4. Comparison Results
4.3 Suggestions for Applying Forecasting Models
Based on the comparison and sign test results, the time-varying extended logistic model statistically significant outperforms than the simple logistic and Gompertz models. Since the time-varying extended logistic model uses more parameters to capture the trend of products, the fit and forecast performance are improved. This research used 22 product datasets to test the performance of the simple logistic, the Gompertz and the time-varying extended logistic model. However, the datasets for LCD panel for notebooks, color-65k mobile phones, servers, and VoIP routers, would not converge when using the time-varying extended logistic model to estimate the coefficients. A similar situation was reported by Meade and Islam (1998). Their research used 25 telecommunications market datasets to compare the performance of seventeen growth curve models. For their study, half of the datasets would not converge when estimating the coefficients of models. Our study showed four products would not converge among 22 products yielding a proportion less than 20%. Therefore, our convergence results are consistent with earlier research.
The four products with data that would not converge provide some insight. These datasets are linear and the curve for the color-65k mobile phone has an obvious jump (Figure 4). Meade and Islam’s (1995) research used telephone data from Sweden to compare the simple logistic, extended logistic, and the local logistic models. They concluded that the extended logistic model had the worst performance. Although the setting of the extended logistic model is different with this research, Meade & Islam’s study serves as a useful example. The growth curve of the Swedish telephone data set is linear. Therefore, the time-varying extended logistic model should not have been used.
If forecasters wish to apply the time-varying extended logistic model, then they should
confirm that the data has an S-shape prior to the forecast.
When using the simple logistic and Gompertz models, the upper limit (L) must be set and then a linear transformation method is applied to calculate parameters using equation (2) and (4). Since only two parameters are estimated, it is easy for the models to converge. However, since the upper limit of the time-varying extended logistic model is dynamic with time, more parameters are needed to capture the trace. Therefore, the linear transformation method used in the simple logistic and Gompertz models cannot be used to estimate the parameters and a nonlinear estimation method must be used. For the cumulative sales volume dataset, five parameters are estimated using nine to thirteen data points which causes an increase in non-convergence for the extended logistics model. Therefore, this research suggests there should be at least fifteen data points in a database to apply the time-varying extended logistic model to reduce the probability of non-convergence.
The problem of the simple logistic and the Gompertz models is because they are limited by the characteristic and the shape of the growth curve. For example, the simple logistic curve is symmetric about the point of inflection, so if the current data points have not reach the point of inflection of the growth curve, then the upper limit can not be estimated and the simple logistic does not predict well. The Gompertz curve is an asymmetric S-curve and the Gompertz model reaches the inflection point before the market penetration has reached half the upper limit (Meade &Islam, 1998). Thus, the Gompertz model may be more suitable for certain types of short lifecycle products than the simple logistic model. As shown in Table 4 and Table 6, the different capacity setting will lead to different forecasting results, and the wrong capacity will lead to an error in prediction. If industrial policy or enterprise decisions are made based on a model using the wrong upper limit, a serious forecast error can be made. However, if
the point of inflection of the growth curve has occurred, the maximum value can be estimated based on the characteristics of the simple logistic and the Gompertz models and then these two models can also be good forecasting models. Based on the discussions, a comparison for the pros and cons of the three models applied in this research is presented in Table 9. Table 9 also outlines the model setting, characteristics, and the timing for applying the three models.
Table 9 The comparison of the time-varying extended logistic, Gompertz, and the simple logistic models
Extended logistic model Simple logistic model Gompertz model
Model setting
( )
z The upper limit is dynamic with time
z Symmetric about the point of inflection
z The point of inflection occurs at half of the upper limit
z Asymmetric about the point of inflection
z The point of inflection occurs at 37.79% of the upper limit
Pros
z A time-varying upper limit
z Suitable for predicting both long and short lifecycle products at early stage
z Only two parameters to estimate, so it is easy to apply
z Suitable for long historical data
z Only two parameters to estimate, so it is easy to apply
z Suitable for long historical data
Cons
z Can be used to the data with an S-shape curve
z More parameters needed
z Need a clear upper limit setting before forecasting
z Need a clear upper limit setting before forecasting
When to apply
z The point of inflection has not occurred and the dataset has more than fifteen data points
z An S-shape curve (no obvious jumps or a flat curve)
z The extended logistic model cannot reach convergent results
z The point of inflection occurred
z The extended logistic model cannot reach convergent results
z The point of inflection occurred
Note z Can shorten the time interval of data to create more data points