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Coursera: Machine Learning (Week 8) [Assignment Solution] - Andrew NG

programming assignment k means clustering and pca

Recommended Machine Learning Courses: Coursera: Machine Learning    Coursera: Deep Learning Specialization Coursera: Machine Learning with Python Coursera: Advanced Machine Learning Specialization Udemy: Machine Learning LinkedIn: Machine Learning Eduonix: Machine Learning edX: Machine Learning Fast.ai: Introduction to Machine Learning for Coders
  • ex7.m - Octave/MATLAB script for the first exercise on K-means
  • ex7 pca.m - Octave/MATLAB script for the second exercise on PCA
  • ex7data1.mat - Example Dataset for PCA
  • ex7data2.mat - Example Dataset for K-means
  • ex7faces.mat - Faces Dataset
  • bird small.png - Example Image
  • displayData.m - Displays 2D data stored in a matrix
  • drawLine.m - Draws a line over an exsiting figure
  • plotDataPoints.m - Initialization for K-means centroids
  • plotProgresskMeans.m - Plots each step of K-means as it proceeds
  • runkMeans.m - Runs the K-means algorithm
  • submit.m - Submission script that sends your solutions to our servers
  • [*] pca.m - Perform principal component analysis
  • [*] projectData.m - Projects a data set into a lower dimensional space
  • [*] recoverData.m - Recovers the original data from the projection
  • [*] findClosestCentroids.m - Find closest centroids (used in K-means)
  • [*] computeCentroids.m - Compute centroid means (used in K-means)
  • [*] kMeansInitCentroids.m - Initialization for K-means centroids
  • Video - YouTube videos featuring Free IOT/ML tutorials

projectData.m :

Recoverdata.m :, findclosestcentroids.m :, check-out our free tutorials on iot (internet of things):.

computeCentroids.m :

Kmeansinitcentroids.m :, 37 comments.

Hi there! Your site has been really helpful. I hope you continue this good work :) In addition, I do have a suggestion for a more vectorized implementation approach for computeCentroids.m: idx_vec = (1:K) == idx; centroids(1:K, :) = (idx_vec' * X)./(sum(idx_vec))'; Cheers!

programming assignment k means clustering and pca

Thanks for suggestion. Glad to know that my solution were helpful to you. I will try/implement your suggestion.

programming assignment k means clustering and pca

in findClosestCentroids [~, idx(i)] what does "~" means here?

min function in MATLAB returns output as [minimum_value, index]. Since we are only interested in the index value, We can ignore storing minimum_value to any variable. If you put the first place blank as [ ,idx(i)]. (To ignore assigning minimum_value to any variable), It will throw an error. So, it can be achieved by replacing the variable by "~" (tilde) character. It is also known as Argument Placeholder.

How about course coursera machine learning andrew ng ?

It's really awesome course to begin studying Machine Learning. This course provide through concepts and enough hands-on practices.

Why are we summing up X values? sqrt((X(i,:)-centroids(j,:)).^2); --> Here

To find closest centroid, we have to calculate euclidean distance and find the minimum value. and This is the formula for calculating Euclidean distance for every sample with Centroid.

Thank you for the solutions,they all were really helpful.

Thank you very much for the appreciation.

What is the meaning of this bro ? " [~,idx(i)]" I also want to know the difference between " [~,idx(i)]" and "idx(i)". aren't they same anyway as idx is single column vector?

"~" is a placeholder. You can put any variable name instead of "~" but that is consume memory. But if you don't want use that value again anywhere then no need to use variable for that. So use can use "~" there. min function returns two values. 1st is the minimum value and 2nd is the index of the minimum value. Here, We want to use only 2nd value so we stored it in "idx(i)" variable and used "~" for 1st value as we don't need it for further use.

I am still getting an error at line 135 'ex7.m' saying there is an error in findClosestCentroids.m. Matrix Dimensions must agree.

A little shorter: function idx = findClosestCentroids(X, centroids) %FINDCLOSESTCENTROIDS computes the centroid memberships for every example % idx = FINDCLOSESTCENTROIDS (X, centroids) returns the closest centroids % in idx for a dataset X where each row is a single example. idx = m x 1 % vector of centroid assignments (i.e. each entry in range [1..K]) % % Set K K = size(centroids, 1); m = size(X,1); % You need to return the following variables correctly. idx = zeros(m, 1); % ====================== YOUR CODE HERE ====================== % Instructions: Go over every example, find its closest centroid, and store % the index inside idx at the appropriate location. % Concretely, idx(i) should contain the index of the centroid % closest to example i. Hence, it should be a value in the % range 1..K % % Note: You can use a for-loop over the examples to compute this. % dist2centroid = zeros(m,K); for ik = 1 : K cent2matrix = repmat(centroids(ik,:),[m 1]); dist2centroid(:,ik) = sum((X - cent2matrix).^2,2); end [~,idx] = min(dist2centroid,[],2) ; % ============================================================= end

how's this code shorter than the given above in the blog post? It is almost same code.

"for"-loops are not optimized in Matlab, so eliminating "for" loops is always a goal in writing Matlab scripts. As a final version, would prefer everything in a single line: dist2centroid(:,ik) = sum((X - repmat(centroids(ik,:),[m 1])).^2,2) but it may be confusing for some viewers.

Ok. Thank you very much for your detailed explanation.

No, thank you for your effort. Great work!

Little shorter: function centroids = computeCentroids(X, idx, K) %COMPUTECENTROIDS returns the new centroids by computing the means of the %data points assigned to each centroid. % centroids = COMPUTECENTROIDS(X, idx, K) returns the new centroids by % computing the means of the data points assigned to each centroid. It is % given a dataset X where each row is a single data point, a vector % idx of centroid assignments (i.e. each entry in range [1..K]) for each % example, and K, the number of centroids. You should return a matrix % centroids, where each row of centroids is the mean of the data points % assigned to it. % % You need to return the following variables correctly. centroids = zeros(K, size(X,2)); % ====================== YOUR CODE HERE ====================== % Instructions: Go over every centroid and compute mean of all points that % belong to it. Concretely, the row vector centroids(i, :) % should contain the mean of the data points assigned to % centroid i. % % Note: You can use a for-loop over the centroids to compute this. % for ik = 1:K centroids(ik,:) = mean(X(idx == ik,:)); end % ============================================================= end

The extra array-variable of dynamic length, "idx_i" is not created, and Matlab is quite sensitive to it. Also, "find" is not a logical operation, which is more expensive than "==" logical.

I got the following error after I use the same code as you posted: error: computeCentroids: =: nonconformant arguments (op1 is 1x3, op2 is 0x1) error: called from computeCentroids at line 30 column 21 runkMeans at line 55 column 15 ex7 at line 135 column 16 Any ideas?

Thank you for your effort is it necessaty that we initialize temp = zeros(K,1); what is its purpose thank you

If create a matrix in MATLAB and increase the size of that matrix in each iteration then internally MATLAB creates a new matrix of updated size and create older matrix. This is very costly operation. So, To solve the issue, the best practise is to create a Zero matrix of its maximum size and then update the values in each iteration.

Ah get it, thank you again

It really helped macha, Thank you.

I came up with this solution but I don't know where exactly is the error But maybe it would be helpful --------------------------- result = zeros(size(X,1), K); for i=1:K result(:,i) = (sum(X - centroids(i,:),2)).^2; endfor [minval, idx] = min(result, [], 2);

this is regarding solution for findClosestCentroids

findClosestCentroids temp(j)=sqrt(sum((X(i,:)-centroids(j,:)).^2)); why .^2 and then sqrt?? i tried deleting both, it worked

okay i made a mistake, i understand it now,, thanks

do u know a vectorized implementation for computeCentroids?? thanks

%%%%%% WORKING: SOLUTION 2 %%%%%%%% in computeCentroids code above is itself vectorized implementation. Don't get confused with the for loop used in that solution. K-mean is a iterative process/algorithm. That for loop is for k-mean iterations. Vectorized implementation means applying operations simultaneously on all data points instead of applying it only on one data point at a time and using for loop for iterating over all data points. (For loop in above code is different)

I'd like to share a completely vectorized version I came up with for computeCentroids.m: assigned = NaN(m, n*K); characteristic = logical(cat(3, repmat((idx == (1:K)), 1, n))); pool = reshape(repmat(X', 1, K)', m, n*K); assigned(characteristic) = pool(characteristic); centroids = reshape(mean(assigned, 1, "omitnan"), K, n); It's definitely worse than your solution both in terms of complexity and readability, but I thought it was interesting that the for loops can be completely eliminated. Maybe it can be optimized and improved on?

Thank you very much for your solution. It's a new approach & will be helpful for others as well.

execution of a script as a function is not possible

when trying to run Sigma = (1/m)*(X'*X); % n x n [U, S, V] = svd(Sigma); it is important to use sigma with a Capital "S" as sigma will coincide with the use of svd, and the code may fail to run. Nice work!!!

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How to Combine PCA and K-means Clustering in Python?

programming assignment k means clustering and pca

Did you know that you can combine Principal Components Analysis (PCA) and K-means Clustering to improve segmentation results?

In this tutorial, we’ll see a practical example of a mixture of PCA and K-means for clustering data using Python.

Why Combine PCA and K-means Clustering?

There are varying reasons for using a dimensionality reduction step such as PCA prior to data segmentation. Chief among them? By reducing the number of features, we’re improving the performance of our algorithm. On top of that, by decreasing the number of features the noise is also reduced.

In the case of PCA and K-means in particular, there appears to be an even closer relationship between the two.

This paper discusses the exact relationship between the techniques and why a combination of both techniques could be beneficial. In case you’re not a fan of the heavy theory, keep reading. In the next part of this tutorial, we’ll begin working on our PCA and K-means methods using Python.

1. Importing and Exploring the Data Set

We start as we do with any programming task: by importing the relevant Python libraries. In our case they are:

Importing relevant Python libraries.

The second step is to acquire the data which we’ll later be segmenting. We’ll use customer data, which we load in the form of a pandas’ data frame.

Loading the data we will be segmenting.

The data set we’ve chosen for this tutorial comprises 2,000 observations and 7 features.

More specifically, it contains information about 2,000 individuals and has their IDs, as well as geodemographic features, such as Age, Occupation, etc.

Lastly, we take a moment to visualize the raw data set on the two numerical features; Age and Income.

Visualizing raw data on the two numerical features.

The graph represents all points in our current data set, which our K-means algorithm will aim to segment.

Another observation from the graph concerns the domains of the two variables Age and Income. We understand that the domain for Age is from around 20 to 70, whereas for Income it is from around 40,000 to over 300,000. Which points to a vast difference between the range of these values. Therefore, we must incorporate an important step in our analysis, and we must first standardize our data.

Standardization is an important part of data preprocessing, which is why we’ve devoted the entire next paragraph precisely to this topic.

2. Data Preprocessing

Our segmentation model will be based on similarities and differences between individuals on the features that characterize them.

We’ll quantify these similarities and differences.

Well, you can imagine that two persons may differ in terms of ‘Age’. One may be a 20-year-old, while another – 70 years old. The difference in age is 50 years. However, it spans almost the entire range of possible ages in our dataset.

At the same time, the first individual may have an annual income of \$100,000; while the second may have an annual income of \$150,000. Therefore, the difference between their incomes will be \$50,000.

If these numbers were to go into any of our segmentation models as they are, the algorithm would believe that the two differ in terms of one variable by 50; while in terms of another by 50,000. Then, because of the mathematical nature of modeling, it would completely disregard ‘Age’ as a feature. The reason is that the numbers from 20 to 70 are insignificant when compared with the income values around 100K.

Because the model is not familiar with our context. So, it defines one as age, while the other as income.

Therefore, it will place a much bigger weight on the income variable.

It is obvious that we must protect ourselves from such an outcome. What’s more, in general, we want to treat all the features equally. And we can achieve that by transforming the features in a way that makes their values fall within the same numerical range. Thus, the differences between their values will be comparable. This process is commonly referred to as standardization.

For this tutorial, we’ll use a Standard Scaler to standardize our data, which is currently in the df segmentation data frame:

Using Standart Scaler to standardize data.

After data standardization, we may proceed with the next step, namely Dimensionality Reduction.

3. How to Perform Dimensionality Reduction with PCA?

We’ll employ PCA to reduce the number of features in our data set. Before that, make sure you refresh your knowledge on what is Principal Components Analysis .

In any case, here are the steps to performing dimensionality reduction using PCA.

First, we must fit our standardized data using PCA.

Fitting our standardized data using PCA.

Second, we need to decide how many features we’d like to keep based on the cumulative variance plot.

Cumulative variance plot

The graph shows the amount of variance captured (on the y-axis) depending on the number of components we include (the x-axis). A rule of thumb is to preserve around 80 % of the variance. So, in this instance, we decide to keep 3 components.

As a third step, we perform PCA with the chosen number of components.

For our data set, that means 3 principal components:

Performing PCA on three chosen components.

We need only the calculated resulting components scores for the elements in our data set:

The calculated resulting components scores.

We’ll incorporate the newly obtained PCA scores in the K-means algorithm. That's how we can perform segmentation based on principal components scores instead of the original features.

4. How to Combine PCA and K-means Clustering?

As promised, it is time to combine PCA and K-means to segment our data, where we use the scores obtained by the PCA for the fit.

Based on how familiar you are with K-means, you might already know that K-means doesn’t determine the number of clusters in your solution. If you need a refresher on all things K-means, you can read our dedicated blog post .

In any case, it turns out that we ourselves need to determine the number of clusters in a K-means algorithm.

In order to do so, we run the algorithm with a different number of clusters. Then, we determine the Within Cluster Sum of Squares or WCSS for each solution. Based on the values of the WCSS and an approach known as the Elbow method , we make a decision about how many clusters we’d like to keep.

First, however, we must decide how many clustering solutions we’d test.

There is no general ruling on this issue. It really depends on the data. In our case, we test an algorithm with up to 20 clusters.

K-means clustering using the transformed data from the PCA.

The next step involves plotting the WCSS against the number of components on a graph.

Plotting the WCSS to define the number of clusters.

And from this graph, we determine the number of clusters we’d like to keep. To that effect, we use the Elbow-method. The approach consists of looking for a kink or elbow in the WCSS graph. Usually, the part of the graph before the elbow would be steeply declining, while the part after it – much smoother. In this instance, the kink comes at the 4 clusters mark. So, we’ll be keeping a four-cluster solution.

All left to do is to implement it.

Running K-means with four clusters.

Here, we use the same initializer and random state as before. Subsequently, we fit the model with the principal component scores.

And now we’ve come to the most interesting part: analyzing the results of our algorithm.

5. How to Analyze the Results of PCA and K-Means Clustering

Before all else, we’ll create a new data frame. It allows us to add in the values of the separate components to our segmentation data set. The components’ scores are stored in the ‘scores P C A’ variable. Let’s label them Component 1, 2 and 3. In addition, we also append the ‘K means P C A’ labels to the new data frame.

Creating a new data frame and adding the PCA scores and assigned clusters.

We’re all but ready to see the results of our labor.

One small step remains: we should add the names of the segments to the labels.

We create a new column named ‘Segment’ and map the four clusters directly inside it.

Adding the names of the segments to the labels.

6. How to Visualize Clusters by Components?

Let’s finish off by visualizing our clusters on a 2D plane. It's a 2D visualization, so we need to choose two components and use them as axes. The point of PCA was to determine the most important components. This way, we can be absolutely sure that the first two components explain more variance than the third one.

So, let’s visualize the segments with respect to the first two components.

Visualizing the data with respect to the first two components.

The X-axis here is our ‘Component 2’. The y-axis, on the other hand, is the first ‘Component 1’.

We can now observe the separate clusters.

The results of K-means clustering without PCA.

In this instance, only the green cluster is visually separated from the rest. The remaining three clusters are jumbled all together.

However, when we employ PCA prior to using K-means we can visually separate almost the entire data set. That was one of the biggest goals of PCA - to reduce the number of variables by combining them into bigger, more meaningful features.

Not only that, but they are ‘orthogonal’ to each other. This means that the difference between components is as big as possible.

There is some overlap between the red and blue segments. But, as a whole, all four segments are clearly separated. The spots where the two overlap are ultimately determined by the third component, which is not available on this graph.

Combining PCA and K-Means Clustering: Overview

Finally, it is important to note that our data set contained only a few features from the get-go. So, when we further reduced the dimensionality, using ‘P C A’ we found out we only need three components to separate the data.

That’s the reason why even a two-dimensional plot is enough to see the separation.

This might not always be the case. You may have more features and more components respectively. Then you might need a different way to represent the results of PCA.

We hope you’ll find this tutorial helpful and try out a K-means and PCA approach using your own data.

If you’re interested in more practical insights into Python,  check out our step-by-step Python tutorials .

In case you’re new to Python, this comprehensive article on learning Python programming will guide you all the way. From the installation, through Python IDEs, Libraries, and frameworks, to the best Python career paths and job outlook.

How to Combine PCA & k-means Clustering (Example)

In this post, I’ll show the use of combining PCA with (k-means) clustering . The tutorial will contain the following sections:

Let’s just jump right in!

Introduction

Cluster analysis aims to identify hidden patterns or structures within an unlabeled dataset. More specifically, it aims to partition the data into groups such that data points within a group are more similar to each other than they are to data points in other groups.

In the presence of high-dimensional data , selecting relevant variables to be used in clustering could be hard. In that regard, employing PCA before clustering is useful to reduce the dimensionality of your data and discard the noise . PCA also brings another advantage by creating a new set of uncorrelated variables, ensuring each variable holds unique information and has equal significance in determining clusters.

Furthermore, the clustering results can be represented in a reduced dimensional space in relation to the initial principal components , which eases the visualization and interpretation of the results. Without further ado, let’s create our sample data for the demonstration!

Sample Data

As sample data, I will use the built-in mtcars dataset in the R programming language . This dataset contains the fuel consumption (mpg) and 10 aspects of automobile design and performance for 32 automobiles (1973-74 models). See below for the first few rows of the dataset.

mtcars dataset

As seen, the dataset contains 11 continuous and discrete ( ordinal and nominal ) numeric variables. To obtain reliable PCA results, I will exclude the discrete variables and only work with the fuel consumption (mpg), engine displacement (disp), gross horsepower (hp), rear axle ratio (drat), weight (1000 lbs) (wt) and 1/4 mile time (qsec). For alternatives of PCA working with categorical and mixed data, see our tutorial: Can PCA be Used for Categorical Variables?

Please also be aware that the retained variables will be standardized to avoid biased PCA results. See: PCA Using Correlation & Covariance Matrix for further explanation. If you are ready, let’s cluster it!!  

Example: k-means Clustering Combined with PCA

The first step is to perform a PCA to reduce the dimensionality of the data. Then we can decide on the number of components to retain based on the percentage of explained variance per principal component.

proportion explained variance in PCA

Table 2 shows that the first two principal components explain enough variation, with 90% in total. So we can use only these components to explain our data and neglect the rest, then visualize our data in 2D reduced dimensional space via a biplot .

biplot for mtcars dataset

Figure 1 shows that the higher PC1 scores correspond to increased horsepower, engine displacement, and weight but decreased fuel consumption, rear axle ratio, and quarter-mile time. Conversely, higher PC2 scores refer to increased rear axle ratio and horsepower but decreased weight and quarter-mile time. To learn more about how to interpret biplots, see Biplot for PCA Explained .

Knowing what the principal components represent, we can skip to the k-means clustering analysis. But first, we must determine the number of clusters to form. I will use the within-cluster sum of squares measure, which indicates the compactness of the clusters. A lower value indicates better clustering.

As the selection method, I will employ the elbow method, which suggests selecting the number of clusters at which the rate of decrease in the within-cluster sum of squares slows down and forms an elbow shape in the plot of choice. Let’s check the respective plot out!

WCSS plot for k-means clustering

Based on Figure 2, forming 4 clusters is convenient for grouping similar observations. Relatedly, we can run our k-means cluster analysis, grouping our data around four centers, also known as centroids . See the visual of the results below.

cluster plot for mtcars dataset

You see how the data points are grouped based on similar principal component scores in Figure 3. Now we can interpret the clusters in the light of what principal components represent (check the biplot given earlier) as follows.

  • Group 1 represents the cars with high horsepower (hp) but low 1/4 mile time (qsec).
  • Group 2 represents the cars with large engine displacement (disp) and weight (wt) but a low rear axle ratio (drat).
  • Group 3 represents the cars with increased rear axle ratio (drat), fuel consumption (mpg), and 1/4 mile time (qsec) but decreased horsepower (hp), engine displacement (disp) and weight (wt).
  • Group 4 represents the cars with increased 1/4 mile time (qsec) and weight (wt) but decreased rear axle ratio (drat) and horsepower (hp).

If you want to see all information regarding the component-variable relations and the clusters, then you can also visualize a combined plot as given below. But please be aware that this type of graph may not be the best option in the presence of a large dataset.

biplot for mtcars dataset

As you can see, the visualization and interpretation of the clustering results got easier using PCA. For the computation of the shown steps in R and Python, see our tutorials PCA Before k-means Clustering in R and PCA Before k-means Clustering in Python , to be published soon.

Video & Further Resources

Do you need more explanations on how to apply PCA before k-means clustering? Then you should have a look at the following YouTube video of the Statistics Globe YouTube channel.

The YouTube video will be added soon.

Moreover, you could check some of the other tutorials on Statistics Globe:

  • What is a Principal Component Analysis (PCA)?
  • Choose Optimal Number of Components for PCA
  • Can PCA be Used for Categorical Variables?
  • PCA Using Correlation & Covariance Matrix
  • Principal Component Analysis (PCA) in R
  • Visualization of PCA in R
  • Biplot for PCA Explained
  • Biplot of PCA in R
  • PCA Before k-means Clustering in Python
  • PCA Before k-means Clustering in R

You have learned in this tutorial how to combine PCA with k-means clustering in R programming. Let me know in the comments section below if you have additional questions.

Cansu Kebabci R Programmer & Data Scientist

This page was created in collaboration with Cansu Kebabci. Have a look at Cansu’s author page to get more information about her professional background, a list of all his tutorials, as well as an overview of her other tasks on Statistics Globe.

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2 Comments . Leave new

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Well explained. Can I get some reference book name regarding PCA basics to advanced mathematical explanation.

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Hello Kayum,

I am glad that you liked our tutorial. You can visit our main PCA tutorial for the basic mathematical explanation. For more advanced mathematical explanations, you can check the book Applied Multivariate Statistical Analysis” by Richard A. Johnson and Dean W.

Best, Cansu

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Joachim Schork Statistician Programmer

I’m Joachim Schork. On this website, I provide statistics tutorials as well as code in Python and R programming.

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  4. How to Combine PCA and K-means Clustering in Python?

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COMMENTS

  1. Coursera: Machine Learning (Week 8) [Assignment Solution]

    K-means clustering algorithm to compress an image. Principal component analysis to find a low-dimensional representation of face images. I have recently completed the Machine Learning course from Coursera by Andrew NG. While doing the course we have to go through various quiz and assignments.

  2. How to Combine PCA and K-means Clustering in Python?

    How to Combine PCA and K-means Clustering in Python? Elitsa Kaloyanova 29 Jul 2021 7 min read Did you know that you can combine Principal Components Analysis (PCA) and K-means Clustering to improve segmentation results? In this tutorial, we'll see a practical example of a mixture of PCA and K-means for clustering data using Python.

  3. PDF K-means Clustering and Principal Component Analysis

    1 K-means Clustering In this this exercise, you will implement the K-means algorithm and use it for image compression. You will rst start on an example 2D dataset that 1Octave is a free alternative to MATLAB. For the programming exercises, you are free to use either Octave or MATLAB.

  4. Machine Learning Exercise 7

    K-means is an iterative, unsupervised clustering algorithm that groups similar instances together into clusters. The algorithm starts by guessing the initial centroids for each cluster, and...

  5. GitHub

    GitHub - AlfTang/K-means-and-PCA: An exercise on K-means clustering algorithm & Principle Component Analysis, and their application to image compression. It is programming exercise 7 in Machine Learning course by Andrew Ng on Coursera. master Code README GPL-2.0 license K-means-and-PCA

  6. K-means Clustering and Principal Component Analysis in 10 Minutes

    K-means is a centroid-based clustering algorithm that works as follows. Random initialization: place k centroids randomly. Cluster assignment: assign each observation to the closest cluster based on the distance to centroids. Centroid update: move centroids to the means of observations of the same cluster.

  7. PCA Before k-means Clustering in Python (Example)

    Let's first install and import the relevant libraries for our use. We need the PCA, StandardScaler, and KMeans modules to perform PCA and k-means clustering and the Matplotlib, scipy, adjustText, and NumPy libraries for visualization purposes. We will import the pandas library and the data function from pydataset to create our sample data.

  8. K-Means Clustering in Python: A Practical Guide

    The k-means clustering method is an unsupervised machine learning technique used to identify clusters of data objects in a dataset. There are many different types of clustering methods, but k-means is one of the oldest and most approachable.These traits make implementing k-means clustering in Python reasonably straightforward, even for novice programmers and data scientists.

  9. GitHub: Let's build from here · GitHub

    {"payload":{"allShortcutsEnabled":false,"fileTree":{"":{"items":[{"name":"data","path":"data","contentType":"directory"},{"name":"Programming Exercise 1 - Linear ...

  10. Andrew Ng's Machine Learning Course in Python (Kmeans-Clustering, PCA

    We will start by implementing the K-means algorithms. Since K-means is an iterative process that assigns training examples to their closest centroids and then recomputing the centroids, we need two main functions that do just that. import numpy as np import matplotlib.pyplot as plt from scipy.io import loadmat mat = loadmat("ex7data2.mat") X ...

  11. Explaining K-Means Clustering. Comparing PCA and t-SNE dimensionality

    In this example, we are going to compare PCA and t-SNE data reduction techniques prior to running our K-Means clustering algorithm. Let's take a few mins to explain PCA and t-SNE. Principal Component Analysis (PCA) Principal component analysis or (PCA) is a classic method we can use to reduce high-dimensional data to a low-dimensional space.

  12. How to Apply PCA Before k-means Clustering

    1) Introduction 2) Sample Data 3) Example: k-means Clustering Combined with PCA 4) Video & Further Resources Let's just jump right in! Introduction Cluster analysis aims to identify hidden patterns or structures within an unlabeled dataset.

  13. Implementation of Principal Component Analysis(PCA) in K Means Clustering

    Implementation of Principal Component Analysis (PCA) in K Means Clustering A beginner's approach to apply PCA using 2 components to a K Means clustering algorithm using Python and...

  14. What is the relation between k-means clustering and PCA?

    It is a common practice to apply PCA (principal component analysis) before a clustering algorithm (such as k-means). It is believed that it improves the clustering results in practice (noise reduction). However I am interested in a comparative and in-depth study of the relationship between PCA and k-means.

  15. Demonstrating PCA in K-Means Clustering

    Demonstrating PCA in K-Means Clustering. Principal Component Analysis (PCA) is a method employed to reduce the dimensionality of a data set consisting of a large number of interrelated variables ...

  16. EngineerInd/Programming-Assignment-K-Means-Clustering-and-PCA ...

    This commit does not belong to any branch on this repository, and may belong to a fork outside of the repository.

  17. Principal Component Analysis and k-means Clustering to ...

    Figure 4. Interactive 3-D visualization of k-means clustered PCA components. Go ahead, interact with it. Figure 4 was made with Plotly and shows some clearly defined clusters in the data.

  18. PCA| K-means Clustering

    In the method of feature dimension reduction, the Principal Component Analysis is the most classic and practical feature dimension reduction technology, especially in the image recognition field. k-means clustering is a method of vector quantization, originally from signal processing, that is popular for cluster….

  19. Interpretation K-Means clustering with PCA

    -1 I have implemented a K-Means clustering on a dataset in which I have reduced the dimensionality to 2 features with PCA. Now I am wondering how to interprete this analysis since there is any reference on which are the variables on the axis.

  20. GitHub

    A tag already exists with the provided branch name. Many Git commands accept both tag and branch names, so creating this branch may cause unexpected behavior.

  21. Market Segmentation with R (PCA & K-means Clustering)

    They are key to help us interpret the PCA results. When directly working with the PCA loadings can be tricky and confusing, we can rotate these loadings to make interpretation easier. There are multiple rotation methods out there, and we will use a method called "varimax". (Note, this step of rotation is NOT a part of the PCA.

  22. RITIK-12/Programming-Assignment-K-Means-Clustering-and-PCA

    Machine Learning Course - Coursera. Contribute to RITIK-12/Programming-Assignment-K-Means-Clustering-and-PCA development by creating an account on GitHub.

  23. Clustering

    Common shallow-learning algorithms are K-Means, Hierarchical Clustering, Expectation-Maximization (EM) and Density-Based Spatial Clustering of Applications with Noise (DBSCAN). Common...