DS 6030 | Fall 2022 | University of Virginia

Intro

Let’s suppose it’s 2010 and you’ve just finished watching a collection of 4 movies on your handy DVD player. You still have some free time left and you decide that you want to spend the remainder of your time watching more movies. You have 3 other DVDs lying around, but you do not know which one will be good to watch - a travesty! In order to resolve this, you call up some of your closest friends and ask them to rate the DVD options that you have. While you’re at it, you ask your friends to also rate the movies you already watched.

Let’s say there are 4 movies that you already watched:

WM1, WM2, WM3, WM4

And let’s say the 3 new movies you’re deciding between are:

NM1, NM2, NM3

You were able to get a response from 3 of your friends: Hyun, Eric, and Chunru. They weren’t able to provide ratings for all 7 movies. In order to compare, you added your ratings to the table as well. You were then able to summarize all of the responses in a simple table:

	WM1	WM2	WM3	WM4	NM1	NM2	NM3
Hyun	5		3	2	1	4	3
Eric		4		5		2	5
Chunru	3	1		4	5		3
You	4	2	3	5

Given this information, will you be able to decide on which movie to watch next? The answer is yes! Thankfully, it’s 2022 and this concept has been around for a while now. Even more so, this concept is applied in technology through applications that predict which options a user will choose. These applications are called Recommendation Systems.

This tutorial will go through the concept of Recommendation Systems, specifically the technique of Collaborative Filtering. The following will be discussed:

Background of Concepts
Mathematical Theory
Coding Example (R)
Coding Example (Python)

Background of Concepts

Before jumping into the math, it is important to go through some examples of recommendation systems as well as the concept of the Utility Matrix.

Recommendation Systems

We mentioned earlier that recommendation systems are being applied in different technologies nowadays (Ullman et al., 2014). Some of these are:

Movie Recommendations - Like the initial example, recommending movies is one of the most popular examples for recommendation systems. Streaming services such as Netflix, Hulu, Disney, and others all provide recommendations for their users to try.
Product Recommendations - Online shopping is another one of the avenues where recommendation systems are very useful. Users receive suggestions on which products to buy based on what similar users are buying.
Song Recommendations - Another example of recommendation systems are with music streaming services where users receive suggestions for songs that they can add to their respective playlists.

There are definitely more examples but these should be able to give us an idea of the vastness of use cases where recommendation systems can be applied. This technology has been able to push many business forwards by helping out in increasing sales, user interaction, and more.

The Utility Matrix

Recall the table we created earlier for the initial example where the different friends had their ratings for different movies. That table is a good representation of the utility matrix:

	WM1	WM2	WM3	WM4	NM1	NM2	NM3
Hyun	5		3	2	1	4	3
Eric		4		5		2	5
Chunru	3	1		4	5		3
You	4	2	3	5

Typically, recommendation systems have two types of entities: users and items. The utility matrix represents the degree of preference that a user has for certain items. In our example, the degree of preference is measured on a movie rating scale of 1 to 5. It is also important to note that it is assumed that the utility matrix is sparse, meaning there are blank values in the matrix. This is good for replicating the real-world scenario where not all users would have a rating for all of the items (Ullman et al., 2014).

The goal of recommendation systems is to fill out the utility matrix by predicting values for the blank ones. Alternatively, recommendation systems can also predict the values for only some entries in each row which are expected to have higher values. This would make it so that the recommendation system does not have to fill out all of the empty cells but just a large subset of cells instead. Applying techniques such as clustering to the original matrix is also something that be considered so that less values would need to be predicted (Ullman et al., 2014).

Collaborative Filtering

Recommendation systems can be implemented using different techniques, but for this tutorial we will be focusing on collaborative filtering. This technique focuses more on the similarity of users with each other instead of similarity of features between items. This follows the intuition that similar users will tend to prefer similar items and thus the recommendations are based off of this idea (Ullman et al., 2014). You can also think of this in terms of having neighbors, where neighbors are based on similar item preferences. A user will receive recommendations that are influenced by what their neighbors prefer (Gormley, 2017).

Before implementing collaborative filtering recommendation systems, we will first discuss the mathematical theory that works in the background. Specifically, we will discuss how collaborative filtering can be done through cosine similarity or matrix decomposition

Mathematical Theory

Cosine Distance

To measure the similarity of users from the utility matrix, we can use the Cosine Distance method, which calculates the angle between the ratings of different users. In this method, we treat the blank entries as zero. One thing to note about this method is that it latently treats the lack of a rating as disliking the movie (i.e. rating = 0) (Ullman et al., 2014).

Suppose there are two users with vectors of size \(J\) that denote their movie ratings where \(J\) is the number of items in the utility matrix. The two vectors are \(v_{1i}\) and \(v_{2i}\), respectively where \(i \in 1,...,J\). The cosine similarity between the users can then be computed using the following formula:

\[ \frac{\sum_{i=1}^{J} v_{1i}v_{2i}}{\sqrt{\sum_{i=1}^{J} v_{1i}^2}\sqrt{\sum_{i=1}^{J} v_{2i}^2}} \] In this formula, the numerator is the sum of the product of common entries, and the denominator takes the product of the square root of the summation of squared values for each user. Note that this method treats blank entries as zero.

Using our example, we can check the similarity of some of the friends in our initial example. To get the cosine angle between Hyun and Eric, we can plug in the values from the utility matrix into our formula to get this equation:

\[ \frac{(2\times5) + (4\times2) + (3\times5)}{\sqrt{5^2+3^2+2^2+1^2+4^2+3^2}\sqrt{4^2+5^2+2^2+5^2}} = 0.493 \]

We can also check for the cosine angle between Hyun and Chunru using the same method:

\[ \frac{(5\times3) + (2\times4) + (1\times5) + (3\times3)}{\sqrt{5^2+3^2+2^2+1^2+4^2+3^2}\sqrt{3^2+1^2+4^2+5^2+3^2}} = 0.597 \]

Since a larger cosine value implies a smaller angle and therefore a smaller distance, this measure tells us that Hyun is closer to Chunru than to Eric.

These similarity values will then be used as weights for the weighed average that will be used to predict the missing ratings. For example, let’s say you are predicting the rating of user \(u_1\) for a movie denoted as \(r_1\) given similarities \(s_{12}\) and \(s_{13}\) with users \(u_2\) and \(u_3\). The rating of \(u_1\), which we can denote as \(r_1\) will be:

\[ r_1 = \frac{r_2s_{12} + r_3s_{13}}{s_{12} + s_{13}} \]

In general, this will be:

\[ r_i = \frac{\sum_{k=1}^{K}r_ks_{ik}}{\sum_{k=1}^K s_{ik}} \]

\(K\) is the total number of users and \(k \in 1,..,K\) where \(k \neq i\) and \(r_k \neq 0\)

In this weighted average, we do not include the missing values. This is in line with the idea behind collaborative filtering where items need to have ratings for them to be recommended (Gormley, 2017).

In addition, we can normalize ratings by subtracting the average rating of each user. In this case, the movies with low ratings would become negative and those with high ratings would be positive. Users with contrasting ratings on movies would have cosine vectors in opposite directions, while users with similar ratings would have smaller distances between them (Ullman et al., 2014).

In our example, the utility matrix after normalizing the ratings would be:

	WM1	WM2	WM3	WM4	NM1	NM2	NM3
Hyun	2		0	-1	-2	1	0
Eric		0		1		-2	1
Chunru	-0.2	-2.2		0.8	1.8		-0.2
You	0.5	-1.5	-0.5	1.5

Using the same formula earlier but with normalized values, the cosine of the angle between Hyun and Eric is:

\[ \frac{(-1\times1) + (1\times-2)}{\sqrt{2^2+(-1^2)+(-2^2)+1^2}\sqrt{1^2+(-2^2)+1^2}} = -0.387 \]

The cosine of the angle between Hyun and Chunru is:

\[ \frac{(2\times-0.2) + (-1\times0.8) + (-2\times1.8)}{\sqrt{2^2+(-1^2)+(-2^2)+1^2}\sqrt{(-0.2^2)+(-2.2^2)+0.8^2+1.8^2+(-0.2^2)}} = -0.512 \]

Using the normalized values, we find that Hyun is closer to Eric than to Chunru.

We can then proceed to calculate ratings using the weighted average formula as discussed previously.

UV (Matrix) Decomposition

In the cosine distance approach we discussed previously, we treat all blank entries as zero. This may lead to us ignoring important information and thus having imprecise predictions. To alleviate this problem, we can use UV decomposition to approximate the utility matrix as the product of two matrices. This is a useful method that can find good estimates for missing values in a matrix (Shummon Maass, 2019). UV decomposition is an instance of a more general theory called singular-value decomposition or SVD.

Let’s take an example, with a given utility matrix M that has n rows and m columns (i.e., in our example utility matrix there are 4 users and 7 movies, so matrix M is \(4 \times 7\)). We can decompose matrix M as the product of a \(n \times d\) matrix U and a \(d \times m\) matrix V. Afterwards, we can choose matrices U and V whose product matrix UV closely approximates M for the non-blank entries. Note that the dimension d is not important here, as long as the columns in matrix U is the same as the rows in matrix V. Once we obtain these two matrices, we can then derive the corresponding blank entries in the utility matrix M (Ullman et al., 2014).

In our example, we decompose the utility matrix M (\(4 \times 7\)), into two matrices U (\(4 \times 2\)) and V (\(2 \times 7\)):

\[ \begin{bmatrix} 5 & & 3 & 2 & 1 & 4 & 3 \\ & 4 & & 5 & & 2 & 5 \\ 3 & 1 & & 4 & 5 & & 3 \\ 4 & 2 & 3 & 5 & & & \end{bmatrix} = \begin{bmatrix} u_{11} & u_{12} \\ u_{21} & u_{22} \\ u_{31} & u_{32} \\ u_{41} & u_{42} \end{bmatrix} \times \begin{bmatrix} v_{11} & v_{12} & v_{13} & v_{14} & v_{15} & v_{16} & v_{17} \\ v_{21} & v_{22} & v_{23} & v_{24} & v_{25} & v_{26} & v_{27} \end{bmatrix} \] Theoretically, we can find many candidates for matrices U and V, whose product UV is close to the utility matrix M. Because of this, we need a measurement to pick the one that is closest to M. The typical choice for this is the root-mean-square error (RMSE), where the difference being checked is between the non-blank values of M and UV (Ullman et al., 2014).

To find the UV-decomposition matrix with the least RMSE, we start with an arbitrary guess of matrix U and V. For our example, let’s set all entries to be one as the starting values. We then repeatedly adjust one entry in U or V to make the RMSE smaller. Here we use our example to show the process of UV-decomposition.

Below is a summary of the steps that Ullman et al. (2014) discusses in their book:

Step 1 : Start with matrices U and V with all entries as 1:

\[ \begin{bmatrix} 1 & 1 \\ 1 & 1 \\ 1 & 1 \\ 1 & 1 \end{bmatrix} \times \begin{bmatrix} 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 & 1 & 1 & 1 \end{bmatrix} = \begin{bmatrix} 2 & 2 & 2 & 2 & 2 & 2 & 2 \\ 2 & 2 & 2 & 2 & 2 & 2 & 2 \\ 2 & 2 & 2 & 2 & 2 & 2 & 2 \\ 2 & 2 & 2 & 2 & 2 & 2 & 2 \\ \end{bmatrix} \]

Step 2 : Alter u11 to reduce the RMSE as much as possible:

\[ \begin{bmatrix} x & 1 \\ 1 & 1 \\ 1 & 1 \\ 1 & 1 \end{bmatrix} \times \begin{bmatrix} 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 & 1 & 1 & 1 \end{bmatrix} = \begin{bmatrix} x+1 & x+1 & x+1 & x+1 & x+1 & x+1 & x+1 \\ 2 & 2 & 2 & 2 & 2 & 2 & 2 \\ 2 & 2 & 2 & 2 & 2 & 2 & 2 \\ 2 & 2 & 2 & 2 & 2 & 2 & 2 \\ \end{bmatrix} \]

The contribution to the sum of squares from the first row is:

\[ (5-(x+1))^2 + (3-(x+1))^2 + (2-(x+1))^2 + (1-(x+1))^2 + (4-(x+1))^2 + (3-(x+1))^2 \\ = (4-x)^2 + (2-x)^2 + (1-x)^2 + (-x)^2 + (3-x)^2 + (2-x)^2 \]

We want a value of \(x\) that minimizes the sum, so we take the derivative and set that equal to 0: \[ -2((4-x)+ (2-x)+ (1-x)+ (-x) + (3-x) + (2-x)) = 0 \] From this equation we get \(x = 2\).

Afterwards, we can update the U and V matrices with the entry that \(u_{11}=2\).

\[ \begin{bmatrix} 2 & 1 \\ 1 & 1 \\ 1 & 1 \\ 1 & 1 \end{bmatrix} \times \begin{bmatrix} 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 & 1 & 1 & 1 \end{bmatrix} = \begin{bmatrix} 3 & 3 & 3 & 3 & 3 & 3 & 3 \\ 2 & 2 & 2 & 2 & 2 & 2 & 2 \\ 2 & 2 & 2 & 2 & 2 & 2 & 2 \\ 2 & 2 & 2 & 2 & 2 & 2 & 2 \\ \end{bmatrix} \]

Step 3 : Alter \(v_{11}\) to reduce the RMSE as much as possible and update \(v_{11}\)

This is similar to step 2 but for the V matrix instead.

Step 4 : Iterate steps 2 and 3 repeatedly until the RMSE is minimized

This method is useful especially for utility matrices with a manageable size of users and items. To learn more about ways to implement this, Ullman et al. (2014) provides a good explanation in their book.

Now that we’ve discussed the math behind this, let’s move on to some coding!

Coding Example (R)

To apply the concepts we’ve learned so far, let us implement these concepts in R. R has a built-in library recommenderlab, which we will get to later. Before that, let’s first implement a basic cosine similarity solution manually in order to illustrate the point.

library(tidyverse) # functions for data manipulation   
library(recommenderlab) # function for recommendation systems

We will continue to use the initial movie rating example. Let’s figure out which movie you should watch out of the three new movies.

movie_ratings <- data.frame(
  WM1 = c(5, NA, 3, 4), 
  WM2 = c(NA, 4, 1, 2),
  WM3 = c(3, NA, NA, 3),
  WM4 = c(2, 5, 4, 5),
  NM1 = c(1, NA, 5, NA),
  NM2 = c(4, 2, NA, NA),
  NM3 = c(3, 5, 3, NA)
)
rownames(movie_ratings) <- c('Hyun', 'Eric', 'Chunru', 'You')
movie_ratings

We will need a function that computes cosine similarity as it was discussed earlier in this tutorial.

# The two vectors must have the same length
cosine_similarity <- function(vec_1, vec_2) {
  vec_len <- length(vec_1)
  
  # NA values are replaced with 0
  vec_1[is.na(vec_1)] <- 0
  vec_2[is.na(vec_2)] <- 0
  
  # Computing the denominator
  vec_1_denom <- sqrt(sum(vec_1^2))
  vec_2_denom <- sqrt(sum(vec_2^2))
  denominator <- vec_1_denom * vec_2_denom
  
  # Computing the numerator
  tib = tibble(vec_1 = vec_1, vec_2 = vec_2)
  tib <- tib %>% mutate(products = vec_1 * vec_2)
  numerator <- sum(tib$products)
  
  # Return the cosine similarity
  return (numerator / denominator)
}

Let’s check how similar your taste in movies are to the 3 friends who responded. Remember, the higher cosine value means more similarity between two users.

# Extract each person's ratings as vectors
you <- as.numeric(as.vector(movie_ratings['You',]))
hyun <- as.numeric(as.vector(movie_ratings['Hyun',]))
eric <- as.numeric(as.vector(movie_ratings['Eric',]))
chunru <- as.numeric(as.vector(movie_ratings['Chunru',]))


# Compute distance using cosine similarity
similarities <- data.frame(
  cosine_similarity = c(cosine_similarity(you, hyun), cosine_similarity(you, eric), cosine_similarity(you, chunru))
)
rownames(similarities) <- c('Hyun', 'Eric', 'Chunru')
similarities

Based on these results, we can see that your taste in movies is most similar to Hyun’s. We will use these similarity scores as weights for predicting your movie ratings for the three new movies. For the weighted average, we will only include those users who have rated the movie.

# Function for computing the weighted average
movie_rating_weighted_average <- function(movie, friends) {
  denominator <- 0
  numerator <- 0
  for (friend in friends) {
    friend_similarity <- similarities[friend,][1]
    friend_rating <- movie_ratings[friend, movie][1]
    
    # Weighted average will take into account users who actually rated the movie
    if (is.na(friend_rating)) next
    
    denominator <- denominator + friend_similarity
    numerator <- numerator + (friend_similarity * friend_rating)
  }
  
  return (numerator / denominator)
}

friend_names <- c('Hyun', 'Eric', 'Chunru')
new_movies <- c('NM1', 'NM2', 'NM3')
new_movie_predicted_ratings <- tibble()
for (n in new_movies) {
  predicted_rating <- movie_rating_weighted_average(n, friend_names)
  prediction_tibble <- tibble(movie = n, predicted_rating = predicted_rating)
  new_movie_predicted_ratings <- bind_rows(new_movie_predicted_ratings, prediction_tibble)
}
new_movie_predicted_ratings

Based on these predictions, it looks like you will probably like NM3 the most out of all the options.

Now that we were able to get a feel for how collaborative filtering works through cosine similarity, we will now discuss how to make recommendations using the recommenderlab library as Topor (2017) explains in his article.

We will start by converting our data frame to a realRatingMatrix object:

# convert the movie ratings data frame to a matrix
rmat <- as.matrix(movie_ratings)

# convert matrix to a recommenderlab realRatingMatrix
rmat <- as(rmat, "realRatingMatrix")

We can then proceed to creating our Recommender models. The library contains the options we discussed earlier in this tutorial. Hahsler (2022) discusses these options in the package documentation.

The type UBCF stands for User-Based Collaborative Filtering which we can use to apply cosine similarity and normalization. The implementation of recommenderlab also makes use of a k-nearest neighbors algorithm when building this type of recommender as seen in the package source code.

The type SVDF stands for Funk Singular Value Decomposition which implements the UV decomposition technique.

We will create three versions:

Recommender using Cosine Similarity
Recommender using Cosine Similarity with Normalized values
Recommender using UV decomposition

set.seed(6030)
# Non-normalized cosine
reco <- Recommender(rmat, "UBCF", 
      param=list(normalize = NULL, method="Cosine"))

# Normalized centered cosine
reco_centered <- Recommender(rmat, "UBCF", 
      param=list(normalize = "center", method="Cosine"))

# UV decomposition
# The parameter k is the missing dimension for the decomposed matrices. 
# For this library, it cannot be greater than the number of users or the number of items
reco_uv <- Recommender(rmat, "SVDF", 
      param=list(k = 4))

Now we can check the results:

predictions <- predict(reco, rmat, type="ratings")
print('Non-normalized Cosine Similarity')

#> [1] "Non-normalized Cosine Similarity"

predictions@data

#> 4 x 7 sparse Matrix of class "dgCMatrix"
#>          WM1   WM2 WM3 WM4   NM1   NM2  NM3
#> Hyun   .     2.345   .   . .     .     .   
#> Eric   3.981 .       3   . 3.057 .     .   
#> Chunru .     .       3   . .     2.954 .   
#> You    .     .       .   . 3.051 2.984 3.67

predictions_centered <- predict(reco_centered, rmat, type="ratings")
print('Normalized Cosine Similarity')

#> [1] "Normalized Cosine Similarity"

predictions_centered@data

#> 4 x 7 sparse Matrix of class "dgCMatrix"
#>          WM1   WM2   WM3 WM4   NM1   NM2   NM3
#> Hyun   .     1.553 .       . .     .     .    
#> Eric   4.293 .     3.536   . 5.413 .     .    
#> Chunru .     .     2.748   . .     1.627 .    
#> You    .     .     .       . 4.093 2.509 3.801

predictions_uv <- predict(reco_uv, rmat, type="ratings")
print('UV Decomposition')

#> [1] "UV Decomposition"

predictions_uv@data

#> 4 x 7 sparse Matrix of class "dgCMatrix"
#>          WM1   WM2   WM3 WM4   NM1   NM2   NM3
#> Hyun   .     3.008 .       . .     .     .    
#> Eric   4.065 .     4.040   . 4.044 .     .    
#> Chunru .     .     3.246   . .     3.240 .    
#> You    .     .     .       . 3.552 3.542 3.561

As we can see from the results, NM3 has the highest predicted rating for 2 out of the 3 models. This just shows how recommender systems can be built using different distance measures and algorithms that may produce varying results.

The recommenderlab library also provides ways to measure models against each other. To show this, we will need a bigger dataset. We will use a larger dataset of movie ratings from Netflix in 2019. We did some pre-processing on the data to construct it into a utility matrix format. To check that function, you can read the code here and you can also download the csv file here. This file contains data on 500 users and 50 movies.

netflix <- read.csv('unscaled_title_data.csv')
head(netflix)

We will then follow similar steps as earlier, by converting the dataframe into a matrix.

# For this example, we will remove the customer id column
netflix_small <- netflix[, 2:51]

# Filter to users who have made atleast 10 movie ratings
netflix_small <- netflix_small[rowSums(is.na(netflix_small))<40,]

# convert the netflix data frame to a matrix
netflix_rmat <- as.matrix(netflix_small)

# convert matrix to a recommenderlab realRatingMatrix
netflix_rmat <- as(netflix_rmat, "realRatingMatrix")

We can then split the data using the evaluationScheme function. We can also set a goodRating value. Since the ratings are on a scale of 1-5, we will use 3 for this example.

set.seed(6030)

# split the data into the training and the test set:
split_data <- evaluationScheme(
  netflix_rmat, 
  method="split", 
  train=0.8, 
  k=1, 
  given=10, 
  goodRating=3
)

We can then build our models using the training data:

set.seed(6030)
# Non-normalized cosine
netflix_reco <- Recommender(getData(split_data, "train"), "UBCF", 
      param=list(normalize = NULL, method="Cosine"))

# Normalized centered cosine
netflix_reco_centered <- Recommender(getData(split_data, "train"), "UBCF", 
      param=list(normalize = "center", method="Cosine"))

# UV decomposition
netflix_reco_uv <- Recommender(getData(split_data, "train"), "SVDF", 
      param=list(k = 4))

The calcPredictioAccuracy function calculates the Root Mean Square Error (RMSE), Mean Squared Error (MSE), and Mean Absolute Error (MAE) for each of the models. This follows an approach suggested in the recommenderlab package, where we first use the known portion of the test set to make predictions and then calculate the error between those predictions and the test data’s unknown portion (Topor, 2017).

netflix_predictions <- predict(netflix_reco, getData(split_data, "known"), type="ratings")
netflix_predictions_centered <- predict(netflix_reco_centered, getData(split_data, "known"), type="ratings")
netflix_predictions_uv <- predict(netflix_reco_uv, getData(split_data, "known"), type="ratings")

error_results <- rbind(
  cosine = calcPredictionAccuracy(netflix_predictions, getData(split_data, "unknown")),
  cosine_normalized = calcPredictionAccuracy(netflix_predictions_centered, getData(split_data, "unknown")),
  uv = calcPredictionAccuracy(netflix_predictions_uv, getData(split_data, "unknown"))
)
error_results

#>                     RMSE    MSE    MAE
#> cosine            0.9842 0.9686 0.7772
#> cosine_normalized 0.9197 0.8458 0.7072
#> uv                0.9241 0.8540 0.7020

From these results, we can see that the Cosine Normalized model produced the lowest RMSE and MSE values for the Netflix data.

In this section, we discussed a few ways to create and test recommendation models in R, but there are many more for you to explore. If you want to explore other methods, Topor (2017) discusses a lot of interesting variations in his article. The recommenderlab documentation and code repository are also very detailed and good places to start.

Coding Example (Python)

Next up, we will discuss a simple example coded in Python as well to show the steps for a bigger example.

We will be using the same Netflix data we used earlier. The full notebook for this example can be found here.

For this Python example, we will be using the cosine_similarity function from the sklearn library. We will implement collaborative filtering using cosine similarity and normalizing the utility matrix by subtracting the average rating of each user.

First, we have to import the necessary libraries for this example:

import numpy as np
import pandas as pd
import seaborn as sns
import matplotlib.pyplot as plt
from sklearn.metrics.pairwise import cosine_similarity

We will start by reading in the data as before. This file contains 500 users and 50 movies.

rating_df = pd.read_csv('unscaled_title_data.csv')
rating_df.head()

Rating Dataframe

We will then normalize the ratings using the technique we discussed earlier, by subtracting each user rating with the average rating of that user.

scaled_df = rating_df.copy().iloc[:,1:] # remove the customer id column
averages = scaled_df.mean(axis=1).values
for i in range(len(rating_df)):
    # Subtract by mean of each user, not the entire user
    scaled_df.iloc[i,:] = (scaled_df.iloc[i,:] - averages[i]) 

scaled_df = scaled_df.fillna(0) # fillna(0) should come after scaling
scaled_df.insert(loc=0, column='Cust_Id', value=rating_df["Cust_Id"])
scaled_df

Scaled Dataframe After this, we can see the normalized ratings for the 500 users and 50 movies.

Now we will create functions for finding similarities for users:

def indices(lst, item):
    return [i for i, x in enumerate(lst) if x == item]

def return_similarity(current_rating, idx, movie_df):
    similarity_list = []
    for i in range(len(movie_df)):
        if i == idx:
            continue
        another_rating = np.array([movie_df.iloc[i,1:]]) # rating of another user
        curr_similarity = round(cosine_similarity(current_rating, another_rating)[0][0], 3)
        similarity = {
            'user_idx': i,
            'cosine_similarity': curr_similarity
        }
        similarity_list.append(similarity)

    return similarity_list

We also have a function for displaying our results later on:

def plot_rating(expected_dict, idx):
    title = expected_dict.keys()
    rating = expected_dict.values()
    recommended = []
    for score in rating:
        if score > 0:
            recommended.append(True)
        else:
            recommended.append(False)
            
    df = pd.DataFrame({
        "Title": title, 
        "Expected Rating": rating, 
        "Recommended": recommended
    }).sort_values(by = "Expected Rating", ascending = False)
    plt.figure(figsize=(20, 7))
    plt.xticks(rotation=90)
    plt.title("Predicted Rating for Unwatched Movies and Recommendation")
    
    # return the plot and a dataframe of the top 10 recommended movies
    return (sns.barplot(x = "Title", y = "Expected Rating", hue = "Recommended", data = df), df.iloc[:10,:])

This is the main function. We will use weighted averages to recommend movies for a user. The weights used are the similarity scores between users. Only users who have rated the movie in question will be included in the weighted average.

This function will be making predictions on the movies that a user has not watched yet. Since our ratings our normalized, positive values would mean better ratings than the negative ones.

def get_similarity(idx, movie_df):
    current_rating = np.array([movie_df.iloc[idx,1:]]) # rating of the current user
  
    # BASE CASE: if the user has watched all movies, then returns nothing.
    if 0 not in current_rating: 
        print("This user has watched all the movies.")
        return -1

    similarity_list = return_similarity(current_rating, idx, movie_df)
    movie_titles = movie_df.columns[1:]
    
    ratings = current_rating[0].tolist()
    expected_dict = dict()
    for movie in movie_titles:
        rating = movie_df[movie].iloc[idx]
        
        # skip if the user has already seen a movie
        if rating != 0: 
            continue

        numerator = 0
        denominator = 0        
        for similarity in similarity_list:
            other_idx = similarity['user_idx']
            other_similarity = similarity['cosine_similarity']
            other_user_rating = round(movie_df[movie].iloc[other_idx], 3)
            
            # we do not consider the similarity of unwatched users 
            # because it only affects the denominator but not numerator
            if other_user_rating == 0: 
                continue

            # numerator is iteratively adding up each rating x each similarity score
            numerator = numerator + (other_user_rating * other_similarity)
            denominator += other_similarity

        if denominator == 0: # avoid zero division error
            expected_rating = 0
        else:
            expected_rating = round(numerator / denominator, 2)

        expected_dict[movie] = expected_rating

    return plot_rating(expected_dict, idx)

Let’s put these functions to work and make recommendations for some users!

# user at index 1
res1 = get_similarity(1, scaled_df)

User at Index 1 Results

# top movies recommended for user at index 1
res1[1]

User at Index 1 Movie Rankings

User at Index 1 will probably not enjoy the new movies recommended to him, but out of all the movies, The Patriot is at the top of the list.

# user at index 160
res2 = get_similarity(160, scaled_df)

User at Index 160 Results

# top movies recommended for user at index 160
res2[1]

User at Index 160 Movie Rankings

User at Index 160 will also probably not enjoy the new movies recommended to him, but out of all the movies, The Patriot is also at the top of his list.

# user at index 200
res3 = get_similarity(200, scaled_df)

User at Index 200 Results

# top movies recommended for user at index 200
res3[1]

User at Index 200 Movie Rankings

User at Index 200 will probably enjoy the top two new movies recommended to him, which are Spider-Man 2 and Mystic River.

# user at index 440
res4 = get_similarity(440, scaled_df)

User at Index 440 Results

# top movies recommended for user at index 440
res4[1]

User at Index 440 Movie Rankings

User at Index 440 will probably enjoy the top new movie recommended to him, which is The Last Samurai.

# user at index 498
res5 = get_similarity(498, scaled_df)

User at Index 498 Results

# top movies recommended for user at index 498
res5[1]

User at Index 498 Movie Rankings

User at Index 498 will also probably enjoy the top two new movies recommended to him, which are The Last Samurai and Mystic River.

Now, let’s try this out with a bigger dataset. We will use this file that has data on 1000 users and 500 movies.

# 1000 x 500
big_rating_df = pd.read_csv('rating_before_scaling.csv')
big_rating_df.head()

Big Rating Dataframe

We will then normalize the new dataset:

big_scaled_df = big_rating_df.copy().iloc[:,1:]
averages = big_scaled_df.mean(axis=1).values
for i in range(len(big_rating_df)):
    # Subtract by mean of each user, not the entire user
    big_scaled_df.iloc[i,:] = (big_scaled_df.iloc[i,:] - averages[i]) 

big_scaled_df = big_scaled_df.fillna(0) # fillna(0) should come after scaling
big_scaled_df.insert(loc=0, column='Cust_Id', value=big_rating_df["Cust_Id"])
big_scaled_df

We can see the normalized ratings for the 1000 users and 500 movies.

Let’s take a look at some results:

# user at index 1
big_res1 = get_similarity(1, big_scaled_df)

User at Index 1 Results

# top movies recommended for user at index 1
big_res1[1]

User at Index 1 Movie Rankings

User at Index 1 will probably enjoy the new movies recommended to him, especially The Sopranos: Season 1!

# user at index 2
big_res2 = get_similarity(2, big_scaled_df)

User at Index 2 Results

# top movies recommended for user at index 2
big_res2[1]

User at Index 2 Movie Rankings

User at Index 2 will likely enjoy American Splendor!

# user at index 175
big_res3 = get_similarity(175, big_scaled_df)

User at Index 175 Results

# top movies recommended for user at index 175
big_res3[1]

User at Index 175 Movie Rankings

User at Index 175 looks like he will enjoy The Godfather - a classic!

# user at index 440
big_res4 = get_similarity(440, big_scaled_df)

User at Index 440 Results

# top movies recommended for user at index 440
big_res4[1]

User at Index 440 Movie Rankings

User at Index 440 will probably enjoy the top new movie recommended to him, which is Star Wars Episode IV.

# user at index 999
big_res5 = get_similarity(999, big_scaled_df)

User at Index 999 Results

# top movies recommended for user at index 999
big_res5[1]

User at Index 999 Movie Rankings

User at Index 999 looks like he will enjoy Schindler’s List. Interestingly, he also has a lot of Disney movies recommended to him.

Limitation of this Implementation:

Since the quality of rating prediction heavily depends on the ratings of other users, collaborative filtering requires as many observations as possible. A small number of users produces biased results and sparse movie ratings for a movie gives small similarities among users, thus leading to imprecise predictions. On the other hand, since our main function for predicting rating has a time complexity of \(O(n^2)\), it is hard to run predictions for every single user, which takes a huge amount of time. Therefore, the more efficient methodology for rating prediction is desirable. Lastly, since collaborative filtering only compares the similarity between one user and the others, it does not consider the content side of things such as genre, director, and length of a movie. Hence, detailed and tailored recommendation might be difficult in that sense. In this regard, content-based recommendation can be implemented used. Ullman et al. (2014) provides a good explanation of that technique in their book.

In this section, we were able to create a simple Collaborative Filtering Recommendation model using cosine similarity. Go ahead and try out different techniques as well!

Conclusion

After discussing the background concepts, mathematical theory, and different coding implementations of collaborative filtering, we can see that there are different ways to make recommendations. The idea behind it is very intuitive, and the mathematics and coding techniques help out in putting those ideas to action. From the simple hypothetical movie scenario to a bigger example using Netflix data, we were able to witness the capabilities of these recommendation systems. It doesn’t stop there! The topic of recommendation systems has a lot of different branches and methods - all of which are exciting to learn. We have provided the references we have used for this tutorial so that you can explore even more. Happy learning!

References

Gormley, M., (2017, April 19). Lecture: Matrix Factorization and Collaborative Filtering. Machine Learning Department, Carnegie Mellon University. Retrieved December 13, 2022, from https://www.cs.cmu.edu/~mgormley/courses/10601-s17/slides/lecture25-mf.pdf

Hahsler, M., (2022, August 17). recommenderlab: Lab for Developing and Testing Recommender Algorithms. Cran. Retrieved December 13, 2022, from https://cran.r-project.org/web/packages/recommenderlab/

Hahsler, M., (2022, August 17). R package recommenderlab - Lab for Developing and Testing Recommender Algorithms. Github. Retrieved December 13, 2022, from https://github.com/cran/recommenderlab

Netflix, (2019, November 13). Netflix Prize data. Kaggle. Retrieved December 7, 2022, from https://www.kaggle.com/datasets/netflix-inc/netflix-prize-data

Shummon Maas, L., (2019, June 11). Recommendation Systems using UV-Decomposition. Towards Data Science. Retrieved December 13, 2022, from https://towardsdatascience.com/recommendation-systems-using-uv-decomposition-a1d4116be4a1

Topor, J., (2017, June 11). User-Based and Item-Based Collaborative Filtering. RPubs. Retrieved December 13, 2022, from https://rpubs.com/jt_rpubs/285729

Ullman, J., Leskovec, J., Rajaraman, A., (2014, July 4). Mining Massive Datasets: Chapter 9. Cambridge University Press. Retrieved December 7, 2022, from http://infolab.stanford.edu/~ullman/mmds/ch9.pdf and http://infolab.stanford.edu/~ullman/mmds/book.pdf

Collaborative Filtering Recommendation Systems Tutorial

Hyun Ko, Eric Tria, Chunru Zheng

Wed Dec 14 | 12:00pm