tIt appears that you've provided a set of instructions or input format for a machine learning task, particularly clustering using K-Means. Let's break down what each component means: (number of clusters): This is a placeholder for an actual numerical value that represents the desired number of clusters into which you want to divide your training data. In K-Means clustering, you need to specify in advance how many clusters (K) you want the algorithm to find in your data. Training set: The "training set" is your dataset, which contains the data points that you want to cluster. Each data point represents an observation or sample in your dataset. (drop convention): It's not clear from this input what "(drop convention)" refers to. It could be related to a specific data preprocessing or handling instruction, but without additional context or information, it's challenging to provide a precise explanation for this part. In summary, you are expected to provide the number of clusters (K) that you want to discover in your training data, and the training data itself contains the observations or samples that will be used for clustering. The "(drop convention)" part may require further clarification or context to provide a meaningful explanation.Clustering is a fundamental concept in the field of machine learning and data analysis that involves grouping similar data points together based on certain criteria or patterns. It is a technique used to discover inherent structures, relationships, or similarities within a dataset when there are no predefined labels or categories. Clustering is widely employed in various domains, including marketing, biology, image analysis, recommendation systems, and more. In this comprehensive explanation of clustering, we will explore its principles, methods, applications, and key considerations. Table of Contents Introduction to Clustering Key Concepts and Terminology Types of Clustering 3.1. Partitioning Clustering 3.2. Hierarchical Clustering 3.3. Density-Based Clustering 3.4. Model-Based Clustering Distance Metrics and Similarity Measures Common Clustering Algorithms 5.1. K-Means Clustering 5.2. Hierarchical Agglomerative Clustering 5.3. DBSCAN (Density-Based Spatial Clustering of Applications with Noise) 5.4. Gaussian Mixture Models (GMM) Evaluation of Clusters Applications of Clustering 7.1. Customer Segmentation 7.2. Image Segmentation 7.3. Anomaly Detection 7.4. Document Clustering 7.5. Recommender Systems 7.6. Genomic Clustering Challenges and Considerations 8.1. Determining the Number of Clusters (K) 8.2. Handling High-Dimensional Data 8.3. Initial Centroid Selection 8.4. Scaling and Normalization 8.5. Interpretation of Results Best Practices in Clustering Future Trends and Advances Conclusion 1. Introduction to Clustering Clustering, in the context of data analysis and machine learning, refers to the process of grouping a set of data points into subsets,

1. Clustering
Unsupervised learning
introduction
Machine Learning

2. Supervised learning
Training set:
in supervised learning the
output or the label (y) of each
data point is given, and the
classification is perfomed based
on that

3. Unsupervised learning (Clustering)
Training set:
In clustering the division is done
based on the x’s alone and there
will be no output or labels.
Each X(i) consist of many features like x1, x2, x3, ...

4. Aspect Clustering Classification
Goal
Group similar data points into clusters
or groups
Assign data points to predefined classes or labels
Supervision Unsupervised learning Supervised learning
Data Labels No predefined labels for clusters Predefined class labels for each data point
Output Clusters or groups of data points Class labels for each data point
Example
Usage
Customer segmentation, anomaly
detection
Spam email detection, image classification
Evaluation Silhouette score, Davies-Bouldin index Accuracy, precision, recall, F1-score
Examples
K-Means, DBSCAN, hierarchical
clustering
Logistic regression, decision trees, support vector
machines
Distance
Metrics
Used to measure similarity between
data points
Not always required, but may be used for similarity
measurement
Training Data Unlabeled data Labeled data
Applications
Grouping similar items, exploratory
data analysis
Predictive modeling, pattern recognition
Clustering vs. Classification

5. Applications of clustering
Organize computing clusters
Social network
analysis
Image credit: NASA/JPL-Caltech/E. Churchwell (Univ. of Wisconsin, Madison)
Astronomical data analysis
Market segmentation

6. Types of clustering
Hard Clustering
• Each data point belongs
to exactly one cluster.
• Data points are assigned
exclusively to a single
cluster.
• Commonly used in
algorithms like K-Means.
Soft Clustering (Fuzzy
Clustering)
• Data points can belong to
multiple clusters.
• Membership degrees vary,
indicating the strength of
belonging to each cluster.
• Used in algorithms like
Gaussian Mixture Models
(GMM).

7. Soft clustering, or fuzzy clustering, is important for handling uncertainty, complex data
structures, and nuanced relationships by allowing data points to belong to multiple
clusters with varying degrees of membership. It provides a more robust and flexible
approach compared to hard clustering when dealing with real-world data.

8. example on soft clustering
• Customers can belong to multiple segments
simultaneously, indicating varying degrees of affinity to
different product categories or shopping behaviors.
• A customer might be 70% associated with the
"Frequent Shoppers" segment, 40% with the "Discount
Seekers" segment, and 20% with the "Occasional
Shoppers" segment, showing the nuanced nature of
their shopping habits.
• This information enables the business to personalize
marketing strategies, recommending products or
discounts tailored to each customer's multiple
preferences, ultimately improving customer
engagement and sales.

9. Most popular Clustering Algorithms
 K-Means: a clustering algorithm that partitions data into K distinct, non-overlapping clusters based
on similarity, with the goal of minimizing the within-cluster variance
 Hierarchical Clustering: method of cluster analysis that builds a hierarchy of clusters by
successively merging or splitting them based on data similarity, resulting in a tree-like structure called
a dendrogram
 DBSCAN (Density-Based Spatial Clustering of Applications with Noise):
density-based clustering algorithm that identifies clusters in data by considering regions of high data
point density, effectively capturing clusters of varying shapes and sizes while also identifying noise
points. Gaussian Mixture Models (GMM): represent data as a combination of
multiple Gaussian distributions, enabling flexible modeling of complex data
distributions.
 Agglomerative Clustering: hierarchical clustering method that starts with each data point as its
cluster and iteratively merges the closest clusters until a hierarchy of clusters is formed, often
visualized as a dendrogram.

10. Distance Metrics
What Are Distance Metrics?
• Distance metrics are measures used to quantify the similarity or dissimilarity
between data points in a clustering algorithm.
• They play a crucial role in determining how clusters are formed.
Common Distance Metrics:
• Euclidean Distance: Measures the straight-line distance between two points in
Euclidean space. Suitable for data with continuous features.
• Manhattan Distance: Calculates the sum of absolute differences between
corresponding elements of two data points. Applicable to data with grid-like
structures.
• Cosine Similarity: Measures the cosine of the angle between two vectors. Useful
for text data, document clustering, and cases where the magnitude of the data is
not as important as the direction.

11. Clustering
K-means algorithm

12. K-means clustering

13. Step 1
Start with data points
randomly distributed.
Data has two features, but
it can have more

14. Step 2
Decide number of
clusters and randomly
pick cluster centriods
cluster
centriods

15. Step 3
Compute distance from
each data points to all
centroids.
Perform clustering based on
the minimum distance from
centroids by Selecting the data
points that are closer to a
centroid to belong to the
same cluster

16. Step 4
Recalculate new centroids based
on the selected clusters by calculating
the mean of each feature of the data
points.

22. Input:
- (number of clusters)
- Training set
(drop convention)
K-means algorithm

23. Randomly initialize cluster centroids
K-means algorithm
Repeat {
for = 1 to
:= index (from 1 to ) of cluster centroid
closest to
for = 1 to
:= average (mean) of points assigned to cluster
}

24. K-means for non-separated clusters
T-shirt sizing
Height
Weight

25. Clustering
Optimization
objective

26. K-means optimization objective
= index of cluster (1,2,…, ) to which example is currently
assigned
= cluster centroid ( )
= cluster centroid of cluster to which example has been
assigned
Optimization objective:

27. Randomly initialize cluster centroids
K-means algorithm
Repeat {
for = 1 to
:= index (from 1 to ) of cluster centroid
closest to
for = 1 to
:= average (mean) of points assigned to cluster
}

28. Clustering
Random
initialization

29. Randomly initialize cluster centroids
K-means algorithm
Repeat {
for = 1 to
:= index (from 1 to ) of cluster centroid
closest to
for = 1 to
:= average (mean) of points assigned to cluster
}

30. Random initialization
Should have
Randomly pick training
examples.
Set equal to these
examples.

31. Local optima

32. For i = 1 to 100 {
Randomly initialize K-means.
Run K-means. Get .
Compute cost function (distortion)
}
Pick clustering that gave lowest cost
Random initialization

33. Clustering
Choosing the
number of clusters

34. What is the right value of K?

35. Choosing the value of K
Elbow method:
1 2 3 4 5 6 7 8
Cost
function
(no. of clusters)
1 2 3 4 5 6 7 8
Cost
function
(no. of clusters)

36. Choosing the value of K
Sometimes, you’re running K-means to get clusters to use for some
later/downstream purpose. Evaluate K-means based on a metric for
how well it performs for that later purpose.
E.g. T-shirt sizing
Height
Weight
T-shirt sizing
Height
Weight

37. A Simple example k-means (using K=2)

38. 6
4
7
3
5
2
1
6
5
4
3
2
1
0
0 1 2 3 4
Variable 2
5 6 7 8
Variable
1
K= 2
2

39. Step 1:
Initialization: Randomly we choose two centroids (k=2) for two clusters. In this
case the 2 centroid are: m1=(1.0,1.0) and m2=(5.0,7.0).

40. Centroid 1 Centroid 2
1 √( 1 – 1 )² +
( 1 – 1 )² = 0 √( 5 – 1 )² + ( 7 – 1 )² = 7.21
2 √( 1 – 1.5 )² +
( 1 – 2 )² = 1.12 √( 5 – 1.5 )² + ( 7 – 2 )² = 6.10
3 √( 1 – 3 )² +
( 1 – 4 )² = 3.61 √( 5 – 3 )² + ( 7 – 4 )² = 3.61
4 √( 1 – 5 )² +
( 1 – 7 )² = 7.21 √( 5 – 5 )² + ( 7 – 7 )² = 0
5 √( 1 – 3.5 )² +
( 1 – 5 )² = 4.72 √( 5 – 3.5 )² + ( 7 – 5 )² = 2.5
6 √( 1 – 4.5 )² +
( 1 – 5 )² = 5.31 √( 5 – 4.5 )² + ( 7 – 5 )² = 2.06
7 √( 1 – 3.5 )² +
( 1 – 4.5 )² = 4.30 √( 5 – 3.5 )² + ( 7 – 4.5 )² = 2.92
Step 2:

41. Step 2:

42. Step 3:
Centroid 1 Centroid 2
1 √( 1.83 – 1 )² + (
2.33
– 1 )
²
= 1.57 √( 4.12 – 1 )² + ( 5.38 – 1 )² = 5.38
2 √(1.83 – 1.5 )² + (2.33 – 2 )² = 0.47 √( 4.12 – 1.5 )² + ( 5.38 – 2 )² = 4.29
3 √(1.83 – 3 )² + (2.33 – 4 )² = 2.04 √( 4.12 – 3 )² + ( 5.38 – 4 )² = 1.78
4 √(1.83 – 5 )² + (2.33 – 7 )² = 5.64 √( 4.12 – 5 )² + ( 5.38 – 7 )² = 1.84
5 √(1.83 – 3.5 )² + (2.33 – 5 )² = 3.15 √( 4.12 – 3.5 )² + ( 5.38 – 5 )² = 0.73
6 √(1.83 – 4.5 )² + (2.33 – 5 )² = 3.78 √( 4.12 – 4.5 )² + ( 5.38 – 5 )² = 0.54
7 √(1.83 – 3.5 )² + (2.33 – 4.5 )² = 2.74 √( 4.12 – 3.5 )² + ( 5.38 – 4.5 )² = 1.08

43. Therefore, the new clusters are:
{1,2} and {3,4,5,6,7}
=(1.25 , 1.5)
Group 1 =(𝟏+𝟏.𝟓
) , (𝟏+𝟐
)
𝟐 𝟐
Group 2 =(𝟑+𝟓+𝟑.𝟓+𝟒.𝟓+𝟑.𝟓
) , (𝟒+𝟕+𝟓+𝟓+𝟒.𝟓
)
𝟓 𝟓
=(3.9 , 5.1)

44. Step 4:
Centroid 1 Centroid 2
1 √(
1.25
– 1 )² + ( 1.5 – 1 )² = 0.58 √( 3.9 – 1 )² + ( 5.1 – 1 )² =
5.02
2 √(1.2
5
– 1.5 )² + (1.5 – 2 )² = 0.56 √( 3.9 –
1.5
)² + ( 5.1 – 2 )² =
3.92
3 √(1.2
5
– 3 )² + (1.5 – 4 )² = 3.05 √( 3.9 – 3 )² + ( 5.1 – 4 )² =
1.42
4 √(1.2
5
– 5 )² + (1.5 – 7 )² = 6.66 √( 3.9 – 5 )² + ( 5.1 – 7 )² =
2.20
5 √(1.2
5
– 3.5 )² + (1.5 – 5 )² = 4.16 √( 3.9 – 3.5 )² + ( 5.1 – 5 )² = 0.41
6 √(1.2
5
– 4.5 )² + (1.5 – 5 )² = 4.78 √( 3.9 – 4.5 )² + ( 5.1 – 5 )² = 0.61
7 √(1.2
5
– 3.5 )² + (1.5 – 4.5 )² = 3.75 √( 3.9 – 3.5 )² + ( 5.1 – 4.5 )² = 0.72

45. ▶ Therefore, there is no change in the cluster
.
▶ Thus, the algorithm comes to a halt here and final result consist
of 2 clusters {1,2} and
{3,4,5,6,7}.

46. Pros and cons
Advantages of k-means
1.Relatively simple to implement.
2.Scales to large data sets.
3.Guarantees convergence.
4.Easily adapts to new examples.
Disadvantages of k-means
1. Choosing (k) manually.
2. Being dependent on initial values.
3. Scaling with number of dimensions.