Unsupervised Clustering Solution

a The assignment should be submitted in the PDF format through Collob. If you prefer hand-writing QA parts of answers, please convert them (e.g., by scanning or using PhoneApps like o ceLens) into PDF form.




For questions and clari cations, please post on piazza.



Policy on collaboration:



Homework should be done individually: each student must hand in their own answers. It is acceptable, however, for students to collaborate in guring out answers and helping each other solve the problems. We will be assuming that, with the honor code, you will be taking the responsibility to make sure you personally understand the solution to any work arising from such collaboration.




d Policy on late homework: Homework is worth full credit at the midnight on the due date. Each student has three extension days to be used at his or her own discretion throughout the entire course. Your grades would be discounted by 15% per day when you use these 3 late days. You could use the 3 days in whatever combination you like. For example, all 3 days on 1 assignment (for a maximum grade of 55%) or 1 each day over 3 assignments (for a maximum grade of 85% on each). After you’ve used all 3 days, you cannot get credit for anything turned in late.




Unsupervised Learning with Clustering



In this programming assignment, you are required to implement clustering algorithm: K-means Clustering. A ZIP le has been provided (\data sets clustering.zip" ) that includes two di erent datasets. Please follow all instructions for submitting source code.







You are required to submit a source-code le \clustering.py" containing the necessary functions for training and evaluations. The maximum number of iterations to be performed for both algorithms is 1000.




DO NOT use scikit-learn package in this problem and please implement from scratch.




1.1 Data description




We have provided two di erent datasets for clustering tasks.




Dataset 1 : The rst dataset consists of height and weight data for average people and baseball players. First column contains human height (inches) and second column has human weight (lbs), while third column has true labels of samples that will be used only for evaluations.




Dataset 2 : The second dataset is for a speech versus music classi cation task. This dataset has been preprocessed and rst 13 columns contain 13 features extracted from audio les. Last column has true labels of samples that will be used only for evaluations.




1.2 load data




(Q1) You are required to code the following function for loading datasets: X = loadData( leDj)







1
1.3 K-means Clustering




(Q2) Next, code the following function to implement k-means clustering: labels = kmeans(X, k, maxIter)




Here X is the input data matrix, k is the number of clusters and maxIter is the maximum number of the iterations selected by you (max value =1000).




(Q3) Implement k-means clustering for Dataset 1(use rst two columns in the le as input) and use scatter() function in the matplotlib package to visualize the result. The two clusters must be in di erent colors.







(Q4) Implement k knee- nding method for Dataset 1 and k = f1,2,...,6g to select value of k (number of clusters) and plot graph for k versus objective function value (e.g. Slide 99, Lecture 20).







(Q5) Now, code the following function to calculate the purity metric for the evaluation of results: purityMetric = purity(labels, trueLabels)




Use this function to evaluate the results of (Q3)




1.4 How will your code be checked?




We will run the following command: \python clustering.py DatasetDirectoryFullPath" and your code should print the following results:




the scatter plots from (Q3) k knee- nding plot in (Q4)




ALL purityMetric values for results obtained in (Q3)




1.5 Submission




Please submit your source code as "clustering.py" and PDF report containing your written answers via collab. In the report, you should include the following contents:




ALL scatter plots generated in (Q3) k knee- nding plot in (Q4)




ALL purityMetric values for results obtained in (Q3)









Sample Exam Questions:




Each assignment covers a few sample exam questions to help you prepare for the midterm and the nal. (Please do not bother by the information of points in some the exam questions.)







Question: 1. K-means and Gaussian Mixture Models







Run k-means manually for the following dataset, where k = 2. Circles are data points and squares are the initial cluster centers. Draw the cluster centers and the decision boundaries that de ne each cluster. Use as many pictures as you need until convergence.



Note: Execute the algorithm such that if a mean has no points assigned to it, it stays where it is for that iteration.




Answer:








Question: 2. K-means Clustering




There is a set S consisting of 6 points in the plane shown below where a = (0; 0); b = (8; 0); c = (16; 0); d = (0; 6); e = (8; 6); f = (16; 6). Now we run the k-means algorithm on these points with k = 3. The algorithm uses the Euclidian distance metric (i.e. the straight line distance between two points) to assign each point to its nearest centroid. Ties are broken in favor of the centroid to the left/down. We de ne the following two de nitions:




A k-starting con guration is a subset of k staring points from S that form the initial centroids, e.g. fa; b; cg.

A k-partition is a partition of S into k non-empty subsets, e.g. fa; b; eg, fc; dg, ffg is a 3-partition.




Clearly any k-partition induces a set of k centroids in the natural manner. A k-partition is called stable if a repetition of the k-means iteration with the induced centroid leaves it unchanged.










How many 3-starting con gurations are there? (Remember, a 3-starting con guration is just a size 3 subset of the 6 datapoints.



Answer: C63 = 20




Fill in the following table: Answer:







An example 3-starting con guration
The number of unique




that can arrive at the 3-partition after




starting con gurations
3-partition
Is it stable?
0 or more iterations of k-means
that can arrive at the




(or write \none" if no such 3-starting




3-partition.




con guration)














fa,b,eg,fc,dg,ffg
N
none
0
fa,bg,fd,eg,fc,fg
Y
fb,c,eg
4
fa,dg,fb,eg,fc,fg
Y
fa,b,cg
8
fag,fdg,fb,c,e,fg
Y
fa,b,dg
2
fa,bg,fdg,fc,e,fg
Y
none
0
fa,b,dg,fcg,fe,fg
Y
fa,c,fg
1





Question: 3. Decision Trees The following dataset will be used to learn a decision tree for




predicting whether a person is happy (H) or sad (S) based on the color of their shoes, whether they wear a wig and the number of ears they have.










[2 points] What is H(EmotionjW ig = Y ) (where H is entropy)? Answer: Answer: 1






[2 points] What is H(EmotionjEars = 3)?
Answer: Answer: 0







[3 points] Which attribute would the decision-tree building algorithm choose to use for the root of the tree (assume no pruning)?



Answer: Answer: Color










[3 points] Draw the full decision tree that would be learned from this data (assume no pruning). Answer: Answer: Color is root node, predict sad if green, happy if red, and 50/50 split if blue.




































The next two parts do not use the previous example, but are still about decision tree classi ers.




[3 points] Assuming that the output attribute can take two values (i.e. has arity 2) what is the maximum training set error (expressed as a percentage) that any dataset could possibly have? Answer: Answer: 50%





[3 points] Construct an example dataset that achieves this maximum percentage training set er-ror (it must have two or fewer inputs and ve or fewer records).



Answer: Answer: x : f0; 0; 1; 1g, y : f0; 1; 0; 1g







Question: 4. Decision Trees




The following dataset will be used to learn a decision tree for predicting whether a mushroom is edible or not based on its shape, color, and odor.







[4 points] What is the entropy H(EdiblejOdor = 1 or Odor = 3)? Answer: Answer: 1






[4 points] Which attribute would the ID3 algorithm choose to use for the root of the tree (no pruning)? Answer: Answer: Odor






[4 points] Draw the full decision tree that would be learned for this data (no pruning).



Answer: see gure:






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