【Practical Exercise】Implementing a Recommendation Algorithm in MATLAB

# 2.1 User Similarity Calculation

User similarity calculation is a core step in collaborative filtering recommendation algorithms, aiming to quantify the degree of sim***mon methods for calculating user similarity include cosine similarity and Pearson correlation coefficient.

### 2.1.1 Cosine Similarity

Cosine similarity is a method of similarity calculation based on the vector space model, which measures the directional similarity of two vectors. For two user vectors `u` and `v`, the cosine similarity is defined as:

cos(u, v) = (u · v) / (||u|| ||v||)

Where `u · v` represents the dot product of vectors `u` and `v`, and `||u||` and `||v||` represent the magnitudes of vectors `u` and `v` respectively. Cosine similarity ranges from -1 to 1, where 1 indicates perfect similarity, -1 indicates perfect opposition, and 0 indicates no correlation.

### 2.1.2 Pearson Correlation Coefficient

The Pearson correlation coefficient is a method of similarity calculation based on statistics, which measures the degree of linear correlation between two variables. For two user vectors `u` and `v`, the Pearson correlation coefficient is defined as:

r(u, v) = (cov(u, v)) / (σ(u) σ(v))

Where `cov(u, v)` represents the covariance between vectors `u` and `v`, and `σ(u)` and `σ(v)` represent the standard deviations of vectors `u` and `v` respectively. The Pearson correlation coefficient ranges from -1 to 1, where 1 indicates perfect positive correlation, -1 indicates perfect negative correlation, and 0 indicates no correlation.

# 2. Collaborative Filtering-Based Recommendation Algorithms

Collaborative filtering recommendation algorithms are based on user behavior data. They predict a user's preference for unrated items by analyzing the similarity between users or items. Collaborative filtering algorithms are divided into two main methods: user-based and item-based approaches.

### 2.1 User Similarity Calculation

User similarity calculation is the core of user-based and item-based recommenda***mon methods for user similarity calculation include cosine similarity and Pearson correlation coefficient.

#### 2.1.1 Cosine Similarity

Cosine similarity is a measure of the similarity between two vectors. It determines the similarity by calculating the cosine of the angle between the two vectors. Cosine similarity ranges from -1 to 1, where -1 indicates complete dissimilarity, 0 indicates orthogonality, and 1 indicates complete similarity.

For two users u and v, the cosine similarity calculation formula is:

sim(u, v) = cos(θ) = (u · v) / (||u|| ||v||)

Where u and v are the rating vectors of users u and v, `u · v` represents the dot product, and `||u||` and `||v||` represent the magnitudes of the vectors.

#### 2.1.2 Pearson Correlation Coefficient

The Pearson correlation coefficient is a measure of the linear correlation between two variables. It determines the correlation by calculating the covariance and standard deviations between the two variables. The Pearson correlation coefficient ranges from -1 to 1, where -1 indicates complete negative correlation, 0 indicates no correlation, and 1 indicates complete positive correlation.

For two users u and v, the Pearson correlation coefficient calculation formula is:

sim(u, v) = r(u, v) = (cov(u, v)) / (σu σv)

Where `cov(u, v)` represents the covariance between u and v, and `σu` and `σv` represent the standard deviations of u and v respectively.

### 2.2 Item-Based Recommendation Algorithms

Item-based reco***mon item-based recommendation algorithms include item-based collaborative filtering and item-based latent semantic models.

#### 2.2.1 Item-Based Collaborative Filtering

Item-based collaborative filtering algorithms predict user preferences for unrated items by calculating item-item similarity. They determine the relevance between items by analyzing user ratings for different items.

For two items i and j, the item-based collaborative filtering similarity calculation formula is:

sim(i, j) = cos(θ) = (i · j) / (||i|| ||j||)

Where i and j are the rating vectors for items i and j, `i · j` represents the dot product, and `||i||` and `||j||` represent the magnitudes of the vectors.

#### 2.2.2 Item-Based Latent Semantic Models

Item-based latent semantic models calculate item-item similarity by representing items as low-dimensional vectors. They learn the latent features of items by analyzing user ratings for different items.

For two items i and j, the item-based latent semantic model similarity calculation formula is:

sim(i, j) = cos(θ) = (q_i · q_j) / (||q_i|| ||q_j||)

Where `q_i` and `q_j` are the low-dimensional vector representations of items i and j, `q_i · q_j` represents the dot product, and `||q_i||` and `||q_j||` represent the magnitudes of the vectors.

# 3.1 Text Similarity Calculation

In content-based recommendation algorithms, text similarity calculation is a key step in measuring the similarity between two text objects. There are many text similarity calculation methods, among which cosine similarity and TF-IDF similarity are two commonly used methods.

#### 3.1.1 Cosine Similarity

Cosine similarity is a similarity calculation method based on the vector space model. It measures similarity by calculating the cosine value of the angle between two vectors. For two text objects, they can be represented as vectors where each element represents the weight of a word. The weight of a word can be its term frequency, TF-IDF value, or other measures.

The cosine similarity calculation formula is:

similarity = cosine(vector1, vector2) = (vector1 · vector2) / (||vector1|| * ||vector2||)

Where `vector1` and `vector2` are the transposes of two text vectors, `·` represents the dot product, and `||vector||` represents the magnitude of the vector.

#### 3.1.2 T