## 4 The Implementation of Trends and Results
Furthermore, all model coefficients are statistically significant, with \(p\)-values of 0.0000, confirming the robustness of the estimates. The standard deviation of the residuals is 18.823, reflecting the degree of variation in the model predictions. Although the quadratic model is more consistent with the observed sales patterns of product B, its predictive ability remains limited, especially when predicting long-term fluctuations in demand. \[Tr_{t}\ =\ 151.2000-1.9663\cdot x_{t}+0.0092\cdot x_{t}^{2} \tag{23}\]  
Figure 7: Predicted sales interval for product B. Prediction interval 95% for B (18.567; 206.930). From a product life cycle and nonlinear trend perspective, sales of product B initially show a gradual increase during the first year, reaching a peak in December 2020. Two data points (\(t\) = 1 (January 2020) and \(t\) = 12 (December 2020) lie outside the prediction interval, and one point (\(t\) = 32 (August 2022) lies at the lower boundary of the prediction interval, indicating deviations from the expected trend. Following a period of fluctuation, the sales of product B have experienced a significant decline, indicating a decline in product attractiveness. Given this downward trend, product B either requires innovation to regain market interest or should be discontinued to optimize resource allocation and market competitiveness. The sales of product C are also described by three types of trends, namely linear, nonlinear and Gompertz trend. The linear sales trend of product C (Figure 8) is described by the equation:  
\[T_{T_{t}}\ =\ 37.5310+0.9264\cdot x_{t} \tag{24}\]  
The red line shows the predicted Y for any X value. The blue dashed lines show the 95% prediction interval. Black bullets show the quantities of products sold. Figure 9 presents the quadratic trend of product C sales. t trends shows that the quadratic trend is suitable at the level of 96.86% and the linear trend at the level of only 70.41%. For the quadratic trend, the model coefficients are statistically significant, with \(p\)-values of 0.0000. The standard deviation of the residuals for the quadratic trend is 3.402, which reflects the degree of variation in the model predictions. \[T_{T_{t}}\ =\ 14.6695+3.1388\cdot x_{t}-0.0363\cdot x_{t}^{2} \tag{25}\]  
The quadratic trend is more consistent with the observed values of product C sales. The peak of product C sales was reached at time \(t\) = 29 (May 2022), specifically the number of products sold was 8125 pieces. Subsequently, a slight decrease in product C sales can be seen. The exponential trend presented by the Gompertz curve:  
\[T_{T_{t}}\ =\ 78.8736\cdot 0.1083^{0.8695^{\prime}} \tag{26}\]  
Figure 8: Linear sales trend for product C.  
Based on the values of the Akaike Information Criterion (AIC) and Bayesian Schwarz Information Criterion (BIC) indicators, it is possible to select a suitable trend for the data under study. The values in Table 3 show that the lowest AIC/BIC values are for the Gompertz trend. The Mann-Kendall analysis (Table 4), when performed for the current dataset, found that, for S (variance) = 124, \(Z_{\rm S}\) = 2.0906. The resulting \(p\)-value of the test is \(p\) = 0.0366, which means that the trend is significant at the 5% and 10% significance levels. However, at the 1% significance level, the data do not show a significant trend. Therefore, with 90% and 95% confidence, it can be concluded that the current sales of product C demonstrate a significant trend. Given that the value of S (variance) is 124, this further supports the presence of a sales trend for product C.  
Figure 10 presents the sales of product C--enclosed terminal blocks. The lowest value of the mean square error (MSE) is 9.0573 precisely at the Gompertz trend, i.e., the deviation