Unsupervised Learning Techniques
1.2 Uses of unsupervised learning
2.2 Three methods
4 Visualise clustering
3 Data visualisation - raw data
4 Constructing the similarity graph
Code
X <- df %>% select(-Player_ID) %>% mutate(across(everything(), as.numeric)) %>% scale()
# Compute k-nearest neighbors adjacency matrix
k <- min(10, nrow(X) - 1) # Ensure k is valid
knn <- get.knn(X, k=k)
adjacency_matrix <- matrix(0, nrow=nrow(X), ncol=nrow(X))
for (i in 1:nrow(X)) {
adjacency_matrix[i, knn$nn.index[i, ]] <- 1
adjacency_matrix[knn$nn.index[i, ], i] <- 1 # Ensure symmetry
}
graph <- graph_from_adjacency_matrix(adjacency_matrix, mode = "undirected", diag = FALSE)
# Improved visualisation
plot(
graph,
vertex.size=10, # Adjust node size
vertex.label=NA,
edge.width=1.5, # Moderate edge width
main="Enhanced KNN Graph of Players",
layout=layout_with_kk(graph), # Kamada-Kawai layout - better spacing
edge.color="darkgrey", # Darken edges for clarity
vertex.color="steelblue" # Improve node visibility
)
7 Clustering results
Code
# Generate synthetic dataset with two moons shape
df_moons <- as.data.frame(mlbench.twonorm(100, d = 2)$x)
names(df_moons) <- c("x", "y")
# Apply K-Means
kmeans_res <- kmeans(df_moons, centers = 2, nstart = 10)
df_moons$kmeans_cluster <- as.factor(kmeans_res$cluster)
# Spectral Clustering
dist_mat <- as.matrix(dist(df_moons))
similarity_matrix <- exp(-dist_mat^2 / (2 * sd(dist_mat)^2))
d <- rowSums(similarity_matrix)
D <- diag(d)
L <- D - similarity_matrix
D_inv_sqrt <- diag(1 / sqrt(d))
L_norm <- D_inv_sqrt %*% L %*% D_inv_sqrt
eigen_decomp <- eigen(L_norm, symmetric = TRUE)
spectral_features <- as.data.frame(eigen_decomp$vectors[, 1:2])
spectral_kmeans <- kmeans(spectral_features, centers = 2, nstart = 10)
df_moons$spectral_cluster <- as.factor(spectral_kmeans$cluster)
# Vis
p1 <- ggplot(df_moons, aes(x = x, y = y, color = kmeans_cluster)) +
geom_point(size = 3) +
ggtitle("K-Means Clustering") +
theme_minimal()
p2 <- ggplot(df_moons, aes(x = x, y = y, color = spectral_cluster)) +
geom_point(size = 3) +
ggtitle("Spectral Clustering") +
theme_minimal()
library(patchwork)
p1 + p2
2 DBSCAN in Action
Code
rm(list=ls())
library(dbscan)
# Create synthetic dataset with two distinct clusters and noise
df_dbscan_demo <- tibble(
x = c(rnorm(50, mean = 2, sd = 0.5), rnorm(50, mean = 6, sd = 0.5), runif(20, min = 0, max = 8)),
y = c(rnorm(50, mean = 2, sd = 0.5), rnorm(50, mean = 6, sd = 0.5), runif(20, min = 0, max = 8))
)
# Apply DBSCAN
dbscan_res_demo <- dbscan(df_dbscan_demo, eps = 0.8, minPts = 5)
df_dbscan_demo$cluster <- as.factor(dbscan_res_demo$cluster)
# Visualising DBSCAN clustering
ggplot(df_dbscan_demo, aes(x = x, y = y, color = cluster)) +
geom_point(size = 3) +
ggtitle("DBSCAN Clustering") +
theme_minimal()
6 K-Means Clustering
7 Spectral Clustering
Code
dist_mat <- as.matrix(dist(df_clusters))
similarity_matrix <- exp(-dist_mat^2 / (2 * sd(dist_mat)^2))
d <- rowSums(similarity_matrix)
D <- diag(d)
L <- D - similarity_matrix
D_inv_sqrt <- diag(1 / sqrt(d))
L_norm <- D_inv_sqrt %*% L %*% D_inv_sqrt
eigen_decomp <- eigen(L_norm, symmetric = TRUE)
spectral_features <- as.data.frame(eigen_decomp$vectors[, 1:3])
spectral_kmeans <- kmeans(spectral_features, centers = 3, nstart = 10)
df_clusters$spectral_cluster <- as.factor(spectral_kmeans$cluster)
ggplot(df_clusters, aes(x = x, y = y, color = spectral_cluster)) +
geom_point(size = 3) +
ggtitle("Spectral Clustering") +
theme_minimal()
8 DBSCAN Clustering
10 Example
Code
11 OPTICS Reachability Plot
- A key feature of OPTICS is the reachability plot (a visual representation of the clustering structure) that helps us intuitively identify the optimal cluster configuration.
- The reachability plot shows the structure of clusters.
- Valleys represent dense regions (clusters), while peaks indicate sparser areas.
12 Summary
Code
library(patchwork)
p1 <- ggplot(df_clusters, aes(x = x, y = y, color = kmeans_cluster)) +
geom_point(size = 3) +
ggtitle("K-Means") +
theme_minimal()
p2 <- ggplot(df_clusters, aes(x = x, y = y, color = spectral_cluster)) +
geom_point(size = 3) +
ggtitle("Spectral Clustering") +
theme_minimal()
p3 <- ggplot(df_clusters, aes(x = x, y = y, color = dbscan_cluster)) +
geom_point(size = 3) +
ggtitle("DBSCAN") +
theme_minimal()
p4 <- ggplot(df_clusters, aes(x = x, y = y, color = optics_cluster)) +
geom_point(size = 3) +
ggtitle("OPTICS") +
theme_minimal()
p1 + p2 + p3 + p4
3.2 Techniques
Commonly-using techniques include PCA (Principal Component Analysis), t-SNE (t-Distributed Stochastic Neighbor Embedding), Manifold Learning and Neural Networks.
4 Scree Plot: variance explained
- The scree plot shows the proportion of variance explained by each PC.
- The elbow method helps decide how many components to retain.
5 PCA Biplot
- Points represent observations, and arrows represent features.
- The direction of an arrow indicates the contribution of a feature to the PCs.
- Closer points indicate greater similarity.
4 t-SNE Visualisation
- Each point represents a player, and their positioning reflects similarity.
- t-SNE tends to cluster similar players together.
5 Comparing PCA vs. t-SNE
Code
# Perform PCA
pca_res <- prcomp(df_sports, scale. = TRUE)
pca_df <- as_tibble(pca_res$x[, 1:2]) %>% rename(Dim1 = PC1, Dim2 = PC2)
# PCA vs. t-SNE plot
p1 <- ggplot(pca_df, aes(x = Dim1, y = Dim2)) +
geom_point(color = "red", size = 3, alpha = 0.7) +
ggtitle("PCA Result") +
theme_minimal()
p2 <- ggplot(tsne_df, aes(x = Dim1, y = Dim2)) +
geom_point(color = "blue", size = 3, alpha = 0.7) +
ggtitle("t-SNE Result") +
theme_minimal()
library(patchwork)
p1 + p2
- PCA maintains global structure but assumes linear relationships.
- t-SNE captures local clusters better but does not preserve distances well.
- Note that t-SNE excels at local structure preservation.
4 Performing UMAP
- UMAP is faster than t-SNE and retains more global structure.
5 Performing Isomap
Code
library(vegan)
library(ggplot2)
# Compute distance matrix
dist_matrix <- dist(df_sports, method = "euclidean")
# Apply Isomap
isomap_res <- isomap(dist_matrix, ndim = 2, k = 5)
# Extract embedded coordinates
isomap_df <- as_tibble(isomap_res$points)
# Rename dimensions
colnames(isomap_df) <- c("Dim1", "Dim2")
# Plot Isomap-transformed data
ggplot(isomap_df, aes(x = Dim1, y = Dim2)) +
geom_point(alpha = 0.7, color = "blue") +
labs(title = "Isomap Dimensionality Reduction",
x = "Dim1",
y = "Dim2") +
theme_minimal()
- Isomap preserves geodesic distances and is useful for manifold structures.
6 Comparing PCA, t-SNE, UMAP, and Isomap
Code
# Perform PCA
pca_res <- prcomp(df_sports, scale. = TRUE)
pca_df <- as_tibble(pca_res$x[, 1:2])%>% rename(Dim1 = PC1, Dim2 = PC2)
p1 <- ggplot(pca_df, aes(x = Dim1, y = Dim2)) +
geom_point(color = "purple", size = 3, alpha = 0.7) +
ggtitle("PCA") +
theme_minimal()
p2 <- ggplot(tsne_df, aes(x = Dim1, y = Dim2)) +
geom_point(color = "blue", size = 3, alpha = 0.7) +
ggtitle("t-SNE") +
theme_minimal()
p3 <- ggplot(umap_df, aes(x = Dim1, y = Dim2)) +
geom_point(color = "green", size = 3, alpha = 0.7) +
ggtitle("UMAP") +
theme_minimal()
p4 <- ggplot(isomap_df, aes(x = Dim1, y = Dim2)) +
geom_point(color = "red", size = 3, alpha = 0.7) +
ggtitle("Isomap") +
theme_minimal()
library(patchwork)
p1 + p2 + p3 + p4
Example
4.1 Overview
6 Visualise the rules
Code
Available control parameters (with default values):
layout = stress
circular = FALSE
ggraphdots = NULL
edges = <environment>
nodes = <environment>
nodetext = <environment>
colors = c("#EE0000FF", "#EEEEEEFF")
engine = ggplot2
max = 100
verbose = FALSE
