# CS 534: Homework #5 SOLUTION

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CS 534: Homework #5
Submission Instructions: The homework is due on Nov 17th at 11:59 PM ET on Gradescope. A
will support Python 3.10. Additional packages and their versions can be found in the
requirements.txt. Please be aware that the use of other packages and/or versions outside of
those in the file may cause your homework to fail some test cases due to incompatible method calls
or the inability to import the module. We have split homework 1 into 2 parts on Gradescope, the
autograded portion and the written answer portion. If either of the two parts is late, then your
homework is late.
1. (5+3+2=10 pts) (Illustrating the “curse of dimensionality”)
(Written) For a hypersphere of radius a in d dimensions, the volume is related to the surface
area of a unit hypersphere (S) as
V =
S × a
d
d
.
(a) Use this result to show (via a formal mathematical proof) that the fraction of the volume
that lies at values of the radius between a − ϵ and a, where 0 < ϵ < a, is given by
f = 1 − (1 − ϵ/a)
d
. Hence, show that for any fixed ϵ, no matter how small, this fraction
tends to 1 as d → ∞.
(b) Evaluate the ratio f numerically by plotting the results for different values of ϵ/a = 0.01
and d = 1, 10, 100, and 1000.
(c) What conclusions can you draw from the plot?
2. (2+3+2+5+2+5+4+3+5+5+3=39 pts) Preprocessing Loan Defaults Using PCA
& NMF
Use the sklearn.decomposition package to perform dimensionality reduction on the
loan default.csv problem from Homework #3. The functions specified should be in the file
‘q2.py’
(a) (Written) How did you partition the loan data to assess the model performance and
choose the hyperparameter (model selection)? This can be repeated from Homework
#3 and Homework #4. You might find it useful to read all of this homework to decide
if you want to change your assessment strategy.
(b) (Written) Pre-process the loan default.csv data from Homework #3 to deal with
categorical data. Specify how you are specifically converting the non-numeric data.
(c) (Code) Write the function preprocess which takes in a pandas dataframe (a direct
result from pd.read csv(loan default.csv)) and outputs a 2D numpy array.
(d) (Written) Based on your data from 2c, your model assessment strategy in (a), and your
functions from Homework #3, perform feature selection to determine which variables
you plan to keep. You should provide the rationale for your feature selection process. You
may find it useful to save the train/test data as a separate file (e.g., loan fs train.csv,
loan fs test.csv) as you will be analyzing its performance below.
(e) (Written) Discuss why you should normalize the data before performing PCA.
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(f) (Code) Write a function run pca(train x, test x) which will take 2 inputs, train x
and test x, both of which are 2D numpy data following the format of 2c, normalizes
both numpy data, determines the number of components to capture 95% variance of
the normalized training data, and returns the principal components and the loadings.
Note that the normalization should be learned from the training data and applied to the
test data. The function should output a tuple in the following order (1) the number of
components as an integer, (2) the principal components as a numpy 2d array of shape
(components × features), (3) the train loadings as a numpy 2d array of shape train
samples × components, and (4) the test loadings as a numpy 2d array of shape test
samples × components.
(g) (Written) Using the PCA function you wrote above in 2f, report how many components
were needed to capture 95% of the variance in the normalized data. Discuss
what characterizes the first 3 principal components (i.e., which original features are
important).
(h) (Written) Using the PCA function you wrote above in 2f, transform your train and test
data using PCA. You might find it useful to save the train/test data as a separate file
(e.g., loan pca train.csv, loan pca test.csv)
(i) (Code) Write a function run nmf(train x, test x, k) which will take 3 inputs.
train x and test x are 2D numpy data following the format of 2c, and k is an integer.
The function should output a tuple in the following order (1) the reconstruction error (a
float), (2) the factors components as a numpy 2d array of shape (k × features), (3) the
as a numpy 2d array of shape test samples × k.
(j) (Written) Using the function above, 2i, determine the optimal number of components.
You should justify your choice via a plot of the reconstruction error as a function of k.
Discuss what characterizes the first 3 factors (i.e., which original features are important).
(k) (Written) Using the NMF function you wrote above in 2i, transform your train and test
data using NMF and the optimal k you found from the previous question. You might
find it useful to save the train/test data as a separate file (e.g., loan nmf train.csv,
loan nmf test.csv)
3. (3+4+6+7+3+20+8=51 pts) Model Bake-off for Predicting Loan Defaults
We will consider many of the models we learned in conjunction with feature selection and
dimensionality reduction and determine the ‘best’ model for this dataset. The functions
specified should be in the file ‘q3.py’. All the models will be evaluated in terms of AUC, F1,
and F2. Note that the input for each of the following functions below should be 4 numpy
arrays train x, train y, test x, and test y. Both train x and train y will be numpy
2d arrays that are obtained from 2b, 2d, 2h, and 2k. train y and test y are 1d numpy
arrays associated with the features. The output for each of the following function should
be a dictionary with the following keys: ‘train-auc’, ‘train-f1’, ‘train-f2’, ‘val-auc’, ‘val-f1’,
‘val-f2’, ‘test-auc’, ‘test-f1’, ‘test-f2’, and ‘params’. Note that train and validation should be
determined according to your model selection strategy (i.e., if it is k-fold CV it should be the
average) and the test metrics should be calculated using all the training data. The ‘params’
key should contain as the value the dictionary that is passed into a hyper-parameter optimizer
within sklearn.model selection such as GridSearchCV. Your dictionary can also include
other things that might be useful for the other parts of the questions.
2
(a) (Code) Write a function build logr that builds an unregularized logistic regression
model. Note that the value in ‘params’ can be an empty dictionary, it just needs to exist
(b) (Code) Write a function build dt that builds a decision tree accordingly. You might
find it useful to base your parameter space on Homework #3 results. Note that the
value ‘params’ should at least contain the parameters that were tested in Homework #3
(i.e., max depth and min samples leaf).
(c) (Code) Write a function build rf that builds a random forest accordingly. You might
find it useful to limit your parameter space to consider only the number of estimators,
max depth, and minimum number of samples per leaf.
(d) (Code) Write a function build svm that will build a kernel-based SVM. You should
only consider linear and polynomial (of a small degree) kernel as RBF will be extremely
expensive.
(e) (Code) Write a function build nn that builds a neuarl network accordingly. You might
find it useful to base your parameter space on Homework #4 results.
(f) (Written) For each of these 5 models above, build a model for each of the 4 datasets
(2b, 2d, 2h, 2k and report the results in a Table (you should have 20 model+dataset
combinations across the 3 metrics). Note that for the original dataset, you may find it
useful to preprocess it using Standardization or MinMaxScaling.
(g) (Written) Using the results from 3f, compare and contrast the various models with
respect to interpretability, computation time, and predictive performance. If you were
the loan officer using this model, which one would you use to guide whether or not to
give a loan?
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