# Patent Keyword Analysis Using Time Series and Copula Models
## 3 Patent Analysis Model Using Copula Directional Dependence via Integer-Valued GARCH  
Patent-keyword matrix is a popular data structure for technology analysis based on patent document data. In this paper, we also use this data structure for the proposed methods. First we collected the patent documents from the popular patent databases in the world. Next, we transformed the patent data into the patent-keyword matrix using various text mining techniques such as text document collections (corpus). The matrix consists of patent (row) and keyword (column), and each element of the matrix represents the occurred frequency of each keyword in each patent. In this paper, we add another column to the matrix. The column is time factor (year), because our model analyzes the patent keywords over time. Figure 1 shows a patent-keyword matrix with the time factor developed by our research. In Figure 1, each frequency value of the matrix is an integer. So we need a statistical approach to integer value analysis that considers the time factor. In this paper, we apply time series analysis and count data analysis to build the integer-valued analysis with time for patent data analysis. Integer-valued time series data are commonly observed in real data applications as well as technology analysis. Dealing with integer-valued time series using ARMA (autoregressive moving average) models and GARCH (generalized autoregressive conditional heteroscedastic) models which are used for real-valued data is not suitable for accommodating integer-valued time series in patent analysis. Integer-valued time series are non-negative and their variance often changes over time, which means that the assumption of homogeneous variance seems to be improper for such a time series. The integer-valued GARCH (INGARCH) model was proposed by [PERSON] et al. (2006) [PERSON] et al. (2006). This model is used for integer-valued time series analysis. Also, the INGARCH model with Poisson deviates is an analogue of the GARCH model with normal deviates. \(X_{t}\) and \(F_{t\text{-}1}\) are integer-valued time series data at time \(t\) and information set up to time \(t\)-\(1\), and then the INGARCH(p,q) model is represented by a Poisson distribution as follows [PERSON] et al. (2006). \[X_{t}|F_{t-1}\sim P(m_{i}) \tag{1}\]  
Figure 1: Patent-keyword matrix with a time factor. where m is the parameter of the Poisson distribution, and this is defined as follows [20]:  
\[m_{i}=\alpha_{0}+\sum_{i=1}^{p}\alpha_{i}X_{t-i}+\sum_{j=1}^{q}\beta_{i}m_{t-j} \qquad\Big{(}\alpha_{0}>0,\ \alpha_{i}\geq 0,\ \beta_{j}\geq 0,\ p\geq 1,\ q\geq 0 \Big{)}. \tag{2}\]  
The INARCH(p) model is the INGARCH(p,q) model with q = 0. The conditional distribution of \(X_{t}\) is Poisson in the INGARCH(p,q) model, and then the conditional mean depends on the past values of the time series and its own past values. In addition, the INGARCH(1,1) model is shown as follows [20]:  
\[X_{t}|F_{t-1}\sim P(\alpha_{0}+\alpha_{1}X_{t-1}+\beta_{1}m_{t-1})\qquad\quad( \alpha_{0}>0,\ \alpha_{1}+\ \beta_{1}<1). \tag{3}\]  
Therefore, the conditional mean at time \(t\) E(\(X_{t}|F_{t-1}\)) is dependent on the value at time \(t\)-1 \(X_{t-1}\) and on its own value at time \(t\)-1 \(m_{t-1}\). In our research, the integer-valued GARCH model is combined with the copula directional dependence for patent keywords analysis over time. Copula is a multivariate function distributed on [0, 1] uniformly. The first proposed Copula by [PERSON] (1959) was a two dimensional function based on bivariate distribution as follows [21]:  
\[\rm{F(x,y)=C(F(x),\ F(y))=C(u_{1},u_{2})} \tag{4}\]  
where C(\(\cdot\)) represents the Copula function, and F(x,y) is a bivariate cumulative distribution of x and y. F(x) and F(\(y\)) are cumulatively marginal distributions of x and y, respectively. The range of F(\(\cdot\)) is [0,1].