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Spring 2022: Int. to Statistical Machine Learning
发布时间：2022-02-09
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Problem Set - 3
Please read all of the guidelines carefully before submitting the problem set. (Unless specified) each question is 20 points and there are 100 points in total.
Due date: Friday, February 11, 11:59 PM. Late submissions will be accepted with a penalty!
(10% reduction per day – no submissions accepted two days after the deadline .)
Guidelines – Before You Start
1) You should complete the problem set on your own. Discussing ideas is fine; but, sharing answers and sharing code will be considered as plagiarism.
2) You will be using the Python programming language. You need to write your codes in an empty .ipynb file.
3) Make sure that you provide many comments to describe your code and the variables that you created.
4) Please use LaTeX or MS Word to submit your written responses (hand-written responses will not be graded).
5) For some of the coding exercises, you may need to do a little bit of “Googling” or review the documentation.
Deliverables:
1) The code of the problem set in .ipynb format (one file)
2) Short answers written with LaTeX or MS Word and exported in .pdf format (one file)
Questions
In this homework, you have three questions. The first question is worth 20 points. The remaining two questions are each worth 40 points.
1) Read the following highly-cited article by Fearon and Laitin (2003):
https://cisac.fsi.stanford.edu/publications/ethnicity_insurgency_and_civil_war
Answer the following:
a. What is the paper about? (Please write a paragraph – max. 250 words)
b. How many observations do authors have in their dataset? What does each observation represent? (=What is the unit of analysis?)
c. What is the identification strategy of the authors? (=What are the different regression equations they are running?) Please write down in form of equations and explain. Identify the independent and dependent variables.
d. What do the coefficient values listed in Table 1 represent? (theoretically speaking)
e. Which independent variables have positive coefficients? Which independent variables have negative coefficients? Which ones are statistically significant?
f. Thinking about the range of your independent variables, which variables do you think have a greater impact on the dependent variable(s)?
2) Build a two-class logistic regression model from scratch. You will need to work on the following:
a. Implement the sigmoid function from scratch and call it sigmoid_f
b. Implement the hypothesis function from scratch and call it classifier_f
c. Implement the entropy function as your cost function and call it binary_loss_f
d. Implement gradient descent for logistic regression and call it gradient_f
e. Combining the functionalities of what you have coded above, create an optimizer function and call it optimizer_f. Note: You should find out the input and output to the functions above by reviewing the class notes and the textbook; in other words, this will be part of the challenge! If needed, use 265as your random seed.
Let’s test your code on a dataset. Load the Breast Cancer Wisconsin Dataset provided
by sklearn:https://scikit- learn.org/stable/modules/generated/sklearn.datasets.load_breast_cancer.html#sklearn .datasets.load_breast_cancer
Now, do the following:
a. Set the target column as your Yvariable.
b. Set all other numeric variables (excluding index) as your X matrix.
c. Apply 0- 1 normalization on both the X matrix and Y vector.
d. Run logistic regression by using the code you have written (no need to do train/test split). Set the maximum number of iterations to 10,000.
e. Report the final equation you have obtained for logistic regression.
f. Also indicate which coefficients are positively associated and which coefficients are negatively associated with the target variable. Rank them from positive to negative. Interpret the results.
3) Implement the three following cross-validation algorithms from scratch:
a. Leave-one-out cross-validation
b. K-fold cross-validation
c. Train-test split cross-validation
Test your results on the California Housing Dataset:
https://scikit- learn.org/stable/modules/generated/sklearn.datasets.fetch_california_housing.html#skl earn.datasets.fetch_california_housing
Now, do the following:
a. Implement the cross-validation algorithms from scratch.
b. Choose the following features from the dataset as your X matrix: MedInc, HouseAge, AveRooms, AveBedrms, Population, AveOccup, Latitude, Longitude
c. Choose the following feature from the dataset as your Y matrix: MedHouseVal
d. Apply 0 – 1 normalization on X and Y.
e. Apply the cross-validation algorithms that you implemented to train your model. (For splitting your data always use 265 as your random number or seed value) . Note: You cannot use the pre-packaged algorithms for splitting the data. To split the data please do the following:
a. Install the random package written for Python
b. Set (the initial) random.seed() to 265
c. Create a list of integers that will function as your index numbers: list(range(0,len(name_of_dataset))
d. Pick one integer for Train-Test Split CV from the list your created in c) to split 70% of your data as training set and the remaining 30% as the test set.
e. For the K-fold CV, set k = 5. Please divide the dataset into 5 quasi-equal portions starting from index 0.
f. For LOOCV, start the training by randomly picking a feature vector associated with an index in your dataset (Reminder: random seed is 265)
– you will need to run the model on every point.
f. Using scikit’s sklearn.linear_model.LinearRegression, predict the house prices by using all of the data in your X matrix. Compare different techniques of CV. Which CV provides the lowest MSE? Why? Interpret the results.
