Homework #1

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Directions:

• Homework must be completed individually. Any guidance or help received from mentors, classmates, online resources other than course materials (including AI/LLMs) must be acknowledged.
• Clearly organize your responses so that each question (1, 2, 3, 4) and sub-question (A, B, C, etc.) are clearly evident.
• Please submit a single Jupyter notebook displaying all output and recording textual answers using neatly formatted markdown chunks.
• Your submission should be made via Canvas no later than 11:59pm on the assigned due-date.

Question #1 (Weighted Averages and Python Skills)

Consider the weighted average: \[ \bar{x}_w = \frac{\sum_{i=1}^{n} w_i x_i}{\sum_{i=1}^{n}w_i}\]

where \(w_i = e^{\theta x_i}\)

• Part A: Create a Python function that accepts, \(\mathbf{x}\), a vector of numerical values and \(\theta\), a user-specified constant, and returns the weighted average using the definitions given above.
• Part B: Show empirically that as \(\theta\) becomes large the weighted average calculation is dominated by the maximum value in the data.
• Part C: For x = [1,2,3,4,5,6], use an iterative approach to find a value of \(\theta\) where the weighted average is within 0.001 of the data’s maximum value.

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Question #2 (Data preparation and Python skills)

In this question you will use data from an experiment where subjects received various combinations of alcohol (coded P for placebo or M for active alcohol) and cannabis (coded X for placebo, Y for low THC, and Z for high THC) before performing a simulated drive in an advanced simulator. You will need to work with two data sources:

1. The lane departures key, a CSV file found at the URL : https://remiller1450.github.io/data/lane_departures_key.csv that records each instance of a lane departure (any part of the subject’s vehicle going outside the boundaries of their lane).
2. The time-series data for each subject, which is stored in this zipped folder.

• Part A: Create a single pandas dataFrame containing every lane departure from the zipped folder. This data frame should have 27 columns and a large number of rows.
• Part B: Using the lane departures key file, attach dosing information to the dataFrame you created in Part A by noting that each run of the driving simulator is given a unique DAQ name. Your approach should preserve the number of rows in the dataFrame from Part A.
• Part C - Find the maximum absolute lateral distance (the variable lat_dist) of each lane departure using grouped summarization to remove the time component. Hint: each DAQ is a unique identifier of a drive, but not an individual lane departure. The variable instance must be combined with the DAQ to provide a unique identifier.
• Part D - Create a plot displaying side-by-side boxplots of the distribution of maximum absolute lateral distances for each experimental condition (combination of the alcohol and cannabis conditions).

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Question #3 (KNN concepts and practice)

The table below provides a training data set consisting of 6 observations, 3 predictors, and a categorical outcome:

Observation	X1	X2	X3	Y
1	0	3	0	Red
2	2	0	0	Red
3	0	1	3	Red
4	0	1	2	Green
5	-1	0	1	Green
6	1	1	1	Red

Suppose we’re interested in using \(k\)-nearest neighbors to predict the outcome of a test data-point at \(\{X_1=0, X_2=0, X_3=0\}\).

You should answer the following questions using a calculator or basic Python functions. You should not use any functions in sklearn. Additionally, you do not need to perform any standardization/scaling.

• Part A: Calculate the euclidean distance between each observation and the test data-point of \(\{X_1=0, X_2=0, X_3=0\}\).
• Part B: What is the predicted class of \(Y\) for the test data-point if uniform weighting and \(k=1\) (one neighbor) are used? Why?
• Part C: What is the predicted probability that \(Y=\text{Green}\) if uniform weighting and \(k=3\) (three neighbors) are used? Why?
• Part D: Using \(k=3\), will the predicted probability of \(Y=\text{Green}\) be higher if distance weighting is used instead of uniform weighting? Briefly explain (you do not need to perform the calculation).
• Part E: Re-scale each data-point by subtracting that feature’s mean and dividing by its standard deviation (ie: what StandardScaler() does) and re-calculate the distance between the test data-point and each observation. What is the predicted probability of this data-point being “Green” using uniform weighting and three neighbors? How does this compare to Part C?

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Question #4 (Application using sklearn)

For this question you should use the dataset available here:

https://remiller1450.github.io/data/Ozone.csv

These data document daily Ozone concentrations in New York City in 1973. Ozone is a pollutant that is has been linked to numerous health problems. The goal of this application is develop methods for accurately predicting the Ozone concentration on future dates using that date’s expected solar radiation, wind speed, and temperature.

• Part A: Separate the outcome from the predictors (dropping the “Day” column), then perform an 80-20 train-test split using random_state=3. Next, create a pre-processing pipeline that performs standardization before applying a KNN regressor model.
• Part B: Create a sequence of values of \(k\) from 5 to 35 by increments of 5. Calculate the RMSE of the model from Part A on the training data for each of these values of \(k\) and display the results on a line graph showing \(k\) on the x-axis and the training RMSE on the y-axis.
• Part C: Now consider using a decision tree model for these data. Create a sequence of maximum depths from 1 to 10, then calculate the training RMSE of each model and display the results on a line graph showing the tree depth on the x-axis and the training RMSE on the y-axis.
• Part D: Using the graphs from Parts B and C, which model do you think might be most effective on the test set? Or is it difficult to tell? Briefly explain your reasoning.
• Part E: Report the RMSE on the test set for each model considered in Parts B and C. How do these results compare with the model you identified in Part D?
• Part F: Suppose you are asked to recommend one of the two models evaluated in Part E to a city planner who cares about both interpretability of the model’s decisions and its overall performance. Provide a recommendation for one of the two models you evaluated accompanied by a 3-5 sentence justification of the model written in a way that the city planner (who is not familiar with machine learning or algorithms) would understand.