Objects
Most clustering methods require calculating a “distance” between two objects.
Let \(\pmb{a} = (a_1, a_2, \ldots, a_n)\) be one object and \(\pmb{b} = (b_1, b_2, \ldots, b_n)\) be another object.
We will assume both objects are composed of real numbers.
Euclidean
Euclidean distance is the shortest spatial distance between two objects in Euclidean space.
Euclidean distance is calculated as:
\[d(\pmb{a}, \pmb{b}) = \sqrt{\sum_{i=1}^n \left(a_i - b_i \right)^2}\]
Manhattan
Manhattan distance is sometimes called taxicab distance. If you picture two locations in a city, it is the distance a taxicab must travel to get from one location to the other.
Manhattan distance is calculated as:
\[d(\pmb{a}, \pmb{b}) = \sum_{i=1}^n \left| a_i - b_i \right|\]
dist()
A distance matrix – which is the set of values resulting from a distance measure applied to all pairs of objects – can be obtained through the function dist().
Default arguments for dist():
> str(dist)
function (x, method = "euclidean", diag = FALSE, upper = FALSE,
p = 2)
The key argument for us is method= which can take values method="euclidean" and method="manhattan" among others. See ?dist.
Distance Matrix data1
> sub_data1 <- data1[1:4, c(1,2)]
> sub_data1
x y
1 2.085818 2.248086
2 1.896636 1.369547
3 2.097729 2.386383
4 1.491026 2.029814
> mydist <- dist(sub_data1)
> print(mydist)
1 2 3
2 0.8986772
3 0.1388086 1.0365293
4 0.6335776 0.7749019 0.7037257
> (sub_data1[1,] - sub_data1[2,])^2 %>% sum() %>% sqrt()
[1] 0.8986772
Strategy
Hierarchical clustering is a hierarchical agglomerative, bottom-up clustering method that strategically joins objects into larger and larger clusters, until all objects are contained in a single cluster.
Hierarchical clustering results are typically displayed as a dendrogram.
The number of clusters does not necessarily need to be known or chosen by the analyst.
Algorithm
The algorithm for hierarchical clustering works as follows.
- Start with each object assigned as its own cluster.
- Calculate a distance between all pairs of clusters.
- Join the two clusters with the smallest distance.
- Repeat steps 2–3 until there is only one cluster.
At the very first iteration of the algorithm, all we need is some distance function (e.g., Euclidean or Manhattan) to determine the two objects that are closest. But once clusters with more than one object are present, how do we calculate the distance between two clusters? This is where a key choice called the linkage method or criterion is needed.
Linkage Criteria
Suppose there are two clusters \(A\) and \(B\) and we have a distance function \(d(\pmb{a}, \pmb{b})\) for all objects \(\pmb{a} \in A\) and \(\pmb{b} \in B\). Here are three ways (among many) to calculate a distance between clusters \(A\) and \(B\):
\begin{eqnarray}
\mbox{Complete: } & \max \{d(\pmb{a}, \pmb{b}): \pmb{a} \in A, \pmb{b} \in B\} \\
\mbox{Single: } & \min \{d(\pmb{a}, \pmb{b}): \pmb{a} \in A, \pmb{b} \in B\} \\
\mbox{Average: } & \frac{1}{|A| |B|} \sum_{\pmb{a} \in A} \sum_{\pmb{b} \in B} d(\pmb{a}, \pmb{b})
\end{eqnarray}
hclust()
The hclust() function produces an R object that contains all of the information needed to create a complete hierarchical clustering.
Default arguments for hclust():
> str(hclust)
function (d, method = "complete", members = NULL)
The primary input for hclust() is the d argument, which is a distance matrix (usually obtained from dist()). The method argument takes the linkage method, which includes method="complete", method="single", method="average", etc. See ?hclust.
Hierarchical Clustering of data1
Standard hclust() Usage
> mydist <- dist(data1, method = "euclidean")
> myhclust <- hclust(mydist, method="complete")
> plot(myhclust)
as.dendrogram()
> plot(as.dendrogram(myhclust))
Modify the Labels
> library(dendextend)
> dend1 <- as.dendrogram(myhclust)
> labels(dend1) <- data1$true_clusters
> labels_colors(dend1) <-
+ c("red", "blue", "gray47")[as.numeric(data1$true_clusters)]
> plot(dend1, axes=FALSE, main=" ", xlab=" ")
Color the Branches
> dend2 <- as.dendrogram(myhclust)
> labels(dend2) <- rep(" ", nrow(data1))
> dend2 <- color_branches(dend2, k = 3, col=c("red", "blue", "gray47"))
> plot(dend2, axes=FALSE, main=" ", xlab=" ")
Cluster Assignments (\(K = 3\))
> est_clusters <- cutree(myhclust, k=3)
> est_clusters
[1] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
[30] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2
[59] 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
[88] 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
[117] 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
[146] 3 3 3 3 3
> est_clusters <- factor(est_clusters)
> p <- data1 %>%
+ mutate(est_clusters=est_clusters) %>%
+ ggplot()
> p + geom_point(aes(x=x, y=y, color=est_clusters))
Cluster Assignments (\(K = 3\))
Cluster Assignments (\(K = 2\))
> (data1 %>%
+ mutate(est_clusters=factor(cutree(myhclust, k=2))) %>%
+ ggplot()) + geom_point(aes(x=x, y=y, color=est_clusters))
Cluster Assignments (\(K = 4\))
> (data1 %>%
+ mutate(est_clusters=factor(cutree(myhclust, k=4))) %>%
+ ggplot()) + geom_point(aes(x=x, y=y, color=est_clusters))
Cluster Assignments (\(K = 6\))
> (data1 %>%
+ mutate(est_clusters=factor(cutree(myhclust, k=6))) %>%
+ ggplot()) + geom_point(aes(x=x, y=y, color=est_clusters))
Linkage: Complete (Default)
> data1 %>% dist() %>% hclust(method="complete") %>%
+ as.dendrogram() %>% plot(axes=FALSE)
Linkage: Average
> data1 %>% dist() %>% hclust(method="average") %>%
+ as.dendrogram() %>% plot(axes=FALSE)
Linkage: Single
> data1 %>% dist() %>% hclust(method="single") %>%
+ as.dendrogram() %>% plot(axes=FALSE)
Linkage: Ward
> data1 %>% dist() %>% hclust(method="ward.D") %>%
+ as.dendrogram() %>% plot(axes=FALSE)
Hierarchical Clustering of data2
as.dendrogram()
> mydist <- dist(data2, method = "euclidean")
> myhclust <- hclust(mydist, method="complete")
> plot(as.dendrogram(myhclust))
Modify the Labels
> library(dendextend)
> dend1 <- as.dendrogram(myhclust)
> labels(dend1) <- data2$true_clusters
> labels_colors(dend1) <-
+ c("red", "blue")[as.numeric(data2$true_clusters)]
> plot(dend1, axes=FALSE, main=" ", xlab=" ")
Color the Branches
> dend2 <- as.dendrogram(myhclust)
> labels(dend2) <- rep(" ", nrow(data2))
> dend2 <- color_branches(dend2, k = 2, col=c("red", "blue"))
> plot(dend2, axes=FALSE, main=" ", xlab=" ")
Cluster Assignments (\(K = 2\))
> (data2 %>%
+ mutate(est_clusters=factor(cutree(myhclust, k=2))) %>%
+ ggplot()) + geom_point(aes(x=x, y=y, color=est_clusters))
Cluster Assignments (\(K = 3\))
> (data2 %>%
+ mutate(est_clusters=factor(cutree(myhclust, k=3))) %>%
+ ggplot()) + geom_point(aes(x=x, y=y, color=est_clusters))
Cluster Assignments (\(K = 4\))
> (data2 %>%
+ mutate(est_clusters=factor(cutree(myhclust, k=4))) %>%
+ ggplot()) + geom_point(aes(x=x, y=y, color=est_clusters))
Cluster Assignments (\(K = 5\))
> (data2 %>%
+ mutate(est_clusters=factor(cutree(myhclust, k=6))) %>%
+ ggplot()) + geom_point(aes(x=x, y=y, color=est_clusters))
Strategy
K-means clustering is a top-down, partitioning cluster analysis method that assigns each object to one of \(K\) clusters based on the distance between each object and the cluster centers, called centroids.
This is an iterative algorithm with potential random initial values.
The value of \(K\) is typically unknown and must be determined by the analyst.
Centroid
A centroid is the coordinate-wise average of all objects in a cluster.
Let \(A\) be a given cluster with objects \(\pmb{a} \in A\). Its centroid is:
\[\overline{\pmb{a}} = \frac{1}{|A|} \sum_{\pmb{a} \in A} \pmb{a}\]
Algorithm
The number of clusters \(K\) must be chosen beforehand.
- Initialize \(K\) cluster centroids.
- Assign each object to a cluster by choosing the cluster with the smalllest distance (e.g., Euclidean) between the object and the cluster centroid.
- Calculate new centroids based on the cluster assignments from Step 2.
- Repeat Steps 2–3 until convergence.
Notes
The initialization of the centroids is typically random, so often the algorithm is run several times with new, random initial centroids.
Convergence is usually defined in terms of neglible changes in the centroids or no changes in the cluster assignments.
kmeans()
K-means clustering can be accomplished through the following function:
> str(kmeans)
function (x, centers, iter.max = 10L, nstart = 1L, algorithm = c("Hartigan-Wong",
"Lloyd", "Forgy", "MacQueen"), trace = FALSE)
x: the data to clusters, objects along rowscenters: either the number of clusters \(K\) or a matrix giving initial centroidsiter.max: the maximum number of iterations allowednstart: how many random intial \(K\) centroids, where the best one is returned
fitted()
The cluster centroids or assigments can be extracted through the function fitted(), which is applied to the output of kmeans().
The input of fitted() is the object returned by kmeans(). The key additional argument is called method.
When method="centers" it returns the centroids. When method="classes" it returns the cluster assignments.
K-Means Clustering of data1
> km1 <- kmeans(x=data1[,-3], centers=3, iter.max=100, nstart=5)
> est_clusters <- fitted(km1, method="classes")
> est_clusters
[1] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
[30] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 3 3 3 3 3 3 3 3
[59] 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
[88] 3 3 3 3 3 3 3 3 3 3 3 3 3 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
[117] 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
[146] 2 2 2 2 2
Centroids of data1
> centroids1 <- fitted(km1, method="centers") %>% unique()
> centroids1
x y
1 1.943184 2.028062
3 2.042872 4.037987
2 4.015934 2.962279
> est_clusters <- fitted(km1, method="classes")
> data1 %>% mutate(est_clusters = factor(est_clusters)) %>%
+ group_by(est_clusters) %>% summarize(mean(x), mean(y))
Source: local data frame [3 x 3]
est_clusters mean(x) mean(y)
(fctr) (dbl) (dbl)
1 1 1.943184 2.028062
2 2 4.015934 2.962279
3 3 2.042872 4.037987
Cluster Assignments (\(K = 3\))
> est_clusters <- factor(est_clusters)
> ggplot(data1) + geom_point(aes(x=x, y=y, color=est_clusters))
Cluster Assignments (\(K = 2\))
Cluster Assignments (\(K = 6\))
K-Means Clustering of data2
> km2 <- kmeans(x=data2[,-3], centers=2, iter.max=100, nstart=5)
> est_clusters <- fitted(km2, method="classes")
> est_clusters
[1] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
[30] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
[59] 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
[88] 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
[117] 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
Cluster Assignments (\(K = 2\))
> est_clusters <- factor(est_clusters)
> ggplot(data2) + geom_point(aes(x=x, y=y, color=est_clusters))
Cluster Assignments (\(K = 3\))
Cluster Assignments (\(K = 5\))
Principal Component Analysis
Goal
For a given set of variables, principal component analysis (PCA) finds (constrained) weighted sums of the variables that capture the maximum level of variation in the data.
Specifically, the first principal component is the weighted sum of the variables that results in a component with the highest variation.
This component is then regressed out of the data, and the second principal component is obtained on the resulting residuals.
This process is repeated until there is no variation left in the data.
Procedure
Suppose we have \(p\) variables, each with \(n\) observations:
\begin{eqnarray}
\pmb{x_1} & = & (x_{11}, x_{12}, \ldots, x_{1n}) \\
\pmb{x_2} & = & (x_{21}, x_{22}, \ldots, x_{2n}) \\
\ & \vdots & \ \\
\pmb{x_p} & = & (x_{p1}, x_{p2}, \ldots, x_{pn})
\end{eqnarray}
Consider all possible weighted sums of these variables
\[\tilde{\pmb{x}} = \sum_{i=1}^{p} w_i \pmb{x_i}\]
where we constrain \(\sum_{i=1}^{p} w_i^2 = 1\).
Procedure
The first principal component (PC1) is the set of weights that produces a vector \(\tilde{\pmb{x}}\) that maximizes
\[\tilde{x}_{1}^2 + \tilde{x}_{2}^2 + \cdots \tilde{x}_{n}^2.\]
Once this is found, then a least squares linear regression of each \(\pmb{x}_i\) on \(\tilde{\pmb{x}}\) (with no intercept) is performed and the residuals are obtained. The next principal component (PC2) is then calculated using the same procedure, and it is regressed out from the to obtain a new set of residuals for calculating PC3.
This iterative process is repeated to obtain up to min\((p, n)\) PCs.
Singular Value Decomposition
Singular value decomposition (SVD) is a numerical matrix decomposition that can find all PCs at once.
This is an advanced topic.
See Principal Component Analysis by I.T. Jolliffe for a thorough account of PCA, including a clear explanation on the relationship between PCA and SVD (in Chapter 1).
Mean Centering and Covariance
PCA can be motivated and derived in terms of covariance matrices.
We will not cover that here, but it is definitely worth learning.
One thing we want to note is that it is usually the case that one centers each variable by its sample mean before performing PCA (i.e., subtract the variable’s sample mean from each observation on that variable). This allows the optimization to be about maximizing sample variance of the component and provides the underlying connection to covariances.
My PCA Function
> pca <- function(x, space=c("rows", "columns"),
+ center=TRUE, scale=FALSE) {
+ space <- match.arg(space)
+ if(space=="columns") {x <- t(x)}
+ x <- t(scale(t(x), center=center, scale=scale))
+ s <- svd(x)
+ loading <- s$u
+ colnames(loading) <- paste0("Loading", 1:ncol(loading))
+ rownames(loading) <- rownames(x)
+ pc <- diag(s$d) %*% t(s$v)
+ rownames(pc) <- paste0("PC", 1:nrow(pc))
+ colnames(pc) <- colnames(x)
+ pve <- s$d^2 / sum(s$d^2)
+ if(space=="columns") {pc <- t(pc); loading <- t(loading)}
+ return(list(pc=pc, loading=loading, pve=pve))
+ }
How It Works (Output)
Output is a list with the following items:
pc: a matrix of all possible PCsloading: the weights or “loadings” that determined each PCpve: the proportion of variation explained by each PC
Note that the rows or columns of pc and loading have names to let you know on which dimension the values are organized.
Application to Weather Data
> mypca <- pca(weather_data, space="rows")
>
> names(mypca)
[1] "pc" "loading" "pve"
> dim(mypca$pc)
[1] 50 50
> dim(mypca$loading)
[1] 2811 50
> mypca$pc[1:3, 1:3]
11 16 18
PC1 5747.5990 7492.4124 7628.1715
PC2 -766.4347 -1269.4797 285.8742
PC3 -196.7599 -365.3963 -1097.7858
> mypca$loading[1:3, 1:3]
Loading1 Loading2 Loading3
AG000060611 -0.015172744 0.013033849 -0.011273121
AGM00060369 -0.009439176 0.016884418 -0.004611284
AGM00060425 -0.015779138 0.007026312 -0.009907972
PC1 vs Time
> day_of_the_year <- as.numeric(colnames(weather_data))
> data.frame(day=day_of_the_year, PC1=mypca$pc[1,]) %>%
+ ggplot() + geom_point(aes(x=day, y=PC1), size=2)
PC2 vs Time
> data.frame(day=day_of_the_year, PC2=mypca$pc[2,]) %>%
+ ggplot() + geom_point(aes(x=day, y=PC2), size=2)
PC1 vs PC2
> data.frame(PC1=mypca$pc[1,], PC2=mypca$pc[2,]) %>%
+ ggplot() + geom_point(aes(x=PC1, y=PC2), size=2)
PC Biplots
Sometimes it is really informative to plot a PC versus another PC (as in the previous slide). This is called a PC biplot.
It is possible that interesting subgroups or clusters of observations will emerge.
This does not appear to be the case in the weather data set, however, due to what we observe on the next slide.
Proportion of Variance Explained
> data.frame(Component=1:length(mypca$pve), PVE=mypca$pve) %>%
+ ggplot() + geom_point(aes(x=Component, y=PVE), size=2)
PCs Reproduce the Data
We can multiple the loadings matrix by the PCs matrix to reproduce the data:
> # mean centered weather data
> weather_data_mc <- weather_data - rowMeans(weather_data)
>
> # difference between the PC projections and the data
> # the small sum is just machine imprecision
> sum(abs(weather_data_mc - mypca$loading %*% mypca$pc))
[1] 3.572175e-08
Loadings
The sum of squared weights – i.e., loadings – equals one for each component:
> sum(mypca$loading[,1]^2)
[1] 1
>
> apply(mypca$loading, 2, function(x) {sum(x^2)})
Loading1 Loading2 Loading3 Loading4 Loading5 Loading6
1 1 1 1 1 1
Loading7 Loading8 Loading9 Loading10 Loading11 Loading12
1 1 1 1 1 1
Loading13 Loading14 Loading15 Loading16 Loading17 Loading18
1 1 1 1 1 1
Loading19 Loading20 Loading21 Loading22 Loading23 Loading24
1 1 1 1 1 1
Loading25 Loading26 Loading27 Loading28 Loading29 Loading30
1 1 1 1 1 1
Loading31 Loading32 Loading33 Loading34 Loading35 Loading36
1 1 1 1 1 1
Loading37 Loading38 Loading39 Loading40 Loading41 Loading42
1 1 1 1 1 1
Loading43 Loading44 Loading45 Loading46 Loading47 Loading48
1 1 1 1 1 1
Loading49 Loading50
1 1
Pairs of PCs Have Correlaton Zero
PCs by contruction have sample correlation equal to zero:
> cor(mypca$pc[1,], mypca$pc[2,])
[1] 1.858194e-16
> cor(mypca$pc[1,], mypca$pc[3,])
[1] 9.762562e-17
> cor(mypca$pc[1,], mypca$pc[12,])
[1] -7.921241e-17
> cor(mypca$pc[5,], mypca$pc[27,])
[1] -2.43523e-16
> # etc...
