Lab 9: Unsupervised Learning

Lab 9: Unsupervised Learning
Due April 18 by midnight

Starting point code

You are encouraged to work with a partner on this lab. Please read the following instructions carefully.

First you need to run setup63 to create a git repository for the lab.
If you want to work alone do:
```
setup63-bryce labs/09 none
```
If you want to work with a partner, then one of you needs to run the following command while the other one waits until it finishes:
```
setup63-bryce labs/09 partnerUsername
```
Once the script finishes, the other partner should run it on their account.
For the next step, only one partner should copy over the starting point code:
```
cd ~/cs63/labs/09
cp /home/bryce/public/cs63/labs/09/* .
```
Whether you are working alone or with a partner, you should now add, commit, and push these files as shown below:
```
git add *
git commit -m "lab9 start"
git push
```
If you are working with a partner, your partner can now pull the changes in:
```
cd ~/cs63/labs/09
git pull
```

You are now ready to start the lab.

Introduction

In this lab you will explore unsupervised learning, specifically bottom-up hierarchical clustering and k-means. You have been given a file called unsupervisedLearning.py that defines a base class with some common methods that read in data and calculate distance. You will be modifying the starting point files: hierarchicalCluster.py and kmeans.py.

Some sample data files have been provided. The UnsupervisedLearning class expects the data to be in the following format:

A points file should contain one data point per line, where each value is separated by white space.
A labels file should contain one label per line, which will be interpreted as a string. Each label should be unique.
The lengths of these two files should be equal.

The data provided includes:

digits.hiddens and digits.labels
This data was generated from a neural network that was trained to recognize hand-written digits (like the one you explored in lab5). After training was complete, the weights were frozen, and the data was passed through the network again. For each digit, the hidden layer's activation was saved. These 10-dimension points are saved in the digits.hiddens file. There are 26 different examples of each digit hand-written by person a-z. The digits.labels file has labels a0-a9, b0-b9, ..., z0-z9.
subset.hiddens and subset.labels
This data is a subset of the digits data, just including samples from person a-c.
small.points and small.labels
This data is provided as a simple testing set to help your verify that your code is working properly. The three dimension points and their associated labels are shown below.
```
0.9 0.8 0.1 a1
0.9 0.1 0.6 b1
0.1 0.1 0.1 c1
0.8 0.9 0.2 a2
0.8 0.0 0.5 b2
0.0 0.0 0.2 c2
0.9 0.9 0.3 a3
0.2 0.0 0.0 c3
0.5 0.5 0.5 d1
```
The a points are close to (1,1,0); the b points are close to (1,0,0.5); the c points are close to (0,0,0), and the d point is an outlier. A "good" clustering of these points would group the a's, the b's, and the c's separately.

Once you have implemented these unsupervised methods, you will pick your own data set to explore with them and analyze the results using the latex template provided in analysis.tex.

Understanding the base class

Execute the base class defined in unsupervisedLearning.py. Make sure you understand what class variables it creates. Note that it converts the points into numpy arrays to more efficiently calculate Euclidean distances.

Hierarchical Clustering

You will construct a tree from the bottom up. Let's walk through the steps using the small data set described above.

Use a dictionary that maps a cluster's label to the cluster's average point. The cluster's label will represent the tree structure that has been created so far. The leaves will simply be the point labels. Internal nodes will be tuples of the form:

(clNUM, leftBranch, rightBranch)

Begin by placing each point from the data set in its own cluster. Given that there are nine points in this data set you will have nine initial clusters.
Calculate the distances between all possible pairs of clusters. Find the pair with the minimum distance. Remove each of these from the dictionary and form a new cluster with this pair adding it to the dictionary. The two closest clusters are a2 and a3. A new cluster called cl0 is formed. The value associated with this cluster is the average of the points from the two original clusters. The current set of cluster names is now:
```
('cl0', 'a3', 'a2')
a1
b1
b2
c3
c2
c1
d1
```
Repeat this process with the current set of clusters. The next two closest are b1 and b2. A new cluster called cl1 is added.
```
('cl0', 'a3', 'a2')
('cl1', 'b1', 'b2')
a1
c3
c2
c1
d1
```

Continuing again, we will have:

('cl0', 'a3', 'a2')
('cl1', 'b1', 'b2')
a1
('cl2', 'c2', 'c1')
c3
d1

The next iteration will merge a leaf with an existing cluster:

('cl3', ('cl0', 'a3', 'a2'), 'a1')
('cl1', 'b1', 'b2')
('cl2', 'c2', 'c1')
c3
d1

Iterating again we see that all of the original points have been merged with a cluster, except for d, which was the outlier.
```
('cl4', ('cl2', 'c2', 'c1'), 'c3')
('cl3', ('cl0', 'a3', 'a2'), 'a1')
('cl1', 'b1', 'b2')
d1
```

This merging process continues until only a single cluster remains in the dictionary.

('cl7', ('cl4', ('cl2', 'c2', 'c1'), 'c3'), ('cl6', ('cl3', ('cl0',
'a3', 'a2'), 'a1'), ('cl5', ('cl1', 'b1', 'b2'), 'd1')))

This tuple reflects the tree structure, but it can be hard to see the structure in this format. Using the plotTree method will generate a picture like this:

Implement this bottom-up process now in the file hierarchicalCluster.py. Calculating distances is the bottleneck of this process, so be sure to store all calculations in the self.distances dictionary. Then you can check this first whenever a distance is needed. If it's there, use it, if not generate it and store it.

Once you can successfully create a hierarchical cluster for the small data. Try the subset data next, and finally the full digits data. Drawing the tree for the full data will take some time. Based on the final tree, which digits seem to be easier (and harder) for the network to classify?

Optional Extensions

On Monday, we discussed several other variations on agglomerative hierarchical clustering. In particular, we considered alternatives for measuring the similarity of (or distance between) two points, and alternatives for aggregating point similarity into cluster similarity. You are encouraged to test some of these alternatives. As a first step, try replacing Euclidean distance with Manhattan distance. Remember, Manhattan distance is equivalent to the 1-norm, so we can add a ManhattanDist function to unsupervisedLearning.py that will look very similar to the EuclideanDist function that has already been defined. To use the new function, you'll need to replace the following line from UnsupervisedLearning.__init__:

self.dist = self.EuclideanDist  -->  self.dist = self.ManhattanDist

You can also try out clustering with other distance or similarity measures. Agglomerative clustering also gives you options for how to define the most-similar clusters. So far, you have implemented average link clustering, but you may want to try out single link or complete link. To implement these, you will need to implement a new function to take the place of averageLinkClosest, and update the following line from HierarchicalClustering.__init__:

self.closestClusters = self.averageLinkClosest

K-Means

You will need to grab some additional starting point files.

cd ~/cs63/labs/09
cp /home/bryce/public/cs63/labs/09/kmeans.py .
cp /home/bryce/public/cs63/labs/09/random.* .

Then add, commit, and push them to your git repo. If you are working with a partner, be sure that they do a git pull to get the new files.

git add kmeans.py random.points random.labels
git commit -m "kmeans start"
git push

You will implement k-means in the kmeans.py file. The additional data files, random.points and random.labels, consist of 200 2D points that were generated in two overlapping clumps. This data will be useful for testing k-means.

Implementing k-means

The k-means algorithm partitions the data into k clusters, where each cluster is defined by a center point. The goal is to minimize the distance between each centroid and the data points that are closest to it. The algorithm begins with an initial partitioning and then iteratively improves it until no further progress can be made.

The following data structures will be helpful in implementing the algorithm:

self.centroids
A dictionary that maps cluster names 'c1'-'ck' to the current center points represented as numpy arrays.
self.members
A dictionary that maps cluster names 'c1'-'ck' to a list of the points (as numpy arrays) that have been assigned to that cluster.
self.labels
A dictionary that maps points (converted to tuples) to the cluster name 'c1' - 'ck' where they are currently assigned.

The following pseudocode describes the algorithm:

init k centroids to be random points from the data set
init labels to be an empty dictionary
while True
   set updateFlag to False
   set error to 0
   in members, init each cluster's member list to be empty
   # E step
   for each point in the data set 
      assign point to the closest centroid
      error += (distance from centroid)^2
      if point is being assigned for the first time or has moved clusters:
         update labels dictionary
         set updateFlag to True
   if not updateFlag
      break
   # M step
   for each cluster
      update the centroid to the average of its assigned points

There is a rare edge case that also needs handling, where one of the clusters will end up with no points assigned to it, and therefore no centroid. If this happens, you should re-initialize that centroid to a random data point, and run a new E step to reassign the data points.

Testing k-means on the small data set

Below is a trace of one execution of k-means on the small.points data that uses the showClusters method to print information about each cluster and its members. In the first iteration, points c3, a2, and a3 were selected as the initial centroids.

ITERATION 1
Current error: 2.8477734056
--------------------
Center: c2 Length: 4
Center point: 0.200 0.000 0.000 c3
0.100 0.100 0.100 c1
0.800 0.000 0.500 b2
0.000 0.000 0.200 c2
0.200 0.000 0.000 c3
--------------------
Center: c1 Length: 3
Center point: 0.800 0.900 0.200 a2
0.900 0.800 0.100 a1
0.800 0.900 0.200 a2
0.500 0.500 0.500 d1
--------------------
Center: c0 Length: 2
Center point: 0.900 0.900 0.300 a3
0.900 0.100 0.600 b1
0.900 0.900 0.300 a3

In the next iteration, the error drops, and several of the points have shifted clusters (a3, b2, and d1).

ITERATION 2
Current error: 2.72339291187
--------------------
Center: c2 Length: 3
Center point: 0.275 0.025 0.200
0.100 0.100 0.100 c1
0.000 0.000 0.200 c2
0.200 0.000 0.000 c3
--------------------
Center: c1 Length: 3
Center point: 0.733 0.733 0.267
0.900 0.800 0.100 a1
0.800 0.900 0.200 a2
0.900 0.900 0.300 a3
--------------------
Center: c0 Length: 3
Center point: 0.900 0.500 0.450
0.900 0.100 0.600 b1
0.800 0.000 0.500 b2
0.500 0.500 0.500 d1

In the final iteration error drops again, but none of the points have moved clusters, so the program ends.

ITERATION 3
Current error: 1.46750698081
--------------------
Center: c2 Length: 3
Center point: 0.100 0.033 0.100
0.100 0.100 0.100 c1
0.000 0.000 0.200 c2
0.200 0.000 0.000 c3
--------------------
Center: c1 Length: 3
Center point: 0.867 0.867 0.200
0.900 0.800 0.100 a1
0.800 0.900 0.200 a2
0.900 0.900 0.300 a3
--------------------
Center: c0 Length: 3
Center point: 0.733 0.200 0.533
0.900 0.100 0.600 b1
0.800 0.000 0.500 b2
0.500 0.500 0.500 d1
--------------------------------------------------
Centers have stabilized after 3 iterations
Final error: 1.46750698081

Remember that each run of k-means will be different. The algorithm is very sensitive to the initial conditions. The final assignments found in this particular run are the most common, but other assignments will also occur.

Testing k-means on the random data set

With two dimensional data we can plot the movement of the centroids and the assignment of points to clusters. Using the plotCenters method we can watch how the clusters evolve over time. After each iteration a plot will appear. The plot shows the centroids as x's and the members as circles. Each cluster is color coded. You'll need to close the plot to continue to the next iteration.

Below is one run with k=2 using the random.points data that required 8 iterations to stabilize. Notice that the green cluster was initially quite small and over time grew downwards.

Try different different values for k on the random data. Recall that one of the issues with using k-means is that it is not always clear how to appropriately set k. Larger k will reduce error, but may create clusters that are over-fitted to the data.

Be sure to discuss how you chose k when you applied k-means to your own data set.

Explore a data set of your choice

Find a data set that you are interested in exploring using unsupervised learning. Here are some places you can look for inspiration:

Apply your implementations of both hierarchical clustering and k-means to your data set. This may require you to do some pre-processing to put the data into the necessary format.

Write up your findings in the provided LaTeX template lab9.tex. To compile it do: pdflatex lab9.tex. To view the result do: xdg-open lab9.pdf.

Submitting your code

To submit your code, you need to use git to add, commit, and push the files that you modified. Be sure to include the data files you analyzed. Also print out your pdf writeup and hand it in at the start of lab on Tuesday, April 19.

cd ~/cs63/labs/09
git add *.py yourDataFiles
git commit -m "final version"
git push