An Evaluation of the Extended Logistic, Simple Logistic, and Gompertz Models for Forecasting Short Lifecycle Products and Services


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1 An Evaluaton of the Extended Logstc, Smple Logstc, and Gompertz Models for Forecastng Short Lfecycle Products and Servces Charles V. Trappey a,1, Hsnyng Wu b a Professor (Management Scence), Natonal Chao Tung Unversty, Hsnchu, Tawan. b PhD student, Natonal Chao Tung Unversty, Hsnchu, Tawan. Abstract. Many successful technology forecastng models have been developed but lttle research has explored the relatonshp between sample set sze and forecast predcton accuracy. Ths research studes the forecast accuracy of large and small data sets usng the smple logstcl, Gompertz, and the extended logstc models. The performance of the models were evaluated usng the mean absolute devaton and the root mean square error. A tme seres dataset of four electronc products and servces were used to evaluate the model performance. The result shows that the extended logstc model fts large and small datasets better than the smple logstc and Gompertz models. The fndngs also show that that the extended logstc model s well suted to predct market growth wth lmted hstorcal data as s typcally the case for short lfecycle products and servces. Keywords. Extended logstc model, Technology forecastng 1 Introducton Wth the rapd ntroducton of new technologes, electronc products and servces are often replaced wthn a year. The product lfe cycle for electronc goods, whch used to be about ten years n the 1960 s, fell to about 5 years n the 1980 s and s now less than two years for cell phones and computer games. As product lfe cycles become shorter, less data becomes avalable for analyss. Gven ths market stuaton, t s mportant to use smaller data sets to forecast future trends of new electronc products and servces as they are ntroduced. A product lfe cycle s typcally dvded nto four stages: ntroducton, growth, maturty and declne stage [3]. At ntroducton stage, the product s new to the market and the product awareness has not been bult, so the feature of ths stage s 1 Professor (Management Scence), Natonal Chao Tung Unversty, 1001 Ta Hsueh Road, Hsnchu, Tawan 300; Tel: +886 (3) ; Fax: +886 (3) ; Emal:
2 2 C. Trappey, HY. Wu slow sales growth. The growth stage s a perod of rapd sales growth snce the product s accepted wdely by the market and the sales volume wll boost. As the sales growth begn to slow down, the mature stage start. Therefore, the product lfe cycle curve can be llustrated wth Sshaped curve from ntroducton stage to begnnng of mature stage. Snce lfecycle curve grow as sgmod curve, growth curve method could be appled to forecast the future trend of products. Growth curves are wdely used n technology forecastng [4, 610] and are referred to as Sshaped curves. Technology product growth follows ths curve snce the ntal growth s often very slow (e.g., a new product replacng a mature product), followed by rapd exponental growth when barrers to adopton fall, whch then falls off as a lmt to market share s approached. The lmt reflects the saturaton of the marketplace wth the product or the replacement of the product wth another. The curve also models an nflecton or break pont where growth ends and declne begns. Many growth curve models have been developed to forecast the adopton rate of technology based products wth the smple logstc curve and the Gompertz curve the most frequently referenced [1, 6]. However, when usng these two models to forecast market share growth, care must be taken to set the upper lmt of the curve correctly or the predcton wll become naccurate. Settng the upper lmt to growth can dffcult and ambguous. If the product s a necessty or wll lkely be popular for decades, then the upper lmt can be set to 100% of the market share. Ths means that the product wll be completely replaced only after everyone n the market has purchased the product. However, when marketers consder new technology products such as a computer game or a new model cell phone, the value for the upper lmt to market share growth can be dffcult to estmate. That s, a computer game can be quckly replaced by another game after only reachng 10% market share. In order to avod the problem of estmatng the market share growth capacty for the smple logstc and the Gompertz models, Meyer and Ausubel [8] proposed the extended logstc model. Under ths model, the capacty (or upper lmt) of the curve s not constant but s dynamc over tme. Meyer and Ausubel also proposed that technology nnovatons do not occur evenly through tme but nstead appear n clusters or nnovaton waves. Thus, they formulated an extended logstcs model whch s a smple logstcs model wth a carryng capacty K ( t) that s tself a logstcs functon of tme. Therefore, the saturaton or celng value becomes dynamc and can model pulses from blogstc growth. Blogstc growth represents growth where market share seems to reach a lmt but then growth s suddenly rejuvenated or reborn and begns agan. Therefore, the researchers extend the constant capacty ( K ) of the smple logstc model to the carryng capacty ( K() t ). Ths study apples ths dea to the study of electroncs products wth K() t representng an extended logstc model. The proposton of ths research s that the extended logstc model wth a tmevaryng capacty feature wll be better than the smple logstc model and the Gompertz model whch requre a set constant capacty. Therefore, ths research studes the forecast accuracy of large and small data sets usng the smple logstc, Gompertz, and extended logstc models. A tmeseres dataset descrbng the
3 An Evaluaton of the Extended Logstc, Smple Logstc, and Gompertz Models for Forecastng Short Lfecycle Products and Servces 3 Tawan market growth rates for two types of electronc products and two types of servces were used to evaluate model performance. 2 Technologcal Forecastng Models 2.1 Smple Logstc Curve Model Most bologcal growth follows an Sshape curve or logstc curve whch best models growth and declne over tme [8]. Snce the adopton of technology and technology based products s smlar to bologcal growth, the smple logstc model s wdely used for technology forecastng. L The model of smple logstc curve s expressed as y = T 1 bt + ae where L s the upper bound of yt, a descrbes the locaton of the curve, and b controls the shape of the curve. To estmate the parameters for a and b, the equaton of the smple logstc model s transformed nto lnear functon usng natural logarthms. The lnear model s expressed as Yt = ln ( yt L yt ) = ln( a) + bt and the parameter a and b are then estmated usng a smple lnear regresson. 2.2 Gompertz Model The Gompertz model was frst used to calculate mortalty rates n 1825 but has snce been appled to technology forecastng [5]. Although the Gompertz curve s smlar to the smple logstc curve, t s not symmetrcal about the nfllecton pont whch occurs at t = ( ln() b k). Gompertz s model reaches the pont of nflcton = bt ae early n the growth trend and s expressed as y t = Le, where L s the upper bound whch should be set before estmatng the parameters a and b. Smlar to the parameters of the smple logstc model, natural logarthms are used to transform the orgnal Gompertz model to lnear equatony = ln ln L y = ln a and then the parameters are estmated. t ( ( )) ( ) bt 2.3 Extended Logstc Model t The smple logstc model and the Gompertz model assume that the capacty of technology adopton s fxed and there s an a upper bound to growth for these models. However, the adopton of new technology s seldom constant and changes over tme. As shown by Meyer & Ausubel [8], the orgnal form of smple logstc df model = y df by 1 s extended to = by dt k () 1 y, where k s the upper dt k t lmt of the logstc curve and k ( t) s the tmevaryng capacty and s the functon whch s smlar to logstc curve.
4 4 C. Trappey, HY. Wu Ths research uses the extended logstc model and assumes that k t = 1 D e, where the value of D can be any number and the value of A () at larger than zero. The settng of k( t) comes from Chen s study [2]. Ths research also assumes that the capacty wll fluctuate wth tme and the saturaton level of a product could be 100% but may also just be 60% or 80% because some new products could be ntroduced to the market and substtute older products. Thus, a product may not acheve the 100% penetraton of the market and may be replaced earler than expected. Fnally, the extended logstc model can be expressed ( t) at k 1 D e as yt = = bt bt 1+ C e 1+ C e tme, and a, b, C, D are the parameters computed usng a nonlnear estmaton method provded by a statstc software package lke SYSTAT., where k ( t) s the capacty that s fluctuate wth 2.4 The Measurements of Ft and Forecast Performance There are many statstcs used to measure forecast accuracy. In ths study, the Mean Absolute Devaton (MAD) and Root Mean Square Error (RMSE) are used to measure performance. The mathematcal representatons are shown below: n f fˆ = MAD = 1 n n ( f fˆ ) 2 = 1 RMSE = n where f s the actual value at tme t, fˆ s the estmate at tme t, and n s the number of observatons. These measurements are based on the resduals, whch represent the dstance between real data and predcted data. Consequently, f the values of these measurements are small, then the ft and predcted performance s acceptable. 3 Data Collecton and Data Analyss The Tawan market growth rate for color TVs, telephones, Asymmetrc Dgtal Subscrber Lnes (ASDL) and mobl nternet subscrbers were collected to test the forecast accuracy of the smple logstc model, the Gompertz model and the extended logstc model. ADSL s a type of data communcatons technology whch enables the use of the Internet over copper telephone lnes. ADSL users must sgn up for accounts so the applcaton data models the growth rate of adopton of the technology servce. Mobl nternet subscrbers also apply for accounts and the subscrptons for WAP, GPRS, and PHS represent ths class of data servce.
5 An Evaluaton of the Extended Logstc, Smple Logstc, and Gompertz Models for Forecastng Short Lfecycle Products and Servces 5 There are 31 data ponts for color TV set purchases (from 1974 to 2004) and 35 data ponts for telephone purchases (1964 to 2004). Twenty sx quarterly data ponts for ADSL subscrpton were collected from the second quarter n 2000 to the thrd quarter n 2006 and 19 quarterly data ponts for moble nternet subscrpton were collected from the thrd quarter n 2000 to the thrd quarter n Fgure 1 shows the saturaton rate for the four products. The growth rate for color TVs and telephones follows as Sshape curve and the curves for ADSL and moble nternet subscrptons also have basc sgmod curve. As can be seen n Fgure 1, t spent 20 years ( ) for telephone gettng nto the mature stage whose characterstc s slower growth rate than that at the growth stage n product lfecycle. Further, color TV also spent 10 years ( ) enterng the mature stage. However, for ADSL and moble nternet subscrptons, t only took 5 years to begn the mature stage. Therefore, the data for color TV and telephones were categorzed as long lfe cycle products, whle the data for ADSL and moble nternet were as short lfe cycle products. These four data are ftted to the models after removng the last fve data ponts. The last 5 data ponts were reserved to test the predcton accuracy of the models. Further, to compare the performance of the three models usng large and small data sets, the data for color TV and telephones were used to represent longer lfecycles (larger data sets) snce these two products have more than 30 years hstorc data. Compared to color TV and telephones, the data for ADSL and moble nternet subscrptons rely on small data sets of less than 6 years and represent shorter lfecycles. Fgure 1. Market growth for Color TVs, telephones, ADSL subscrpton, and moble Internet subscrpton Saturaton (%) Color TV Telephone ADSL Moble nternet Tme
6 6 C. Trappey, HY. Wu 4 Analyss The frst step s to ft the data to the smple logstc, Gompertz, and the extended logstc models. After reservng the last fve data ponts, the coeffcents of the models and the statstcs for MAD and RMSE are computed. The second step uses the derved models to forecast the fve data ponts and compare the forecast wth the true observatons and compare. Table 1 summarzes the ft and predcton performance of the three models. The evaluaton rule s that the smaller the value for MAD and RMSE, then the better the predcton performance. Therefore, the results show that the extended logstc model has the best ft and predcton performance for both long and short data sets. Thus, the extended logstc model s sutable for predctng both long and short lfecycle products. The smple logstc and the Gompertz model requre that the values for the upper lmt be set correctly, otherwse the accuracy of the models can be suffer. For the long data sets, the upper lmt for color TVs and telephones s easy to see and s set at 100%. However, for the short data sets, the capacty for ADSL and moble Internet servce s dffcult to determne. Thus, for the smple logstc and the Gompertz model, dfferent upper lmts are set for the short data sets. The upper lmt for the saturaton rate of ADSL s set to 100%, 50% or 30% whereas and the upper bound for moble Internet s set to ether 100% or 50%. As shown n Table 1, the extended logstc model wth tmevaryng capacty yelds the best ft and predcton performance. Table 1. Performance measures for the extended logstc, Gompertz, and the smple logstc models Data length Model Long data Short data Statstc Ft Forecast Extended Gompertz Smple Extended Gompertz Smple logstc logstc logstc logstc MAD Color TV RMSE Phone MAD RMSE ADSL MAD % RMSE ADSL MAD % RMSE ADSL MAD % RMSE Moble MAD Internet 100% RMSE
7 An Evaluaton of the Extended Logstc, Smple Logstc, and Gompertz Models for Forecastng Short Lfecycle Products and Servces 7 Moble MAD Internet 50% RMSE Dscusson and Concluson Ths study compares the ft and predcton performance of the smple logstc, Gompertz, and the extended logstc models for four electronc products. Snce the smple logstc and Gompertz curves requre the correct settng of upper lmts for accurate market growth rate predctons, these two models may not be sutable for short lfe cycle products wth lmted data. Therefore, to solve ths problem, the extended logstc model proposed by Meyer and Ausubel was used n ths study. Snce the extended logstc model estmates the tmevaryng capacty from the data, t tends to perform better for long data sets and for short data sets. Besdes, the extended logstc model s also sutable for predctng both long and short lfecycle products. There are lmtatons to the use of the extended logstc model. For example, Meade & Islam [6] use telephone data from Sweden to compare the smple logstc, extended logstc, and the local logstc models. They concluded that the extended logstc model had the worst performance. However, the growth curve of ther data set was resembled a lnear curve, not lke sgmod curve. Snce the capacty of extended logstc model s tmevaryng and s a logstcs functon of tme, t s not sutable for the data wth lnear curve. Therefore, we suggest that f forecasters wsh to apply the extended logstc model, then they should confrm that the data s not lnear. Ths research s concerned wth accurate market predctons for short lfe cycle, small data set problems. For ths study, we conclude that the extended logstc model s most sutable and research s planned to conduct a more rgorous and systematc testng of the model as proposed by Meade and Islam [8]. 5 Reference [1] Bengsu M, Nekhl R. Forecastng emergng technologes wth the ad of scence and technology databases. Technologcal Forecastng and Socal Change 2006;73: [2] Chen CP. The Test of Technologcal Forecastng Models: Comparson between Extended Logstc Model, FsherPry Model, and Gompertz Model. Master thess, Natonal Chao Tung Unversty, [3] Kotlor P. Marketng management. 11 th ed. 2003, New Jersey: Prentce Hall. [4] Levary RR, Han D. Choosng a technologcal forecastng method. Industral Management 1995;37:14. [5] Martno JP. Technologcal Forecastng for Decson Makng. 3 rd ed. 1993, New York: McGrawHll. [6] Meade N, Islam T. Forecastng wth growth curves: An emprcal comparson. Internatonal Journal of Forecastng 1995;11: [7] Meade N, Islam T. Technologcal forecastngmodel selecton, model stablty, and combnng models. Management Scence 1998;44:1115.
8 8 C. Trappey, HY. Wu [8] Meyer PS, Ausubel JH. Carryng Capacty A Model wth Logstcally Varyng Lmts. Technologcal Forecastng and Socal Change 1999;61: [9] Meyer PS, Yung JW, Ausubel JH. A Prmer on Logstc Growth and Substtuton. Technologcal Forecastng and Socal Change 1999;61: [10] Ra LP, Kumar N. Development and applcaton of mathematcal models for technology substtuton. PRANJANA 2003;6:
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