## 3 Background
### Specification of the Gompertz Curve  
The Gompertz curve is described by an equation that depends on two parameters, a and b, which influence its shape. Parameter a represents the maximum value that sales tend to reach, while parameter b characterizes the rate of growth or saturation. One of the key properties of the Gompertz curve is its ability to predict both the initial and final behavior of the system. It is often used to forecast sales growth and to analyze market dynamics as well as the adoption of new technologies [23; 24].  
The Gompertz curve depends on two parameters, \(\alpha_{0}\) and \(\alpha_{1}\), which influence its shape. Parameter \(\alpha_{0}\) represents the maximum sales level that the trend is expected to reach, acting as the upper asymptote, while parameter \(\alpha_{1}\) characterizes the rate of growth or the speed at which the process, such as sales or market penetration, progresses over time. These parameters define the asymmetrical S-shape of the Gompertz curve, making it particularly useful for modeling growth processes where initial adoption is slow, followed by a phase of rapid expansion, and ultimately stabilizing as market saturation is reached [23; 24].  
The Gompertz trend belongs to the group of S-curves and has the following form:  
\[Tr_{t}\ =\ k\,\alpha_{0}^{a_{1}^{t}}+\varepsilon_{t} \tag{16}\]  
where  
\(\alpha_{0}\), \(\alpha_{1}\), \(k\) are unknown parameters,  
\(k\)--the unknown boundary parameter \(k\) gives, for example, the upper limit of sales of goods at its saturation, \(t\) = 1, 2,..., n is a time variable,  
\(\varepsilon_{t}\) is a random error.  
To perform parameter estimation, we first linearize the function [23; 24]:  
\[lnTr_{t}\ =\ \ln k+\alpha_{1}^{t}\cdot ln\alpha_{0} \tag{17}\]where  
\[Tr_{t}^{\prime}\ =\ lnTr_{t},\ k^{\prime}\ =\ lnk,\ a_{0}^{\prime}\ =\ ln\alpha_{0} \tag{18}\]  
The parameters of the modified exponential trend can be calculated using the previously mentioned partial sum method. Unlike the logistic trend, the function's progression is asymmetrical, with most of the values occurring after the inflection point (Figure 3). The curve has an upper positive asymptote at parameter \(k\).  
When assessing the suitability of a model, priority should be given to substantive considerations supplemented by statistical criteria. We usually examine the size of the differences \(y_{t}\ =\ T_{tr}\) (the actual values of the variable \(Tr_{t}\) at time \(t\)) and \(\hat{y}_{t}\) (the adjusted values of the variable \(Tr_{t}\) at time \(t\)), i.e., the residuals \(y_{t}-\hat{y}_{t}\) at time \(t\) = 1, 2,..., n. The accuracy of the model can be evaluated using basic measures of accuracy (absolute and relative measures), for example, using the Mean Square Error. The Mean Square Error (MSE) expresses the average squared difference between the estimated values and the actual values. It is calculated using the following formula [23]:  
\[MSE\ =\ \frac{1}{n}.\sum\left(Tr-\widehat{Tr}_{t}\right)^{2} \tag{19}\]  
where  
\(Tr\) is the actual value,  
\(\widehat{Tr}_{t}\) is the estimated value.  
\(n\) is the total number of observations.  
It can be stated that the Mean Square Error (MSE) estimates the amount of error in a statistical model; for example, how accurately the regression curve fits the observed (actual) values. For example, a company has a model that predicts sales revenue for a one-year period. At the end of the year, the company has actual sales figures. Subsequently, it is possible to calculate the MSE, which ultimately determines how well the model predicts (describes) the outcome. This evaluation helps in assessing the accuracy and reliability of the forecasting model, and the lower the MSE, the better the model's predictions align with actual results.  
Figure 3: Logistics trend [23].