Analyzing Movie Ratings via SVD

• 25 million ratings
• from 162,000 users
• on 62,000 movies

• 7.1 million ratings
• from 9,663 users
• on 3,790 movies

1. To work with a different type of data than images or temperatures (here we will be working with ratings). Applying the tools you have learned in this module to different domains will help solidify your learning, help you see connections, and potentially get you excited for your module 1 project.
2. To see how SVD can be used to examine the important trends in your data (since we had lots of practice with using the EVD on the overnight).
3. To have some fun!

load('movielens25m.mat');

sizeOfMovies = size(movies)

sizeOfMovies = 1×2

3790 3

% the cell array `movies` is 3706 by 3. Each of the 3706 entries correspopnds to a

% particular movie, and along the second dimension the entries correspond to the movie ID,

% the movie title, and the movie genre

%

% Here we extract the information about the first movie in the dataset

[movieId, movieTitle, movieGenre] = movies{1,:}

movieId = int64 1

movieTitle = 'Toy Story (1995)'

movieGenre = 'Adventure|Animation|Children|Comedy|Fantasy'

ratingsSize = size(ratings)

ratingsSize = 1×2

9663 3790

% the matrix `ratings` is 6040 by 3706 and encodes the rating that a

% particular user (row) gave to a particular movie (column). The ratings

% are 1, 2, 3, 4, or 5 stars or the special value NaN (not a number) if the

% user didn't rate that particular movie.

%

% Let's look at the ratings that were given to the first movie in the

% dataset, which as we saw is Toy Story. We can do this using the histc

% function (we'll ignore missing values in this analysis)

possibleRatings = [0.5:0.5:5];

nRatings = histc(ratings(1,:), possibleRatings);

figure;

bar(possibleRatings, nRatings);

xlabel('Rating');

ylabel('Number of Users');

title(['Ratings for ', movieTitle])

anacondaIndex = 850;

[movieId, movieTitle, movieGenre] = movies{anacondaIndex,:}

movieId = int64 1499

movieTitle = 'Anaconda (1997)'

movieGenre = 'Action|Adventure|Thriller'

nRatings = histc(ratings(anacondaIndex,:), possibleRatings);

figure;

bar(possibleRatings, nRatings);

xlabel('Rating');

ylabel('Number of Users');

title(['Ratings for ', movieTitle])

Cleaning up the Data

Framing the Problem Using SVD

Next, let's think about how SVD might help us to analyze this dataset. Suppose we compute the SVD of the matrix ratingsMeanCentered. Let's use to refer to the first left singular vector, to refer to the first right singular vector, and to refer to the first singular value (let's assume that the first pair of singular vectors has the largest singular vector).

Exercise

1. What are the sizes of and ? What do each of the dimensions of correspond to? How about each dimension of ?
2. In 15.3.8 we talked about compressing the original matrix down to m + n + 1 values. If we think of as the compressed version of ratings data, how would we reconstruct the ratings data using (you essentially did this already, we're hoping you can recall this fact from earlier and apply it here).
3. We can think of as encoding the dominant trend that explains the ratings of each movie. For this dataset, what might this correspond to?
4. We can think of as encoding the dominant trend that explains the ratings by each usere. For this dataset, what might this correspond to? Keep in mind we have already subtracted out the mean of each user. It might be helpful to expand your formula from problem 2 to see how interact with each other.

Examining the Right Singular Vectors

Exercise

Looking at Large and Small Values

Examining the Left Singular Vectors

Exercise

Looking at Large and Small Values

Next Steps

1. We didn't really look at how you would use the SVD to make recommendations. It turns out the SVD can be used to come up with good guesses for the missing values in the original ratings matrix (the NaNs) and you can then provide recommendations based tailored for a praticular user.
2. We didn't quantify how well the svd predicted the ratings. In order to do that, you could divide the ratings into a training and test set and see how well your SVD model can predict the test ratings (i.e., a rating set that wasn't used to compute the SVD).
3. We filled in the missing values with the means of each movie, but there are variants of SVD that can handle the missing values directly (they do entail tradeoffs). You could inverstigate how one of those methods would work on this data.