# Emergence of Nano-Dentistry as a Reality of Contemporary Dentistry
## 3 Results
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Year** & \(x\) **(Year)** & \(y\) **(Number of Papers)** & \(\hat{y}\) **(Number of Papers (Model))** & \(u\) **(Residuals)** \\ \hline
2006 & 6 & 2 & 2.1 & 0.1 \\
2007 & 7 & 4 & 2.9 & \(-1.1\) \\
2008 & 8 & 3 & 3.8 & 0.8 \\
2009 & 9 & 9 & 4.9 & \(-4.1\) \\
2010 & 10 & 6 & 6.1 & 0.1 \\
2011 & 11 & 7 & 7.4 & 0.4 \\
2012 & 12 & 4 & 8.9 & 4.9 \\
2013 & 13 & 6 & 10.6 & 4.6 \\
2014 & 14 & 16 & 12.3 & \(-3.7\) \\
2015 & 15 & 14 & 14.2 & 0.2 \\
2016 & 16 & 26 & 16.3 & \(-9.7\) \\
2017 & 17 & 17 & 18.5 & 1.5 \\
2018 & 18 & 18 & 20.8 & 2.8 \\
2019 & 19 & 22 & 23.3 & 1.3 \\
2020 & 20 & 34 & 26 & \(-8\) \\
2021 & 21 & 32 & 28.7 & \(-3.3\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Results of modeling using power model (9). Figure 7: Residual plot of power model (9). After substituting the variables, we obtained the linear model  
\[\hat{y}^{*}=a_{0}^{*}+a_{1}^{*}\times x. \tag{13}\]  
To estimate the coefficients, we used the least squares equations derived in [14; 15]. The coefficient of determination of the linearized model is 0.846, which is a good indicator [14; 15]. It indicates good fit of the chosen regression to data. The model as a whole is significant, but the coefficient \(a_{0}\) is not statistically significant. Then we returned to the form of model (10) by inversely substituting variables  
\[a_{0}=e^{a_{0}^{*}},\ a_{1}=e^{a_{1}^{*}}. \tag{14}\]  
We obtained the model as follows  
\[\hat{y}=0.992\times 1.1863^{x}. \tag{15}\]  
Full results of the modeling are presented in Table 3.  
The residual plot of the exponential regression is given in Figure 8.  
Figure 8: Residual plot of exponential model (15). \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Year** & \(x\) **(Year)** & \(y\) **(Number of Papers)** & \(\hat{y}\) **(Number of Papers (Model))** & \(u\) **(Residuals)** \\ \hline
2006 & 6 & 2 & 2.8 & 0.8 \\
2007 & 7 & 4 & 3.3 & \(-0.7\) \\
2008 & 8 & 3 & 3.9 & \(0.9\) \\
2009 & 9 & 9 & 4.6 & \(-4.4\) \\
2010 & 10 & 6 & 5.5 & \(-0.5\) \\
2011 & 11 & 7 & 6.5 & \(-0.5\) \\
2012 & 12 & 4 & 7.7 & 3.7 \\
2013 & 13 & 6 & 9.1 & 3.1 \\
2014 & 14 & 16 & 10.8 & \(-5.2\) \\
2015 & 15 & 14 & 12.9 & \(-1.1\) \\
2016 & 16 & 26 & 15.3 & \(-10.7\) \\
2017 & 17 & 17 & 18.1 & 1.1 \\
2018 & 18 & 18 & 21.5 & 3.5 \\
2019 & 19 & 22 & 25.5 & 3.5 \\
2020 & 20 & 34 & 30.2 & \(-3.8\) \\
2021 & 21 & 32 & 35.8 & 3.8 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Results of simulation using exponential model (15). As it can be seen from Figure 8, the amplitude of the residuals is approximately constant for all points of the model and their distribution is random. However, there is an outlier for 2016. According to [14; 15] this is a serious drawback of the chosen model. It makes such the model not very good for use. As it was mentioned above, the coefficient \(a_{0}^{*}\) is insignificant in the linearized model. This is probably due to the fact that the coefficient \(a_{0}\) is close to one, and the logarithm of one is zero. Due to the statistical feature of the \(t\)-test any coefficient close to zero cannot be considered significant. That is why we had reason to consider both coefficients of model (15) significant. To check this assumption, we tried a linearized exponential model without an intercept  
\[\hat{y}^{*}=a_{1}^{*}\times x. \tag{16}\]  
However, such model is not good due to the increase in the amplitude of the residuals with the rise of the independent variable, as shown in Figure 9.  
As can be seen from the analysis, simple models have drawbacks in their application to fit the dependence of the number of publications on year. The use of complex models is not reasonable due to the relatively small set of observations (16 observations) and possible overfitting [14; 19]. That is, the model will describe this data set well for the cost of losing of generalization properties. Therefore, we decided to focus on model (15) and chose it as the one that has the least shortcomings among the considered models.