Cluster Analysis of Bio-Contaminating Algae in the Port Jackson Bay

For clustering, I use data from Roberts et al. 2008 on biological contaminants in Port Jackson Bay.
Machine Learning
Cluster Analysis
R
Author
Affiliation

Linus Ghanadan

MEDS

Published

April 1, 2024

Context

This project was completed for my Machine Learning class, taken as part of my Master’s program at UC Santa Barbara. Provided with data and questions, I carried out this analysis using appropriate machine learning techniques.

Question

What are the major clusters of bio-contaminating algae in the Port Jackson Bay, in terms of the expected ranges of metal content across five types of metal (Cd, Cr, Cu, Mn, and Ni)?

Analysis Summary

Implemented K-means clustering algorithm with data on metal contents (Cd, Cr, Cu, Mn, and Ni) in two species of co-occurring algae at 10 sample sites around Port Jackson Bay to plot the Total Within Sum of Square (TWSS) for different numbers of clusters and determine the optimal number of clusters that indicate distinct types of bio-contaminating algae. In addition, calculated Euclidean distance matrix to build hierarchical clustering model and inspect the resulting dendrogram for outlier points.

Data

We’ll use data from Roberts et al. 2008 on biological contaminants in Port Jackson Bay (located in Sydney, Australia). The data are measurements of metal content in algae at 10 sample sites around the bay.

Setup & import data

# set seed
set.seed(123)
# load packages
library(tidyverse) 
library(tidymodels)
library(cluster) # cluster analysis
library(factoextra) # cluster visualization
# load data
metals_df <- readr::read_csv("../../data/2024-4-1-post-data/harbour_metals.csv")

# select only columns for pollutant variables
metals_df <- metals_df[, 4:8]

K-means clustering

Find optimal k value & build model

# find optimal number of clusters using elbow method
fviz_nbclust(metals_df, kmeans, method = "wss")

The elbow method indicates that k=3 is the ideal number of clusters, as at this point, we see a very significant drop in the Total Within Sum of Square compared to k=2, followed by a very insignificant drop at k=4.

# build k-means cluster model with optimal number of clusters (3)
kmeans_model <- kmeans(metals_df, centers = 3, nstart = 25)

Inspect clustering model

# plot the clusters
fviz_cluster(kmeans_model, geom = "point", data = metals_df) + ggtitle("k=3")

When we connect the outermost points in each cluster, we see that there is overlap between the far right side of cluster 2 and the far left side of cluster 3. The points near this boundary might have been assigned this way just because of the other nearby points that belonged to each cluster, or this could indicate that we should have chosen a larger number of clusters.

# inspect model object
kmeans_model
K-means clustering with 3 clusters of sizes 10, 22, 28

Cluster means:
        Cd       Cr        Cu        Mn    Ni
1 0.796000 6.530000 127.09000 126.28200 2.304
2 1.076818 6.180455  70.28455  66.79864 2.385
3 1.593214 3.462857  22.39893  18.85250 1.870

Clustering vector:
 [1] 2 2 1 1 1 1 3 2 2 2 2 2 2 2 1 2 2 1 3 3 3 3 3 3 3 3 2 2 2 2 3 3 3 3 3 3 1 3
[39] 2 3 2 2 3 3 3 3 3 3 2 1 1 2 1 2 2 3 3 3 3 3

Within cluster sum of squares by cluster:
[1] 13159.695 22633.706  7511.094
 (between_SS / total_SS =  80.4 %)

Available components:

[1] "cluster"      "centers"      "totss"        "withinss"     "tot.withinss"
[6] "betweenss"    "size"         "iter"         "ifault"

Looking at the specific values, ‘size’, which give the number of points allocated to each of the three clusters, stands out because cluster 2 and cluster 3 both have more than double the number of points as cluster 1. In addition, our ‘withinss’ values, which give the within-cluster variation for each of the three clusters, stands out because cluster 3 has over five times more variation than cluster 1 and over three times more variation than cluster 2.

Hierarchical clustering

Calculate Euclidean distance matrix & build model

# calculate distance matrix
dist_matrix <- dist(metals_df, method = "euclidean")

Each value in this matrix tells us the Euclidean distance between a set of two points in our data. There are 1770 rows in the table, one for each unique set of points.

# build clustering model
hc_model <- hclust(dist_matrix)

Inspect clustering model

# plot the dendrogram of clustering model
plot(as.dendrogram(hc_model), main = "Hierarchical Clustering Dendrogram")

The dendrogram looks as expected. 51 is clearly an outlier point because starting from the bottom of the dendrogram and moving up, it is by far the last point to be assigned to a cluster that includes any other points besides itself, and when this assignment does occur, the point is about 80 distance units away from the centroid of the cluster it is assigned to. Comparatively, the point with the next highest distance away from the centroid of its initial assignment to a cluster containing more than just itself is 15, at about 40 distance units.

Conclusion

Our clustering analysis indicates that there are 3 major clusters of biological contaminating algae. For each cluster of algae, there are differing expected ranges of metal content across the five types of metal looked at (Cd, Cr, Cu, Mn, and Ni).

Citation

BibTeX citation:
@online{ghanadan2024,
  author = {Ghanadan, Linus},
  title = {Cluster {Analysis} of {Bio-Contaminating} {Algae} in the
    {Port} {Jackson} {Bay}},
  date = {2024-04-01},
  url = {https://linusghanadan.github.io/blog/2024-4-1-post/},
  langid = {en}
}
For attribution, please cite this work as:
Ghanadan, Linus. 2024. “Cluster Analysis of Bio-Contaminating Algae in the Port Jackson Bay.” April 1, 2024. https://linusghanadan.github.io/blog/2024-4-1-post/.