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Cs345 cl

The document discusses various techniques for clustering data, including hierarchical clustering, k-means algorithms, and distance measures. It provides examples of how different types of data like documents, customer purchases, DNA sequences can be represented as vectors and clustered. Key clustering approaches described are hierarchical agglomerative clustering using different linkage criteria, k-means clustering and its variant BFR for large datasets.

K mean-clustering algorithm

k-means clustering aims to partition n observations into k clusters in which each observation belongs to the cluster with the nearest mean, serving as a prototype of the cluster. This results in a partitioning of the data space into Voronoi cells.

K mean-clustering

K-means clustering is an algorithm that partitions n observations into k clusters by minimizing the within-cluster sum of squares. It assigns each observation to the cluster with the nearest mean. The algorithm works by iteratively updating cluster means and reassigning observations until convergence is reached. Some key applications of k-means clustering include machine learning, data mining, image segmentation, and choosing color palettes. However, it has weaknesses such as dependency on initial conditions and requirement to pre-specify the number of clusters k.

k-mean-clustering.ppt

K-means clustering is an algorithm that partitions n observations into k clusters, where each observation belongs to the cluster with the nearest mean. It works by assigning each observation to a cluster whose mean yields the least within-cluster sum of squares, then recalculating the mean of each cluster. This process repeats until cluster means converge. K-means clustering is commonly used in data mining applications and has been applied to areas like image segmentation, vector quantization, and astronomy.

k-mean-Clustering impact on AI using DSS

K-means clustering is an algorithm that partitions n observations into k clusters by minimizing the within-cluster sum of squares. It assigns each observation to the cluster with the nearest mean. The algorithm works by iteratively updating cluster means and reassigning observations until convergence is reached. Some key applications of k-means clustering include machine learning, data mining, image segmentation, and choosing color palettes. However, it has weaknesses such as dependency on initial cluster means and sensitivity to outliers.

AI-Lec20 Clustering I - Kmean.pptx

This document discusses k-means clustering, an unsupervised machine learning algorithm. It begins by distinguishing between supervised and unsupervised learning. It then defines clustering as classifying objects into groups where objects within each group share common traits. The document proceeds to describe hierarchical and partitional clustering algorithms. It focuses on k-means clustering, explaining how it works by iteratively assigning objects to centroids to minimize intra-cluster distances. Examples are provided to illustrate the k-means algorithm steps. Weaknesses and applications of k-means clustering are also summarized.

K means clustering

K-means clustering is an algorithm that groups data points into k clusters based on their attributes and distances from initial cluster center points. It works by first randomly selecting k data points as initial centroids, then assigning all other points to the closest centroid and recalculating the centroids. This process repeats until the centroids are stable or a maximum number of iterations is reached. K-means clustering is widely used for machine learning applications like image segmentation and speech recognition due to its efficiency, but it is sensitive to initialization and assumes spherical clusters of similar size and density.

K Means Clustering Algorithm | K Means Clustering Example | Machine Learning ...

This K-Means clustering algorithm presentation will take you through the machine learning introduction, types of clustering algorithms, k-means clustering, how does K-Means clustering work and at least explains K-Means clustering by taking a real life use case. This Machine Learning algorithm tutorial video is ideal for beginners to learn how K-Means clustering work.
Below topics are covered in this K-Means Clustering Algorithm presentation:
1. Types of Machine Learning?
2. What is K-Means Clustering?
3. Applications of K-Means Clustering
4. Common distance measure
5. How does K-Means Clustering work?
6. K-Means Clustering Algorithm
7. Demo: k-Means Clustering
8. Use case: Color compression
- - - - - - - -
About Simplilearn Machine Learning course:
A form of artificial intelligence, Machine Learning is revolutionizing the world of computing as well as all people’s digital interactions. Machine Learning powers such innovative automated technologies as recommendation engines, facial recognition, fraud protection and even self-driving cars.This Machine Learning course prepares engineers, data scientists and other professionals with knowledge and hands-on skills required for certification and job competency in Machine Learning.
- - - - - - -
Why learn Machine Learning?
Machine Learning is taking over the world- and with that, there is a growing need among companies for professionals to know the ins and outs of Machine Learning
The Machine Learning market size is expected to grow from USD 1.03 Billion in 2016 to USD 8.81 Billion by 2022, at a Compound Annual Growth Rate (CAGR) of 44.1% during the forecast period.
- - - - - -
What skills will you learn from this Machine Learning course?
By the end of this Machine Learning course, you will be able to:
1. Master the concepts of supervised, unsupervised and reinforcement learning concepts and modeling.
2. Gain practical mastery over principles, algorithms, and applications of Machine Learning through a hands-on approach which includes working on 28 projects and one capstone project.
3. Acquire thorough knowledge of the mathematical and heuristic aspects of Machine Learning.
4. Understand the concepts and operation of support vector machines, kernel SVM, naive bayes, decision tree classifier, random forest classifier, logistic regression, K-nearest neighbors, K-means clustering and more.
5. Be able to model a wide variety of robust Machine Learning algorithms including deep learning, clustering, and recommendation systems
- - - - - - -

Cs345 cl

The document discusses various techniques for clustering data, including hierarchical clustering, k-means algorithms, and distance measures. It provides examples of how different types of data like documents, customer purchases, DNA sequences can be represented as vectors and clustered. Key clustering approaches described are hierarchical agglomerative clustering using different linkage criteria, k-means clustering and its variant BFR for large datasets.

K mean-clustering algorithm

k-means clustering aims to partition n observations into k clusters in which each observation belongs to the cluster with the nearest mean, serving as a prototype of the cluster. This results in a partitioning of the data space into Voronoi cells.

K mean-clustering

K-means clustering is an algorithm that partitions n observations into k clusters by minimizing the within-cluster sum of squares. It assigns each observation to the cluster with the nearest mean. The algorithm works by iteratively updating cluster means and reassigning observations until convergence is reached. Some key applications of k-means clustering include machine learning, data mining, image segmentation, and choosing color palettes. However, it has weaknesses such as dependency on initial conditions and requirement to pre-specify the number of clusters k.

k-mean-clustering.ppt

K-means clustering is an algorithm that partitions n observations into k clusters, where each observation belongs to the cluster with the nearest mean. It works by assigning each observation to a cluster whose mean yields the least within-cluster sum of squares, then recalculating the mean of each cluster. This process repeats until cluster means converge. K-means clustering is commonly used in data mining applications and has been applied to areas like image segmentation, vector quantization, and astronomy.

k-mean-Clustering impact on AI using DSS

K-means clustering is an algorithm that partitions n observations into k clusters by minimizing the within-cluster sum of squares. It assigns each observation to the cluster with the nearest mean. The algorithm works by iteratively updating cluster means and reassigning observations until convergence is reached. Some key applications of k-means clustering include machine learning, data mining, image segmentation, and choosing color palettes. However, it has weaknesses such as dependency on initial cluster means and sensitivity to outliers.

AI-Lec20 Clustering I - Kmean.pptx

This document discusses k-means clustering, an unsupervised machine learning algorithm. It begins by distinguishing between supervised and unsupervised learning. It then defines clustering as classifying objects into groups where objects within each group share common traits. The document proceeds to describe hierarchical and partitional clustering algorithms. It focuses on k-means clustering, explaining how it works by iteratively assigning objects to centroids to minimize intra-cluster distances. Examples are provided to illustrate the k-means algorithm steps. Weaknesses and applications of k-means clustering are also summarized.

K means clustering

K-means clustering is an algorithm that groups data points into k clusters based on their attributes and distances from initial cluster center points. It works by first randomly selecting k data points as initial centroids, then assigning all other points to the closest centroid and recalculating the centroids. This process repeats until the centroids are stable or a maximum number of iterations is reached. K-means clustering is widely used for machine learning applications like image segmentation and speech recognition due to its efficiency, but it is sensitive to initialization and assumes spherical clusters of similar size and density.

K Means Clustering Algorithm | K Means Clustering Example | Machine Learning ...

This K-Means clustering algorithm presentation will take you through the machine learning introduction, types of clustering algorithms, k-means clustering, how does K-Means clustering work and at least explains K-Means clustering by taking a real life use case. This Machine Learning algorithm tutorial video is ideal for beginners to learn how K-Means clustering work.
Below topics are covered in this K-Means Clustering Algorithm presentation:
1. Types of Machine Learning?
2. What is K-Means Clustering?
3. Applications of K-Means Clustering
4. Common distance measure
5. How does K-Means Clustering work?
6. K-Means Clustering Algorithm
7. Demo: k-Means Clustering
8. Use case: Color compression
- - - - - - - -
About Simplilearn Machine Learning course:
A form of artificial intelligence, Machine Learning is revolutionizing the world of computing as well as all people’s digital interactions. Machine Learning powers such innovative automated technologies as recommendation engines, facial recognition, fraud protection and even self-driving cars.This Machine Learning course prepares engineers, data scientists and other professionals with knowledge and hands-on skills required for certification and job competency in Machine Learning.
- - - - - - -
Why learn Machine Learning?
Machine Learning is taking over the world- and with that, there is a growing need among companies for professionals to know the ins and outs of Machine Learning
The Machine Learning market size is expected to grow from USD 1.03 Billion in 2016 to USD 8.81 Billion by 2022, at a Compound Annual Growth Rate (CAGR) of 44.1% during the forecast period.
- - - - - -
What skills will you learn from this Machine Learning course?
By the end of this Machine Learning course, you will be able to:
1. Master the concepts of supervised, unsupervised and reinforcement learning concepts and modeling.
2. Gain practical mastery over principles, algorithms, and applications of Machine Learning through a hands-on approach which includes working on 28 projects and one capstone project.
3. Acquire thorough knowledge of the mathematical and heuristic aspects of Machine Learning.
4. Understand the concepts and operation of support vector machines, kernel SVM, naive bayes, decision tree classifier, random forest classifier, logistic regression, K-nearest neighbors, K-means clustering and more.
5. Be able to model a wide variety of robust Machine Learning algorithms including deep learning, clustering, and recommendation systems
- - - - - - -

Designing a Minimum Distance classifier to Class Mean Classifier

This document describes designing and implementing a minimum distance to class mean classifier. It begins with an introduction to how minimum distance classification works by calculating the mean of each class and assigning unknown samples to the closest class mean. It then describes the experimental design which includes plotting training data, classifying test data using linear discriminant functions, drawing the decision boundary, and calculating accuracy. The implementation section provides code details for these steps. It finds an accuracy of 85% for classifying test data and concludes the classifier performs well for linear datasets but has lower accuracy for nonlinear problems due to only modeling linear decision boundaries.

Hierarchical clustering

Hierarchical clustering is a method of partitioning a set of data into meaningful sub-classes or clusters. It involves two approaches - agglomerative, which successively links pairs of items or clusters, and divisive, which starts with the whole set as a cluster and divides it into smaller partitions. Agglomerative Nesting (AGNES) is an agglomerative technique that merges clusters with the least dissimilarity at each step, eventually combining all clusters. Divisive Analysis (DIANA) is the inverse, starting with all data in one cluster and splitting it until each data point is its own cluster. Both approaches can be visualized using dendrograms to show the hierarchical merging or splitting of clusters.

K mean-clustering

This document provides an introduction to k-means clustering, including:
1. K-means clustering aims to partition n observations into k clusters by minimizing the within-cluster sum of squares, where each observation belongs to the cluster with the nearest mean.
2. The k-means algorithm initializes cluster centroids and assigns observations to the nearest centroid, recomputing centroids until convergence.
3. K-means clustering is commonly used for applications like machine learning, data mining, and image segmentation due to its efficiency, though it is sensitive to initialization and assumes spherical clusters.

11-2-Clustering.pptx

This document discusses two types of clustering algorithms: partitional and hierarchical clustering. It provides details on K-means, a popular partitional clustering algorithm, including the pseudocode and an example. It also discusses hierarchical clustering, including different cluster distance measures, the agglomerative algorithm, and provides an example of applying the agglomerative approach. Evaluation of K-means performance using sum of squared errors is also covered.

[PPT]

The document discusses various techniques for clustering and dimensionality reduction of web documents. It introduces machine learning clustering methods like k-means clustering and discusses challenges like handling different cluster sizes and shapes. It also covers dimensionality reduction methods like principal component analysis (PCA) and locality-sensitive hashing that can be used to cluster high dimensional web document datasets by reducing their dimensionality.

overviewPCA

Principal component analysis (PCA) is a dimensionality reduction technique that identifies important contributing components in big data. It works by transforming the data to a new coordinate system such that the greatest variance by some projection of the data lies on the first coordinate (called the first principal component), the second greatest variance on the second coordinate, and so on. Specifically, PCA analyzes the covariance matrix of the data to obtain its eigenvectors and eigenvalues. It then chooses principal components corresponding to the largest eigenvalues, which identify the directions with the most variance in the data.

Enhance The K Means Algorithm On Spatial Dataset

The document describes an enhancement to the standard k-means clustering algorithm. The enhancement aims to improve computational speed by storing additional information from each iteration, such as the closest cluster and distance for each data point. This avoids needing to recompute distances to all cluster centers in subsequent iterations if a point does not change clusters. The complexity of the enhanced algorithm is reduced from O(nkl) to O(nk) where n is points, k is clusters, and l is iterations.

Unsupervised Learning in Machine Learning

The document discusses various unsupervised learning techniques including clustering algorithms like k-means, k-medoids, hierarchical clustering and density-based clustering. It explains how k-means clustering works by selecting initial random centroids and iteratively reassigning data points to the closest centroid. The elbow method is described as a way to determine the optimal number of clusters k. The document also discusses how k-medoids clustering is more robust to outliers than k-means because it uses actual data points as cluster representatives rather than centroids.

MSE.pptx

The document discusses linear and non-linear support vector machine (SVM) algorithms. SVM finds the optimal hyperplane that separates classes with the maximum margin. For linear data, the hyperplane is a straight line, while for non-linear data it is a higher dimensional surface. The data points closest to the hyperplane that determine its position are called support vectors. Non-linear SVM handles non-linear separable data by projecting it into a higher dimension where it may become linearly separable.

Cluster analysis

Clustering is the process of grouping objects into clusters based on similarities. There are different types of clustering including hierarchical, k-means, and two stage clustering. Factor analysis reduces a large number of variables into fewer factors that capture maximum common variance. Classification sorts data into predefined categories or classes while clustering does not predefine categories, allowing structure in the data to determine the grouping. Clustering and classification are both used for data analysis but differ in how groups are determined.

2012 mdsp pr09 pca lda

This document contains the course calendar for a machine learning course covering topics like Bayesian estimation, Kalman filters, particle filters, hidden Markov models, Bayesian decision theory, principal component analysis, independent component analysis, and clustering algorithms. The calendar lists 15 classes over the semester, the topics to be covered in each class, and any dates where there will be no class. It also includes lecture plans and slides on principal component analysis, linear discriminant analysis, and comparing PCA and LDA.

Pattern Recognition - Designing a minimum distance class mean classifier

1) The document describes designing a minimum distance to class mean classifier to classify unlabeled sample vectors given labeled samples clustered into two classes.
2) It involves plotting the sample points, calculating class means, classifying test points using minimum distance to class mean, and drawing the decision boundary.
3) Accuracy is reported to decrease from 75% to 65% to 55% as the number of sample points increases from 10 to 20 to 30, showing that classification accuracy decreases with more complex datasets, though training samples remain accurately classified.

Clustering

Clustering algorithms are used to group similar data points together. K-means clustering aims to partition data into k clusters by minimizing distances between data points and cluster centers. Hierarchical clustering builds nested clusters by merging or splitting clusters based on distance metrics. Density-based clustering identifies clusters as areas of high density separated by areas of low density, like DBScan which uses parameters of minimum points and epsilon distance.

Project

The document describes a project analyzing clustering of galaxies using the Shapley Galaxy dataset. The dataset contains information on 4215 galaxies. The author applies hierarchical clustering algorithms and Gaussian mixture modeling to determine the optimal number of clusters in the data. Single, complete, and average linkage clustering are applied but do not clearly show clustering structure. Gaussian mixture modeling with the EM algorithm and BIC criterion indicates the best fit model has 8 clusters. Scatter plots are presented for solutions with 1 to 8 clusters.

Traveling Salesman Problem in Distributed Environment

In this paper, we focus on developing parallel algorithms for solving the traveling salesman problem (TSP) based on Nicos Christofides algorithm released in 1976. The parallel algorithm
is built in the distributed environment with multi-processors (Master-Slave). The algorithm is installed on the computer cluster system of National University of Education in Hanoi,
Vietnam (ccs1.hnue.edu.vn) and uses the library PJ (Parallel Java). The results are evaluated and compared with other works.

TRAVELING SALESMAN PROBLEM IN DISTRIBUTED ENVIRONMENT

The document describes developing a parallel algorithm for solving the traveling salesman problem (TSP) based on Christofides' algorithm. It discusses implementing Christofides' algorithm in a distributed environment using multiple processors. The parallel algorithm divides the graph vertices and distance matrix across slave processors, which calculate the minimum spanning tree in parallel. The master processor then finds odd-degree vertices, performs matching, and finds the Hamiltonian cycle to solve TSP. The algorithm is tested on a computer cluster using graphs of 20,000 and 30,000 nodes, showing improved runtime over the sequential algorithm.

Vectorise all the things

Have you found that your code works beautifully on a few dozen examples, but leaves you wondering how to spend the next couple of hours after you start looping through all of your data? Are you only familiar with Python, and wish there was a way to speed things up without subjecting yourself to learning C?
In this talk, you'll see some simple tricks, borrowed from linear algebra, which can give you significant performance gains in your data science code, and how you can implement these in NumPy. We'll start exploring an inefficient implementation of a machine learning algorithm that relies heavily on loops and lists. Throughout the talk, we'll iteratively replace bottlenecks with NumPy vectorized operations and learn the linear algebra that makes these methods work. You'll see how straightforward it can be to make your code many times faster, all without losing readability or needing to understand complex coding concepts.

Kakuro: Solving the Constraint Satisfaction Problem

This work was done as a part of the project for the course CS 271: Introduction to Artificial Intelligence (http://www.ics.uci.edu/~kkask/Fall-2014%20CS271/index.html), taught in Fall 2014.

Fessant aknin oukhellou_midenet_2001:comparison_of_supervised_self_organizing...

The document compares using Euclidean distance versus Mahalanobis distance for supervised self-organizing maps in a rail defect classification application. It finds that using Mahalanobis distance, which accounts for variance differences across input dimensions, leads to better classification results, especially when using a partitioning approach that separates the multi-class problem into binary sub-problems. Specifically, the Mahalanobis distance improved classification rates from 92.8% to 94.1% for a global classifier, and from 90% to 95% for a partitioning classifier.

Lect4

The document discusses various clustering algorithms and concepts:
1) K-means clustering groups data by minimizing distances between points and cluster centers, but it is sensitive to initialization and may find local optima.
2) K-medians clustering is similar but uses point medians instead of means as cluster representatives.
3) K-center clustering aims to minimize maximum distances between points and clusters, and can be approximated with a farthest-first traversal algorithm.

How to Use Payment Vouchers in Odoo 18.

Every business, big or small, deals with outgoing payments. Whether it’s to suppliers for inventory, to employees for salaries, or to vendors for services rendered, keeping track of these expenses is crucial. This is where payment vouchers come in – the unsung heroes of the accounting world.

Fabular Frames and the Four Ratio Problem

Digital, interactive art showing the struggle of a society in providing for its present population while also saving planetary resources for future generations. Spread across several frames, the art is actually the rendering of real and speculative data. The stereographic projections change shape in response to prompts and provocations. Visitors interact with the model through speculative statements about how to increase savings across communities, regions, ecosystems and environments. Their fabulations combined with random noise, i.e. factors beyond control, have a dramatic effect on the societal transition. Things get better. Things get worse. The aim is to give visitors a new grasp and feel of the ongoing struggles in democracies around the world.
Stunning art in the small multiples format brings out the spatiotemporal nature of societal transitions, against backdrop issues such as energy, housing, waste, farmland and forest. In each frame we see hopeful and frightful interplays between spending and saving. Problems emerge when one of the two parts of the existential anaglyph rapidly shrinks like Arctic ice, as factors cross thresholds. Ecological wealth and intergenerational equity areFour at stake. Not enough spending could mean economic stress, social unrest and political conflict. Not enough saving and there will be climate breakdown and ‘bankruptcy’. So where does speculative design start and the gambling and betting end? Behind each fabular frame is a four ratio problem. Each ratio reflects the level of sacrifice and self-restraint a society is willing to accept, against promises of prosperity and freedom. Some values seem to stabilise a frame while others cause collapse. Get the ratios right and we can have it all. Get them wrong and things get more desperate.

Designing a Minimum Distance classifier to Class Mean Classifier

This document describes designing and implementing a minimum distance to class mean classifier. It begins with an introduction to how minimum distance classification works by calculating the mean of each class and assigning unknown samples to the closest class mean. It then describes the experimental design which includes plotting training data, classifying test data using linear discriminant functions, drawing the decision boundary, and calculating accuracy. The implementation section provides code details for these steps. It finds an accuracy of 85% for classifying test data and concludes the classifier performs well for linear datasets but has lower accuracy for nonlinear problems due to only modeling linear decision boundaries.

Hierarchical clustering

Hierarchical clustering is a method of partitioning a set of data into meaningful sub-classes or clusters. It involves two approaches - agglomerative, which successively links pairs of items or clusters, and divisive, which starts with the whole set as a cluster and divides it into smaller partitions. Agglomerative Nesting (AGNES) is an agglomerative technique that merges clusters with the least dissimilarity at each step, eventually combining all clusters. Divisive Analysis (DIANA) is the inverse, starting with all data in one cluster and splitting it until each data point is its own cluster. Both approaches can be visualized using dendrograms to show the hierarchical merging or splitting of clusters.

K mean-clustering

This document provides an introduction to k-means clustering, including:
1. K-means clustering aims to partition n observations into k clusters by minimizing the within-cluster sum of squares, where each observation belongs to the cluster with the nearest mean.
2. The k-means algorithm initializes cluster centroids and assigns observations to the nearest centroid, recomputing centroids until convergence.
3. K-means clustering is commonly used for applications like machine learning, data mining, and image segmentation due to its efficiency, though it is sensitive to initialization and assumes spherical clusters.

11-2-Clustering.pptx

This document discusses two types of clustering algorithms: partitional and hierarchical clustering. It provides details on K-means, a popular partitional clustering algorithm, including the pseudocode and an example. It also discusses hierarchical clustering, including different cluster distance measures, the agglomerative algorithm, and provides an example of applying the agglomerative approach. Evaluation of K-means performance using sum of squared errors is also covered.

[PPT]

The document discusses various techniques for clustering and dimensionality reduction of web documents. It introduces machine learning clustering methods like k-means clustering and discusses challenges like handling different cluster sizes and shapes. It also covers dimensionality reduction methods like principal component analysis (PCA) and locality-sensitive hashing that can be used to cluster high dimensional web document datasets by reducing their dimensionality.

overviewPCA

Principal component analysis (PCA) is a dimensionality reduction technique that identifies important contributing components in big data. It works by transforming the data to a new coordinate system such that the greatest variance by some projection of the data lies on the first coordinate (called the first principal component), the second greatest variance on the second coordinate, and so on. Specifically, PCA analyzes the covariance matrix of the data to obtain its eigenvectors and eigenvalues. It then chooses principal components corresponding to the largest eigenvalues, which identify the directions with the most variance in the data.

Enhance The K Means Algorithm On Spatial Dataset

The document describes an enhancement to the standard k-means clustering algorithm. The enhancement aims to improve computational speed by storing additional information from each iteration, such as the closest cluster and distance for each data point. This avoids needing to recompute distances to all cluster centers in subsequent iterations if a point does not change clusters. The complexity of the enhanced algorithm is reduced from O(nkl) to O(nk) where n is points, k is clusters, and l is iterations.

Unsupervised Learning in Machine Learning

The document discusses various unsupervised learning techniques including clustering algorithms like k-means, k-medoids, hierarchical clustering and density-based clustering. It explains how k-means clustering works by selecting initial random centroids and iteratively reassigning data points to the closest centroid. The elbow method is described as a way to determine the optimal number of clusters k. The document also discusses how k-medoids clustering is more robust to outliers than k-means because it uses actual data points as cluster representatives rather than centroids.

MSE.pptx

The document discusses linear and non-linear support vector machine (SVM) algorithms. SVM finds the optimal hyperplane that separates classes with the maximum margin. For linear data, the hyperplane is a straight line, while for non-linear data it is a higher dimensional surface. The data points closest to the hyperplane that determine its position are called support vectors. Non-linear SVM handles non-linear separable data by projecting it into a higher dimension where it may become linearly separable.

Cluster analysis

Clustering is the process of grouping objects into clusters based on similarities. There are different types of clustering including hierarchical, k-means, and two stage clustering. Factor analysis reduces a large number of variables into fewer factors that capture maximum common variance. Classification sorts data into predefined categories or classes while clustering does not predefine categories, allowing structure in the data to determine the grouping. Clustering and classification are both used for data analysis but differ in how groups are determined.

2012 mdsp pr09 pca lda

This document contains the course calendar for a machine learning course covering topics like Bayesian estimation, Kalman filters, particle filters, hidden Markov models, Bayesian decision theory, principal component analysis, independent component analysis, and clustering algorithms. The calendar lists 15 classes over the semester, the topics to be covered in each class, and any dates where there will be no class. It also includes lecture plans and slides on principal component analysis, linear discriminant analysis, and comparing PCA and LDA.

Pattern Recognition - Designing a minimum distance class mean classifier

1) The document describes designing a minimum distance to class mean classifier to classify unlabeled sample vectors given labeled samples clustered into two classes.
2) It involves plotting the sample points, calculating class means, classifying test points using minimum distance to class mean, and drawing the decision boundary.
3) Accuracy is reported to decrease from 75% to 65% to 55% as the number of sample points increases from 10 to 20 to 30, showing that classification accuracy decreases with more complex datasets, though training samples remain accurately classified.

Clustering

Clustering algorithms are used to group similar data points together. K-means clustering aims to partition data into k clusters by minimizing distances between data points and cluster centers. Hierarchical clustering builds nested clusters by merging or splitting clusters based on distance metrics. Density-based clustering identifies clusters as areas of high density separated by areas of low density, like DBScan which uses parameters of minimum points and epsilon distance.

Project

The document describes a project analyzing clustering of galaxies using the Shapley Galaxy dataset. The dataset contains information on 4215 galaxies. The author applies hierarchical clustering algorithms and Gaussian mixture modeling to determine the optimal number of clusters in the data. Single, complete, and average linkage clustering are applied but do not clearly show clustering structure. Gaussian mixture modeling with the EM algorithm and BIC criterion indicates the best fit model has 8 clusters. Scatter plots are presented for solutions with 1 to 8 clusters.

Traveling Salesman Problem in Distributed Environment

In this paper, we focus on developing parallel algorithms for solving the traveling salesman problem (TSP) based on Nicos Christofides algorithm released in 1976. The parallel algorithm
is built in the distributed environment with multi-processors (Master-Slave). The algorithm is installed on the computer cluster system of National University of Education in Hanoi,
Vietnam (ccs1.hnue.edu.vn) and uses the library PJ (Parallel Java). The results are evaluated and compared with other works.

TRAVELING SALESMAN PROBLEM IN DISTRIBUTED ENVIRONMENT

The document describes developing a parallel algorithm for solving the traveling salesman problem (TSP) based on Christofides' algorithm. It discusses implementing Christofides' algorithm in a distributed environment using multiple processors. The parallel algorithm divides the graph vertices and distance matrix across slave processors, which calculate the minimum spanning tree in parallel. The master processor then finds odd-degree vertices, performs matching, and finds the Hamiltonian cycle to solve TSP. The algorithm is tested on a computer cluster using graphs of 20,000 and 30,000 nodes, showing improved runtime over the sequential algorithm.

Vectorise all the things

Have you found that your code works beautifully on a few dozen examples, but leaves you wondering how to spend the next couple of hours after you start looping through all of your data? Are you only familiar with Python, and wish there was a way to speed things up without subjecting yourself to learning C?
In this talk, you'll see some simple tricks, borrowed from linear algebra, which can give you significant performance gains in your data science code, and how you can implement these in NumPy. We'll start exploring an inefficient implementation of a machine learning algorithm that relies heavily on loops and lists. Throughout the talk, we'll iteratively replace bottlenecks with NumPy vectorized operations and learn the linear algebra that makes these methods work. You'll see how straightforward it can be to make your code many times faster, all without losing readability or needing to understand complex coding concepts.

Kakuro: Solving the Constraint Satisfaction Problem

This work was done as a part of the project for the course CS 271: Introduction to Artificial Intelligence (http://www.ics.uci.edu/~kkask/Fall-2014%20CS271/index.html), taught in Fall 2014.

Fessant aknin oukhellou_midenet_2001:comparison_of_supervised_self_organizing...

The document compares using Euclidean distance versus Mahalanobis distance for supervised self-organizing maps in a rail defect classification application. It finds that using Mahalanobis distance, which accounts for variance differences across input dimensions, leads to better classification results, especially when using a partitioning approach that separates the multi-class problem into binary sub-problems. Specifically, the Mahalanobis distance improved classification rates from 92.8% to 94.1% for a global classifier, and from 90% to 95% for a partitioning classifier.

Lect4

The document discusses various clustering algorithms and concepts:
1) K-means clustering groups data by minimizing distances between points and cluster centers, but it is sensitive to initialization and may find local optima.
2) K-medians clustering is similar but uses point medians instead of means as cluster representatives.
3) K-center clustering aims to minimize maximum distances between points and clusters, and can be approximated with a farthest-first traversal algorithm.

Designing a Minimum Distance classifier to Class Mean Classifier

Designing a Minimum Distance classifier to Class Mean Classifier

Hierarchical clustering

Hierarchical clustering

K mean-clustering

K mean-clustering

11-2-Clustering.pptx

11-2-Clustering.pptx

[PPT]

[PPT]

overviewPCA

overviewPCA

Enhance The K Means Algorithm On Spatial Dataset

Enhance The K Means Algorithm On Spatial Dataset

Unsupervised Learning in Machine Learning

Unsupervised Learning in Machine Learning

MSE.pptx

MSE.pptx

Cluster analysis

Cluster analysis

2012 mdsp pr09 pca lda

2012 mdsp pr09 pca lda

Pattern Recognition - Designing a minimum distance class mean classifier

Pattern Recognition - Designing a minimum distance class mean classifier

Clustering

Clustering

Project

Project

Traveling Salesman Problem in Distributed Environment

Traveling Salesman Problem in Distributed Environment

TRAVELING SALESMAN PROBLEM IN DISTRIBUTED ENVIRONMENT

TRAVELING SALESMAN PROBLEM IN DISTRIBUTED ENVIRONMENT

Vectorise all the things

Vectorise all the things

Kakuro: Solving the Constraint Satisfaction Problem

Kakuro: Solving the Constraint Satisfaction Problem

Fessant aknin oukhellou_midenet_2001:comparison_of_supervised_self_organizing...

Fessant aknin oukhellou_midenet_2001:comparison_of_supervised_self_organizing...

Lect4

Lect4

How to Use Payment Vouchers in Odoo 18.

Every business, big or small, deals with outgoing payments. Whether it’s to suppliers for inventory, to employees for salaries, or to vendors for services rendered, keeping track of these expenses is crucial. This is where payment vouchers come in – the unsung heroes of the accounting world.

Fabular Frames and the Four Ratio Problem

Digital, interactive art showing the struggle of a society in providing for its present population while also saving planetary resources for future generations. Spread across several frames, the art is actually the rendering of real and speculative data. The stereographic projections change shape in response to prompts and provocations. Visitors interact with the model through speculative statements about how to increase savings across communities, regions, ecosystems and environments. Their fabulations combined with random noise, i.e. factors beyond control, have a dramatic effect on the societal transition. Things get better. Things get worse. The aim is to give visitors a new grasp and feel of the ongoing struggles in democracies around the world.
Stunning art in the small multiples format brings out the spatiotemporal nature of societal transitions, against backdrop issues such as energy, housing, waste, farmland and forest. In each frame we see hopeful and frightful interplays between spending and saving. Problems emerge when one of the two parts of the existential anaglyph rapidly shrinks like Arctic ice, as factors cross thresholds. Ecological wealth and intergenerational equity areFour at stake. Not enough spending could mean economic stress, social unrest and political conflict. Not enough saving and there will be climate breakdown and ‘bankruptcy’. So where does speculative design start and the gambling and betting end? Behind each fabular frame is a four ratio problem. Each ratio reflects the level of sacrifice and self-restraint a society is willing to accept, against promises of prosperity and freedom. Some values seem to stabilise a frame while others cause collapse. Get the ratios right and we can have it all. Get them wrong and things get more desperate.

The Impact of Generative AI and 4th Industrial Revolution

This infographic explores the transformative power of Generative AI, a key driver of the 4th Industrial Revolution. Discover how Generative AI is revolutionizing industries, accelerating innovation, and shaping the future of work.

1：1制作加拿大麦吉尔大学毕业证硕士学历证书原版一模一样

原版一模一样【微信：741003700 】【加拿大麦吉尔大学毕业证硕士学历证书】【微信：741003700 】学位证，留信认证（真实可查，永久存档）offer、雅思、外壳等材料/诚信可靠,可直接看成品样本，帮您解决无法毕业带来的各种难题！外壳，原版制作，诚信可靠，可直接看成品样本。行业标杆！精益求精，诚心合作，真诚制作！多年品质 ,按需精细制作，24小时接单,全套进口原装设备。十五年致力于帮助留学生解决难题，包您满意。
本公司拥有海外各大学样板无数，能完美还原海外各大学 Bachelor Diploma degree, Master Degree Diploma
1:1完美还原海外各大学毕业材料上的工艺：水印，阴影底纹，钢印LOGO烫金烫银，LOGO烫金烫银复合重叠。文字图案浮雕、激光镭射、紫外荧光、温感、复印防伪等防伪工艺。材料咨询办理、认证咨询办理请加学历顾问Q/微741003700
留信网认证的作用:
1:该专业认证可证明留学生真实身份
2:同时对留学生所学专业登记给予评定
3:国家专业人才认证中心颁发入库证书
4:这个认证书并且可以归档倒地方
5:凡事获得留信网入网的信息将会逐步更新到个人身份内，将在公安局网内查询个人身份证信息后，同步读取人才网入库信息
6:个人职称评审加20分
7:个人信誉贷款加10分
8:在国家人才网主办的国家网络招聘大会中纳入资料，供国家高端企业选择人才

Optimizing Net Interest Margin (NIM) in the Financial Sector (With Examples).pdf

NIM is calculated as the difference between interest income earned and interest expenses paid, divided by interest-earning assets.
Importance: NIM serves as a critical measure of a financial institution's profitability and operational efficiency. It reflects how effectively the institution is utilizing its interest-earning assets to generate income while managing interest costs.

Seeman_Fiintouch_LLP_Newsletter_Jun_2024.pdf

The Impact of the 2024 Indian
Election Beyond Borders
01. Investment Gyan
02. Market Update
03. Inspiration investment story

Tdasx: In-Depth Analysis of Cryptocurrency Giveaway Scams and Security Strate...

Tdasx: In-Depth Analysis of Cryptocurrency Giveaway Scams and Security Strategies

1比1复刻(ksu毕业证书)美国堪萨斯州立大学毕业证本科文凭证书原版一模一样

原版定制【微信:bwp0011】《(ksu毕业证书)美国堪萨斯州立大学毕业证本科文凭证书》【微信:bwp0011】成绩单 、雅思、外壳、留信学历认证永久存档查询，采用学校原版纸张、特殊工艺完全按照原版一比一制作（包括：隐形水印，阴影底纹，钢印LOGO烫金烫银，LOGO烫金烫银复合重叠，文字图案浮雕，激光镭射，紫外荧光，温感，复印防伪）行业标杆！精益求精，诚心合作，真诚制作！多年品质 ,按需精细制作，24小时接单,全套进口原装设备，十五年致力于帮助留学生解决难题，业务范围有加拿大、英国、澳洲、韩国、美国、新加坡，新西兰等学历材料，包您满意。
【业务选择办理准则】
一、工作未确定，回国需先给父母、亲戚朋友看下文凭的情况，办理一份就读学校的毕业证【微信bwp0011】文凭即可
二、回国进私企、外企、自己做生意的情况，这些单位是不查询毕业证真伪的，而且国内没有渠道去查询国外文凭的真假，也不需要提供真实教育部认证。鉴于此，办理一份毕业证【微信bwp0011】即可
三、进国企，银行，事业单位，考公务员等等，这些单位是必需要提供真实教育部认证的，办理教育部认证所需资料众多且烦琐，所有材料您都必须提供原件，我们凭借丰富的经验，快捷的绿色通道帮您快速整合材料，让您少走弯路。
留信网认证的作用:
1:该专业认证可证明留学生真实身份
2:同时对留学生所学专业登记给予评定
3:国家专业人才认证中心颁发入库证书
4:这个认证书并且可以归档倒地方
5:凡事获得留信网入网的信息将会逐步更新到个人身份内，将在公安局网内查询个人身份证信息后，同步读取人才网入库信息
6:个人职称评审加20分
7:个人信誉贷款加10分
8:在国家人才网主办的国家网络招聘大会中纳入资料，供国家高端企业选择人才
【关于价格问题（保证一手价格）】
我们所定的价格是非常合理的，而且我们现在做得单子大多数都是代理和回头客户介绍的所以一般现在有新的单子 我给客户的都是第一手的代理价格，因为我想坦诚对待大家 不想跟大家在价格方面浪费时间
对于老客户或者被老客户介绍过来的朋友，我们都会适当给一些优惠。

Detailed power point presentation on compound interest and how it is calculated

Detailed information about compund interest

在线办理(GU毕业证书)美国贡萨加大学毕业证学历证书一模一样

学校原件一模一样【微信：741003700 】《(GU毕业证书)美国贡萨加大学毕业证学历证书》【微信：741003700 】学位证，留信认证（真实可查，永久存档）原件一模一样纸张工艺/offer、雅思、外壳等材料/诚信可靠,可直接看成品样本，帮您解决无法毕业带来的各种难题！外壳，原版制作，诚信可靠，可直接看成品样本。行业标杆！精益求精，诚心合作，真诚制作！多年品质 ,按需精细制作，24小时接单,全套进口原装设备。十五年致力于帮助留学生解决难题，包您满意。
本公司拥有海外各大学样板无数，能完美还原。
1:1完美还原海外各大学毕业材料上的工艺：水印，阴影底纹，钢印LOGO烫金烫银，LOGO烫金烫银复合重叠。文字图案浮雕、激光镭射、紫外荧光、温感、复印防伪等防伪工艺。材料咨询办理、认证咨询办理请加学历顾问Q/微741003700
【主营项目】
一.毕业证【q微741003700】成绩单、使馆认证、教育部认证、雅思托福成绩单、学生卡等！
二.真实使馆公证(即留学回国人员证明,不成功不收费)
三.真实教育部学历学位认证（教育部存档！教育部留服网站永久可查）
四.办理各国各大学文凭(一对一专业服务,可全程监控跟踪进度)
如果您处于以下几种情况：
◇在校期间，因各种原因未能顺利毕业……拿不到官方毕业证【q/微741003700】
◇面对父母的压力，希望尽快拿到；
◇不清楚认证流程以及材料该如何准备；
◇回国时间很长，忘记办理；
◇回国马上就要找工作，办给用人单位看；
◇企事业单位必须要求办理的
◇需要报考公务员、购买免税车、落转户口
◇申请留学生创业基金
留信网认证的作用:
1:该专业认证可证明留学生真实身份
2:同时对留学生所学专业登记给予评定
3:国家专业人才认证中心颁发入库证书
4:这个认证书并且可以归档倒地方
5:凡事获得留信网入网的信息将会逐步更新到个人身份内，将在公安局网内查询个人身份证信息后，同步读取人才网入库信息
6:个人职称评审加20分
7:个人信誉贷款加10分
8:在国家人才网主办的国家网络招聘大会中纳入资料，供国家高端企业选择人才

Tdasx: In-Depth Analysis of Cryptocurrency Giveaway Scams and Security Strate...

Tdasx: In-Depth Analysis of Cryptocurrency Giveaway Scams and Security Strategies

OAT_RI_Ep20 WeighingTheRisks_May24_Trade Wars.pptx

How will new technology fields affect economic trade?

Accounting Information Systems (AIS).pptx

An accounting information system (AIS) refers to tools and systems designed for the collection and display of accounting information so accountants and executives can make informed decisions.

快速办理(美国Fordham毕业证书)福德汉姆大学毕业证学历证书一模一样

学校原件一模一样【微信：741003700 】《(美国Fordham毕业证书)福德汉姆大学毕业证学历证书》【微信：741003700 】学位证，留信认证（真实可查，永久存档）原件一模一样纸张工艺/offer、雅思、外壳等材料/诚信可靠,可直接看成品样本，帮您解决无法毕业带来的各种难题！外壳，原版制作，诚信可靠，可直接看成品样本。行业标杆！精益求精，诚心合作，真诚制作！多年品质 ,按需精细制作，24小时接单,全套进口原装设备。十五年致力于帮助留学生解决难题，包您满意。
本公司拥有海外各大学样板无数，能完美还原。
1:1完美还原海外各大学毕业材料上的工艺：水印，阴影底纹，钢印LOGO烫金烫银，LOGO烫金烫银复合重叠。文字图案浮雕、激光镭射、紫外荧光、温感、复印防伪等防伪工艺。材料咨询办理、认证咨询办理请加学历顾问Q/微741003700
【主营项目】
一.毕业证【q微741003700】成绩单、使馆认证、教育部认证、雅思托福成绩单、学生卡等！
二.真实使馆公证(即留学回国人员证明,不成功不收费)
三.真实教育部学历学位认证（教育部存档！教育部留服网站永久可查）
四.办理各国各大学文凭(一对一专业服务,可全程监控跟踪进度)
如果您处于以下几种情况：
◇在校期间，因各种原因未能顺利毕业……拿不到官方毕业证【q/微741003700】
◇面对父母的压力，希望尽快拿到；
◇不清楚认证流程以及材料该如何准备；
◇回国时间很长，忘记办理；
◇回国马上就要找工作，办给用人单位看；
◇企事业单位必须要求办理的
◇需要报考公务员、购买免税车、落转户口
◇申请留学生创业基金
留信网认证的作用:
1:该专业认证可证明留学生真实身份
2:同时对留学生所学专业登记给予评定
3:国家专业人才认证中心颁发入库证书
4:这个认证书并且可以归档倒地方
5:凡事获得留信网入网的信息将会逐步更新到个人身份内，将在公安局网内查询个人身份证信息后，同步读取人才网入库信息
6:个人职称评审加20分
7:个人信誉贷款加10分
8:在国家人才网主办的国家网络招聘大会中纳入资料，供国家高端企业选择人才

Machine Learning in Business - A power point presentation.pptx

Power point presentation on machine learning.

Ending stagnation: How to boost prosperity across Scotland

A toxic combination of 15 years of low growth, and four decades of high inequality, has left Britain poorer and falling behind its peers. Productivity growth is weak and public investment is low, while wages today are no higher than they were before the financial crisis. Britain needs a new economic strategy to lift itself out of stagnation.
Scotland is in many ways a microcosm of this challenge. It has become a hub for creative industries, is home to several world-class universities and a thriving community of businesses – strengths that need to be harness and leveraged. But it also has high levels of deprivation, with homelessness reaching a record high and nearly half a million people living in very deep poverty last year. Scotland won’t be truly thriving unless it finds ways to ensure that all its inhabitants benefit from growth and investment. This is the central challenge facing policy makers both in Holyrood and Westminster.
What should a new national economic strategy for Scotland include? What would the pursuit of stronger economic growth mean for local, national and UK-wide policy makers? How will economic change affect the jobs we do, the places we live and the businesses we work for? And what are the prospects for cities like Glasgow, and nations like Scotland, in rising to these challenges?

South Dakota State University degree offer diploma Transcript

办理美国SDSU毕业证书制作南达科他州立大学假文凭定制Q微168899991做SDSU留信网教留服认证海牙认证改SDSU成绩单GPA做SDSU假学位证假文凭高仿毕业证GRE代考如何申请南达科他州立大学South Dakota State University degree offer diploma Transcript

Discover the Future of Dogecoin with Our Comprehensive Guidance

Learn in-depth about Dogecoin's trajectory and stay informed with 36crypto's essential and up-to-date information about the crypto space.
Our presentation delves into Dogecoin's potential future, exploring whether it's destined to skyrocket to the moon or face a downward spiral. In addition, it highlights invaluable insights. Don't miss out on this opportunity to enhance your crypto understanding!
https://36crypto.com/the-future-of-dogecoin-how-high-can-this-cryptocurrency-reach/

STREETONOMICS: Exploring the Uncharted Territories of Informal Markets throug...

Delve into the world of STREETONOMICS, where a team of 7 enthusiasts embarks on a journey to understand unorganized markets. By engaging with a coffee street vendor and crafting questionnaires, this project uncovers valuable insights into consumer behavior and market dynamics in informal settings."

真实可查(nwu毕业证书)美国西北大学毕业证学位证书范本原版一模一样

原版定制【微信:bwp0011】《(nwu毕业证书)美国西北大学毕业证学位证书》【微信:bwp0011】成绩单 、雅思、外壳、留信学历认证永久存档查询，采用学校原版纸张、特殊工艺完全按照原版一比一制作（包括：隐形水印，阴影底纹，钢印LOGO烫金烫银，LOGO烫金烫银复合重叠，文字图案浮雕，激光镭射，紫外荧光，温感，复印防伪）行业标杆！精益求精，诚心合作，真诚制作！多年品质 ,按需精细制作，24小时接单,全套进口原装设备，十五年致力于帮助留学生解决难题，业务范围有加拿大、英国、澳洲、韩国、美国、新加坡，新西兰等学历材料，包您满意。
【业务选择办理准则】
一、工作未确定，回国需先给父母、亲戚朋友看下文凭的情况，办理一份就读学校的毕业证【微信bwp0011】文凭即可
二、回国进私企、外企、自己做生意的情况，这些单位是不查询毕业证真伪的，而且国内没有渠道去查询国外文凭的真假，也不需要提供真实教育部认证。鉴于此，办理一份毕业证【微信bwp0011】即可
三、进国企，银行，事业单位，考公务员等等，这些单位是必需要提供真实教育部认证的，办理教育部认证所需资料众多且烦琐，所有材料您都必须提供原件，我们凭借丰富的经验，快捷的绿色通道帮您快速整合材料，让您少走弯路。
留信网认证的作用:
1:该专业认证可证明留学生真实身份
2:同时对留学生所学专业登记给予评定
3:国家专业人才认证中心颁发入库证书
4:这个认证书并且可以归档倒地方
5:凡事获得留信网入网的信息将会逐步更新到个人身份内，将在公安局网内查询个人身份证信息后，同步读取人才网入库信息
6:个人职称评审加20分
7:个人信誉贷款加10分
8:在国家人才网主办的国家网络招聘大会中纳入资料，供国家高端企业选择人才
【关于价格问题（保证一手价格）】
我们所定的价格是非常合理的，而且我们现在做得单子大多数都是代理和回头客户介绍的所以一般现在有新的单子 我给客户的都是第一手的代理价格，因为我想坦诚对待大家 不想跟大家在价格方面浪费时间
对于老客户或者被老客户介绍过来的朋友，我们都会适当给一些优惠。

How to Use Payment Vouchers in Odoo 18.

How to Use Payment Vouchers in Odoo 18.

Fabular Frames and the Four Ratio Problem

Fabular Frames and the Four Ratio Problem

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- 1. Dendogram Hierarchical Clustering : Its slow :: complicated :: repeatable :: not suited for big data sets. Lets take 6 simple Vectors. 6 Vectors Using Euclidean Distance lets compute the Distance Matrix. Euclidean Distance = sqrt( (x2 -x1)**2 + (y2-y1)**2 )
- 2. Using Euclidean Distance lets compute the Distance Matrix. Euclidean Distance = sqrt( (x2 -x1)**2 + (y2-y1)**2 ) Distance Matrix Complete Link Clustering: Considers Max of all distances. Leads to many small clusters. Distance Matrix: Diagonals will be 0 and values will be symmetric. Stage 0
- 3. Step a: The shortest distance in the matrix is 1 and the vectors associated with that are C & D So the first cluster is C — D Distance between other vectors and CD A to CD = max(A->C, A->D) = max(25,24) = 25 B to CD = max(B-<C, B->D) = max(21,20) = 21 and similarly find for E -> CD & F -> CD Stage 1
- 4. Step b : Now 2 is the shortest distance and the vectors associated with that are E & F Second cluster is E — F A to EF = max(A->E, A->F) = max(9,7) = 9 CD to EF = max(CD->E, CD->F) = max(15,17) = 17 Step c : Now 4 is the shortest distance and vectors associated are A & B. Third cluster is A — B CD to AB = max(CD -> A, CD ->B) = max(25,21) = 25 EF to AB = max(EF -> A, EF ->B) = max(9,5) = 9
- 5. Step d : Now 9 is the shortest distance and vectors associated are AB and EF. Fourth cluster is AB — EF CD to ABEF = max(CD->AB, CD->EF) = max(25,18) = 25 Step e : Last cluster is CD — ABEF
- 6. Let’s take a sample of 5 students: Creating a Proximity Matrix First, we will create a proximity matrix which will tell us the distance between each of these points. S ince we are calculating the distance of each point from each of the other points, we will get a square matrix of shape n X n (where n is the number of observations). Let’s make the 5 x 5 proximity matrix for our example:
- 7. Step 1: First, we assign all the points to an individual cluster: Different colors here represent different clusters. You can see that we have 5 different clusters for the 5 points in our data. Step 2: Next, we will look at the smallest distance in the proximity matrix and merge the points with the smallest distance. We then update the proximity matrix: Here, the smallest distance is 3 and hence we will merge point 1 and 2:
- 8. Let’s look at the updated clusters and accordingly update the proximity matrix: Here, we have taken the maximum of the two marks (7, 10) to replace the marks for this cluster. Instead of the maximum, we can also take the minimum value or the average values as well. Now, we will again calculate the proximity matrix for these clusters:
- 9. Step 3: We will repeat step 2 until only a single cluster is left. So, we will first look at the minimum distance in the proximity matrix and then merge the closest pair of clusters. We will get the merged clusters as shown below after repeating these steps:
- 10. How should we Choose the Number of Clusters in Hierarchical Clustering? Let’s get back to our teacher-student example. Whenever we merge two clusters, a dendrogram will record the distance between these clusters and represent it in graph form. Let’s see how a dendrogram looks like: We have the samples of the dataset on the x-axis and the distance on the y- axis. Whenever two clusters are merged, we will join them in this dendrogram and the height of the join will be the distance between these points.
- 11. Let’s build the dendrogram for our example: Take a moment to process the above image. We started by merging sample 1 and 2 and the distance between these two samples was 3 (refer to the first proximity matrix in the previous section).
- 12. Let’s plot this in the dendrogram: Here, we can see that we have merged sample 1 and 2. The vertical line represents the distance between these samples. S imilarly, we plot all the steps where we merged the clusters and finally, we get a dendrogram like this:
- 13. Now, we can set a threshold distance and draw a horizontal line (Generally, we try to set the threshold in such a way that it cuts the tallest vertical line). Let’s set this threshold as 12 and draw a horizontal line: The number of clusters will be the number of vertical lines which are being intersected by the line drawn using the threshold. In the above example, since the red line intersects 2 vertical lines, we will have 2 clusters. One cluster will have a sample (1,2,4) and the other will have a sample (3,5).