Personalized recommendation system based on social tags in the era of Internet of Things

3.1 Label-based personalized Markov chain transfer matrix estimation

For each user u, the information contained in its historical behavior, which equal to the set of tags, is ℬ u = ( B 1 u , … , B t u ) , and   B t u ⊆ T denotes the set of tags of the items that user u queries or browses at moment t. The historical labels of all users are ℬ ≔ { ℬ u 1 , … , ℬ u m } .

Usually, a Markov chain with step number k is defined as follows:

where X t , … , X t − k are random variables and x t − k are their specific instances. To simplify the transfer probability matrix, the number of steps of the Markov chain k is set as 1, The simplified Markov chain is expressed as follows:

Similarly, the Markov chain for each user is expressed through the transfer matrix between labels, but here the Markov chain is individualized as follows:

Based on the label information given by the user’s last state, the likelihood of user preference for a label is mined, which can be interpreted as the average value in the transfer matrix from the user’s last query to the query for this label.

This means that for each user u, a personalization transfer matrix A u ∈ ℝ m × m is created [12]. Let label t _l transfer to label t _i be a state pair (t _l , t _i ), and t l ∈ B t − 1 u , t i ∈ B t u , C t l , t i u denotes the number of occurrences of state pair (t _l , t _i ) in user u’s historical label set B ^u . For each user u, the transfer probability formula for label t _l to label t _i is noted as follows:

3.2 User’s interest degree model for tags

1. Constructing user’s interest degree on tags

In the study of two dimensional relationship, based on the Markov chain model according to the known set of labels contained in the user’s most recent behavior and the user’s personalized label transfer matrix A ^u , refer to equation (3.4) to construct the probability matrix I(u,t) of the user’s preference for the label in the next behavior, the user’s interest degree matrix for the tag.

Due to the sparsity of user’s historical behavior data, the matrix I(u,t) also has data sparsity. Therefore, while introducing the idea of collaborative filtering, the similarity of tags is combined to complement the matrix of users’ interest in tags, and if there are no data sparsity, then this step is not necessary.

If it is believed that different labels on the same item have a certain degree of similarity, then when two labels appear in the label set of many items at the same time, it can be considered that the two labels have a greater degree of similarity. The number of labeling tags is also considered, and hence, the Cosine coefficient is used for calculation, which is calculated as follows:

where n t , i , n t ′ , i denotes the number of times item i is labeled t as well as label t′, and N ( t ) , N ( t ′ ) denotes the set of items labeled with label t as well as label t′.

In summary, the degree of user interest in the labels after completing the sparse values is calculated as follows:

Through Markov chain and collaborative filtering ideas based on labels to discover the set of labels preferred by users, the size of the label set T should need to be adjusted accordingly to the actual situation.

4.1 Top-N recommendation

The user satisfaction model for items based on tags is constructed based on the number of times users label the items. However, in the tag-based item recommendation problem, the item does not contain a single tag, but a set of tags. Existing ranking algorithms usually assume that the relationships between the tags contained in the set of item tags are independent of each other. However, the relationships among the labels are mutually influential.

Consider an item I r ≔ { t i 1 , ⋯ , t i k } containing multiple labels, and S _i1…ik denotes the satisfaction of user u with item I r labeled with labels t i 1 , ⋯ , t i k . Let p u ( t i 1 , ⋯ , t i k | k ) be the probability of user u querying label t i 1 , ⋯ , t i k , and the user will query k different labels, k ∈ [ 1 , l ] . When user u queries according to the set of tags, user u’s total expected satisfaction can be expressed as S _total, defined as follows:

Since the number of tags used when users actively use tags to query items is mostly only 1 or 2. Therefore, in this article, we only consider the first and second terms in S _total and ignore other higher-order terms. At this point, S _total is represented as follows:

In this article, two probabilities will be defined: p ( t i 1 | k = 1 ) and p ( t i 1 , t i 2 | k = 2 ) . The probability p ( t i 1 | k = 1 ) represents the probability that user u queries t _i1, which equals to p ( t i 1 | k = 1 ) = p ( t i 1 | u ) , which can be calculated by using a recommendation model such as memory-based collaborative filtering [13] or matrix decomposition. For the probability p ( t i 1 , t i 2 | k = 2 ) , assuming that users query different tags independently of each other, a simple approach is to define this probability as p ( t i 1 , t i 2 | k = 2 ) =   p ( t i 1 | u ) ⋅ p ( t i 2 | u ) . However, as mentioned earlier, the user’s interest in tags may be influenced by other tags, which means that the influence between tags cannot be ignored. Therefore, in the model of this article, its joint probability is defined by equal probability as p ( t i 1 , t i 2 | k = 2 ) = 1 2   ( p ( t i 2 | t i 1 ) ⋅ p ( t i 1 | u ) +   p ( t i 1 | t i 2 ) ⋅ p ( t i 2 | u ) ) , where p ( t i 1 | t i 2 ) is the transfer probability from t _i2 to t _i1, which can be obtained from historical behavioral data and is calculated as follows:

The probabilities p ( t i 1 | k = 1 ) and p ( t i 1 , t i 2 | k = 2 ) are defined only based on the historical behavior data of the users. However, a more complex definition of these probabilities can be given by integrating additional information. For example, the probability of querying two tags can be increased (or decreased) if they are known to be complementary (or substitutable) to each other. In addition, other methods (e.g., classification or textual information) can be used to compute query probabilities. The definition of satisfaction S _i1i2,…,ik is given. Usually, the more labels a user queries, the higher satisfaction the user obtains. Therefore, in this article, we set S _i1 = 1, S _i1i2 = 2, …, S _i1i2,…,ik = k in the project, and other satisfaction functions can be defined according to different application requirements. Thus, the expected satisfaction S _total can be re-expressed as follows:

where O I r represents all ordered pairs < t i , t j > in   I r , and β = P ( k = 1 ) P ( k = 2 ) is the user-related parameter. However, it is very difficult to set β for each user; therefore, in this article, β is set to 1 because it provides the best effect for the satisfaction model and thus improves the accuracy of the recommendation.

4.2 Complementary Top-N recommendations

As mentioned earlier, items with T tags can be matched. At this point, if the data have sparsity, sometimes it may not be matched or only a few items with T tags can be matched, so when T < N , this section needs to complete the N − T satisfaction of the recommendation to the user by the user’s interest in the item in a collaborative filtering way.

This section uses labels to calculate the similarity between items, mainly based on the principle that the more times item i and item i′ are labeled with the same label by users, the more similar item i and item i′ are. Also, considering that only concern about whether the label is labeled, n t , i and n t , i '   denote whether item i and item i′ are labeled with label t, being labeled takes the value of 1, otherwise 0. N ( i ) , N ( i ′ ) denote the set of labels of item i and item i′, respectively, and use the Jaccard coefficient for the calculation, and the calculation formula is expressed as follows:

Completing the user’s satisfaction with the item, obtaining the user u’s satisfaction with the recommended item i according to S a t ( u , i ) , calculating the similarity between the recommended item i and the new item i′ to be recommended, and finding the user u’s satisfaction with the new item i′ to be recommended, we obtain:

Finally, use user’s satisfaction with the item S a t ′ ( u , r ) to complement user’s Top-N recommendation for the item. After building a Markov chain model of users’ interest in tags and constructing a model of users’ interest in items by T tags, we use the correlations between tags to construct a model of users’ satisfaction ranking of items.