# Patent Keyword Analysis Using Time Series and Copula Models
## 3 Patent Analysis Model Using Copula Directional Dependence via Integer-Valued GARCH
So \(u_{1}\) and \(u_{2}\) are part of a uniform distribution as follows:  
\[u_{1},\ u_{2}\sim Uniform(0,1). \tag{5}\]  
Copula provides the dependent process between the random variables of x and y by using a marginal distribution with removing influences. To control the variables with heteroskedasticity, we used the bounded time series model based on beta regression analysis [22]. Using this model, we directly interpreted the regression parameters with an original response scale. In this paper, we propose Copula directional dependence via the integer-valued GARCH based beta regression model. In the proposed model for patent keyword analysis, \(Y_{t}\) is a response variable (keyword) on the unit interval at time t, and \(x_{t}\) is a covariate vector (keyword vector). In the beta regression model, \(Y_{t}|x_{t}\) follows the beta distribution as follows [23]:  
\[\rm{f(}y_{t}|\alpha,\beta)=\frac{1}{B(\alpha,\beta)}y_{t}^{\alpha-1}(1-y_{t}) ^{\beta-1}\ \ 0<y_{t}<1,\ \alpha>0,\ \beta>0 \tag{6}\]  
where \(\alpha\) and \(\beta\) are the parameters of the beta distribution for mean and precision, and \(B(\alpha,\beta)\) is the beta function based on gamma function \(\Gamma(\cdot)\) as follows [23]:  
\[B(\alpha,\beta)=\frac{\Gamma(\alpha)\Gamma(\beta)}{\Gamma(\alpha,\beta)}. \tag{7}\]  
In our research, the dependence of \(y_{t}\) on \(x_{t}\) was obtained by a logit mode of mean parameter(\(\mu_{t}\)) as follows:  
\[\rm{logit}(\mu_{t})=x_{t}^{T}\beta_{x} \tag{8}\]  
where \(x_{t}^{T}\) is a transpose of \(x_{t}\), and \(\beta_{x}\) is coefficient vector. Our proposed copula directional dependence by integer-valued GARCH model, which is based on nonlinear logit model, is more flexible than the current available approaches to directional dependence by linear regression type model [24]. To verify the accuracy of the model, we used two measures: Akaike information criterion (AIC) and Bayes information criterion (BIC). The AIC is defined as follows [25]:\[\text{AIC}=\text{nlog}\Big{(}\frac{SSE}{n}\Big{)}+2P \tag{9}\]  
where n and P are sample size and model parameters. SSE represents the error sum of squares. We determined the best model with the smallest AIC value. Next, the BIC was represented as follows [25]:  
Figure 1:0  
We also selected the best model with the smallest BIC value. In next section, we carried out various experiments to illustrate the validity of our research models using the AIC and BIC measures. Figure 2 shows the proposed patent keyword analysis process using time series and copula models. Once the target technology for analysis was selected, we collected the patent documents related to the target technology from all the patent databases in the world. Using the collected patent document data, we constructed a patent-keyword matrix, and the row and column of this matrix were the patent and the keyword. The value of this matrix was the occurred frequency of each keyword in a patent. Next we extracted the core keywords related to target technology from the matrix. To understand the technological structure of the target domain we first performed a visualization using a count time series and autocorrelation function plots. For more detailed technology analysis, we used the INGARCH model and estimated the parameters of this model. For the evaluation of the final model, we used AIC and BIC measures. Based on the final model after the evaluation, we used copula direction dependence to find the technology relation structure for the target technology. The results of our study can be used for the R&D planning of countries and companies. A company can also use the results to develop new innovative products, and conduct technology transfer and commercialization.