# Patent Analysis Using Bayesian Data Analysis and Network Modeling
## 3 Proposed Method
### BART Modeling  
Next, we analyze the drone patent data using statistical models. Thus, we use multiple linear regression and BART for the patent analysis of drone technology. In the multiple regression model of Equation (3), the response variable \(Y\) is the keyword drone and the other keywords except drone are used as explanatory variables \(\big{(}X_{1},X_{2},\ldots,X_{p}\big{)}\). \[Y=\beta_{0}+\beta_{1}X_{1}+\beta_{2}X_{2}+\cdots+\beta_{p}X_{p}+\varepsilon, \,\mathrm{i}=1,2,\ldots,n \tag{3}\]  
where \(Y\) and \(\big{(}X_{1},X_{2},\ldots,X_{p}\big{)}\) represent the frequency values of patent keywords occurring in each patent document. \(\varepsilon\) is a noise term distributed to normal distribution with mean = 0 and variance = \(\sigma^{2}\). Moreover, \(\beta_{0}\) is an intercept of regression equation, and \(\big{(}\beta_{1},\beta_{2},\ldots,\beta_{p}\big{)}\) are the parameters corresponding to input keywords (explanatory variables). To find \(X_{i}\) that has a statistically significant effect on \(Y\), we perform the following hypothesis test. \[H_{0}:\beta_{i}=0\ \ \ vs.\ \ \ H_{1}:\beta_{i}\
eq 0 \tag{4}\]  
\(H_{0}\) represents the null hypothesis in that the regression coefficient \(\beta_{i}\) of \(X_{i}\) becomes 0. This means that the explanatory variable \(X_{i}\) has no significant effect on \(Y\). On the other hand, \(H_{1}\) represents the alternative hypothesis in which the regression coefficient \(\beta_{i}\) is not zero. If the null hypothesis is rejected, the alternative hypothesis is adopted, and the explanatory variable \(X_{i}\) explains \(Y\) statistically significantly. For the hypothesis testing, we use \(t\)-statistic defined in Equation (5) to verify the statistical significance of keyword \(X_{i}\) with respect to \(Y\) (keyword drone) [27]. \[t_{i}=\frac{b_{i}-\beta_{i}}{\text{SE}(b_{i})} \tag{5}\]  
where \(b_{i}\) is the estimator of \(\beta_{i}\) computed by minimizing the sum of squares of the error in the model of Equation (3). \(SE(b_{i})\) is standard error (SE) of \(b_{i}\) corresponding to keyword \(X_{i}\). We can reject or accept \(H_{0}\) by the value of \(t\)-statistics of (5). The larger the value, the more statistically significant the explanatory power of the corresponding keyword for drone. We used the statistic to find the significance probability (\(p\)-value) and used this value to determine the statistical significance of each keyword. For example, at the 95% confidence level, we determine a keyword with \(p\)-value less than 0.05 as statistically significant, that is, we reject \(H_{0}\)[28]. In this paper, we selected the explanatory variables (keywords) with \(p\)-value less than 0.05 in the multiple regression model. Lastly, we extracted the important keywords for drone technology using BART modeling. Compared to the existing classification and regression tree models that rely on only ensemble methods, BART is based on Bayesian probability distribution as well as ensemble of trees [29]. Therefore, BART is an regression approach based on nonparametric Bayesian [30]. This finds unknown function \(f\) using recursive partitioning of predictor space. In this paper, the dimension of predictor space is \(p\) and the size of this dimension were equal to the number of explanatory variables (keywords). In our BART model, we define the fundamental equation as (6). \[Y=\mathrm{f}(X)+\varepsilon \tag{6}\]  
The response variable \(Y\) is the keyword drone, and the explanatory variables are represented by \(X=\big{(}X_{1},X_{2},\ldots,X_{p}\big{)}=\big{(}Keyword_{1},Keyword_{2}, \ldots,Keyword_{p}\big{)}\). The error term \(\varepsilon\) is followed to normal distribution \(N\big{(}0,\sigma^{2}\big{)}\). We make inference about model \(\mathrm{f}(\cdot)\)Using sum of \(m\) distinct regression trees, we expressed the model of Equation (6) as Equation (7) as follows [29; 30; 31; 32]. \[Y=\text{Tree}_{1}^{M}(X)+\text{Tree}_{2}^{M}(X)+\dots+\text{Tree}_{m}^{M}(X)+\varepsilon \tag{7}\]  
where \(\text{Tree}_{i}\) is \(i\)th regression tree.