1.Plot the original data.
install.packages("mlbench")
library(mlbench)
set.seed(111)
obj <- mlbench.spirals(100,1,0.025)
my.data <- 4 * obj$x
plot(my.data)
plot(my.data)
URL 'http://cran.r-project.org/src/contrib/mlbench_2.1-1.tar.gz'를 시도하고 있습니다 Content type 'application/x-gzip' length 920768 bytes (899 Kb) 열린 URL ================================================== downloaded 899 Kb * installing *source* package ‘mlbench’ ... ** 패키지 ‘mlbench’ 가 성공적으로 압축해제 되었고, MD5 sums 가 확인되었습니다 ** libs gcc -std=gnu99 -I/root/sage-5.8/local/lib/R/include -DNDEBUG -fpic -g -O2 -c waveform.c -o waveform.o gcc -std=gnu99 -shared -o mlbench.so waveform.o -L/root/sage-5.8/local/lib/R//lib -lR 다음 부분에 설치 /root/sage-5.8/local/lib/R/library/mlbench/libs ** R ** data ** inst ** preparing package for lazy loading ** help *** installing help indices ** building package indices ** testing if installed package can be loaded * DONE (mlbench) 다운로드된 소스 패키지들은 다음에 위치해 있습니다 ‘/tmp/RtmpkiZRRz/downloaded_packages’ URL 'http://cran.r-project.org/src/contrib/mlbench_2.1-1.tar.gz'를 시도하고 있습니다 Content type 'application/x-gzip' length 920768 bytes (899 Kb) 열린 URL ================================================== downloaded 899 Kb * installing *source* package ‘mlbench’ ... ** 패키지 ‘mlbench’ 가 성공적으로 압축해제 되었고, MD5 sums 가 확인되었습니다 ** libs gcc -std=gnu99 -I/root/sage-5.8/local/lib/R/include -DNDEBUG -fpic -g -O2 -c waveform.c -o waveform.o gcc -std=gnu99 -shared -o mlbench.so waveform.o -L/root/sage-5.8/local/lib/R//lib -lR 다음 부분에 설치 /root/sage-5.8/local/lib/R/library/mlbench/libs ** R ** data ** inst ** preparing package for lazy loading ** help *** installing help indices ** building package indices ** testing if installed package can be loaded * DONE (mlbench) 다운로드된 소스 패키지들은 다음에 위치해 있습니다 ‘/tmp/RtmpkiZRRz/downloaded_packages’  
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2. Find the similarity matrix S (if two data are very close, then the similarity will be closer to 1.)
s <- function(x1, x2, alpha=1) {
exp(- alpha * norm(as.matrix(x1-x2), type="F"))
}
make.similarity <- function(my.data, similarity) {
N <- nrow(my.data)
S <- matrix(rep(NA,N^2), ncol=N)
for(i in 1:N) {
for(j in 1:N) {
S[i,j] <- similarity(my.data[i,], my.data[j,])
}
}
S
}
S <- make.similarity(my.data, s)
round(S[1:12,1:12],2)
[,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12] [1,] 1.00 0.06 0.74 0.63 0.10 0.09 0.57 0.03 0.08 0.47 0.39 0.32 [2,] 0.06 1.00 0.07 0.04 0.21 0.28 0.05 0.01 0.53 0.05 0.06 0.04 [3,] 0.74 0.07 1.00 0.61 0.09 0.09 0.67 0.04 0.08 0.60 0.52 0.41 [4,] 0.63 0.04 0.61 1.00 0.06 0.06 0.72 0.04 0.05 0.57 0.44 0.40 [5,] 0.10 0.21 0.09 0.06 1.00 0.78 0.06 0.01 0.39 0.06 0.05 0.04 [6,] 0.09 0.28 0.09 0.06 0.78 1.00 0.06 0.01 0.51 0.06 0.06 0.04 [7,] 0.57 0.05 0.67 0.72 0.06 0.06 1.00 0.06 0.05 0.79 0.61 0.55 [8,] 0.03 0.01 0.04 0.04 0.01 0.01 0.06 1.00 0.01 0.07 0.08 0.10 [9,] 0.08 0.53 0.08 0.05 0.39 0.51 0.05 0.01 1.00 0.05 0.05 0.04 [10,] 0.47 0.05 0.60 0.57 0.06 0.06 0.79 0.07 0.05 1.00 0.78 0.68 [11,] 0.39 0.06 0.52 0.44 0.05 0.06 0.61 0.08 0.05 0.78 1.00 0.75 [12,] 0.32 0.04 0.41 0.40 0.04 0.04 0.55 0.10 0.04 0.68 0.75 1.00 [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12] [1,] 1.00 0.06 0.74 0.63 0.10 0.09 0.57 0.03 0.08 0.47 0.39 0.32 [2,] 0.06 1.00 0.07 0.04 0.21 0.28 0.05 0.01 0.53 0.05 0.06 0.04 [3,] 0.74 0.07 1.00 0.61 0.09 0.09 0.67 0.04 0.08 0.60 0.52 0.41 [4,] 0.63 0.04 0.61 1.00 0.06 0.06 0.72 0.04 0.05 0.57 0.44 0.40 [5,] 0.10 0.21 0.09 0.06 1.00 0.78 0.06 0.01 0.39 0.06 0.05 0.04 [6,] 0.09 0.28 0.09 0.06 0.78 1.00 0.06 0.01 0.51 0.06 0.06 0.04 [7,] 0.57 0.05 0.67 0.72 0.06 0.06 1.00 0.06 0.05 0.79 0.61 0.55 [8,] 0.03 0.01 0.04 0.04 0.01 0.01 0.06 1.00 0.01 0.07 0.08 0.10 [9,] 0.08 0.53 0.08 0.05 0.39 0.51 0.05 0.01 1.00 0.05 0.05 0.04 [10,] 0.47 0.05 0.60 0.57 0.06 0.06 0.79 0.07 0.05 1.00 0.78 0.68 [11,] 0.39 0.06 0.52 0.44 0.05 0.06 0.61 0.08 0.05 0.78 1.00 0.75 [12,] 0.32 0.04 0.41 0.40 0.04 0.04 0.55 0.10 0.04 0.68 0.75 1.00 |
3. Compute an adjacency matrix A based on S.
This is usually done by applying a k-nearest neighboor filter to build a representation of a graph connecting just the closest dataset points.
make.affinity <- function(S, n.neighboors=2) {
N <- length(S[,1])
if (n.neighboors >= N) { # fully connected
A <- S
} else {
A <- matrix(rep(0,N^2), ncol=N)
for(i in 1:N) { # for each line
# only connect to those points with larger similarity
best.similarities <- sort(S[i,], decreasing=TRUE)[1:n.neighboors]
for (s in best.similarities) {
j <- which(S[i,] == s)
A[i,j] <- S[i,j]
A[j,i] <- S[i,j] # to make an undirected graph, ie, the matrix becomes symmetric
}
}
}
A
}
A <- make.affinity(S, 3) # use 5 neighboors (includes self)
round(A[1:12,1:12],2)
[,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12] [1,] 1.00 0 0.74 0.63 0.00 0.00 0.00 0 0 0.00 0.00 0.00 [2,] 0.00 1 0.00 0.00 0.00 0.00 0.00 0 0 0.00 0.00 0.00 [3,] 0.74 0 1.00 0.00 0.00 0.00 0.67 0 0 0.00 0.00 0.00 [4,] 0.63 0 0.00 1.00 0.00 0.00 0.72 0 0 0.00 0.00 0.00 [5,] 0.00 0 0.00 0.00 1.00 0.78 0.00 0 0 0.00 0.00 0.00 [6,] 0.00 0 0.00 0.00 0.78 1.00 0.00 0 0 0.00 0.00 0.00 [7,] 0.00 0 0.67 0.72 0.00 0.00 1.00 0 0 0.79 0.00 0.00 [8,] 0.00 0 0.00 0.00 0.00 0.00 0.00 1 0 0.00 0.00 0.00 [9,] 0.00 0 0.00 0.00 0.00 0.00 0.00 0 1 0.00 0.00 0.00 [10,] 0.00 0 0.00 0.00 0.00 0.00 0.79 0 0 1.00 0.78 0.00 [11,] 0.00 0 0.00 0.00 0.00 0.00 0.00 0 0 0.78 1.00 0.75 [12,] 0.00 0 0.00 0.00 0.00 0.00 0.00 0 0 0.00 0.75 1.00 [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12] [1,] 1.00 0 0.74 0.63 0.00 0.00 0.00 0 0 0.00 0.00 0.00 [2,] 0.00 1 0.00 0.00 0.00 0.00 0.00 0 0 0.00 0.00 0.00 [3,] 0.74 0 1.00 0.00 0.00 0.00 0.67 0 0 0.00 0.00 0.00 [4,] 0.63 0 0.00 1.00 0.00 0.00 0.72 0 0 0.00 0.00 0.00 [5,] 0.00 0 0.00 0.00 1.00 0.78 0.00 0 0 0.00 0.00 0.00 [6,] 0.00 0 0.00 0.00 0.78 1.00 0.00 0 0 0.00 0.00 0.00 [7,] 0.00 0 0.67 0.72 0.00 0.00 1.00 0 0 0.79 0.00 0.00 [8,] 0.00 0 0.00 0.00 0.00 0.00 0.00 1 0 0.00 0.00 0.00 [9,] 0.00 0 0.00 0.00 0.00 0.00 0.00 0 1 0.00 0.00 0.00 [10,] 0.00 0 0.00 0.00 0.00 0.00 0.79 0 0 1.00 0.78 0.00 [11,] 0.00 0 0.00 0.00 0.00 0.00 0.00 0 0 0.78 1.00 0.75 [12,] 0.00 0 0.00 0.00 0.00 0.00 0.00 0 0 0.00 0.75 1.00 |
4. There is the need of a degree matrix.
D <- diag(apply(A, 1, sum)) # sum rows
round(D[1:8,1:8],1)
[,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [1,] 2.4 0.0 0.0 0.0 0.0 0.0 0.0 0.0 [2,] 0.0 2.6 0.0 0.0 0.0 0.0 0.0 0.0 [3,] 0.0 0.0 2.4 0.0 0.0 0.0 0.0 0.0 [4,] 0.0 0.0 0.0 2.4 0.0 0.0 0.0 0.0 [5,] 0.0 0.0 0.0 0.0 2.5 0.0 0.0 0.0 [6,] 0.0 0.0 0.0 0.0 0.0 2.5 0.0 0.0 [7,] 0.0 0.0 0.0 0.0 0.0 0.0 3.2 0.0 [8,] 0.0 0.0 0.0 0.0 0.0 0.0 0.0 2.3 [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [1,] 2.4 0.0 0.0 0.0 0.0 0.0 0.0 0.0 [2,] 0.0 2.6 0.0 0.0 0.0 0.0 0.0 0.0 [3,] 0.0 0.0 2.4 0.0 0.0 0.0 0.0 0.0 [4,] 0.0 0.0 0.0 2.4 0.0 0.0 0.0 0.0 [5,] 0.0 0.0 0.0 0.0 2.5 0.0 0.0 0.0 [6,] 0.0 0.0 0.0 0.0 0.0 2.5 0.0 0.0 [7,] 0.0 0.0 0.0 0.0 0.0 0.0 3.2 0.0 [8,] 0.0 0.0 0.0 0.0 0.0 0.0 0.0 2.3 |
5. Find the Laplacian matrix and normalized Laplacian matrix.
U <- D - A
round(U[1:12,1:12],1)
[,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12] [1,] 1.4 0.0 -0.7 -0.6 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 [2,] 0.0 1.6 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 [3,] -0.7 0.0 1.4 0.0 0.0 0.0 -0.7 0.0 0.0 0.0 0.0 0.0 [4,] -0.6 0.0 0.0 1.4 0.0 0.0 -0.7 0.0 0.0 0.0 0.0 0.0 [5,] 0.0 0.0 0.0 0.0 1.5 -0.8 0.0 0.0 0.0 0.0 0.0 0.0 [6,] 0.0 0.0 0.0 0.0 -0.8 1.5 0.0 0.0 0.0 0.0 0.0 0.0 [7,] 0.0 0.0 -0.7 -0.7 0.0 0.0 2.2 0.0 0.0 -0.8 0.0 0.0 [8,] 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.3 0.0 0.0 0.0 0.0 [9,] 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.5 0.0 0.0 0.0 [10,] 0.0 0.0 0.0 0.0 0.0 0.0 -0.8 0.0 0.0 1.6 -0.8 0.0 [11,] 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 -0.8 1.5 -0.8 [12,] 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 -0.8 1.5 [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12] [1,] 1.4 0.0 -0.7 -0.6 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 [2,] 0.0 1.6 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 [3,] -0.7 0.0 1.4 0.0 0.0 0.0 -0.7 0.0 0.0 0.0 0.0 0.0 [4,] -0.6 0.0 0.0 1.4 0.0 0.0 -0.7 0.0 0.0 0.0 0.0 0.0 [5,] 0.0 0.0 0.0 0.0 1.5 -0.8 0.0 0.0 0.0 0.0 0.0 0.0 [6,] 0.0 0.0 0.0 0.0 -0.8 1.5 0.0 0.0 0.0 0.0 0.0 0.0 [7,] 0.0 0.0 -0.7 -0.7 0.0 0.0 2.2 0.0 0.0 -0.8 0.0 0.0 [8,] 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.3 0.0 0.0 0.0 0.0 [9,] 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.5 0.0 0.0 0.0 [10,] 0.0 0.0 0.0 0.0 0.0 0.0 -0.8 0.0 0.0 1.6 -0.8 0.0 [11,] 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 -0.8 1.5 -0.8 [12,] 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 -0.8 1.5 |
evL <- eigen(U, symmetric=TRUE)
plot(1:10, rev(evL$values)[1:10], log="y")
plot(1:10, rev(evL$values)[1:10], log="y")
경고 메시지가 손실되었습니다 In xy.coords(x, y, xlabel, ylabel, log) : 1 y value <= 0 omitted from logarithmic plot 경고 메시지가 손실되었습니다 In xy.coords(x, y, xlabel, ylabel, log) : 1 y value <= 0 omitted from logarithmic plot 경고 메시지가 손실되었습니다 In xy.coords(x, y, xlabel, ylabel, log) : 1 y value <= 0 omitted from logarithmic plot 경고 메시지가 손실되었습니다 In xy.coords(x, y, xlabel, ylabel, log) : 1 y value <= 0 omitted from logarithmic plot  
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7. Find the corresponding eigenvectors of k smallest eigenvalues of each matrix and use k-means clustering to find the appropriate clusters.
k <- 2
Z <- evL$vectors[,(ncol(evL$vectors)-k+1):ncol(evL$vectors)]
km <- kmeans(Z, centers=k)
plot(my.data, col=km$cluster)
plot(my.data, col=km$cluster)
 
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