A Piecewise Smooth Approach to Modeling Innovation Adoption Under Time-Varying External Influences
DOI:
https://doi.org/10.13052/jrss0974-8024.1916Keywords:
Adoption rates, bass model, innovation diffusion process, piecewise smooth functionAbstract
Innovation diffusion modeling plays a crucial role in understanding how new technologies, products, or ideas spread through a population over time. Classical approaches such as the Bass model assume smooth and continuous adoption patterns, which often fail to capture abrupt changes caused by market dynamics, technological disruptions, or policy interventions. This study develops a piecewise smooth diffusion framework that extends the Bass innovation diffusion model to incorporate random shifts across different time intervals. The framework introduces modulation functions that allow both gradual transitions and abrupt perturbations in adoption rates, thereby reflecting the non-linear dynamics of real-world diffusion. Stability analysis is conducted to examine the robustness of the system. The model is applied to historical datasets on cassette sales, compact discs, and physical video records. Empirical evaluation demonstrates that the piecewise approach provides superior fitting accuracy compared with standard Bass formulations, while also reducing parameter estimation errors. The findings highlight the value of modeling random shifts in diffusion processes, offering new insights for understanding technology substitution and for designing adaptive marketing and policy strategies.
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Agarwal, A., Giraud-Carrier, F. C., & Li, Y. (2018). A mediation model of green supply chain management adoption: the role of internal impetus. International journal of production economics, 205, 342–358.
Anand, A., Garg, K., & Singh, O. (2025). Transition time based modelling for sustainable innovation adoption. Sustainable Futures, 9, 100759.
Bai, J., & Perron, P. (2003). Computation and analysis of multiple structural change models. Journal of applied econometrics, 18(1), 1–22.
Bass, F. M., Krishnan, T. V., & Jain, D. C. (1994). Why the Bass Model Fits without Decision Variables. Marketing Science, 13(3), 203–223. http://www.jstor.org/stable/183674.
Bass, F.M. (1969) ‘A new product growth for model consumer durables’, Management Science, Vol. 15, No. 5, pp. 215–227.
Bemmaor, A. C., & Lee, J. (2002). The impact of heterogeneity and ill-conditioning on diffusion model parameter estimates. Marketing Science, 21(2), 209–220.
Boswijk, H. P., & Franses, P. H. (2005). On the econometrics of the Bass diffusion model. Journal of Business & Economic Statistics, 23(3), 255–268. https://doi.org/10.1198/073500104000000604.
Chandrasekaran, D., & Tellis, G. J. (2008). Global takeoff of new products: culture, wealth, or vanishing differences?. Marketing Science, 27(5), 844–860.
Coccia, M. (2020). How does institutional change of democratization affect the origin and diffusion of technological innovation across countries?. Journal of Economic and Social Thought, 7(2), 60–91.
Cosguner, K., & Seetharaman, P. (2022). Dynamic Pricing for New Products Using a Utility-Based Generalization of the Bass Diffusion Model. Manag. Sci., 68, 1904–1922. https://doi.org/10.1287/mnsc.2021.4257.
Darke, P. R., & Chung, C. M. (2005). Effects of pricing and promotion on consumer perceptions: it depends on how you frame it. Journal of retailing, 81(1), 35–47.
Fibich, G., & Golan, A. (2023). Diffusion of new products with heterogeneous consumers. Mathematics of Operations Research, 48(1), 257–287.
Golder, P. N., & Tellis, G. J. (2004). Growing, growing, gone: Cascades, diffusion, and turning points in the product life cycle. Marketing Science, 23(2), 207–218.
Guidolin, M., & Mortarino, C. (2010). Cross-country diffusion of photovoltaic systems: Modelling choices and forecasts for national adoption patterns. Technological Forecasting and Social Change, 77, 279–296. https://doi.org/10.1016/J.TECHFORE.2009.07.003.
Gurov, S. V., & Utkin, L. V. (2012). Load-share reliability models with the piecewise constant load. International Journal of Reliability and Safety, 6(4), 338. https://doi.org/10.1504/IJRS.2012.049599.
Gurov, S. V., & Utkin, L. V. (2014). A continuous extension of a load-share reliability model based on a condition of the residual lifetime conservation. European J. of Industrial Engineering, 8(3), 349. https://doi.org/10.1504/EJIE.2014.060995.
Guseo, R., & Guidolin, M. (2015). Heterogeneity in diffusion of innovations modelling: A few fundamental types. Technological Forecasting and Social Change, 90, 514–524. https://doi.org/10.1016/j.techfore.2014.02.023.
Guseo, R., Dalla Valle, A., & Guidolin, M. (2007). World Oil Depletion Models: Price effects compared with strategic or technological interventions. Technological Forecasting and Social Change, 74(4), 452–469. https://doi.org/10.1016/j.techfore.2006.01.004.
Huang, C. Y., & Lyu, M. R. (2011). Estimation and analysis of some generalized multiple change-point software reliability models. IEEE Transactions on reliability, 60(2), 498–514.
Irshad, M. S., Anand, A., & Bisht, M. (2019). Modelling popularity dynamics based on YouTube viewers and subscribers. International Journal of Mathematical, Engineering and Management Sciences, 4(6), 1508.
Jha, P. C., Gupta, A., & Kapur, P. K. (2008). Bass model revisited. Journal of Statistics and Management Systems, 11(3), 413-437.
Kapur, P. K., Panwar, S., & Singh, O. (2019). Modeling two-dimensional technology diffusion process under dynamic adoption rate. Journal of Modelling in Management, 14(3), 717–737. https://doi.org/10.1108/JM2-06-2018-0088.
Kapur, P. K., Singh, V. B., Anand, S. and Yadavalli, V. S. S. (2007). An innovation diffusion model incorporating change in the adoption rate. Management Dynamics: Journal of the Southern African Institute for Management Scientists, 16,1:34–41.
Kim, T., & Hong, J. (2015). Bass model with integration constant and its applications on initial demand and left-truncated data. Technological Forecasting and Social Change, 95, 120–134. https://doi.org/10.1016/J.TECHFORE.2015.02.009.
Kumar, R., Sharma, A., & Agnihotri, K. (2020). Bifurcation behaviour of a nonlinear innovation diffusion model with external influences. International Journal of Dynamical Systems and Differential Equations. https://doi.org/10.1504/ijdsde.2020.10031334.
Lee, S., Trimi, S., & Kim, C. (2013). Innovation and imitation effects’ dynamics in technology adoption. Industrial Management & Data Systems, 113(6), 772–799. https://doi.org/10.1108/IMDS-02-2013-0065.
Li, L., & Rao, M. (2023). The impact of government intervention on innovation efficiency of green technology – a threshold effect analysis based on environmental taxation and government subsidies. Frontiers in Energy Research, 11, 1197158.
Liang, L. (2021). Novel Optimization-Based Parameter Estimation Method for the Bass Diffusion Model. SAGE Open, 11. https://doi.org/10.1177/21582440211026954.
Mahajan, V., Muller, E., & Bass, F. M. (1990). New product diffusion models in marketing: A review and directions for research. Journal of marketing, 54(1), 1–26.
Massiani, J., & Gohs, A. (2015). The Choice of Bass Model Coefficients to Forecast Diffusion for Innovative Products: An Empirical Investigation for New Automotive Technologies. Department of Economics. https://doi.org/10.2139/ssrn.2903850.
Min-Hi, H. (2006). Modeling Diffusion Process Under Abrupt Changes of External Factors. Journal of the Korean Operations Research and Management Science Society, 31(2), 15–26.
Niu, S.-C. (2006). A Piecewise-Diffusion Model of New-Product Demands. Operations Research, 54(4), 678–695. https://doi.org/10.1287/opre.1060.0287.
Niu, J., Jin, S., Chen, G., & Geng, X. (2024). How Can Price Promotions Make Consumers More Interested? An Empirical Study from a Chinese Supermarket. Sustainability. https://doi.org/10.3390/su16062512.
Oliinyk, V., Kozmenko, O., Wiebe, I., & Kozmenko, S. (2018). Optimal Control over the Process of Innovative Product Diffusion: The Case of Sony Corporation. Economics & Sociology. https://doi.org/10.14254/2071-789X.2018/11-3/16.
Øverby, H., Audestad, J., & Szalkowski, G. (2022). Compartmental market models in the digital economy – extension of the Bass model to complex economic systems. Telecommunications Policy. https://doi.org/10.1016/j.telpol.2022.102441.
Panwar, S., Kapur, P. K., & Singh, O. (2019). Modeling Technological Substitution by Incorporating Dynamic Adoption Rate. International Journal of Innovation and Technology Management, 16(01), 1950010. https://doi.org/10.1142/S021987701950010X.
Peng, C., Liu, G., & Wang, L. (2016). Piecewise modelling and parameter estimation of repairable system failure rate. SpringerPlus, 5(1), 1477. https://doi.org/10.1186/s40064-016-3122-4.
Peres, R., Muller, E., & Mahajan, V. (2010). Innovation diffusion and new product growth models: A critical review and research directions. International journal of research in marketing, 27(2), 91–106.
Pti. (2024, January 22). EVs development, adoption to play major role in India’s transition to low carbon economy: DPIIT Secy. The Economic Times.
Rahman, M. M., & Thill, J. C. (2023). What drives people’s willingness to adopt autonomous vehicles? A review of internal and external factors. Sustainability, 15(15), 11541.
Ramírez-Solís, E., & Rodriguez-Marin, M. (2022). Diffusion Model for Mexican SMEs to Support the Success of Innovation. Sustainability. https://doi.org/10.3390/su141610305.
Rogers, E.M. (2010) Diffusion of Innovations. 4th Edition, Simon and Schuster, New York.
Saeed, K., & Xu, J. (2020). Understanding diffusion of information systems-based services: evidence from mobile banking services. Internet Res., 30, 1281–1304. https://doi.org/10.1108/intr-01-2019-0008.
Schweidel, D. A., & Fader, P. S. (2009). Dynamic changepoints revisited: An evolving process model of new product sales. International Journal of Research in Marketing, 26(2), 119–124. https://doi.org/10.1016/j.ijresmar.2008.12.005.
Sony game and network services: net sales & operating income FY 2012-2023. (2024, May 14). Statista. https://www.statista.com/statistics/323452/sony-net-sales-and-operating-income-game-network-services/.
Takahashi, C., De Figueiredo, J., & Scornavacca, E. (2024). Investigating the diffusion of innovation: A comprehensive study of successive diffusion processes through analysis of search trends, patent records, and academic publications. Technological Forecasting and Social Change. https://doi.org/10.1016/j.techfore.2023.122991.
U.S. Music Revenue Database – RIAA. (2023, July 12). RIAA. https://www.riaa.com/u-s-sales-database/.
Van den Bulte, C., & Stremersch, S. (2004). Social contagion and income heterogeneity in new product diffusion: A meta-analytic test. Marketing Science, 23(4), 530–544.
Wang, Y., Pei, L., & Wang, Z. (2017). The NLS-based Grey Bass Model for Simulating New Product Diffusion. International Journal of Market Research, 59, 655–669. https://doi.org/10.2501/IJMR-2017-045.
Yu, J. R., & Tseng, F.-M. (2016). Fuzzy Piecewise Logistic Growth Model for Innovation Diffusion: A Case Study of the TV Industry. International Journal of Fuzzy Systems, 18(3), 511–522. https://doi.org/10.1007/s40815-015-0066-8.
Zhang, C., Tian, Y., & Fan, L. (2020). Improving the Bass model’s predictive power through online reviews, search traffic and macroeconomic data. Annals of Operations Research, 295, 881–922. https://doi.org/10.1007/s10479-020-03716-3.
Zhao, H., Yao, X., Liu, Z., & Yang, Q. (2021). Impact of pricing and product information on consumer buying behavior with customer satisfaction in a mediating role. Frontiers in psychology, 12, 720151.
