# A Personalized Learning Path Recommendation Method Incorporating Multi-Algorithm
## 3 Methodology
### Related Work
#### 3.1.1 Cognitive Level Based on Fuzzy-CDF  
The cognitive level refers to the mastery of knowledge points, including the knowledge that is mastered and the knowledge that is not mastered. In cognitive psychology, the modelling process of learners' mastery of knowledge points is referred to as the cognitive diagnostic model [8]. In terms of personalized learning, typical cognitive diagnostic models include the DINA [9] model and the DINO [10] model, which mainly reflect learners' understanding of the knowledge points that they have learned and explain whether learners have mastered the current knowledge points. However, these models are secondary graded cognitive diagnostic models, namely, they only have two states: mastering and not mastering. To obtain the cognitive level of learners using a more fine-grained approach, this paper adopts a fuzzy cognitive diagnosis model, Fuzzy-CDF [11], to obtain learners' knowledge mastery. The output of this model is a continuous value between 0 and 1, which can more accurately obtain the learners' mastery of knowledge points. The Fuzzy-CDF model starts from the learner's potential characteristics, determines the learner's skill proficiency, calculates the examinee's mastery of the problem, and generates the examinee's observable score of the problem by considering the error and guess factors.  
Fuzzy-CDF assumes that the set of students who master all (part) of the knowledge and skills required for Problem \(i\) is the intersection (union) of fuzzy sets related to skills [12]. Fuzzy-CDF uses fuzzy numbers to quantify subjective, qualitative, and uncertain information. It is assumed that students' knowledge ability level is their membership degree in the fuzzy set corresponding to the knowledge ability. Therefore, in objective questions, the mastery degree \(\eta_{ji}\) of learner \(j\) on test question \(i\) is the fuzzy intersection of learner \(j\)'s cognition of knowledge points. In subjective questions, \(\eta_{ji}\) is the fuzzy sum of learners' cognition of knowledge points. The specific calculation is presented as follows:  
\[\eta_{ji}=\underset{1\leq k\leq K,\eta_{ik}=1}{\cap}\mu_{k}(j) \tag{1}\]  
\[\eta_{ji}=\underset{1\leq k\leq K,\eta_{ik}=1}{\cup}\mu_{k}(j) \tag{2}\]  
where \(q_{ik}=1\) indicates whether test question \(i\) has examined knowledge point \(k\) and \(K\) indicates the total number of knowledge points examined in test question \(i\). Whether a learner correctly answers a question is not only related to the extent to which the learner has mastered the knowledge points investigated in the test but also may be affected by the factors of guess and error, resulting in the inability to accurately obtain the learner's true ability. Therefore, we can use the 4-parameter logistic model to estimate the error and guess parameters, and we can intervene in the acquisition of learners' abilities. The specific calculation is presented as follows:  
\[P(X_{ij}=1|\theta_{i};a_{j},b_{j},c_{j},d_{j})=c_{j}+(d_{j}-c_{j})\frac{e^{1.7 a_{j}(\theta_{i}-b_{j})}}{1+e^{1.7a_{j}(\theta_{i}-b_{j})}} \tag{3}\]\(a_{j}\), \(b_{j}\), \(c_{j}\) and \(d_{j}\) are the discrimination, difficulty, guess, and error parameters, respectively. The real scores of the learners on the objective and subjective questions are obtained. The specific calculation is presented as follows:  
\[P(R_{ji}=1\big{|}\eta_{ji},s_{i},g_{i})=(1-s_{i})\eta_{ji}+g_{i}(1-\eta_{ji}) \tag{4}\]  
\[P(R_{ji}=0\big{|}\eta_{ji},s_{i},g_{i})=N(R_{ji}\big{|}(1-s_{i})\eta_{ji}+g_{i} (1-\eta_{ji}),\sigma^{2}) \tag{5}\]  
where \(R_{ji}\) indicates learner \(j\)'s score on test Item \(i\) and \(s_{i}\) and \(g_{i}\) indicate learner \(j\)'s mistakes and guesses, respectively, on test Item \(i\).