 20 6.6290417 [22,] 19 21 18.7730146

34. Divisive At the start all rows are considered to be a single group At each stage a group is divided into two groups based on the average dissimilarities The process stops when all rows are in separate clusters