Due: 10/1 at 11:59pm.

In this assignment you will explore ridge regression applied to the task of predicting wine quality. You will use the wine quality dataset from the UCI machine learning repository, and compare accuracy obtained using ridge regression to the results from a recent publication (if you have trouble accessing that version of the paper, here's a link to a preprint. The wine data is composed of two datasets - one for white wines, and one for reds. Perform all your analyses on both.

Implement ridge regression and functions for computing the following measures of error:

• The Root Mean Square Error (RMSE).
• The Maximum Absolute Deviation (MAD).

For a hypothesis $h$, they are defined as follows:

$$RMSE(h) = \sqrt{\frac{1}{N}\sum_{i=1}^N (y_i - h(\mathbf{x}_i))^2}$$

and

$$MAD(h) = \frac{1}{N}\sum_{i=1}^N |y_i - h(\mathbf{x}_i)|.$$

Compute these measures of error for ridge regression applied to the wine dataset over a range of the regularization parameter, $\lambda$ (choose values on a logarithmic scale, e.g. 0.01, 0.1, 1, 10, 100, 1000) and plot the results (use a fixed test set for computing them!) The features for the wine dataset are not standardized, so make sure you do this, especially since we are going to consider the magnitude of the weight vector (recall that standardization entails subtracting the mean and then dividing by the standard deviation for each feature; you can use the Numpy statistics module to perform the required calculations). What is the potential advantage of MAD over RMSE?

In addition to RMSE and MAD, plot the Regression Error Characteristic (REC) curve of a representative classifier. REC curves are described in the following paper. What can you learn from this curve that you cannot learn from RMSE or MAD?

Compare the results that you are getting with the published results in the paper.

As we discussed in class, the magnitude of the weight vector can be interpreted as a measure of feature importance. Train a ridge regression classifier on a subset of the dataset that you reserved for training. We will explore the relationship between the magnitude of weight vector components and their relevance to the classification task in several ways. Each feature is associated with a component of the weight vector. It can also be associated with the correlation of that feature with the vector of labels. Create a scatter plot of the weight vector component against the Pearson correlation coefficient of a feature against the labels (again, you can use the Numpy statistics module to compute it). What can you conclude from this plot? The paper ranks features according to their importance using a different approach. Compare your results with what they obtain.

Next, perform the following experiment: Incrementally remove the feature with the lowest absolute value of the weight vector and retrain the ridge regression classifier. Plot RMSE and MAD as a function of the number of features that remain on the test set which you have set aside.

Submit your report via Canvas. Python code can be displayed in your report if it is succinct (not more than a page or two at the most) or submitted separately. The latex sample document shows how to display Python code in a latex document. Code needs to be there so we can make sure that you implemented the algorithms and data analysis methodology correctly. Canvas allows you to submit multiple files for an assignment, so DO NOT submit an archive file (tar, zip, etc).

Here is what the grade sheet will look like for this assignment. A few general guidelines for this and future assignments in the course:

• Always provide a description of the method you used to produce a given result in sufficient detail such that the reader can reproduce your results on the basis of the description (UNLESS the method has been provided in class or is there in the book). Your code needs to be provided in sufficient detail so we can make sure that your implementation is correct. The saying that “the devil is in the details” holds true for machine learning, and is sometimes the makes the difference between correct and incorrect results. If your code is more than a few lines, you can include it as an appendix to your report, or submit it as a separate file. Make sure your code is readable!
• You can provide results in the form of tables, figures or text - whatever form is most appropriate for a given problem.
• In any machine learning paper there is a discussion of the results. There is a similar expectation from your assignments that you reason about your results. For example, for the learning curve problem, what can you say on the basis of the observed learning curve?
• Write succinct answers. We will take off points for rambling answers that are not to the point, and and similarly, if we have to wade through a lot of data/results that are not to the point.

Grading sheet for assignment 2

Part 1:  50 points.
(20 points):  Plots of MAD and RMSE as a function of lambda are generated correctly.
(20 points):  REC curves are generated correctly
( 5 points):  discussion of REC curves
( 5 points):  Discussion of the MAD and RMSE plots and comparison of results to the published ones.

Part 2:  40 points.
(30 points):  Weight vector analysis
(10 points):  Comparison to the published results

Report structure, grammar and spelling:  10 points
(10 points):  Heading and subheading structure easy to follow and clearly divides report into logical sections.  
              Code, math, figure captions, and all other aspects of the report are well-written and formatted.
              Grammar, spelling, and punctuation.