What are 3 clusters and their centers after one iteration? Show the detailed steps, same as questions b and c.
What are 3 clusters and their centers after one iteration? Show the detailed steps, same as questions b and c.
Database System Concepts
7th Edition
ISBN:9780078022159
Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Chapter1: Introduction
Section: Chapter Questions
Problem 1PE
Related questions
Question
- What are 3 clusters and their centers after one iteration? Show the detailed steps, same as questions b and c.
- What are 3 clusters and their centers after two iterations?
- What are 3 clusters and their centers when the clustering converges?
- .How many iterations are required for the clusters to converge?
![All data in \( X \) were plotted in Figure 1. The centers of 3 clusters were initialized as:
- \( \vec{c}_1 = (6.2, 3.2) \) (red)
- \( \vec{c}_2 = (6.6, 3.7) \) (green)
- \( \vec{c}_3 = (6.5, 3.0) \) (blue).
The matrix \( X \) is as follows:
\[
X = \begin{bmatrix}
5.9 & 3.2 \\
4.6 & 2.9 \\
6.2 & 2.8 \\
4.7 & 3.2 \\
5.5 & 4.2 \\
5.0 & 3.0 \\
4.9 & 3.1 \\
6.7 & 3.1 \\
5.1 & 3.8 \\
6.0 & 3.0 \\
\end{bmatrix}
\]
This set of coordinates represents data points that are part of a clustering analysis. Each row corresponds to a data point in a two-dimensional space. The cluster centers \( \vec{c}_1 \), \( \vec{c}_2 \), and \( \vec{c}_3 \) are used for initializing the clustering process, with each center being assigned a distinct color for differentiation.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F54632a43-84a1-4aed-a94c-629242c9b8d2%2Facc22bdd-eb22-44a1-a858-9878bfb63f31%2Feu7bklp_processed.jpeg&w=3840&q=75)
Transcribed Image Text:All data in \( X \) were plotted in Figure 1. The centers of 3 clusters were initialized as:
- \( \vec{c}_1 = (6.2, 3.2) \) (red)
- \( \vec{c}_2 = (6.6, 3.7) \) (green)
- \( \vec{c}_3 = (6.5, 3.0) \) (blue).
The matrix \( X \) is as follows:
\[
X = \begin{bmatrix}
5.9 & 3.2 \\
4.6 & 2.9 \\
6.2 & 2.8 \\
4.7 & 3.2 \\
5.5 & 4.2 \\
5.0 & 3.0 \\
4.9 & 3.1 \\
6.7 & 3.1 \\
5.1 & 3.8 \\
6.0 & 3.0 \\
\end{bmatrix}
\]
This set of coordinates represents data points that are part of a clustering analysis. Each row corresponds to a data point in a two-dimensional space. The cluster centers \( \vec{c}_1 \), \( \vec{c}_2 \), and \( \vec{c}_3 \) are used for initializing the clustering process, with each center being assigned a distinct color for differentiation.
![### Implementing K-Means Clustering Manually
**Figure 1:** Scatter plot of datasets and the initialized centers of 3 clusters
The figure above illustrates a scatter plot containing various data points represented by blue triangles and the initial centers of three clusters denoted by colored circles: red, green, and blue. Each point's coordinates are labeled for reference.
#### Cluster Initialization:
- **Red Cluster Center**: Located at (6.2, 3.2)
- **Green Cluster Center**: Located at (6.6, 3.7)
- **Blue Cluster Center**: Located at (6.5, 3.0)
#### Data Points:
- Points such as (4.6, 2.9), (5.1, 3.8), and (6.2, 2.8) are depicted as blue triangles scattered across the plot.
#### Task:
Given the input matrix \( X \) where each row represents a different data point, perform k-means clustering using the Euclidean distance as the distance function. Here, \( k \) is chosen as 3, indicating the number of clusters to be formed.
#### Euclidean Distance Formula:
The Euclidean distance \( d \) between two vectors \( \vec{p} \) and \( \vec{q} \) in \( R^n \) is given by:
\[
d = \sqrt{\sum_{i=1}^{n} (p_i - q_i)^2}
\]
This mathematical formula helps in calculating the distance between points, essential for assigning them to the nearest cluster center during the k-means clustering process.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F54632a43-84a1-4aed-a94c-629242c9b8d2%2Facc22bdd-eb22-44a1-a858-9878bfb63f31%2F9nnbwpn_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Implementing K-Means Clustering Manually
**Figure 1:** Scatter plot of datasets and the initialized centers of 3 clusters
The figure above illustrates a scatter plot containing various data points represented by blue triangles and the initial centers of three clusters denoted by colored circles: red, green, and blue. Each point's coordinates are labeled for reference.
#### Cluster Initialization:
- **Red Cluster Center**: Located at (6.2, 3.2)
- **Green Cluster Center**: Located at (6.6, 3.7)
- **Blue Cluster Center**: Located at (6.5, 3.0)
#### Data Points:
- Points such as (4.6, 2.9), (5.1, 3.8), and (6.2, 2.8) are depicted as blue triangles scattered across the plot.
#### Task:
Given the input matrix \( X \) where each row represents a different data point, perform k-means clustering using the Euclidean distance as the distance function. Here, \( k \) is chosen as 3, indicating the number of clusters to be formed.
#### Euclidean Distance Formula:
The Euclidean distance \( d \) between two vectors \( \vec{p} \) and \( \vec{q} \) in \( R^n \) is given by:
\[
d = \sqrt{\sum_{i=1}^{n} (p_i - q_i)^2}
\]
This mathematical formula helps in calculating the distance between points, essential for assigning them to the nearest cluster center during the k-means clustering process.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, computer-science and related others by exploring similar questions and additional content below.Recommended textbooks for you
Database System Concepts
Computer Science
ISBN:
9780078022159
Author:
Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher:
McGraw-Hill Education
Starting Out with Python (4th Edition)
Computer Science
ISBN:
9780134444321
Author:
Tony Gaddis
Publisher:
PEARSON
Digital Fundamentals (11th Edition)
Computer Science
ISBN:
9780132737968
Author:
Thomas L. Floyd
Publisher:
PEARSON
Database System Concepts
Computer Science
ISBN:
9780078022159
Author:
Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher:
McGraw-Hill Education
Starting Out with Python (4th Edition)
Computer Science
ISBN:
9780134444321
Author:
Tony Gaddis
Publisher:
PEARSON
Digital Fundamentals (11th Edition)
Computer Science
ISBN:
9780132737968
Author:
Thomas L. Floyd
Publisher:
PEARSON
C How to Program (8th Edition)
Computer Science
ISBN:
9780133976892
Author:
Paul J. Deitel, Harvey Deitel
Publisher:
PEARSON
Database Systems: Design, Implementation, & Manag…
Computer Science
ISBN:
9781337627900
Author:
Carlos Coronel, Steven Morris
Publisher:
Cengage Learning
Programmable Logic Controllers
Computer Science
ISBN:
9780073373843
Author:
Frank D. Petruzella
Publisher:
McGraw-Hill Education