Gompertz curve

The different curves obtained by increasing the number of terms in the Taylor expansion are represented in Figure 3.3 on top of the Gompertz curve itself. The exponential growth model can thus be now justified not only because it fits well the data but also because it can be seen as a first approximation to the Gompertz growth model, which is endowed with a mechanistic interpretation, namely, competition between the catabolic and anabolic processes. 

In practice most distribution curves are not symmetrical about the median but are inherently skewed. The effect of an advertising campaign is usually to increase the rate of sales in the early years. It may also increase the level of mature demand for the product, but this mature demand must be asymptotic to a finite upper limit of sales c. Such a curve is positively skewed since (x - Xl) < (xy - x ). This situation can often be approximated by the Gompertz curve defined by Eq. (9-96) ... 

The underlying frequency distribution curve of the Gompertz curve may be obtained by diflFerentiation of Eq. (9-96) to give... 

From equations (18) and (19) we find or = 191 days in case the two highest points mentioned above are omitted (S = 222 days in case they are included). Comparing the value 191 days with the value a = 161 days based on the Gompertz death-rate curve (670-day life span) and the value = 258 days inferred from Dr. Fiokers data, we see that the theoretical curve fits the experimental values about as well as would be expected from the Gompertz curve, and somewhat better than would have been expected on the basis of Dr. Finkel s acceptance region. 

Gompertz is a United Kingdom statistician and mathematician who discovered a type of curve used to predict the population growth in 1820. American scholar R. Prescott firstly applied this curve to forecasting market development in 1922 [3, 5, 6]. Here is Gompertz curve equation in general form ... 

K, a and b are parameters, t is time parameter. Gompertz curve is double index, also called double exponential curve. According to different values of a and b, the curve has four types shown in Fig. 3.2. 

This study focuses on the numerical example of using Gompertz curve approximation. 

The calculated value of (log Y, — log F, i)/(log F, i log F, 2) varies between [0.085941, 1.0444288], approximately tending to constant 0.86148976 shown in Table 3.1. This shows the raw data have a growth curve characteristics. Gompertz curve can be used for approximate calculation of dynamic equation (3.4) parameter K. The following Sanwa method is used to estimate value of K. 

Yang YQ, Wu LH, Zhang HE (1996) Comparisons between logistic curve and gompertz curve. J Biomath 11(3) 166-168... 

Feng LY, Wu XD (2003) Application of gompertz curve in commodity life-cycle study. J Inner Mong Univ Financ Econ 4 1—3... 

Winsor, C. P. (1932). The Gompertz curve as a growth curve. Proceedings of the National Academy of Sciences, USA, 18,1-8. 

For the record, there are constancies in these equations. The Gompertz curve implies a constant rate of decline in the increments to the logarithm of the observed value. Logistic growth connotes a constant rate of dechne in the decrements to the reciprocal of the observed value. These features become evident when the equations are converted to the form of a modified exponential curve. 

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