# Zero-Inflated Patent Data Analysis Using Generating Synthetic Samples
## 4 Results
### Simulation Study with Zero-Inflated Data  
We obtained our simulation data by generating random numbers from multivariate Poisson distribution. The process for the generation of multivariate generalized Poisson data consists of two steps as follows [14, 29, 30, 31, 32, 33]:  
(Step1)  
\[X_{i}=C_{i}+C_{0},\ C_{i}\ \sim\ Poisson(\lambda_{i}),\ i=1,\ 2,\ 3,\ 4\]  
\[(X_{1},X_{2},X_{3},X_{4})\ \sim\ Multivariate\ Poisson(\lambda_{1},\lambda_{2}, \lambda_{3},\lambda_{4}),\ \lambda_{i}>0\]  
(Step2)  
\[P(X_{1}=x_{1},X_{2}=x_{2},X_{3}=x_{3},X_{4}=x_{4})=\sum_{k=0}^{\min(x_{1},x_{2},x_{3},x_{4})}p_{0}(k)\prod_{i=1}^{4}p_{i}(x_{i}-k)\]  
\[x_{1},x_{2},x_{3},x_{4}=0,1,2,3,\]  
where \(p_{i}\) is the probability function of the Poisson distribution with mean = \(\lambda_{i}\). In our simulation study, we generated four variables \((X_{1},X_{2},X_{3},X_{4})\) by Steps 1 and 2. In this simulation study, we used the R data language and a package provided by R for simulation data generation [14, 31]. We performed two simulations with different means and correlation structures. Table 3 shows the parameter settings. The data size of each simulation is 20,000. The correlation coefficients in simulation 1 are larger than those in simulation 2. In addition, the mean values of two simulation data are different. Table 4 illustrates the descriptive statistics of simulation data. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Simulation** & **Statistic** & **Variable** & **X1** & **X2** & **X3** & **X4** \\ \hline \multirow{3}{*}{Data 1} & Mean & & 1.0 & 3.0 & 4.0 & 2.0 \\ \cline{2-6}  & Correlations & X2 & & 0.352 & 0.265 & 0.342 \\  & Correlations & X3 & & & 0.152 & 0.352 \\  & & X4 & & & & 0.265 \\ \hline \multirow{3}{*}{Data 2} & Mean & & 1.0 & 1.0 & 2.0 & 2.0 \\ \cline{2-6}  & Correlations & X2 & & 0.200 & \(-\)0.300 & 0.002 \\ \cline{1-1}  & Correlations & X3 & & & 0.001 & 0.003 \\ \cline{1-1}  & & X4 & & & & 0.004 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Correlation structure for simulation data. Table 4 represents descriptive statistics of the variables with variances and correlation coefficients from each simulation datapoint. It shows the difference of statistics between two simulation data such as the parameter settings. Next, we illustrate the results of performance evaluation between comparative methods using simulation datasets. Table 5 shows the simulation result using the highly correlated simulation data 1. In our simulation study, we used four variables \((X_{1},X_{2},X_{3},X_{4})\) to build a linear model as:  
\[X_{1}=b_{1}+b_{2}X_{2}+b_{3}X_{3}+b_{4}X_{4}. \tag{9}\]  
The \(X_{1}\) is a response variable and the others are explanatory variables in the simulation model. As in Table 2, Table 5 shows the parameter estimations of each explanatory variable with their standard errors and AIC for each method as well. We knew the AIC of the zero-inflated models are smaller than that of models with original and noise-added data from Table 2. In addition, the synthetic sample had the smallest AIC among the comparative models. It is confirmed that the model by synthetic samples also has the best performance \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Simulation** & **Statistic** & **Variable** & **X1** & **X2** & **X3** & **X4** \\ \hline \multirow{6}{*}{Data 1} & Min. & 0 & 0 & 0 & 0 \\  & Median & 1 & 3 & 4 & 2 \\  & Mean & 1.004 & 3.000 & 3.992 & 1.994 \\  & Max. & 7 & 12 & 14 & 10 \\ \cline{2-6}  & Variances & X1 & 0.993 & 0.348 & 0.263 & 0.335 \\  & (diagonal) & X2 & & 3.026 & 0.145 & 0.355 \\  & \& X3 & & & 4.000 & 0.256 \\  & Correlations & X4 & & & & 2.001 \\ \hline \multirow{6}{*}{Data 2} & Min. & 0 & 0 & 0 & 0 \\  & Median & 1 & 1 & 2 & 2 \\  & Mean & 1.011 & 1.015 & 1.995 & 2.006 \\  & Max. & & 7 & 7 & 11 & 10 \\ \cline{2-6}  & Variances & X1 & 1.010 & 0.185 & \(-\)0.318 & \(-\)0.005 \\  & (diagonal) & X2 & & 1.022 & \(-\)0.009 & \(-\)0.004 \\  & \& X3 & & & 2.024 & 0.012 \\  & Correlations & X4 & & & & 2.009 \\ \hline \hline \end{tabular}
\end{table}