## 3 Background
### Product Life Cycle Model
Time series data at different time intervals are derived from each other; for instance, monthly series are created from daily observations, and quarterly series are constructed from monthly data. When selecting an appropriate time series analysis method, factors such as the objective of the analysis, its economic relevance, and the type of time series must be considered, as there is no universally applicable model. Time series emerge as a result of both significant and minor influencing factors. These data can be analyzed using both descriptive and inductive approaches, providing valuable insights for decision-making and strategic planning. The classical time series model is based on the decomposition of data into four key components:  
* Trend (development). * Seasonal component. * Cyclical component. * Random (residual) component. Among these, the trend is the most fundamental characteristic of time series development. It represents the long-term trajectory of a variable, reflecting its underlying directional movement over time. The simplest conceptualization of a trend is a linear increase or decrease, represented by a straight line fitted to actual data points. However, more complex mathematical curves can be used for smoothing time series data, providing a more accurate representation of real-world patterns. The most commonly used trend models include [22]  
1. Linear trend--represents a constant rate of increase or decrease over time: \[Tr_{t}\ =\ a_{0}+a_{1}\cdot t\] (1)
2. Quadratic Trend--captures acceleration or deceleration in the trend, allowing for curvature in the trajectory: \[Tr_{t}\ =\ a_{0}+a_{1}\cdot t+a_{2}\cdot t^{2}\] (2)
3. Hyperbolic Trend--models trends that initially change rapidly but gradually level off over time: \[Tr_{t}\ =\ a_{0}+\frac{a_{1}}{t}\] (3)
4. Exponential Trend--suitable for processes exhibiting exponential growth or decay: \[Tr_{t}\ =\ a\cdot b^{t}\] (4)
5. Modified Exponential Trend--a variation in the exponential trend that incorporates an additional constant to adjust for baseline effects: \[Tr_{t}\ =\ k+a\cdot b^{t}\] (5)
6. Logistics Trend--commonly used for modeling constrained growth, such as product adoption or population saturation: \[Tr_{t}\ =\ \frac{k}{1+a\cdot b^{t}}\] (6)
7. Gompertz trend--a sigmoid-shaped trend model that is asymmetrical, often applied in forecasting sales, population growth, and biological processes: \[Tr_{t}\ =\ k\cdot a_{0}^{a_{1}^{t}}\] (7)  
The most common approach to describing the product life cycle is through trends with asymptotic bounds, specifically S-curves or sigmoidal curves. The S-curve function serves as a universal mathematical model for various growth processes characterized by an initial slow phase, followed by rapid growth, and finally, a stabilization phase. S-curves are trend functions commonly used to model the sales trajectory of a new product. Initially, sales grow slowly due to limited consumer awareness. Over time, as a result of advertising and market penetration, sales accelerate before eventually stabilizing at a constant level, reflecting market saturation. This pattern is best represented by a modified exponential trend, which can be applied to different transformations of the time series. When applied to the logarithmic transformation of sales data \(lny_{t}\), it results in the Gompertz S-curve. Alternatively, when applied to the reciprocal transformation \(\frac{1}{y_{t}}\) it yields the logistic trend or Pearl-Reed trend, both of which effectively capture the dynamics of product adoption and market saturation. Non-parametric tests are commonly employed for trend analysis, including the Mann-Kendall test, Mann-Whitney-Wilcoxon test, Pettitt test, Standard Normal Homogeneity Test (SNHT), Cumulative Sums (CUSUM) test, and the Two-Sample Kolmogorov-Smirnov test, among others. These tests are particularly valuable when they not only assess the statistical significance of a change point but also identify the precise timing of the change.