I will use {ordinal} clm() (and other cool R packages such as {text2vec} as well) here to develop a hybrid content-based, collaborative filtering, and (obviously) model-based approach to solve the recommendation problem on the MovieLens 100K dataset in R. All R code used in this project can be obtained from the respective GitHub repository; the chunks of code present in the body of the post illustrate the essential steps only. The MovieLens 100K dataset can be obtained from the GroupLens research laboratory of the Department of Computer Science and Engineering at the University of Minnesota. The first part of the study introduces the new approach and refers to the feature engineering steps that are performed by the OrdinalRecommenders_1.R script (found on GitHub). The second part, to be published soon, relies on the R code in OrdinalRecommenders_3.R and presents the model training, cross-validation, and analyses steps. The OrdinalRecommenders_2.R script encompasses some tireless for-looping in R (a bad habit indeed) across the dataset only in order to place the information from the dataset in the format needed for the modeling phase. The study aims at (a) the demonstration of the improvement in predicted ratings for recommending on a well-known dataset, and (b) attempts to shedd light on the importance of various types of information in the work of recommendation engines. Consequently, the code is not suited for use in production; additional optimizations are straightforward, simple, and necessary as well.

### Introduction

Avoiding the formal exposition of the problem altogether, the recommendation problem in the context of contemporary online markets can expressed in the following way: given (A) the information on the user's past ratings of various items (i.e. the user-item ratings; for example, the ratings on a 5-point Likert type scale of the movies or TV shows that she or he has watched in the past, the ratings on similar scales of the items purchased, and similar), (B) the information on the user-item ratings of other users, (C) the attributes of users and/or items (c.f. the user's gender, geographical location, age, occupation, and similar; the genre of the movie, product type, and similar for items), provide a prediction of the user's rating of a novel, previously unassessed item. The predictions are then used to select the novel items (or lists of items) that a particular user would - by assumption - enjoy to assess, and eventually purchase, in the future. The cognitive systems used for making such predictions are known as recommendation engines, or recommender systems, and are widely used nowadays across the Internet business.

It is generally recognized that recommendation engines can be grouped in two broad categories: (1) content-based systems, or (2) collaborative filtering systems, with the later additionally described as (1.1) neighbourhood or (1.2) model-based methods [1]. Content based systems (CF) rely on a typical description of items over feature vectors, and then recommend novel items to users by computing some similarity metric between them and the items that the user has already rated. Collaborative filtering systems, on the other hand, rely on the assumption that the covariations between the ratings (a) from different users over the same set of items, or (b) to different items from the same set of users, implicitly carry the information on item and user attributes. The information about the true user and item attributes are thus latent in CF. In the neighbourhood (or memory) variant of CF, these approaches use the user-item ratings directly and proceed by aggregating the ratings of similar users (user-based CF) and/or similar items (item-based CF) in order to provide an estimate of the user's rating of a novel item, typically weighting the known ratings of similar users/items by the respective proximity/similarity measures. In the model-based variant of CF, the systems tries to discover the structure of the latent factors from the observed behavioural measures (ie. known user-item ratings), and then a myriad of machine learning algorithms and statistical approaches can be used to train a predictive model for the novel user-item ratings of novel items.

In this blog post, I will introduce a new hybrid approach to recommender systems with the following characteristics. First, the system is both content-based and CF. The content-based component of the system encompasses two matrices: the user-user and the item-item proximity matrices, both obtained from applying the relevant distance metric over a set of features that characterize users and items, respectively. The CF component of the system, relies on the typical user-user and item-item similarity matrices computed from the known, past user-item ratings, providing for a memory component of the recommender. Second, the system is model-based in the sense that it combines the information from these four matrices in an ordinal logistic regression model to predict the ratings of novel items on a 5-point Likert scale. The underlying logic of this approach is to try to express the recommendation problem as a discrete choice problem, where the alternatives are provided by ordinal information from a 5-point Likert scale, while the decision makers and the choice context are described by a set of content-based and memory-based features, all referring to some degree of closeness between the unknown user/item combination with the known user/item in two spaces: the attribute (feature) space and the neighbourhood (memory) space. Because the CF approaches rely on users-user (or item-item) proximity, the question of how many users (or items) that are similar to the context of the present, novel user-item rating should be used in prediction arises naturally. By expressing the recommender problem as a regression problem essentially, we become able to judge the significance of additional user-user and/or item-item similarity information from the change in regression coefficients obtained by progressively adding more and more information to nested ordinal logistic models and training them. Thus the new approach also provides and excellent opportunity for a comparison between content-based information and memory-based information (the later as being typically used in CF): which one is more important? Given the assumption that the former is implicitly present in the later, i.e. that it presents latent information in the CF approach, does the usage of manifest memory-based information on user-item ratings completely eliminates the need for a content-based description? If not, how the two should be combined? I will demonstrate how the present approach resolves this dilemma while resulting in highly improved recommendations on a well-known and widely used MovieLens dataset, at the same time reducing drastically the number of features that are needed to build the relevant prediction model.

### Motivation

Consider the following user-item ratings matrix, encompassing 10 items (columns) and three users (rows):

In this rating matrix, the ratings of user1 and user2 have a maximal Pearson correlation of 1; however, the ratings of user2 and user3 have a linear correlation of .085, sharing around .007 % of variance only. Upon a closer look, user1 and user2 have only three common ratings (for items i1, i4, and i5), while user2 and user3 have rated exactly the same items (i1, i3, i4, i5, i8, i9, and i10). Imagine that the items rated by user2 and user 3 are all Sci-Fi movies; we can imagine these two users chating and challenging each other's views on the contemporary Sci-Fi production on online forums or fan conferences, right? They do not have to have perfectly similar tastes, right, but in fact they are similar: simply because they both enjoy Sci-Fi. The Jaccard distance measures the proximity of two vectors A and B, both given over binary features (e.g. present vs. not present), by computing the ratio of (a) the difference between the union of A and B and the intersection of A and B to the (b) union of A and B; it ranges from zero to one and can be viewed as a proportion of common features that the two vectors share in the total number of features present in both. The Jaccard similarity coefficient is given by one minus the Jaccard distance; we will use the {proxy} dist() function to compute these from our ratings matrix:

User2 and User3 who have provided the ratings for the same items exactly have a Jaccard similarity index of 1; they both have the Jaccard similarity index of .375 with User1. Then who is the user similar to User2 - the one whose ratings could be used to approximate the experiences of User2 with previously unseen items: (a) User1, with whom he has a memory-based Pearson correlation of 1 on three shared items, or (b) User3, with whom he has a Pearson correlation of only .085, but with whom he had shared the same experiences and thus achieved a maximum level of Jaccard similarity?

The value of the linear correlation coefficient (or cosine similarity, or any other similar metric) will not preserve the information on the number of shared items; that information is imlicitly present in the Type I Error test (for example), not in the extent of covariance itself. The Jaccard distance, on the other hand, neglects the covariances from the rating scales altogether, preserving only the information about the extent of the shared ratings. If we think about the items rated by a particular user, neglecting the particular rating values, we understand that we can describe some aspects of the user's taste by binarizing his or her ratings; we can know, for example, the frequency distribution of the item types that he or she has decided to assess at all. If we additionally compute a distance or similarity matrix over that information for all users, we express the content-based approach in terms of the attributes that describe a particular use by its distance or similarity from other users in terms of the very same attributes - in a content-based description space. We can add the avaialable demographic information to the binary vectors before computing the similarities too and thus enrich the representation. And we can do the same for items, of course. As we already have the similarity matrices from linear correlations between the paired users' ratings at our disposal, we can combine the information from these two approaches - the content-based and the memory, or neighborhood-based - and try to predict the unobserved user-item ratings from a unified model.

The details of feature engineering along these lines in R are provided in the respective section below.

### Dataset

The MovieLens 100K dataset [2] comprises 100,000 ratings (Likert type 1 to 5 scale) from 943 users on 1682 movies; additionally, there are at least 20 movies rated per user. Some demographic information for the users is present - age, gender, occupation, zip - as well as the genre and the release dates for movies. The dataset can be downloaded from the GroupLens research laboratory of the Department of Computer Science and Engineering at the University of Minnesota. Historically, it is among the most popular datasets used to test the new approaches to the development of recommendation engines. MovieLens 100K can be also obtained from Kaggle and Datahub. Our OrdinalRecommenders_1.R script reads in the components of this dataset and computes the desired user-user and item-item distance and similarity matrices following some data reshaping.

### Feature engineering

Part 1.A of the OrdinalRecommenders_1.R script is used merely to read the original MovieLens 100K dataset files, rename the columns present there, and save as .csv files; you will need to run it too in order to use the code from part 1.B and the remaining scripts. In Part 1.B we begin to operate over the following three data.frames: (1) the ratingsData, which comprises the ratings on 5-point scales for all user-item pairs from the dataset, (2) usersData, which comprises demographic information on users (Gender, Age, Occupation, Zip-code), and (3) moviesData, which comprises all item information (Title, Release Date, and Genre). Part 1.B of this R script performs the essentially feature engineering operations in order to produce the following four matrices:

Matrix P2 - the User-User Proximity Matrix: the Jaccard distances between the users, computed from the binarized features representing whether a particular user has or has not rated a particular item only. This matrix plays the role of the content-based representation of users:

Note that I've missed to use the user demographics available in MovieLens 100K. In this project, I will not use them at all, but the general ratio is: because I am going after an ordinal logistic regression model, I can use these information as regressors there. Also, as I have computed the user-user distance by looking at what movies ratings they have in common, presumeably capturing some information on shared tastes, I could have done the some for the items, by inspecting what movies have been rated by the same users. However, a hunch told me that I might be end up using redundand feature representations in the same model; I've decided to describe the movie similarities from content-based features directly (genre and the decade of production), and user similarities from content-based features that encompass the similarity in tastes (the common movie ratings among them).

The following two similarity matrices are those typically used in user and item-based CF; they comprise Pearson correlation coefficients computed directly from the ratings matrix for 1681 items (note: one movie was 'unknown', so I have removed it from the dataset and cleaned up from the ratings matrix accordingly) and 943 users. Given the moderate size of the desired correlation matrices, I have used the {base} cor() function on pairwise complete observations to compute them. For digging out correlation or cosine matrices from problems larger than the MovieLens 100K, you might want to take a look at the functions provided by my colleague Stefan Nikolić in his "Recommender Systems: Matrix operations for fast calculation of similarities" (supporting the computations for CF in an order of magnitude lesser time than the popular {recommenderlab} R package). The code produces the remaining two matrices:

Matrix S1 - the User-User Similarity Matrix: the pairwise complete Pearson correlation coefficients between the users' ratings from the ratings matrix directly. This matrix plays the role of the memory-based representation of users:

Matrix S2 - the Item-Item Similarity Matrix: the pairwise complete Pearson correlation coefficients between the item' ratings from the ratings matrix directly. This matrix plays the role of the memory-based representation of items:

### Summary

The idea is to combine the content-based with memory-based information in a unified model to solve the recommendation problem for online recommender systems. I have used the Jaccard distances to describe the content-based proximity between users, based on what items they have rated or not; on the other hand, the Jaccard distances were also used to described the proximity of items in a content-based space derived from the "essentialistic" item features like movie genres and the decade of their production. Additionally, the typical Pearson correlation similarity matrices - computed directly from the ratings matrix - were produced for both users and items. The two user-user and item-item matrices share the same respective dimensionality. In the next step, the idea is to use the neighborhoods of different size from these matrices (say, take p similar users from the content-based proximity and the memory-based user-user similarity matrices, and q similar items from the content-based proximity and the memory-based item-item matrices) as regressors in an ordinal logistic model to predict the ordinal 5-point ratings of novel user-item combinations. By inspecting the regression coefficients from these predictive models while varying the size of the neighborhood sets of users and items whose ratings will be used as regressors, I hope to be able to provide some conclusion on the relative importance of content-based and memory-based information for building efficient recommendation engines. We'll see. But before skipping to OrdinalRecommenders_3.R to play with the beautiful {ordinal} models, you might wish to run OrdinalRecommenders_2.R that will prepare the first 100 neighbours from the proximity and similarity matrices for you. As I told you, there's a for loop there - and a bad one.

**References**

[1] Desrosiers, C. & Karypis, G. (2010). A comprehensive survey of neighborhood-based

recommendation methods. In Ricci, F., Rokach, L., Shapira, B. & Kantor, P. B. (Eds), Recommender Systems Handbook. Springer: US. DOI:10.1007/978-0-387-85820-3.

[2] F. Maxwell Harper and Joseph A. Konstan. 2015. The MovieLens Datasets: History and Context. ACM Transactions on Interactive Intelligent Systems (TiiS) 5, 4, Article 19 (December 2015), 19 pages. DOI=http://dx.doi.org/10.1145/2827872.