Description
Homework Overview
Data analytics and machine learning both revolve around using computational models to capture relationships
between variables and outcomes. In this assignment, you will code and fit a range of well-known models from
scratch and learn to use a popular Python library for machine learning.
In Q1, you will implement the famous PageRank algorithm from scratch. PageRank can be thought of as a
model for a system in which a person is surfing the web by choosing uniformly at random a link to click on at
each successive webpage they visit. Assuming this is how we surf the web, what is the probability that we
are on a particular webpage at any given moment? The PageRank algorithm assigns values to each webpage
according to this probability distribution.
In Q2, you will implement Random Forests, a very common and widely successful classification model, from
scratch. Random Forest classifiers also describe probability distributions—the conditional probability of a
sample belonging to a particular class given some or all of its features.
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Finally, in Q3, you will use the Python scikit-learn library to specify and fit a variety of supervised and
unsupervised machine learning models.
Q1 [20 pts] Implementation of Page Rank Algorithm
Note: You must use Python 3.7.x for this question.
Technology PageRank Algorithm
Graph
Python 3.7.x
Allowed Libraries NA
Max allowed
runtime
5 minutes
Deliverables [Gradescope]
Q1.py [12 pts]: your modified implementation
simplified_pagerank_iter{n}.txt: 2 files (as given below) containing the top
10 node IDs (w.r.t the PageRank values) and their PageRank values for n
iterations via the simplified_pagerank command
o simplified_pagerank_iter10.txt [2 pts]
o simplified_pagerank_iter25.txt [2 pts]
personalized_pagerank_iter{n}.txt: 2 files (as given below) containing the
top 10 node IDs (w.r.t the PageRank values) and their PageRank values for
n iterations via the personalized_pagerank command
o personalized_pagerank_iter10.txt [2 pts]
o personalized_pagerank_iter25.txt [2 pts]
In this question, you will implement the PageRank algorithm in Python for a large graph network dataset.
The PageRank algorithm was first proposed to rank web pages in search results. The basic assumption is
that more “important” web pages are referenced more often by other pages and thus are ranked higher. The
algorithm works by considering the number and “importance” of links pointing to a page, to estimate how
important that page is. PageRank outputs a probability distribution over all web pages, representing the
likelihood that a person randomly surfing the web (randomly clicking on links) would arrive at those pages.
As mentioned in the lectures, the PageRank values are the entries in the dominant eigenvector of the modified
adjacency matrix in which each column’s values adds up to 1 (i.e., “column normalized”), and this eigenvector
can be calculated by the power iteration method that you will implement in this question, which iterates through
the graph’s edges multiple times to update the nodes’ PageRank values (“pr_values” in pagerank.py) in each
iteration.
For each iteration, the PageRank computation for each node in the network graph is
𝑃𝑅𝑡+1(𝑣𝑗) = (1 − 𝑑) × 𝑃𝑑(𝑣𝑗) + 𝑑 × ∑
𝑃𝑅𝑡
(𝑣𝑖
)
𝑜𝑢𝑡 𝑑𝑒𝑔𝑟𝑒𝑒(𝑣𝑖
)
𝑣𝑖
for each edge (𝑣𝑖
, 𝑣𝑗) from 𝑣𝑖
to 𝑣𝑗
, where
𝑣𝑗
is node 𝑗
𝑣𝑖
is node 𝑖 that points to node 𝑗
𝑜𝑢𝑡 𝑑𝑒𝑔𝑟𝑒𝑒(𝑣𝑖
) is the number of links going out of node 𝑣𝑖
𝑃𝑅𝑡+1(𝑣𝑗) is the pagerank value of node 𝑗 at iteration 𝑡 + 1
𝑃𝑅𝑡(𝑣𝑖) is the pagerank value of node 𝑖 at iteration 𝑡
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𝑑 is the damping factor; set it to the common value of 0.85 that the surfer would continue to follow links
𝑃𝑑(𝑣𝑗) is the probability of random jump that can be personalized based on use cases
Tasks
You will be using the “network.tsv” graph network dataset in the hw4-skeleton/Q1 folder, which contains about
1 million nodes and 3 million edges. Each row in that file represents a directed edge in the graph. The edge’s
source node id is stored in the first column of the file, and the target node id is stored in the second column.
Note: your code must NOT make any assumptions about the relative magnitude between the node ids of an
edge. For example, suppose we find that the source node id is smaller than the target node id for most edges
in a graph, we must NOT assume that this is always the case for all graphs (i.e., in other graphs, a source
node id can be larger than a target node id).
You will complete the code in submission.py (guidelines also provided in the file).
● Step 1: in calculate_node_degree(), calculate and store each node’s out-degree and the graph’s
maximum node id.
A node’s out-degree is its number of outgoing edges. Store the out-degree in class variable
“node_degree”.
max_node_id refers to the highest node id in the graph. For example, suppose a graph
contains the two edges (1,4) and (2,3), in the format of (source,target), the max_node_id
here is 4. Store the maximum node id to class variable max_node_id.
● Step 2: implement run_pagerank()
For simplified PageRank algorithm, where Pd( vj ) = 1/(max_node_id + 1) is provided as
node_weights in the script and you will submit the output for 10 and 25 iteration runs for a
damping factor of 0.85. To verify, we are providing the sample output of 5 iterations for a
simplified PageRank (simplified_pagerank_iter5_sample.txt). For personalized PageRank, the
Pd( ) vector will be assigned values based on your 9-digit GTID (e.g., 987654321) and you will
submit the output for 10 and 25 iteration runs for a damping factor of 0.85.
● The beginning of the main function in submission.py describes how to run the algorithm and
generate output files. Note: When comparing your output for simplified_pagerank for 5 iterations with
the given sample output, the absolute difference must be less than 5%, i.e.,
Absolute((SampleOutput – YourOutput)/SampleOutput) must be less than 0.05.
Q2 [50 pts] Random Forest Classifier
Technology Python 3.7.x
Allowed Libraries Do not modify the import statements; everything you need to complete this question
has been imported for you. You may not use other libraries for this assignment.
Max runtime 300 seconds
Deliverables [Gradescope]
Q2.ipynb [45 pts]: your solution as a Jupyter notebook, developed by
completing the provided skeleton code
o 10 points are awarded for 2 utility functions, 5 points for entropy() and 5 points
for information_gain()
o 35 points are awarded for successfully implementing your random forest
Random Forest Reflection [5 pts]: multiple-choice question completed on
Gradescope.
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Q2.1 – Random Forest Setup [45 pts]
Note: You must use Python 3.7.x for this question.
You will implement a random forest classifier in Python via a Jupyter notebook. The performance of the
classifier will be evaluated via the out-of-bag (OOB) error estimate using the provided dataset pimaindians-diabetes.csv, a comma-separated (csv) file in the Q2 folder. The dataset was derived from the
National Institute of Diabetes and Digestive and Kidney Diseases. You must not modify the dataset. Each
row describes one person (a data point, or data record) using 9 columns. The first 8 are attributes. The 9th is
the label, and you must NOT treat it as an attribute. You will perform binary classification on the dataset to
determine if a person has diabetes.
Important:
1. Remove all “testing” code that renders output, or Gradescope will crash. For instance, any additional
print, display, and show statements used for debugging must be removed.
2. You may only use the modules and libraries provided at the top of the notebook file included in the
skeleton for Q2 and modules from the Python Standard Library. Python wrappers (or modules) must
NOT be used for this assignment. Pandas must NOT be used — while we understand that they are
useful libraries to learn, completing this question is not critically dependent on their functionality. In
addition, to make grading more manageable and to enable our TAs to provide better, more consistent
support to our students, we have decided to restrict the libraries accordingly.
Essential Reading
Decision Trees. To complete this question, you will develop a good understanding of how decision trees
work. We recommend that you review the lecture on the decision tree. Specifically, review how to construct
decision trees using Entropy and Information Gain to select the splitting attribute and split point for the
selected attribute. These slides from CMU (also mentioned in the lecture) provide an excellent example of
how to construct a decision tree using Entropy and Information Gain. Note: there is a typo on page 10,
containing the Entropy equation; ignore one negative sign (only one negative sign is needed).
Random Forests. To refresh your memory about random forests, see Chapter 15 in the Elements of
Statistical Learning book and the lecture on random forests. Here is a blog post that introduces random
forests in a fun way, in layman’s terms.
Out-of-Bag Error Estimate. In random forests, it is not necessary to perform explicit cross-validation or use
a separate test set for performance evaluation. Out-of-bag (OOB) error estimate has shown to be
reasonably accurate and unbiased. Below, we summarize the key points about OOB in the original article
by Breiman and Cutler.
Each tree in the forest is constructed using a different bootstrap sample from the original data. Each bootstrap
sample is constructed by randomly sampling from the original dataset with replacement (usually, a bootstrap
sample has the same size as the original dataset). Statistically, about one-third of the data records (or data
points) are left out of the bootstrap sample and not used in the construction of the kth tree. For each data
record that is not used in the construction of the kth tree, it can be classified by the kth tree. As a result, each
record will have a “test set” classification by the subset of trees that treat the record as an out-of-bag sample.
The majority vote for that record will be its predicted class. The proportion of times that a record’s predicted
class is different from the true class, averaged over all such records, is the OOB error estimate.
While splitting a tree node, make sure to randomly select a subset of attributes (e.g., square root of the number
of attributes) and pick the best splitting attribute (and splitting point of that attribute) among these subsets of
attributes. This randomization is the main difference between random forest and bagging decision trees.
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Starter Code
We have prepared some Python starter code to help you load the data and evaluate your model. The starter
file name is Q2.ipynb has three classes:
● Utililty: contains utility functions that help you build a decision tree
● DecisionTree: a decision tree class that you will use to build your random forest
● RandomForest: a random forest class
What you will implement
Below, we have summarized what you will implement to solve this question. Note that you must use
information gain to perform the splitting in the decision tree. The starter code has detailed comments on
how to implement each function.
1. Utililty class: implement the functions to compute entropy, information gain, perform splitting,
and find the best variable (attribute) and split-point. You can add additional methods for convenience.
Note: Do not round the output or any of your functions.
2. DecisionTree class: implement the learn() method to build your decision tree using the utility
functions above.
3. DecisionTree class: implement the classify() method to predict the label of a test record
using your decision tree.
4. RandomForest class: implement the methods _bootstrapping(), fitting(), voting() and
user().
5. get_random_seed(), get_forest_size(): implement the functions to return a random seed
and forest size (number of decision trees) for your implementation.
Important:
1. You must achieve a minimum accuracy of 70% for the random forest.
2. Your code must take no more than 5 minutes to execute (which is a very long time, given the low
program complexity). Otherwise, it may time out on Gradescope. Code that takes longer than 5
minutes to run likely means you need to correct inefficiencies (or incorrect logic) in your program. We
suggest that you check the hyperparameter choices (e.g., tree depth, number of trees) and code logic
when figuring out how to reduce runtime.
3. The run()function is provided to test your random forest implementation; do NOT modify this
function.
As you solve this question, consider the following design choices. Some may be more straightforward to
determine, while some maybe not (hint: study lecture materials and essential reading above). For example:
● Which attributes to use when building a tree?
● How to determine the split point for an attribute?
● How many trees should the forest contain?
● You may implement your decision tree using a data structure of your choice (e.g., dictionary, list,
class member variables). However, your implementation must still work within the DecisionTree
Class Structure we have provided.
● Your decision tree will be initialized using DecisionTree(max_depth=10), in the
RandomForest class in the jupyter notebook.
● When do you stop splitting leaf nodes?
● The depth found in the learn function is the depth of the current node/tree. You may want a check
within the learn function that looks at the current depth and returns if the depth is greater than or
equal to the max depth specified. Otherwise it is possible that you continually split on nodes and
create a messy tree. The max_depth parameter should be used as a stopping condition for when
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your tree should stop growing. Your decision tree will be instantiated with a depth of 0 (input to the
learn() function in the jupyter notebook). To comply with this, make sure you implement the decision
tree such that the root node starts at a depth of 0 and is built with increasing depth.
Note that, as mentioned in the lecture, there are other approaches to implement random forests. For example,
instead of information gain, other popular choices include the Gini index, random attribute selection (e.g.,
PERT – Perfect Random Tree Ensembles). We decided to ask everyone to use an information gain based
approach in this question (instead of leaving it open-ended), to help standardize students’ solutions to help
accelerate our grading efforts.
Q2.2 – Random Forest Reflection [5 pts]
On Gradescope, answer the following question:
What is the main reason to use a random forest versus a decision tree?
Q3 [30 points] Using Scikit-Learn
Technology Python 3.7.x
Scikit-Learn 0.22
Allowed Libraries Do not modify the import statements; everything you need to complete this question
has been imported for you. You may not use other libraries for this assignment.
Max runtime 15 minutes
Deliverables [Gradescope] Q3.ipynb [30 pts]: your solution as a Jupyter notebook, developed by
completing the provided skeleton code
Scikit-learn is a popular Python library for machine learning. You will use it to train some classifiers to
predict diabetes in the Pima Indian tribe. The dataset is provided in the Q3 folder as pima-indiansdiabetes.csv.
For this problem, you will be utilizing a Jupyter notebook.
Important:
1. Remove all “testing” code that renders output, or Gradescope will crash. For instance, any additional
print, display, and show statements used for debugging must be removed.
2. Use the default values while calling functions unless specific values are given.
3. Do not round off the results except the results obtained for Linear Regression Classifier.
Q3.1 – Data Import [2 pts]
In this step, you will import the pima-indians-diabetes dataset and allocate the data to two separate arrays.
After importing the data set, you will split the data into a training and test set using the scikit-learn function
train_test_split. You will use scikit-learns built-in machine learning algorithms to predict the accuracy of
training and test set separately. Refer to the hyperlinks provided below for each algorithm for more details,
such as the concepts behind these classifiers and how to implement them.
Q3.2 – Linear Regression Classifier [4 pts]
Q3.2.1 – Classification
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Train the Linear Regression classifier on the dataset. You will provide the accuracy for both the test
and train sets. Make sure that you round your predictions to a binary value of 0 or 1. See the Jupyter
notebook for more information. Linear regression is most commonly used to solve regression
problems. The exercise here demonstrates the possibility of using linear regression for classification
(even though it may not be the optimal model choice).
Q3.3 – Random Forest Classifier [10 pts]
Q3.3.1 – Classification
Train the Random Forest classifier on the dataset. You will provide the accuracy for both the test
and train sets. Do not round your prediction.
Q3.3.2 – Feature Importance
You have performed a simple classification task using the random forest algorithm. You have also
implemented the algorithm in Q2 above. The concept of entropy gain can also be used to evaluate
the importance of a feature. You will determine the feature importance evaluated by the random
forest classifier in this section. Sort the features in descending order of feature importance score,
and print the sorted features’ numbers.
Hint: There is a function available in sklearn to achieve this. Also, take a look at argsort()
function in Python numpy. argsort()returns the indices of the elements in ascending order. You
will use the random forest classifier that you trained initially in Q3.3.1, without any kind of
hyperparameter-tuning, for reporting these features.
Q3.3.3 – Hyper-Parameter Tuning
Tune your random forest hyper-parameters to obtain the highest accuracy possible on the dataset.
Finally, train the model on the dataset using the tuned hyper-parameters. Tune the hyperparameters
specified below, using the GridSearchCV function in Scikit library:
‘n_estimators’: [4, 16, 256], ’max_depth’: [2, 8, 16]
Q3.4 – Support Vector Machine [10 pts] –
Q3.4.1 – Preprocessing
For SVM, we will standardize attributes (features) in the dataset using StandardScaler, before
training the model.
Note: for StandardScaler,
● Transform both x_train and x_test to obtain the standardized versions of both.
● Review the StandardScaler documentation, which provides details about standardization and
how to implement it.
Q3.4.2 – Classification
Train the Support Vector Machine classifier on the dataset (the link points to SVC, a particular
implementation of SVM by Scikit). You will provide the accuracy on both the test and train sets.
Q3.4.3. – Hyper-Parameter Tuning
Tune your SVM model to obtain the highest accuracy possible on the dataset. For SVM, tune the
model on the standardized train dataset and evaluate the tuned model with the test dataset. Tune
the hyperparameters specified below, using the GridSearchCV function in Scikit library:
‘kernel’:(‘linear’, ‘rbf’), ‘C’:[0.01, 0.1, 1.0]
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Note: If GridSearchCV takes a long time to run for SVM, make sure you standardize your data
beforehand using StandardScaler.
Q3.4.4. – Cross-Validation Results
Let’s practice obtaining the results of cross-validation for the SVM model. Report the rank test score
and mean testing score for the best combination of hyper-parameter values that you obtained. The
GridSearchCV class holds a cv_results_ dictionary that helps you report these metrics easily.
Q3.5 – Principal Component Analysis [4 pts]
Performing Principal Component Analysis based dimensionality reduction is a common task in many data
analysis tasks, and it involves projecting the data to a lower-dimensional space using Singular Value
Decomposition. Refer to the examples given here; set parameters n_component to 8 and svd_solver to
full. See the sample outputs below.
1. Percentage of variance explained by each of the selected components. Sample Output:
[6.51153033e-01 5.21914311e-02 2.11562330e-02 5.15967655e-03
6.23717966e-03 4.43578490e-04 9.77570944e-05 7.87968645e-06]
2. The singular values corresponding to each of the selected components. Sample Output:
[5673.123456 4532.123456 4321.68022725 1500.47665361
1250.123456 750.123456 100.123456 30.123456]
Use the Jupyter notebook skeleton file called Q3.ipynb to write and execute your code.
As a reminder, the general flow of your machine learning code will look like:
1. Load dataset
2. Preprocess (you will do this in Q3.2)
3. Split the data into x_train, y_train, x_test, y_test
4. Train the classifier on x_train and y_train
5. Predict on x_test
6. Evaluate testing accuracy by comparing the predictions from step 5 with y_test.
Here is an example. Scikit has many other examples as well that you can learn from.
