Growth Models for the Forecasting of New Product Market Adoption

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1 Growth odels for the Forecasting of New Product arket Adoption LADEN SOKELE Appropriate forecasting of product market adoption enables optimal planning of resources, investments, revenue, marketing and sales. Quantitative forecasting methods for the new product adoption rely on the S-shaped sigmoidal) growth model such as the logistic, the Richards and Bass growth models. This paper presents adaptations of these models by introducing explanatory and marketing variables which are suitable for the forecasting prior to product launch or in the early phases of the product life cycle. laden Sokele is Assistant Director in Croatian Telecom Inc, Zagreb 1 Introduction Growth models are widely used in quantitative research in order to understand the forces that influence growth in the sense of its dynamics, market capacities as well as forecasting of growth in the future. Particularly, diffusion of innovation and new technology, market adoption of consumer durables, number of users of subscription services for example: telecommunications services) and allocation of restricted resources at the beginning of a product life cycle PLC) have sigmoidal growth. Such products/ services that do not include repeaales are denoted products, and the cumulative volume of adopted products by users, number of users. Growth models based on the logistic, Richards and Bass models involving explanatory and marketing variables will be presented and examined for the forecasting application in cases of new product market adoption. 2 Growth Forecasting Growth forecasting relies on the basic principle: The growth model will be valid in the perceivable future, and the forecasting result could be obtainable by extrapolation of observed values sequentially through time and supplementary information. In general, this principle is valid only for stable markets where internal forces remain the same eg. same markeegment boundaries, competition, cause-and-effect among products, etc) and without change of external influences eg. technology, macroeconomics, purchasing power, regulatory, etc changes). This type of forecasting belongs to the quantitative time series methods [1, 11]. For time series growth model f t i ; α 1, α 2,..., α k ) based on a set of k parameters {α i }, at least k known data points t i ; Nt i )) are needed for the complete parameter estimation. In cases when exactly k data points are available, parameters α i are solutions of a system of equations 1): Nt i ) f t i ; α 1, α 2,..., α k ) =, i = 1,..., k 1) System 1) is usually a nonlinear system, so iterative numerical methods for its solution are needed to be performed eg. Newton method). In cases when k or more data points are available, the weighted leasquares method can be used for parameter estimation to adjust the parameters of a model so as to best fit a data set. The objective is to minimize the sum of squared difference between data points and model evaluated points: n S = w i [Nt i ft i ; α 1,α 2,..., α k )] 2 2) i=1 where w i are weights. When weights are equal to 1 w i = 1), the method is called ordinary leasquares method OLS). inimisation of 2) can be done by software tools such as icrosoft Excel solver. Analytically, values of parameters are resulting from solution of the system of equations 3): S α j =, j =1,..., k 3) By use of leasquares method, obtained values for parameters are statistically smoothed; ie. the influence on parameter values due to particular measurement errors such as unanticipated seasonal variation, uncertain measure, etc) is reduced. For forecasting purposes, parameter estimation is usually focused on the time interval near the last observed data point. Thus, weights in equation 2) can be set to higher value for the most recent data points than for data points in far history. For example, the geometric series for weights w i = 1 q n i, q > 1 4) 144 ISSN Telenor ASA 29 Telektronikk 3/4.28

2 leads to the following weights: 1 for the last known point) t n, 1/q for t n-1 the penultimate known point), 1/q 2 for t n-2, etc. In some forecasting cases, model f t i ; α 1, α 2,..., α k ) is modified to include the fixed value of the last data point t n, Nt n )). Therefore, the model has one parameter less, because α k is obtained from the equation: Nt n ft n ; α 1,α 2,..., α k )= The above mentioned simplification is used only when it is certain that the last data point is obtained with negligible measurement error. 5) Furthermore, relationships between model parameters and explanatory and marketing variables can be used for forecasting purposes, aiming at additional reduction of the number of unknown parameters in the growth model f t i ; α 1, α 2,..., α k ), eg. including information of exact time when product introduction starts, time and value of anticipated sales maximum, market capacity, product price, advertising expenditures, etc. This approach is also suitable for applying results of judgmental forecasting in cases when little or no historical data points are available. In addition, the model can have auxiliary parameters which adjust the model to practical requirements. Auxiliary parameters do not need additional data points for their determination. In general, grouping of forecast results for specific markeegments eg. separate for residential segment, for business segment and/or for segments related to specific life-styles, etc) yields better forecasting accuracy than aggregate forecasting performed for the whole market. Due to the measurement errors of input data, associated uncertainties of estimated model parameters can be represented by a confidence interval. Consequently, forecasting results can be represented by a prediction interval between pessimistic and optimistic values. The range depends on determined confidence level, which is typically 95. Besides that, a sensitivity analysis of model parameters and/or explanatory variables should be deployed to examine what effect their variations have upon the forecasting result. 1 1 a = A a = -A a = A a =.5 A 5 5 B-15 B-1 B-5 B B+5 B+1 B b = B b = B-5 = C =.7 C 5 5 B-15 B-1 B-5 B B+5 B+1 B+15 B-15 B-1 B-5 B B+5 B+1 B+15 B-15 B-1 B-5 B B+5 B+1 B+15 Figure 1 Effect of logistic model parameter change on the form of S-curve Cases from top left to bottom right): positive and negative growth rate parameter; 5 decrease of growth rate parameter; decrease of time shift parameter for 5 time units eg. years); and 3 decrease of market capacity parameter Telektronikk 3/4.28 ISSN Telenor ASA

3 Lt) L t) GR Salest 1,t 2 )=Lt 2 Lt 1 ) t 2 t 1 ) L t2 + t 1 2 ) 9) e aδṯ 1 /2 I The maximum of L't), as well as the time point t I when Lt) has inflexion are obtained from the solution of equation L''t) =, where L''t) is the second derivative of Lt): a/4 Figure 2 Characteristic values and points of the logistic growth model 3 Logistic Growth odel The logistic model Lt) describes growth in the number of users observed over time in a closed market, without the impact of any other product. The model is defined by three parameters: market capacity, a growth rate parameter and b time shift parameter. To emphasise the model s dependence on its parameters, it is convenient to indicate the model as Lt;, a, b) [11]: Lt;,a,b) =Lt) = Figure 1 shows the effects of change of parameters a, b and on the form of S-curve. 6) 3.1 Characteristics of the Logistic Growth odel The logistic model is a widely used growth model with many useful properties for technological and market development forecasting. odel 6) is the solution of differential equation 7) which consists of exponential growth term and negative feedback term [5]. In the beginning, the growth of the logistic model is identical to exponential growth, but later negative feedback slows the gradient of growth as Lt) approaches the market capacity limit : dlt) dt = alt) 1 Lt) ) Exponential growth 1+e at b) First derivative of Lt) is given in 8): L t) = dlt) dt Negative feedback = a e at b) [ 1+e at b) ] 2 7) 8) Contrary to S-shaped cumulative adoption Lt), adoption per period sales) is a bell-shaped curve see Figure 2), and it is proportional to the first derivative L't) of cumulative adoption: b e aδṯ 1)/2 Time L t) = d2 Lt) dt 2 = = a2 e at b) [1 e at b)] [ 1+e at b) ] 3 From 1) follows that Lt) has inflexion for t I = b, which is for a > the maximum of L't), too see Figure 2): max L t) = a 4 a>; t = b 1) 11) The value of the logistic model at point of inflexion is see Figure 2) Lb) = / 2. Accordingly, maximum of sales occurs at t = b = t I when penetration is 5. For t 1 and t 2 near b from 9) follows thaales in time interval [t 1, t 2 ] can be approximated by: max Salest 1,t 2 ) t 2 t 1 ) a 4 12) Asymptotes of logistic growth for positive and negative parameter a can be summarized in 13): { Lt) = a> lim t lim t + Lt) = a< { Lt) = a< Lt) = a> The growth rate GR for time interval Δ t is: GR = Lt Lt ) Lt ) 13) 14) For positive a, the growth rate is always positive and the maximum of growth rate is when t > see Figure 2, grey columns): When t = b = t I, a sales peak occurs. The growth rate at this point in time is half of its maximum value: 15) Above described characteristics of the logistic growth model with its explanatory attributes can be used as helpful input for estimation or assessment of model parameters for forecasting purposes. = = 1+e at b) 1+e at b 1 max GR = e a 1 a>; GR = ea 1 2 t = b t 146 ISSN Telenor ASA 29 Telektronikk 3/4.28

4 3.2 Logistic odel through two Fixed Points The modification of model 6), which has embedded values of two data points, u. ) and t e, v. ) is shown in Figure 3. For this case, it is suitable to define new parameters and instead of a and b; as well as to introduce two auxiliary parameters u and v. New parameters have explanatory meaning: time when the product perceivably starts with penetration level u, period needed until penetration grows to level v, eg. characteristic duration from productart to product maturity [8, 12]. Parameters a and b in 6) should be substituted with expressions 16) and 17), which are dependent on input parameters u, v and : a = 1 b = + [ ) 1 ln u 1 16) 17) Conditions that must be satisfied for equations 16) and 17) are < u < v < 1. This modified model Lt;,,, u, v) needs three parameters:, and to be determined. Values of auxiliary parameters u and v are not needed to be determined, they just allow the forecasting practitioner to choose the levels of starting and ending the penetration he/she wants to deal with. In the case of symmetrical u and v, ie. u = 1 v, equations become simpler: a = 2 ) 1 ln u 1 18) b = + 19) 2 Therefore model Lt;,,, u) has only one auxiliary parameter u: Lt;,,, u) = ln )] 1 v 1 ln 1 u 1) ln 1 u 1 ln 1 v 1) 1+ 1 u 1) 1 2t )/ 2) v u A condition that must be satisfied for model 2) is <u < 1. Used simplification gives a framework for the forecasting of new product adoption when little or no data are available. In Table 1 are presented resulting models for typical values of characteristic duration for products, product families and basic technologies according to model 2), but uniformed on the same natural logarithm base e. 3.3 Logistic odel Through one Fixed Point odification which has embedded value of one data point t p, Nt p )) in model 6) is called local logistic model LLt): LLt;,a,t p,nt p )) = = Lt) Figure 3 The logistic growth model defined via parameters,,, u and v Δ t Nt p ) Nt p )+[ Nt p )] e at tp) 21) Local logistic model is useful for forecasting from the last observed point t > t p. The idea is that it is better to start forecasting from a known base rather than to rely on an anticipated but un-modelled reversion to a historical trend. [4] t e Time u = 5, v = 95 u = 1, v = 9 = 2 years = 5 years = 1 years = 15 years Nt) = Nt) = Nt) = Nt) = Nt) = 1+e ) 1+e ) 1+e ) 1+e.589 5) Nt) = 1+e ) Nt) = 1+e.439 5) 1+e ) Nt) = 1+e ) Note: Characteristic duration according to [6] can be assumed as follows: products consist of units sold that have a typical life cycle of 6 to 1 quarters; product families consist of related products that have a typical business cycle of 5 years, and basic technologies consist of a set of related product families that have a typical cycle of 1 to 15 years. Table 1 Logistic model framework for forecasting of new product adoption prior to launch Telektronikk 3/4.28 ISSN Telenor ASA

5 1 c=.2 c = 1 = D 1 c=.2 c = 1 = D/ Figure 4 Richards model for different parameters c and 3.4 Limitations of the Logistic odel Although the logistic model is widely used for forecasting purposes, it is nouitable for modelling the product adoption when the number of users grows fast instantly after the product is introduced. The reason is in the shape of logistic growth that hardly starts to grow up. This problem is visible from the condition for equation 16), ie. it is not possible to model the time point when the product is introduced, and its penetration is u = ), because equations will give infinity value for parameter a. This deficiency is solved with the Bass model. The second main deficiency is fixed inflexion point I b, /2), which is not crucial for most forecasting purposes, but it is solved with the Richards growth model, which is sometimes called the four-parameter logistic model [7]: Rt;,a,b,c) = [ 1+e at b) ] c 22) with parameters market capacity, a growth rate parameter, b time shift parameter, and c shape parameter which determines position of the inflexion point. Rt) has inflexion for t = t I : t I = b + ln c a R t I )= 23) inimal value of Rt I ) / arises for c > and cannot be smaller than e minimal vertical position of an inflexion point). For c = 1, the Richards model is identical to the logistic model and Rt I ) / =.5. aximal value is without restriction, ie. Rt I ) / > 1 for c > : e 1 < Rt ) c I) c 24) = < 1 1+c Based on reparameterization shown in Section 3.2, the Richards model through two fixed points, u. ) and +, 1 u). ); with condition that < u < 1, has the form see Figure 4): Rt;,,, c, u) = [ ) Like the logistic model, the Richards model cannot model the time point when the product is introduced, ie. when Nt) = because only for t > Rt) approaches. The Richards model through one fixed point t p, Nt p )) has the following form: and could be called the local Richards model due to the similarity with the local logistic model. The model is useful for forecasting from the last known data point t > t p. 26) 4 The Bass model The best known model for a full description of the genesis and extensions of new product market adoption when interaction with other products can be neglected) is the Bass model. In distinction from logistic and Richards model, Bass model Bt) introduces the effect of innovators via coefficient of innovation p in differential equation of growth 27), which makes iuitable for modelling market adoption immediately after product is introduced. odel considers a population of adopters who are both innovators with a constant propensity to purchase) and imitators whose propensity to purchase is influenced by the amount of previous purchasing) [2, 3, 11]. = 1+ 1/ c u 1 ) LRt;,a,c,t p,nt p )) = = [ 1+ dbt) dt 1/ c u 1 1/ c 1 u 1 ) ] c Nt 1 p) e at tp) c = qbt) 1 Bt) Effect of imitators Logistic growth) 1 ts ] c ) + p Bt)) Effect of innovators 27) 148 ISSN Telenor ASA 29 Telektronikk 3/4.28

6 Solution of the differential equation 27) gives the Bass diffusion model 28) defined by four parameters: market capacity, p coefficient of innovation, p >, q coefficient of imitation, q and time when the product is introduced, B ) =. To emphasise the model s dependence on its parameters, it is convenient to indicate the model as Bt;, p, q, ), t : Bt;,p,q, )= 1 e p+q)) 1+ q p e p+q)) 28) The Bass model has the shape of an S-curve, identical to the simple logistic growth model, buhifted down on the y-axis. Figure 5 shows the effects of different values of parameters p and q on the form of S-curve, with fixed values for and. 4.1 Characteristics of the Bass odel The Bass model has many common characteristics with the logistic growth model. First derivative of Bt) is given in 29): B t) = dbt) dt = p + q)2 e p+q)) = ] p 2 [1+ qp e p+q)) Adoption per period sales) is a bell-shaped curve see Figure 6), and it is proportional to the first derivative B't) of cumulative adoption: 29) Salest 1,t 2 )= = Bt 2 Bt 1 ) t 2 t 1 ) B t2 + t 1 2 3) aximum of B't), as well as the time point when Bt) has inflexion, is obtained from the solution of equation B''t) =, where B''t) is the second derivative of Bt): B t) = d2 Bt) p + q)3 dt 2 = p ) q p e p+q) 1 e p+q)) ] 3 [1+ qp e p+q)) and maximum of B't) also occurs for t = t I, when it has the value: The value of the Bass model at point of inflexion is see Figure 6): ) 31) From 31) follows that Bt) has inflexion for t = t I : t I = + 1 ) q 32) p + q ln B t I )= p max B p + q)2 t) = t = t I 4q q p) Bt I )= 2q 33) 34) p=.26, q=.236 p=.139, q=.15 p=.53, q=.473 p=.279, q= p=.4, q=.374 p=.149, q=.2 p=.8, q=.747 p=.298, q= Figure 5 Effects of different values of parameters p and q. Chosen values are explained in Section 4.2 Telektronikk 3/4.28 ISSN Telenor ASA

7 p s.5 p+q) 2 4q q-p) 2q I s <.5 q-p) 2q t I I +1 q p +2 t I +1 q > p +2 Bt) dbt)/dt Bt) dbt)/dt Figure 6 Characteristic values and points of the Bass growth model In cases when q < p, the inflexion point and maximum of B't) occurs before the productarts t I < ), and the value of the Bass model at that point is negative according to 34), therefore interior maximum of B't) occurs at t =. Similarly, in cases when q = p, inflexion point and maximum of B't) occurs when the productarts t I = ). For q > p, the sales peak occurs in conventional sense of a PLC t I > ). The above mentioned is summarized in 35): max B t) = { p+q)2 4q q>p, t= t I p q p, t = 35) Accordingly, a maximum of sales occurs when penetration is q p) / 2q in cases when q > p at t = t I ), and in cases when q p, maximum of sales occurs at t = when penetration is, which is summarized in 36): max Salest 1,t 2 ) { t 2 t 1 ) p+q)2 4q q>p; t 1 and t 2 are near t I t 2 t 1 ) p q p, t 1 and t 2 are near 36) The growth rate GR for time interval is always positive: GR = Bt Bt ) Bt ) Due to the fact that the Bass model starts from, B ) =, the growth rate for t > goes to infinity. Above described characteristics of the Bass model with its explanatory attributes can be used as helpful input for estimation or assessment of model parameters for forecasting purposes. dependent while shape Bass model S-curve. Namely, the value of a characteristic duration of a product is provided only indirectly through the values of p and q parameters see Figure 5). The idea is to replace p and q with two independent explanatory parameters: a parameter that describes the vertical shape of the S-curve s and characteristic duration time to reach certain penetration level measured from ). Expected penetration level at time point + Δ t is v see explanation for and v in Section 3.2). The shape parameter s is chosen in order to encompass the relation between the amplitude of the positive S-curve part and the amplitude of the negative S-curve part. Asymptotes of the Bass model are: lim Bt) = p t q The ratio between negative asymptote and the distance of these asymptotes lays in range,1] which is convenient to choose as the shape parameter s, and which can be measured in percent. In fact, according to the value of s, the S-curve is stretched in the vertical direction on the y-axis) preserving the total market capacity. [9] The distance between these asymptotes is. 1 + p / q), so the shape parameter s is: s = Characteristic values of s are: lim Bt) = t + p/q + p/q = p, p >, q q + p 37) 4.2 The Bass odel with Explanatory Parameters Parameters and are descriptive and can easily be linked with market conditions. Although p as a coefficient of innovation and q as a coefficient of imitation have explanatory features, they are mutually s > s =.5 negative asymptote >, imitation prevails, curve is similar to a simple logistic growth model, q >> p > ), sales peak occurs at time when productarts q = p > ) 15 ISSN Telenor ASA 29 Telektronikk 3/4.28

8 s = 1 negative asymptote >, innovation prevails; curve is similar to an exponential saturation growth model q =, p > ). From 37) follows: p = p + q). s; q = p + q). 1 s) 38) Putting information about penetration level B + Δ t) = v in 28) together with 38) give 39) and 4) p + q = 1 ) 1+ 39) ln v s1 v) Bt;,,, s, v) = = 1 1+1/s 1) 4) Expression 4) is the reparameterized Bass model with explanatory parameters instead of p and q) where: market capacity; time when product is introduced, B ) =, t, Δ t characteristic duration of product, Δ t >, s shape parameter, <s 1; and v penetration at time point + Δ t, v < 1. The model from 4), Bt;,, Δ t, s, v), needs four parameters;,, Δ t and s to be determined. The value of the auxiliary parameter v does not need to be determined, it just allows the forecasting practitioner to choose which level of penetration he/she wants to deal with ie. 9, 95, etc). Special cases of 4): 1+ v s1 v) For v =, the value of model Bt) is zero: Bt;,, Δ t, s, v = ) = 1+ v s1 v) For s >, the Bass model degrades into a simple logistic model: Graph in Shape Shape Characteristic Figure 5 parameter parameter duration to 95 s 1 s 2 penetration Top-left years Top-right years Bottom-left years Bottom-right years Table 2 Bt;,,, s,v) 1+ 1 s v s1 v) s Lt;,a,b) where parameters of logistic growth model a and b are: a = 1 ) ln v,b= ln s s1 v) a For s =.5 the Bass model gets a form: 1 Bt;,,, s =.5,v)= 1+ 2 = 1+ 1+v 1 v 1+v 1 v 1+v 1 v = Lt;2,a,b This curve has a shape of logistic model with double market capacity but vertically shifted down by. Parameters a and b of this halved logistic model are: a = 1 ) 1+v ln, b = 1 v For s = 1 the Bass model degrades into an exponential saturation growth model: ) Bt;,,, s =1,v)= 1 1 v) = s = 2 s = 5 s = 8 = 2 years ) ) ) ) ) ) = 5 years ) ) ) ) ) ) = 1 years ) ) ) ) ) ) = 15 years ) ) ) ) ) ) Table 3 Reparameterized Bass model framework for forecasting new products adoption prior to launch. s shape parameter, characteristic duration time to reach penetration level v measured from ), value for v is chosen for 95 penetration v = 95 ) Telektronikk 3/4.28 ISSN Telenor ASA

9 Table 2 gives the explanation of the chosen values of parameters p and q presented in Figure 5, that are selected according to shape parameter and characteristic duration. Similarly to the framework for forecasting of new products adoption prior to launch presented in Section 3.2, model 4) can be used in cases when little or no data is available by comparison with other similar products histories. 4.3 Bass odel through one Fixed Point Similarly to the concept of the local logistic model described in Section 3.3, the Bass model with explanatory parameters which have embedded value of one data point t p, Nt p )) has the following form: LBt;,,s,t p,nt p )) = = /s 1) Nt p) tp ts s Nt p)) 1+ Nt p) s Nt p)) tp ts 41) and could be called the local Bass model. By default, the local Bass model as well as the Bass model, have an embedded value of starting point, ). The local Bass model is useful for forecasting from the last known data point t > t p. 4.4 Limitations of the Bass odel The Bass model is the most convenient model for market adoption forecasting of new product in sense of flexibility vs. number of free parameters needed to be estimated. Estimation of parameter values when limited data is available can be improved by introducing Bass model with explanatory parameters. Although several generalizations of Bass model expand model usage for later phases of PLC, numerous supplementary parameters demand a large set of known data points, which limits their application for the forecasting purposes. 5 Using the Logistic, Richards and Bass odels for Forecasting Purposes The logistic, Richards and Bass models are commonly used for forecasting of new product market adoption when interaction with other products can be neglected. In general, the logistic and Richards models are nouitable for modelling market adoption immediately after a product is introduced due to the fact that only for t >, Lt) and Rt) approach. In such cases the Bass model is used. The Richards model is more flexible than the logistic and the Bass model in cases of fitting data with asymmetrical position of inflexion point. The Bass and Richards models require finding four parameters instead of the three needed for the logistic model. There are several different circumstances when and how to use the logistic and the Bass model, but in the main there are three cases: Little or no data is available In cases of product market adoption forecasting prior to product launch, an insufficienet of historical data is available. arket capacity should be estimated by market research and/or markeegmentation techniques. Comparison with other similar product histories ie. forecasting by analogy) or judgmental assumption is needed for the following parameters:, Δ t in model: Lt;,, Δ t, u); auxiliary parameter u is usually set at 5 or 1, Δ t, c in model: Rt;,, Δ t, c, u); auxiliary parameter u is usually set at 5 or 1, Δ t, s in model: Bt;,, Δ t, s, v); auxiliary parameter v is usually set at 9 or 95. For the forecasting of new product adoption prior to launch when no historical data are available), the practical framework for the logistic and the Bass model are given in Table 1 and 3, respectively. In addition, it is possible to utilize information about peak of sales and/or growth rate as additional explanatory variables see Section 3.1 for the logistic model and Section 4.1 for the Bass model). Values of obtained parameters are uncertain ie. their confidence cannot be tested), therefore sensitivity analysis of forecasted market adoption depending on change of their values is strongly suggested. Usually, values for these parameters are assumed in interval: optimistic pessimistic. Limited data available A minimal set of historical data is three known data points for the logistic model and four known data points for the Richards and Bass models. When input data have a high level of uncertainty due to errors in measurement, uncorrected seasonal deviation, etc) and/or when observations only at the beginning of PLC are available, sensitivity analysis should be performed. In such cases, to reduce uncertainty of obtained results, it is better to estimate market capacity by market research and/or markeegmentation techniques and treat as a fixed value in models [1]. The minimal set of historical data with assumed is two data points for the logistic model and three data points for the Richards and Bass models. 152 ISSN Telenor ASA 29 Telektronikk 3/4.28

10 odel/equation in text Assumed Number of Fixed point ethod / Parameters known data in model t p, Nt p )) needed to be estimated by it Local logistic / 21)* 4 The latest known OLS /, a Local logistic / 21)* 3 The latest known OLS / a Logistic / 6)* 4 Weighted LS /, a, b Logistic / 6)* 3 Weighted LS / a, b Local Richards / 26)* 5 The latest known OLS /, a, c Local Richards / 26)* 4 The latest known OLS / a, c Richards / 22)* 5 Weighted LS /, a, b, c Richards / 22)* 4 Weighted LS / a, b, c Local Bass / 41)* 5 The latest known OLS /,, s Local Bass / 41)* 4 The latest known OLS /, s Bass / 28)* 5 Weighted LS /, p, q, Bass / 28)* 4 Weighted LS / p, q, * The numbers in brackets represent references to the equations in the paper. Table 4 Framework for parameter estimation in cases of a larger set of known data than minimal In cases when only a minimal set of historical data is available, values of model parameters are obtained from the solution of system 1). For a larger set of known data than minimal, parameter estimation can be done by the OLS method on models that treat the latest known data as a fixed point local logistic, local Richards or local Bass model) or by weighted leasquares method on logistic/richards/bass model with higher weights for the most recent data points, which is summarized in Table 4. Extensive set of input data In cases of mature products, an extensive set of historical market adoption data is known for the whole interval of PLC. The fit of the logistic, Richards or Bass models is usually very strong when the product is sole on the market and can be measured with correlation coefficient R. Due to the fact that an extensive set of data has to be known already, this case has low usability for practical forecasting purposes. However, it could be useful for accurate estimation of model parameters for the certain product and could later be used for forecasting by analogy of a subsequent product or for penetration forecasting of identical products on comparable markets. Parameter estimation can be done by the OLS method on logistic/richards/bass model. References 1 Armstrong, J S eds). Principles of Forecasting: A Handbook for Researchers and Practitioners. Kluwer Academic Publishers, Bass, F. A new product growth for model consumer durables. anagement Science, 15 5), , Bass, F, Gordon, K, Ferguson, T L, Githens, L. DIRECTV: Forecasting Diffusion of a New Technology prior to Product Launch. Interfaces, 31 3), S82-S93, eade, N. A odified Logistic odel Applied to Human Populations. Journal of the Royal Statistical Society. Series A Statistics in Society), 151 3), , eade, N, Islam, T. odelling and forecasting the diffusion of innovation A -year review. International Journal of Forecasting, 22 3), , odis, T. Conquering Uncertainty: Understanding Corporate Cycles and Positioning Your Company to Survive the Changing Environment. New York, Business Week Books cgraw-hill), Richards, F J. A flexible growth curve for empirical use. J. Exp. Bot., 1, 29-3, Telektronikk 3/4.28 ISSN Telenor ASA

11 8 Sokele,, Hudek, V. Extensions of logistic growth model for the forecasting of product life cycle segments. Advances in Doctoral Research in anagement, 1, 77-16, 26. World Scientific Publishing 9 Sokele,. Incorporating arket and Competitive Analysis Insights into your Forecast odels to Improve Accuracy. Proc. of the Telecoms arket Forecasting Conference, IIR, London, Sokele,. Growth models / Logistic Growth odel / Bass odel. Dictionary of Quantitative Research ethods in anagement. London, SAGE Publications. Forthcoming) 12 Stordahl, K. Long-term telecommunication forecasting. Norwegian University of Science and Technology, Trondheim, 26. PhD thesis) 1 Sokele,. Uncertainty of forecasted new service market capacity obtained by logistic model. Proc. of the 28th International Symposium on Forecasting, Nice, 28. laden Sokele is Assistant Director of the Operative Planning and Project anagement Department in Croatian Telecom Inc., Zagreb, Croatia. He graduated in 1983 and received his aster s degree in 1987 at the Faculty of Electrical Engineering and Computing of the University of Zagreb. After graduation he joined Ericsson Nikola Tesla Company and from 1989 he was lecturer at the Faculty of Electrical Engineering in Osijek. He has been with Croatian Telecom Inc. since r. Sokele is author or co-author of more than fifty published scientific and professional papers in the field of telecommunications and related indicators on modelling and forecasting. He is also author of the LOgistic Spline Trend LOST) method and program tool for the prediction of service penetration over time. laden.sokele@t.ht.hr 154 ISSN Telenor ASA 29 Telektronikk 3/4.28