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COMP61021: Representation Learning Ke Chen
Clustering Analysis*
Ke Chen
To carry out this assessed coursework, you will need the Python notebook, clustering.ipynb,
which contains code to get you started with each of the assignments, along with several data files
required by dierent assignments (see the related assignments for details of those data sets) in this
coursework. All above are available in a zipped file on BlackBoard alongside this document.
Clustering analysis in its own right is a classical topic in unsupervised learning to learn a high-level
summary of data, which provides a representation of data from a global perspective. In this course-
work, you are asked to implement state-of-the-art clustering algorithms in Python and apply your
implementations along with the provided Python implementation of clustering algorithms to sev-
eral meaningful synthetic datasets for clustering analysis.
1 K-means Clustering Analysis
K-means is one of the most commonly used clustering analysis algorithm. In this work, you are
asked to apply the providedK-means functions to synthetic datasets to understand its behaviour
and how to use a cluster validity index to decide the proper number of clusters underlying a data
set.
Assignment 1 [3 marks] Apply the built-inK-means function, sklearn.cluster.KMeans, in the
scikit-learn library with Euclidean distance to the 2-D dataset of 3 clusters,kmeans_data_1.npy,
with three dierent initial conditions given in the notebook, clustering.ipynb. In your answer
notebook, (a) implement the function, partition, that produces a partition for a given dataset so
that the data points in this dataset are grouped into clusters based on their closest centriods. All
the points in each cluster are assigned the same label specified by their corresponding centroid; (b)
apply the built-inK-means function to the dataset mentioned above for clustering analysis with 3
given dierent initialisation settings, respectively; and (c) based on the clustering analysis results
achieved in (b), use your partition function implemented in (a) to create 6 scatter plots corre-
sponding to 3 initial partitions and 3 final partitions in order to visualise 6 partitions. You are asked
to display 6 plots in a 2×3grid; the first row shows 3 initial partitions and the second row shows 3 fi-
nal partitions aligned with their corresponding initial partitions shown in the first row. In each scat-
ter plot, you must mark the cluster centroids with the red-coloured “+”, all the data points in a clus-
ter must be marked with the same colour (but dierent from red to allow for seeing their centroid
clearly) and dierent clusters must be indicated by dierent colours. (Hint: to carry out the display
format described above, you may use the built-in function, matplotlib.pyplot.subplot.)
Assignment 2 [3 marks] K-means algorithm cannot be used until the hyperparameter K (the
number of clusters) is set up. In a real application, however, the number of clusters is oen un-
*Assessed Coursework: the deadline and requirements can be found at the end of this document.
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COMP61021: Representation Learning Ke Chen
known. In this circumstance, the scatter-based F-ratio index1 may be applied to decide the num-
ber of clusters. In your answer notebook, (a) implement the scatter-based F-ratio index in Python
where Euclidean distance is used; (b) forK = 2, 3, · · · , 10, run the theK-means built-in function,
sklearn.cluster.KMeans, in the scikit-learn library with Euclidean distance on the dataset,
kmeans_data_2.npy (for each K, you must run K-means with 3 dierent random initialisation
conditions set by yourself), then plot F-ratio index (y-axis) versus K (x-axis) (in this plot, for each
K, use only the least F-ratio index value measured on 3 partitions resulting from dierent initial-
isation conditions), and report the optimal number of clusters in this data set; and (c) display the
final partition corresponding to the optimal number of clusters you find out in (b) with the the same
display format described in Assignment 1.
2 Spectral Clustering Analysis
As a state-of-the-art clustering analysis method, spectral clustering can deal with data of nonlinear
and non-convex manifold with the connectivity criteria. In this work, you are asked to implement a
spectral clustering algorithm in Python and apply your own implementation to synthetic datasets
in order to understand how spectral clustering works on data sets in dierent nature.
Assignment 3 [4marks] In your answer notebook, implement the asymmetric normalised spec-
tral clustering2 algorithm where the fully connected graph is used to generate the Laplacian ma-
trix used in spectral clustering. You are asked to use two functions: gaussian_similarity for
generating the fully connected graph and asymmetric_SC for the asymmetric normalised spec-
tral clustering in your implementation. (Hint: To implement the asymmetric_SC function, you can
use the built-in function, np.linalg.eig, in the numpy library for eigen analysis and two built-
in functions in the scikit-learn library, sklearn.cluster.KMeans for K-means clustering and
pairwise_distances for measuring the distances between data points.)
Assignment4 [3marks]Apply your implementedgaussian_similarity function inAssignment
3 to the dataset, SC_data_1.npy, to generate a Laplacian matrix. In your answer notebook, (a) de-
scribe how you find out an appropriate hyperparameter value in the Gaussian kernel and explicitly
report this value; (b) conduct eigen analysis on the Laplacian matrix, list all the elements of the
eigenvector corresponding to the second smallest eigenvalue or the 1st smallest non-zero eigen-
value in a 4 × 10 table carried out by a proper 2-D array; and (c) with the result achieved in (b),
provide the computational evidence to decide how many clusters in this dataset and display the
resultant partition in a 2-D plot where dierent clusters must be marked in dierent colours.
Assignment 5 [2 marks] Apply your implemented gaussian_similarity and asymmetric_SC
functions in Assignment 3 to the dataset, SC_data_1.npy, with the number of clusters you find
out inAssignment 4, and two other datasets of two clusters, SC_data_2.npy andSC_data_3.npy,
respectively. In your answer notebook, (a) display the 3 resultant partitions of the datasets in 2-D
plots where clusters must be marked in dierent colours, and (b) check whether this clustering
algorithm works for all 3 datasets. If not, identify the dataset(s) it fails and explain why.
1For the definition of the F-ratio index, see“K-means Clustering”lecture note.
2For the algorithmic description of this algorithm, see“Spectral Clustering”lecture note.
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COMP61021: Representation Learning Ke Chen
3 Hierarchical Clustering Analysis
As one of the most important clustering analysis techniques, hierarchical clustering enables one
to achieve all possible clusters at dierent levels for a data set. In this work, you are asked to apply
the commonly-used agglomerative algorithm with dierent cluster-distance metrics to a synthetic
dataset in order to understand how this algorithm works.
Assignment 6 [4 marks] You are asked to use Euclidean distance to produce the initial distance
matrix and further apply the built-in function, scipy.cluster.hierarchy with three dierent
cluster-distance metrics3, single-linkage, complete-linkage and group-average, in thescipy library
to the dataset, HC_data.npy, respectively. In your answer notebook, (a) plot three dendrogram
trees achieved by the use of three cluster-distance metrics in the agglomerative algorithm; (b)
implement the function, get_longest_lifetime, used to find out the longest K-cluster lifetime
from a dendrogram; and (c) use your get_longest_lifetime function implemented in (b) to re-
port the longest K-cluster lifetimes along with their corresponding K (number of clusters) found
in terms of the longest K-cluster lifetime criterion for 3 dendrograms achieved from (a) and fur-
ther display 3 partitions corresponding to the longest K-cluster lifetimes. For visualisation, you
must arrange 3 plots, each with the explicit cluster-distance metric title, in one row in the order of
single-linkage, complete-linkage and group-average from le to right.
4 Ensemble Clustering Analysis
As one of the state-of-the-art clustering analysis techniques, ensemble clustering algorithms al-
low for combining dierent types of clustering algorithms and working on various feature sets of a
data set to reach a synergy for clustering analysis. In this work, you are asked to implement the
evidence-accumulated clustering algorithm and apply your implementation to two benchmark
data sets to understand how this algorithm works in practice.
Assignment7 [2marks] In your answer notebook, implement a function, ensemble_clustering,
for the evidence-accumulated clustering algorithm4 based on K-means and the agglomerative
algorithms in Python where K-means algorithm with Euclidean distance is used to generate ini-
tial partitions. (Hint: To implement the ensemble_clustering function, you can use the built-in
functions, sklearn.cluster.KMeans, in the scikit-learn library for K-means clustering, the
scipy.cluster.hierarchy.linkage and scipy.spatial.distance.squareform in the scipy
library for hierarchical clustering.)
Assignment8 [4marks]Apply yourensemble_clustering function implemented inAssignment
7 to two datasets, SC_data_2.npy and SC_data_3.npy, respectively. In your experiment, you are
asked to figure out how to decide how many initial partitions need to be generated by K-means
algorithm with Euclidean distance and what an appropriate cluster-distance metric is to make the
clustering ensemble algorithm work properly on a given dataset. In your answer notebook, (a) for
each of two datasets, describe how you generate initial partitions and what the cluster-distance
metric that lead to satisfactory results with justification (you need to state the number of initial
3For the definition of cluster-distance metrics, see“Hierarchical Clustering”lecture note.
4For the algorithmic description of this algorithm, see“Clustering Ensemble”lecture note.
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COMP61021: Representation Learning Ke Chen
partitions and the chosen cluster-distance metric explicitly), and (b) plot two dendrogram trees
produced by the cluster ensemble algorithm on two datasets and further display the optimal con-
sensus partitions of two dataset with the longestK-cluster lifetimes achieved, respectively, in the
format of 2-D scatter plot where clusters in a partition must be marked in dierent colours.
Requirement: Before starting working on this assessed coursework, you need to
- download all the files required by this coursework from Blackboard as specified at the be-
ginning of this document; - unzip the file then you should be see a Jupyter notebook file named clustering.ipynb and
one sub-directory named Data (you must keep this directory/sub-directory structure and its
name unchanged when you work on this coursework); - rename clustering.ipynb in the directory as yourfullname_clustering.ipynb. For in-
stance, if your name is“John Smith”, your filename should bejohn_smith_clustering.ipynb.
This file will be your answer notebook to be submitted for marking, so you must include ev-
erything required by the coursework in this Jupyter notebook.
Deliverable: Only your answer notebook, yourfullname_clustering.ipynb, which should in-
clude all your code, output, answers and your interpretation/justification. In this Jupyter note-
book, all assignments have been separated with the clear delimiters. You must put your stu re-
garding an assignment in those cells related to this assignment and, if necessary, create new cells
within the delimiters of this assignment.
Your answer notebook, yourfullname_clustering.ipynb, must be submitted via the Black-
board.
Marking: Marking is on the basis of (1) correctness of results and quality of comments on your
code; (2) rigorous experimentation; (3) how informative and clear your description/answers pre-
sented in your answer book; and (4) your knowledge exhibited, interpretation and justification.
Late Submission Policy: The default university late submission policy (for details, see the of-
ficial document: http://documents.manchester.a…) is ap-
plied to this coursework.
