# Zero-Inflated Patent Data Analysis Using Generating Synthetic Samples
## 3 Methods
Since it has the strength to minimize information loss in the process of synthesis [15; 17; 18; 19; 20; 21], the CART is used efficiently for generating synthetic samples. CART is a decision tree method for predictive modeling tools in data analysis for regression and classification [22]. It partitions the input space recursively and defines regions of divided input space by the objects included in each region [23; 24]. A synthetic sample of count data (frequency of keyword occurred in patent) is generated using regression and classification costs for patent keywords. First, the regression cost is computed as Equation (3) [23];  
\[Cost_{regression}(m)=\sum_{i\in m}(y_{i}-\overline{y})^{2}, \tag{3}\]  
where \(\overline{y}\) is the mean of the response keyword and \(m\) is data in tree node. Second, the classification cost is defined as Equation (4) [23]. \[Cost_{classification}(k)=1-\sum_{k}p_{k}^{2}, \tag{4}\]  
where \(p_{k}\) is the probability of data objects in nodes belonging to class \(k\). This classification cost could be determined as the Gini index [23]. The cost functions are used to generate synthetic samples using CART and the synthesizer is used based on CART to generate the synthetic data from the patent-keyword matrix. The data consist of \(n\) sample units;  
\[\left\{x_{patent}=(x_{1},x_{2},\ldots,x_{n}),y_{patent}=(y_{1},y_{2},\ldots,y_ {n})\ \right\}, \tag{5}\]  
where \(x_{patent}\) is unchanged patent data and \(y_{patent}\) is a synthesized data that is \((n\times p)\) matrix with \(p\) keywords. The row size of the original data and the synthetic data are set to be same. Patent-keyword matrix is assumed as a sample from a population with parameters estimated by synthesizer, \(y_{patent}\). The synthesizer is sampled from \(\text{g}\big{(}Y|x_{patent},\varphi\big{)}\) where \(\text{g}(\cdot)\) is a probability distribution function and \(\varphi\) is a parameter vector. We defined the joint distribution as a sequence of conditional distributions. Figure 4 shows the sequential modeling of the CART method to generate synthetic patent data. Figure 3: Proposed method for zero-inflated data analysis. In Figure 4, a selected keyword is distributed by the conditional probability density and estimated by \(P\big{(}y_{patent}\big{|}x_{patent}\big{)}\). The distributions of the next selected keywords are estimated by conditional distribution given by \(x_{patent}\) and the previous keywords of \(y_{patent}\). Each patent of synthetic samples is generated from the distribution conditional on \(x_{patent}\). The parameters of the conditional distribution are the synthesized values of the previous keywords of \(y_{patent}\). In another way, the synthetic values are generated from the posterior distribution of the parameters. Figure 5 illustrates our proposed methods for generating synthetic samples. In Figure 5, the synthesizer generates a synthetic patent-keyword matrix from original patent-keyword matrix using CART and simple random sampling. Additionally, the synthetic samples are generated by the parameters sampled and estimated from posterior distribution. To evaluate the model performance of zero-inflated data analysis, four popular tests are commonly performed. They are the boundary likelihood ratio test, Vuong test, Bayesian information criterion (BIC), and Akaike information criterion (AIC) [1,5,6]. We used AIC to compare synthetic sample generation to other methods because AIC is based on a log-likelihood function to evaluate the output of statistical models [1,5,6]. The AIC is shown as [22,23,25]:  
\[\text{AIC}=-2(logL-p), \tag{6}\]  
where \(logL\) is a log-likelihood and \(p\) is the number of parameters from a given model. The smaller AIC represents the better model performance. In the next section, we will use AIC to compare the method of synthetic sample generation to others. Figure 4: Sequential modeling of CART method to generate synthetic patent data. Figure 5: Generating synthetic samples from patent-keyword matrix.