A Stochastic Logistic Innovation Diffusion Model Studying the Electricity Consumption in Greece and the United States

In this article, a stochastic innovation diffusion model is proposed, derived from the original logistic growth model assuming that the future remaining growth of the underlying process is not known with certainty but is modeled using an appropriate stochastic process. At any time, the potential adopters of a product are affected by a number of socioeconomic factors that determine their nonuniform behavior, and the way they act is considered to be random. The stochastic model is solved analytically using the theory of reducible stochastic differential equations. The parameter estimators of the model are derived using two procedures for discrete observations of the process. Finally, the model is applied to the data of electricity consumption in Greece and the United States. The prediction of the consumption process is made possible by defining a subdomain such that all possible trajectories of the process should belong within a predefined probability.  1999 Elsevier Science Inc. Introduction In the last 30 years, there has been growing interest in forecasting the market demand of new products. The reason is that, at any time, market demand forecasts are critical to production, distribution, and promotion of these products. Until now, a number of deterministic new product forecasting models have been proposed in order to study the time-dependent aspects of the innovation diffusion process. The most representative are those of Mansfield [1], Floyd [2], Bass [3], Sharif and Kabir [4], Easingwood et al. [5], and Skiadas [6–8]. These models have been applied in the field of marketing management and have been successful in forecasting the growth pattern of new products. One major observation that can be made from reviewing the literature, however, is that the evolution of innovation diffusion models has ignored merely stochastic considerations. The desirability of a stochastic perspective is especially vital given the long-term forecasts that the diffusion models can potentially provide and the existence of several rapidly changing factors in the environment as well as in the interior of the A. N. GIOVANIS and C. H. SKIADAS are in the Department of Production Engineering and Management at the Technical University of Crete, Crete, Greece. Address correspondence to C. H. Skiadas, Department of Production Engineering and Management, Technical University of Crete, Agiou Markou Street, 73132 Chania, Crete, Greece. Technological Forecasting and Social Change 61, 235–246 (1999)  1999 Elsevier Science Inc. All rights reserved. 0040-1625/99/$–see front matter 655 Avenue of the Americas, New York, NY 10010 PII S0040-1625(99)00005-0 236 A. N. GIOVANIS AND C. H. SKIADAS system. The action of these factors, no matter how small they are, can cause a random adoption pattern of the new product. Some authors have proposed models that relax the deterministic nature of the models by introducing parametric stochasticity assuming that the parameters of an aggregate diffusion model follow a stationary stochastic process [9–12]. Another approach to the problem of introducing stochasticity results from the assumption that, unlike the majority of the deterministic models, the future remaining growth of the underlying process is not known with certainty but is modeled using an appropriate stochastic process [13–17]. The resulting stochastic model is represented by an Ito’s stochastic differential equation (SDE), taking into account the internal and/or external fluctuations. In this paper, a stochastic version of the well-known logistic model is solved analytically using the theory of reducible SDEs. Then, the model is used to study the course of electricity consumption growth in Greece and the United States. This paper is organized as follows. In section 1, the stochastic model is formulated, and in Section 2, it is solved analytically by applying the method of reducible SDEs. In Section 3, the parameter estimators of the model are derived using two methods for discrete observations. Finally, in Section 4, the model is applied to electricity consumption data in Greece and the United States, and approximated value subdomains are illustrated, where the possible states of the underlying processes for the forecasting period should belong within a predefined probability. 1. Formulation of the Stochastic Logistic Growth Model A general formulation of a deterministic innovation diffusion model assumes that the current growth rate is modeled as the product of functions of the current size and the remaining growth [18], Eq. (1.1): df(t) dt ~ g(f(t),p) · [h(F) 2 h(f(t))] (1.1) Where df(t)dt 5 the current growth rate of the process F 5 the total population of the potential adopters f(t) 5 the current cumulative number of adopters h(F) 2 h(f(t)) 5 the remaining growth of the process p 5 vector of parameters Under the assumption of a homogeneously mixing population, it is plausible to assume that the probability for a potential adopter to adopt the new product in a small time interval dt is (b · f(t)). In a population of (F 2 f(t)) potential adopters, the number of adoptions in a small time interval dt would be Eq. (1.2): df(t) dt 5 b · f(t) F · (F 2 f(t)) (1.2) where b is called the coefficient of imitation. Eq. (1.2) represents the well-known logistic growth curve and can be derived from Eq. (1.1) for g(f(t),p) 5 (b · f(t))/F,h(F) 5 F and h(f(t)) 5 f(t). Given the above formulation, the deterministic logistic innovation diffusion model assumes that the process takes place in a stable and finite environment, where at any time the remaining growth is known since we know the saturation level and the current size of the process. Many authors [10–12] claim that the environment of the diffusion process is unstable, however, because of the existence of many economic, ELECTRICITY CONSUMPTION IN GREECE AND USA 237 social, and political factors, which affect crucially the behavior of potential adopters of the product. Hence, a more realistic modeling demands the introduction of a random quantity for the representation of the remaining adoptions of the product. Taking into account the assumptions underlying the logistic model as well as the fact that the presence of the aforementioned factors will be clearer and clearer as the system grows (e.g., competitive effects, adopters who do not contribute to the diffusion of the product, new legal restrictions, and so forth), we expect the behavior of the potential adopters of the product to become more random and, accordingly, the infinitesimal variance of the diffusion process to be proportional to the current size of the system. Based on this assumption, the usual form of the stochastic models applied in innovation diffusion and population dynamics is given by the following stochastic equation [10, 11, 12, 19], Eq. (1.3): df(t) dt 5 g(f(t),p) · [h(F) 2 h(f(t))] 1 g(f(t),p) · u(t) (1.3) Hence, if F 2 f(t) denotes the known remaining adoptions and u(t) represents the random fluctuations because of the action of many uncontrollable factors, then we assume that the real remaining adoptions of the product is F 2 f(t) 1 q · u(t), where u(t) is a one-dimensional “white noise” process and q is the parameter controlling the power of the noise. Then Eq. (1.2) is written in the following form: df(t) dt 5 b · f(t) F · (F 2 f(t) 1 q · u(t)) or df(t) dt 5 b · f(t) F · (F 2 f(t)) 1 c · f(t) · u(t) Using the formulation of Ito’s SDEs to the above equation, the stochastic logistic growth model is given by the following Eq. (1.4) [20]: df(t) 5 b · f(t) F · (F 2 f(t)) · dt 1 q · f(t) · dW(t) (1.4) where W(t) is a one-dimensional Wiener process and c 5 (b · q)/F. Eq. (1.4) is a nonlinear autonomous SDE with multiplicative noise satisfying the assumption that the infinitesimal variance of the process is proportional to the current size of the process. 2. Analytic Solution of the Stochastic Logistic Growth Model The theory of reducible SDEs provides a method for the reduction of a certain nonlinear autonomous SDE, Eq. (2.1): dy(t) 5 a(y(t)) · dt 1 b(y(t)) · dW(t) (2.1) to a linear SDE, Eq. (2.2): dx(t) 5 (a1 · x(t) 1 a2) · dt 1 (b1 · x(t) 1 b2) · dW(t) (2.2) by means of a time-independent transformation Xt 5 U(Yt) providing that the initial equation has a unique strong solution. According to some authors [21, 22], two reducibility conditions are given by the following equations, Eq. (2.3): 238 A. N. GIOVANIS AND C. H. SKIADAS