Constructing and Testing AI International Legal Education CouplingEnabling Model
4.1. AI Data Analysis
$$r=\frac{{\displaystyle \sum (X\overline{X})}(Y\overline{Y})}{n\times {\sigma}_{x}\times {\sigma}_{y}}$$
In the formula, ${\sigma}_{x}$ and ${\sigma}_{y}$ represent the standard deviation of the two variables, $n$ is the capacity of the sample, and the meaning of $n$ is the sum of the product of the two standardized scores divided by the sample capacity. The value of $r$ is between −1 and +1: the positive value of $r$ indicates that the correlation between the two is positive, and the negative value of $r$ indicates that the correlation between the two is positive.
$$r=1\frac{6{\displaystyle \sum _{i=1}^{n}{d}_{i}^{2}}}{{n}^{3}n}$$
The application of these two correlation coefficients helps to reveal the relationship between various learning behaviors of learners in the innovative exploration of the coupled empowerment model of AI and international legal education, providing a powerful tool for further analysis.
4.2. AI Knowledge Graph
4.3. AI Intelligent Diagnosis
$$\begin{array}{c}X\to Y\\ {f}_{bp}\hspace{1em}X=\left\{{X}_{1},{X}_{2},{X}_{3},\cdots ,{X}_{15}\right\}\\ Y=\left\{{Y}_{1},{Y}_{2},{Y}_{3}\right\}\end{array}$$
$${O}_{i}^{(1)}=x(i),\hspace{1em}i=1,2,\cdots ,n$$
with ${w}_{ij}^{(2)}$ and $f[\cdot ]$ representing the weight coefficients of the implicit layer of the BP neural network and denoting the mapping function, the computation formulas for its input and output are respectively expressed as follows:
$$ne{t}_{i}^{(2)}(k)={\displaystyle \sum _{j=1}^{m}{w}_{ij}^{(2)}}{O}_{j}^{(1)}(k)$$
$${O}_{i}^{(2)}(k)=f\left[{\mathrm{net}}_{i}^{(2)}(k)\right]$$
with ${w}_{li}^{(3)}$ and $g[\cdot ]$ representing the weight coefficients of the output layer of the BP neural network and denoting the mapping function, the computation formulas for its input and output are respectively expressed as follows:
$$ne{t}_{l}^{(3)}(k)={\displaystyle \sum _{i=1}^{p}{w}_{li}^{(3)}}{O}_{i}^{(2)}(k)$$
$${O}_{l}^{(3)}(k)=g\left[ne{t}_{l}^{(3)}(k)\right]$$
for the p st sample, whose actual and network outputs are ${O}_{p}(k+1)$ and ${O}_{p}^{\prime}(k+1)$, respectively, then the error is given by the following:
$${E}_{p}=\frac{1}{2}{\left[{O}_{p}(k+1){O}_{p}^{\prime}(k+1)\right]}^{2}$$
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