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# CPSC 340 Assignment 1 solution

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CPSC 340 Assignment 1
Instructions
Rubric: {mechanics:5}
Please pay special attention to the section on visualizations as there were no visualizations required in a0.
A note on the provided code: in the code directory we provide you with a file called main.py. This file,
when run with different arguments, runs the code for different parts of the assignment. For example,
python main.py -q 1.1

Category:

## Description

CPSC 340 Assignment 1
Instructions
Rubric: {mechanics:5}
Please pay special attention to the section on visualizations as there were no visualizations required in a0.
A note on the provided code: in the code directory we provide you with a file called main.py. This file,
when run with different arguments, runs the code for different parts of the assignment. For example,
python main.py -q 1.1
runs the code for Question 1.1. At present, this should do nothing, because the code for Question 1.1 still
needs to be written (by you). But we do provide some of the bits and pieces to save you time, so that you can
focus on the machine learning aspects. The file utils.py contains some helper functions. You’re not required
to read/understand the code in there (although you’re welcome to!) and will not need to modify it.
1 Data Exploration
Your repository contains the file fluTrends.csv, which contains estimates of the influenza-like illness percentage over 52 weeks on 2005-06 by Google Flu Trends. Your main.py loads this data for you and stores it in
a pandas DataFrame X, where each row corresponds to a week and each column corresponds to a different
region. If desired, you can convert from a DataFrame to a raw numpy array with X.values().
1.1 Summary Statistics
Rubric: {reasoning:2}
Report the following statistics: The minimum, maximum, mean, median, and mode of all values across the
dataset.
1. The 5%, 25%, 50%, 75%, and 95% quantiles of all values across the dataset.
2. The names of the regions with the highest and lowest means, and the highest and lowest variances.
In light of the above, is the mode a reliable estimate of the most “common” value? Describe another way we
could give a meaningful “mode” measurement for this (continuous) data. Note: the function utils.mode()
will compute the mode value of an array for you.
1.2 Data Visualization
Rubric: {reasoning:3}
Consider the figure below.
1
0 10 20 30 40 50
Weeks
1
2
3
4
5
Illness percentage
Plot A
NE
MidAtl
ENCentral
WNCentral
SAtl
ESCentral
WSCentral
Mtn
Pac
WtdILI
1 6 11 16 21 26 31 36 41 46 51
Weeks
1
2
3
4
5
Illness percentage
Plot B
1 2 3 4 5
Percentages
0
20
40
60
80
100
120
140
160
Frequency
Plot C
1 2 3 4 5
Percentages
0.0
2.5
5.0
7.5
10.0
12.5
15.0
17.5
20.0
Frequency
Plot D
NE
MidAtl
ENCentral
WNCentral
SAtl
ESCentral
WSCentral
Mtn
Pac
WtdILI
0.5 1.0 1.5 2.0
NE
1
2
3
4
5
Mtn
Plot E
0.5 1.0 1.5 2.0
MidAtl
0.5
1.0
1.5
2.0
2.5
ENCentral
Plot F
2
The figure contains the following plots, in a shuffled order:
1. A single histogram showing the distribution of each column in X.
2. A histogram showing the distribution of each the values in the matrix X.
3. A boxplot grouping data by weeks, showing the distribution across regions for each week.
4. A plot showing the illness percentages over time.
5. A scatterplot between the two regions with highest correlation.
6. A scatterplot between the two regions with lowest correlation.
Match the plots (labeled A-F) with the descriptions above (labeled 1-6), with an extremely brief (a few
words is fine) explanation for each decision.
2 Decision Trees
If you run python main.py -q 2, it will load a dataset containing longtitude and latitude data for 400 cities
in the US, along with a class label indicating whether they were a “red” state or a “blue” state in the 2012
election.1 Specifically, the first column of the variable X contains the longitude and the second variable
contains the latitude, while the variable y is set to 1 for blue states and 2 for red states. After it loads the
data, it plots the data and then fits two simple classifiers: a classifier that always predicts the most common
label (1 in this case) and a decision stump that discretizes the features (by rounding to the nearest integer)
and then finds the best equality-based rule (i.e., check if a feature is equal to some value). It reports the
training error with these two classifiers, then plots the decision areas made by the decision stump.
2.1 Splitting rule
Rubric: {reasoning:1}
Is there a particular type of features for which it makes sense to use an equality-based splitting rule rather
than the threshold-based splits we discussed in class?
2.2 Decision Stump Implementation
Rubric: {code:3}
The file decision stump.py contains the class DecisionStumpEquality which finds the best decision stump
using the equality rule and then makes predictions using that rule. Instead of discretizing the data and using
a rule based on testing an equality for a single feature, we want to check whether a feature is above or below
a threshold and split the data accordingly (this is the more sane approach, which we discussed in class).
Create a DecisionStump class to do this, and report the updated error you obtain by using inequalities
instead of discretizing and testing equality.
Hint: you may want to start by copy/pasting the contents DecisionStumpEquality and then make modifications from there.
1The cities data was sampled from http://simplemaps.com/static/demos/resources/us-cities/cities.csv. The election
information was collected from Wikipedia.
3
2.3 Constructing Decision Trees
Rubric: {code:2}
Once your decision stump class is finished, running python main.py -q 2.3 will fit a decision tree of
depth 2 to the same dataset (which results in a lower training error). Look at how the decision tree is stored
and how the (recursive) predict function works. Using the same splits as the fitted depth-2 decision tree,
write an alternative version of the predict function for classifying one training example as a simple program
using if/else statements (as in slide 6 of the Decision Trees lecture). Save your code in a new file called
Note: this code should implement the specific, fixed decision tree which was learned after calling fit on this
particular data set. It does not need to be a learnable model.
2.4 Decision Tree Training Error
Rubric: {reasoning:2}
Running python main.py -q 2.4 fits decision trees of different depths using two different implementations:
first, our own implementation using your DecisionStump, and second, the decision tree implementation from
the popular Python ML library scikit-learn. The decision tree from sklearn uses a more sophisticated splitting
criterion called the information gain, instead of just the classification accuracy. Run the code and look at
the figure. Describe what you observe. Can you explain the results?
Note: we set the random_state because sklearn’s DecisionTreeClassifier is non-deterministic. I’m guessing
this is because it breaks ties randomly.
Note: the code also prints out the amount of time spent. You’ll notice that sklearn’s implementation is
substantially faster. This is because our implementation is based on the O(n
2d) decision stump learning
algorithm and sklearn’s implementation presumably uses the faster O(nd log n) decision stump learning
algorithm that we discussed in lecture.
2.5 Cost of Fitting Decision Trees
Rubric: {reasoning:3}
In class, we discussed how in general the decision stump minimizing the classification error can be found in
O(nd log n) time. Using the greedy recursive splitting procedure, what is the total cost of fitting a decision
tree of depth m in terms of n, d, and m?
Hint: even thought there could be (2m − 1) decision stumps, keep in mind not every stump will need to go
through every example. Note also that we stop growing the decision tree if a node has no examples, so we
may not even need to do anything for many of the (2m − 1) decision stumps.
3 Training and Testing
If you run python main.py -q 3, it will load citiesSmall.pkl. Note that this file contains not only training
data, but also test data, X_test and y_test. After training a depth-2 decision tree it will evaluate the
performance of the classifier on the test data.2 With a depth-2 decision tree, the training and test error are
fairly close, so the model hasn’t overfit much.
2The code uses the ”information gain” splitting criterion; see the Decision Trees bonus slides for more information.
4
3.1 Training and Testing Error Curves
Rubric: {reasoning:2}
Make a plot that contains the training error and testing error as you vary the depth from 1 through 15. How
do each of these errors change with the decision tree depth?
Note: it’s OK to reuse the provided code from part 2.4 as a starting point.
3.2 Validation Set
Rubric: {reasoning:3}
Suppose that we didn’t have an explicit test set available. In this case, we might instead use a validation
set. Split the training set into two equal-sized parts: use the first n/2 examples as a training set and the
second n/2 examples as a validation set (we’re assuming that the examples are already in a random order).
What depth of decision tree would we pick to minimize the validation set error? Does the answer change
if you switch the training and validation set? How could use more of our data to estimate the depth more
reliably?
4 K-Nearest Neighbours
In the citiesSmall dataset, nearby points tend to receive the same class label because they are part of the
same U.S. state. This indicates that a k-nearest neighbours classifier might be a better choice than a decision
tree. The file knn.py has implemented the training function for a k-nearest neighbour classifier (which is to
just memorize the data).
4.1 KNN Prediction
Rubric: {code:3, reasoning:4}
Fill in the predict function in knn.py so that the model file implements the k-nearest neighbour prediction
rule. You should Euclidean distance, and may numpy’s sort and/or argsort functions useful. You can also
use utils.euclidean dist squared, which computes the squared Euclidean distances between all pairs of points
in two matrices.
1. Write the predict function.
2. Report the training and test error obtained on the citiesSmall dataset for k = 1, k = 3, and k = 10.
How do these numbers compare to what you got with the decision tree?
3. Hand in the plot generated by utils.plotClassifier on the citiesSmall dataset for k = 1, using both your
implementation of KNN and the KNeighborsClassifier from scikit-learn.
4. Why is the training error 0 for k = 1?
5. If you didn’t have an explicit test set, how would you choose k?
4.2 Condensed Nearest Neighbours
Rubric: {code:3, reasoning:5}
5
The dataset citiesBig1 contains a version of this dataset with more than 30 times as many cities. KNN
can obtain a lower test error if it’s trained on this dataset, but the prediction time will be very slow. A
common strategy for applying KNN to huge datasets is called condensed nearest neighbours, and the main
idea is to only store a subset of the training examples (and to only compare to these examples when making
predictions). If the subset is small, then prediction will be faster.
The most common variation of this algorithm works as follows:
initialize subset with first training example;
for each subsequent training example do
if the example is incorrectly classified by the KNN classifier using the current subset then
add the current example to the subset;
else
do not add the current example to the subset (do nothing);
end
end
Algorithm 1: Condensed Nearest Neighbours
You are provided with an implementation of this condensed nearest neighbours algorithm in knn.py.
1. The point of this algorithm is to be faster than KNN. Try running the condensed NN on the
citiesBig1 dataset and report how long it takes to make a prediction. What about if you try to use
KNN for this dataset – how long did it take before you panicked and went for CTRL-C… or did it
actually finish?
2. Report the training and testing errors for condensed NN, as well as the number of variables in the
subset, on the citiesBig1 dataset with k = 1.
3. Hand in the plot generated by utils.plotClassifier on the citiesBig1 dataset for k = 1.
4. Why is the training error with k = 1 now greater than 0?
5. If you have s examples in the subset, what is the cost of running the predict function on t test examples
in terms of n, d, t, and s?
6. Try out your function on the dataset citiesBig2. Why are the test error and training error so high
(even for k = 1) for this method on this dataset?
7. Try running a decision tree on citiesBig1 using scikit-learn’s DecisionTreeClassifier with default
hyperparameters. Does it work? Is it fast? Any other thoughts? Overall, do you prefer decicison trees
of (condensed) KNN for this data set? Include the usual plot of the decision surface.
Rubric: {reasoning:8}
Write a short one or two sentence answer to each of the questions below. Make sure your answer is clear
and concise.
1. What is one reason we would want to look at scatterplots of the data before doing supervised learning?
2. What is a reason that the examples in a training and test set might not be IID?
3. What is the difference between a validation set and a test set?
4. What is the main advantage of non-parametric models?
6
5. A standard pre-processing step is “standardization” of the features: for each column of X we subtract
its mean and divide by its variance. Would this pre-processing change the accuracy of a decision tree
classifier? Would it change the accuracy of a KNN classifier?
6. Does increasing k in KNN affect the training or prediction asymptotic runtimes?
7. How does increase the parameter k in k-nearest neighbours affect the two parts (training error and
approximation error) of the fundamental trade-off (hint: think of the extreme values).
8. For any parametric model, how does increasing number of training examples n affect the two parts of