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Cellular Automata and Riccati Equation Models for Diffusion of Innovations Renato Guseo1 , Mariangela Guidolin2 1

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University of Padua, Department of Statistical Sciences, via C. Battisti 241, 35100 Padua, Italy; tel.-fax ++39-049-8274146; e-mail: [email protected] University of Padua, Department of Economics, via del Santo 33, 35100 Padua, Italy; e-mail: [email protected]

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Abstract Innovation diffusion represents a central topic both for researchers and for managers and policy makers. Traditionally, it has been examined using the successful Bass models (BM, GBM), based on an aggregate differential approach, which assures flexibility and reliable forecasts. More recently, the rising interest towards adoptions at the individual level has suggested the use of agent based models, like Cellular Automata models (CA), that are generally implemented through computer simulations. In this paper we present a link between a particular kind of CA and a separable non autonomous Riccati equation, whose general structure includes the Bass models as a special case. Through this link we propose an alternative to direct computer simulations, based on real data, and a new aggregate model, which simultaneously considers birth and death processes within the diffusion. The main results, referred to the closed form solution, the identification and the statistical analysis of our new model, may be both of theoretical and empirical interest. In particular, we examine two applied case studies, illustrating some forecasting improvements obtained. Key words Diffusion models – Technology forecasting – Cellular automata – Riccati equation – Generalized Bass model – NLS – ARMAX

1 Introduction A crucial aspect related to the existence of an innovation is represented by its diffusion process. Diffusion can be considered a growth process describing the evolution over time of an innovation (a new product, service or technology), on

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Renato Guseo, Mariangela Guidolin

the hypothesis that commercial novelties can have their own life cycle, like biologic organisms (see, for instance, Vernon 1966). Many analytical representations of structural change in the economic and technological domain have been built upon biological metaphors with the concepts of mutation, selection, life cycle, adaptation. Indeed, one of most used tools for representing biological growth processes, the logistic or Verhulst (1838) equation, has been also defined almost a “natural law of technological diffusion” (see Devezas 2005). But which are the elements allowing this life cycle to begin and develop? As Hauser et al. (2006) have noticed “the success of innovations depends ultimately on consumers accepting them” and consumers’ behaviour depends on the way they get informed on innovation. Rogers (2003) highlighted that diffusion is “a process through which innovation is communicated through certain channels over time among the members of a social system”. For detailed and up to date reviews on diffusion of innovation in quantitative marketing see Mahajan et al. (2000), Meade and Islam (2006) and Mahajan et al. (2007). The crucial role played by different consumer typologies in determining the destiny of an innovation has justified the use of an adequate and successful evolution of the logistic equation, represented by the Bass model (see Bass 1969) and, among others, its extensions (see Bass et al. 1994). The innovativeness of this aggregate model is partly due to the recognition of two non–overlapping sub-populations of adopters, innovators and imitators, that play different roles having different attitudes towards adoption. In particular, innovators pay attention to advertisement and mass-media information, while imitators adopt thanks to interactions with early adopters, giving rise to a word-of-mouth effect. In a formal way we have that the standard Bass model, BM, is a solution of Equation, z 0 = p(m − z) + q(z/m)(m − z), p, q > 0. (1) The model describes the life-cycle of a product through consumption choices of agents using a cumulative function, z = z(t), that depends on time, t, and on the carrying capacity, m (potential market). Instantaneous adoptions, z 0 , are determined by two additive components. The first one, p(m − z), refers to innovators, that adopt with a rate p. The second one, q(z/m)(m − z), is referred to imitators: they adopt with a rate q, modulated by the ratio (z/m), representing the word-ofmouth effect. If we divide both members of Equation (1) by m and denote F (t) = z(t)/m as a cumulative probability of adoption within time t, function f (t) = F 0 (t) represents the probability density of adoption at time t. Equation (1) can be usefully interpreted as a hazard rate h1 (t) =

z0 z f (t) = = p + q = p + qF (t), 1 − F (t) m−z m

(2)

describing the conditional probability density of an adoption at time t, and resulting from the sum of the probabilities of incompatible events. In fact, parameters p, q and (1 − p − q) may denote the probabilities of selection from three

Cellular Automata and Riccati Equation

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incompatible segments of the system: innovators, imitators and neutrals. The conditional probability of adoption depends upon segment typology: 1 for innovators, F (t) for imitators (note the dynamical strength of word–of–mouth) and 0 for neutrals. We can not observe the agent type, so that, conditionally on time t, adoption probability density is obtained with a simple marginalization: h1 (t) = p·1+q ·F (t)+(1−p−q)·0. The model excludes adoptions of mixed causal type: adoption is due to only one effect, the external one (advertising) or the internal one (word–of–mouth). On the basis of earlier stock data, it is possible to estimate and test the diffusion parameters p and q, and the carrying capacity, m. Despite this apparent simplicity, the Bass model has proven to be a very reliable tool for modelling and forecasting diffusion of innovation processes. A very interesting improvement, by Bass et al. (1994), is based on the generalized version, GBM, by including exogenous effects on the adoption hazard rate by the means of a very general multiplicative intervention function x(t) = x(t, ϑ), ϑ ∈ Rk , that allows a proper statistical evaluation of induced time dependent modifications. We observe that this generality has not been completely appreciated in marketing literature due to a reduced interpretation of x(t) limited to price and advertising incorporated effects that constitute only a special example in Bass et al. (1994). See, for instance, Bass et al. (2000) and Roberts and Lattin (2000). Both the possibility of working with stock data, quite easy to recover, and the need to estimate only 3+k parameters (m, p, q; ϑ) are great advantages related to this kind of aggregate models. The use of aggregate models surely involves some loss of information due to reduction of phenomena, but the forecasting ability is normally linked to specific characteristics such as model simplicity and parametric flexibility. Aggregate models avoid to take into account and describe explicitly interactions between individual actors, and, ignoring spatial correlations, they replace local relations by averaged uniform long-range ones, for instance by a mean field approximation (see Wolfram 1983 and Boccara and Fuk´s 1999). However, the lack of attention towards individual and local levels has been perceived as a shortage especially in the economic context, which, on the contrary, emphasizes conditions and behaviour of single (local) units. Disaggregate diffusion models have been studied in the marketing literature at least since Roberts and Urban (1988). Chatterjee and Eliashberg (1990), in particular, have developed a structural model of consumer adoption by modelling the consumer as a risk averse utility maximizer and their Equations (19) through (21) constitute an explicit derivation of how the micromodelling is aggregated to the diffusion process. Nevertheless, they admit that their (19)-(21) Equations are “cumbersome” so that external approximations are necessary. For a recent review on micromodelling approach within disaggregate diffusion theory see Roberts and Lattin (2000). In the early 1990’s, Mahajan et al. (1990) wrote: “how can the Bass model, which captures diffusion at the aggregate level, be linked to the adoption decisions at the individual level?”. This quite legitimate objection (that nevertheless,

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from an empirical perspective, should reckon with the effective availability of individual data) has found a new chance in agent-based models. More specifically, Cellular Automata models (CA) and Automata Networks (NA) represent a trial to fill in this shortage (see, for instance, Ganguly et al. 2003). Basically, we may define CA models as “simulations of global consequences, based on local interactions between individual members of a population” (see, Goldenberg et al. 2001). The models typically consist of an environment in which the interactions occur between individuals that are defined in terms of their behaviour. The way of transferring information from one individual to another is determined by predefined or stochastic transition rules and the implementation of such models consists of tracking the characteristics of each individual through time. The possibility of modelling the evolution of relationships using CA models is offered by the recent computer–based simulation techniques, that seem to be helpful for the study of emergent behaviour and complex adaptive systems, whose structure comes out in a bottom-up self organizing way rather than a top-down one (see, for instance, Goldenberg et al. 2001). Apparently, CA models represent an interesting opportunity to improve the analysis of diffusion processes, dealing directly with spatially distributed agents interacting between them. Garber et al. (2004) have applied CA based diffusion models to several empirical space and time data sets to aid in early prediction of new product success. The central idea is the dynamic assessment of spatial cross– entropy of adopters distribution. However, it should be considered that the application through simulation– techniques of such a type of models would require very detailed information about a certain population, its environment and the procedural rules (deterministic or stochastic) that define communication between agents. This specific information is the necessary basis of a reliable simulation. In this sense, the utility of simulations perhaps may lie mainly in analyzing particular artificial cases. Indeed, “the better a simulation is for its own purposes, by the inclusion of all relevant details, the more difficult it is to generalize its conclusions” (Maynard Smith 1974) to other cases. As Maynard Smith (1974) pointed out “whereas a good simulation should include as much detail as possible, a good model should include as little as possible”. This interesting distinction between the role of “models” and that of “simulations” allows to clarify better an aspect related to the uncertain availability of information and the need of empirical evaluations and control on the other hand. The case of CA models, that, as we have said, are generally implemented by computer simulations, suggests that the choice of the techniques for applying models may play a key role in determining their scientific validity or their generality. The characterization of linear and non–linear CA, from local to global mapping, often relies (see, for instance, Ganguly et al. 2003) on simulations for various initial configurations in order to achieve the qualitative average dynamics. Independence of the initial conditions is usually based on Lyapunov exponent. More recently, the mean field approximation, which is based on independence of states at any time (see, for instance, Boccara 2004), has allowed a simplified description of emergent pattern formation at macro or aggregate level. The second and more


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interesting characterization of a CA is the so–called inverse problem of deducing the local rules from a given global behaviour. The proposed methodology is based on evolutionary computation techniques, namely Genetic Algorithms, GA, (see Mitchell et al. 1993) and simulated annealing. Some criticism is expressed in Ganguly et al. (2003) with reference to the inherent problem. The convergence speed and accuracy of a constrained GA is really improved if the genetic algorithm evolves within a reduced search space. In this paper we question the use of simulation as the unique tool to implement CA models, thus – implicitly – discussing the theoretical independence of such a type of computer intensive techniques. Under a mean field approximation we introduce a link between a particular class of CA and the Riccati Equation, whose general structure includes, as a special case, the standard Bass model, BM, and the Generalized Bass model, GBM, (see Bass et al. 1994). A nonlinear differential characterization of the CA at aggregate level (macro level) combined with Nonlinear Least Squares (NLS) techniques may orient search algorithms on the basis of local aggregate representation avoiding unrealistic combinatorial search of standard GAs. Some interesting results on this theme are provided by Guinot (2002). The paper is structured as follows. In Section 2 we introduce and generalize a class of CA models proposed by Boccara and Fuk´s (1999) which simultaneously considers birth and death (exit rule) processes in the diffusion. We extend this model, under the mean field approximation, giving a continuous representation, say BFG, of the corresponding discrete time equation which is embedded in a Riccati equation. In Section 3 we solve, in closed form, a quite general separable non autonomous Riccati equation. In Section 4 we apply previous results to a perturbed BFG and illustrate statistical aspects concerned with inference and applications. In Section 5 we examine two different case studies: First, a bank account diffusion process in north–east regions of Italy with a standard exit rule and then, the evolution of annual Italian tricars stock (1950–2002) by introducing a special time dependent exit rule. Section 6 is devoted to final comments and discussion.

2 Cellular Automata Models We introduce here a class of simple CA models for the study of innovation diffusion processes. Dealing with the representation of individual units, a CA model is composed of a grid of cells and each one of them is in a specific state given a finite number of K possible states (e.g. adopter, neutral for K = 2). The possible change of state is governed by a defined transition rule f . This function (deterministic or stochastic) synthesizes the local interactions between a cell i ∈ Z (the set of all integers) and its range of interaction usually a neighborhood N (i) ⊂ Z.

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According to Boccara and Fuk´s (1999), we denote by s(i, t) the state of the cell i at time t and by f (·) the associated transition rule, s(i, t + 1) = f (s(i − v, t), s(i − v + 1, t), · · · , s(i, t), · · · , s(i + v, t)),

(3)

where v is the ray of neighborhood N (i). Function (3), characterizes a Cellular Automaton of ray v and tells that the state of the cell i at time t + 1 depends on the state at time t of the cells belonging to i’s range N (i) of ray v. A wider class of automata, Network Automata, NA, considers function f as i–dependent and with asymmetrical and variable neighborhoods. Function f represents here the local transition rule of a CA. In the simplest version of this model, the number K of possible states is two. For example, cells may be of two types: adopters, s(i, t) = 1, and neutrals, s(i, t) = 0. The paradigmatic specification of function f proposed by Boccara and Fuk´s (1999) considers an evolutive structure by which s(i, t + 1) depends on its previous state and on a transformation of the specific range N (i) defined by a mean value σ(i, t) σ(i, t) =

∞ X n=−∞

s(i + n, t)p(n),

∞ X

p(n) = 1, p(n) ≥ 0,

(4)

n=−∞

where p(n) is a probability distribution. Note that σ(i, t) itself is a probability and represents a kind of neighboring pressure on cell i to adopt. A first tentative model describing f is indirectly defined on the probabilities of both members of Equation (3) as follows. A neutral cell may become an adopter with probability σ(i, t) and its decision to adopt is not reversible. The simpler model is therefore a probabilistic CA with probability distribution p(s(i, t + 1) = 0) = (1 − s(i, t))(1 − σ(i, t)) and p(s(i, t + 1) = 1) = 1 − (1 − s(i, t))(1 − σ(i, t)). Collecting in a unique matrix the probabilities of change we obtain the transition probability matrix µ ¶ µ ¶ P0←0 P0←1 1 − σ(i, t) 0 = . (5) P1←0 P1←1 σ(i, t) 1 If we define with ρ(t) the average density of adopters at time t and with η(t) = 1 − ρ(t) the average density of neutrals, we can calculate ρ(t + 1) and η(t + 1) using the transition probability matrix, with an operator h i that denotes a spatial average, i.e., µ ¶ µ ¶ µ ¶ η(t + 1) hP0←0 i hP0←1 i η(t) = · , (6) ρ(t + 1) hP1←0 i hP1←1 i ρ(t) and therefore ρ(t + 1) = ρ(t) + (1 − ρ(t))hσ(i, t)i.

(7)


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Reasonably, Boccara and Fuk´s (1999) relax an assumption of the previous model by considering the probability of becoming an adopter no longer equal but only proportional to σ(i, t), i.e., P1←0 = qσ(i, t),

0 < q < 1.

(8)

This assumption considers the joint effect of personal preference (probability) to adopt, q, with neighboring pressure (probability) σ(i, t), which are stochastically independent. Moreover, it would be more realistic to consider an adoption as a process that is reversible with a probability r. Note that reversibility is surely not common in consumer durables diffusion but if we refer to “services” this effect may be more effective. For example, consider the registration of a customer within an internet service, the definition of a service provider, the temporal limited registration to a mobile network etc.. The available data usually record “active positions”, i.e., a stock data stream. Simplifying local transition by a spatial averaging of σ(i, t) with the operator h i, Boccara and Fuk´s (1999) obtain the following representation (6), µ ¶ µ ¶ µ ¶ η(t + 1) 1 − qhσ(i, t)i r η(t) = · (9) ρ(t + 1) qhσ(i, t)i 1−r ρ(t) and, therefore, ρ(t + 1) = (1 − r)ρ(t) + qhσ(i, t)i(1 − ρ(t)).

(10)

If we refer to ρ(t) as a mean field approximation, i.e., a limiting behaviour of hσ(i, t)i when the range of interactions tends to infinity for a spreading distribution p(n), we may rewrite Equation (10) as follows ρ(t + 1) = (1 − r)ρ(t) + qρ(t)(1 − ρ(t)).

(11)

In other words, Boccara and Fuk´s (1999) obtain a discrete time equation on a simple CA by eliminating spatial interactions by the means of mean field approximation. This reduction is surely due to the unavailability of individual spatial information. BFG: a non perturbed model with initializing and exit rules We propose an extension, say BFG, of the Boccara and Fuk´s (1999) model (11) in which we change the probability of becoming an adopter from a logistic pattern hP1←0 i = qhσ(i, t)i to a Riccati–like framework in order to recover the initializing aspects of the diffusion, which can not be omitted and are included, among others, in Bass family models, i.e., hP1←0 i = p + qhσ(i, t)i,

(12)

where p denotes the probability of an external information pressure due to media communication effect.

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In Equation (12) we state that the average conditional probability of becoming an adopter, hP1←0 i, refers to two mutually exclusive events: The adoption of an innovator, due to an institutional communication channel, within the CA, characterized by the specific segment dimension (probability) p and the strictly alternative adoption of an imitator whose segment dimension (probability) q is combined with the neighboring pressure σ(i, t). In our model we exclude that a cell (an adopter) can be jointly influenced by internal and external effects. This would mean being simultaneously innovator and imitator. Our CA approach is consistent with the Bass’ one as illustrated in Section 1. Furthermore, we approximate the finite difference ρ(t + 1) − ρ(t) with the prime derivative, ρ0 (t), so that, introducing Equation (12) in Equation (11), the proposed limiting continuous time BFG model is, ρ(t + 1) − ρ(t) ' ρ0 (t) = −rρ(t) + (p + qρ(t))(1 − ρ(t)),

(13)

with hazard rate h2 (t) = p + qρ(t) − r

ρ(t) . 1 − ρ(t)

(14)

Our assumptions differ from those of Goldenberg et al. (2001) and Gaber et al. (2004) in which they consider a time dependent “individual” probability of adoption as prob(t) = 1 − (1 − p)(1 − q)k(t) (15) implying a broader definition of adopting units with some difficulties in interpretation due to the inclusion of mixed causal adoptions. It is easy to verify that prob(t) is a particular cumulative function F (t) so that f (t) ' F (t) − F (t − 1) = (1 − p)(1 − q)k(t)−d(t) [1 − (1 − q)d(t) ] and h3 (t) = f (t)/(1 − F (t)) = [1 − (1 − q)d(t) ]/(1 − q)d(t) quite different from (2) or (14). Equation (12) may allow a more general interpretation of the imitative pattern as denoted by the product qρ(t). Some assessments on this issue are developed in Section 6.

3 A Separable Non Autonomous Riccati Equation In order to solve Equation (13) let us consider a separable non autonomous Riccati equation Z 0

2

y + (ay + by + c)x(t) = 0,

a, b, c ∈ R,

I(t) =

t

x(τ )dτ < ∞, (16) 0

√ 2 where x(t) is an integrable function. √ Let us define ri = (−b ± b − 4ac)/2a ∈ 2 R, i = 1, 2, with a(r2 − r1 ) = b − 4ac > 0 for r2 > r1 so that an equivalent representation of Equation (16) is y 0 + a(y − r1 )(y − r2 )x(t) = 0. Let us denote y˙ = y − r2 and y˙ 0 = y 0 with initial conditions y(0) = 0 or y(0) ˙ = −r2 then 0 previous equation divided by y˙ 2 is equivalent to yy˙˙2 + {a(r2 − r1 ) y1˙ + a}x(t) = 0.


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With a new variable change, yˆ = y1˙ , so that yˆ0 = − yy˙˙2 and initial condition yˆ(0) = − r12 we obtain equation yˆ0 = {a(r2 − r1 )ˆ y + a}x(t) ,

(17)

which may be integrated following, Rfor example, Apostol (1978 p. 31). a(r2 −r1 )

Let us denote G(t) = e

t

0

x(τ )dτ

so that the solution of Equation (17)

is Z t Rτ 1 −a(r2 −r1 ) x(ξ)dξ 0 x(τ )e G(t) + G(t)a dτ r2 0 · ¸ Rt 1 1 1 −a(r2 −r1 ) x(ξ)dξ 0 = − G(t) + G(t)a − e + r2 a(r2 − r1 ) a(r2 − r1 ) G(t)r1 − r2 = . (18) (r2 − r1 )r2

yˆ = −

In terms of the initial variable, y =

1 yˆ

+ r2 , we obtain

r2 (r2 − r1 ) 1 − 1/G(t) y(t) = r2 + = = 1 1 G(t)r1 − r2 r2 − r1 G(t)

1−e 1 r2

−

−a(r2 −r1 )

Rt 0

1 −a(r2 −r1 ) r1 e

x(τ )dτ

Rt 0

x(τ )dτ

. (19)

If limt→∞ I(t) = +∞, we attain a limiting behaviour of y(t), i.e., limt→∞ y(t) = r2 . 4 Statistical Modelling of a Perturbed BFG The proposed BFG model in Equation (13) of Section 2 may be represented within previous Riccati equation in order to incorporate interventions by function x(t). The Equation of a perturbed BFG model is y 0 = {−ry + (p + qy)(1 − y)}x(t),

r, p, q, ∈ R, 0 < p, q, 0 < r,

(20)

and x(t) may describe environmental constraints, marketing mix interventions, managerial strategies, by including ad hoc impulses or systematic control covariates. Therefore, standard BFG model is a special case under x(t) = 1. The wellknown Generalized Bass model, GBM, by Bass et al. (1994) is attained for r = 0. The standard Bass model, BM, by Bass (1969) includes both conditions, x(t) = 1 and r = 0. The factorization of Equation (20) is y 0 + q(y − r1 )(y − r2 )x(t) = 0, p with real roots, for D = (r + p − q)2 + 4pq > 0, equal to ri =

−(r + p − q) ± D , i = 1, 2, 2q

(21)

(22)

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so that we have a(r2 − r1 ) = D and the closed form solution is Rt −D x(τ )dτ 0 1−e Rt y(t) = . 1 1 −D 0 x(τ )dτ − e r2 r1

(23)

Note that limt→+∞ y(t) = r2 so that the absolute scale representation of natural diffusion may be obtained, in a direct way by the means of a scale parameter M , i.e., z(t) = M y(t). Therefore, the asymptotic behaviour is m = lim z(t) = M r2 . t→+∞

(24)

It is easy to prove, following Guseo (2004), ³that the´asymptotic fraction of innovative adoption is Fp,r (t → +∞) = qrp2 log 1 + pq for r2 > 0. Note that in GBM, r = 0, so that r2 = 1 with a carrying capacity ³ m =´M and p an asymptotic innovative adoption fraction Fp (t → +∞) = q log 1 + pq . The perturbed closed form solution (23) is very useful for a statistical approach to correct identification under intervention strategies and to forecasting. This allows the detection of the internal rules generating a CA under a widespread distribution of local influence on individual adoption or withdrawal of a new product or service. The statistical implementation of model (23) may adopt different error structures. In a nonlinear regressive approach, NLS (Nonlinear Least Squares), we consider a particular model for observations, w(t) = z(t) + ε(t), with an i.i.d. residual ε(t). A useful complementary approach for autocorrelated residual errors is based on ARMAX representation with a standard nonlinear estimation as a first step (see e.g. Guseo and Dalla Valle 2005, Guseo 2004 and Guseo et al. 2006). This combined statistical technique, namely NLS–ARMAX, is efficient and simple to implement. Our NLS–ARMAX procedure relies on efficient and statistically sure techniques for nonlinear step identification (e.g. Levenberg–Marquardt) and it is possibly improved if some evidence in residual autocorrelation suggests the second step, ARMAX. Recently, Venkatesan et al. (2004) have proposed evolutionary estimation techniques for the standard Bass model using GAs as a claimed alternative to NLS. They refer to an approximated version of instantaneous Bass model i.e. s(t) = m[F (t) − F (t − 1)] + ε(t), where F (t) = (1 − E(t))/(1 + (q/p)E(t)), E(t) = exp(−(p + q)t) and ε(t) ∼ N (0, σ 2 ). Note that a more suitable approximation of f (t) = F 0 (t) is f (t) ' F (t + 0.5) − F (t − 0.5) and the variance of β parameters estimates in their formula (3) is unusual. Moreover, they introduce a multiplicative version of the error term that obviously implies an intervention on least squares weighting which is ignored in the paper and in particular in their NLS simulations. The starting values are randomly uniformly selected within a hyper–rectangle. This is not a current and common strategy and reduces NLS performances under simulations. The countercheck is offered by their section 3.3 where a good starting point (by GA) gives a NLS estimate indistinguishable from GA’s. It is well–known


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that Newton–like procedures (e.g. Levemberg–Marquardt, Gauss–Newton, etc.) are efficient in terms of convergence speed in a suitable neighborhood of solution. Superiority of GAs with respect to NLS procedures is questionable. 5 Two case studies Here we propose two applied case studies to test the performance of our perturbed BFG model. Through these applications we will try to illustrate the innovativeness of this model, which jointly considers birth and death processes within the diffusion. We believe that this may represent a useful choice for improving forecasting of diffusion processes and an effective tool for managing the life cycle of an innovation under the effect of exit rules. As we stated in Section 1, an innovation may be both a new product and a new service. For this reason in the first application we propose the case of a new bank current account diffusion with a standard exit rule which is statistically significant even if with a limited effect. In the second one we consider the life cycle of Italian tricars with a diffusion that is affected by a strong time dependent exit rule. This example highlights larger perspectives in extending current BFG with various exit rate structures. Both cases do not present individual data but only stock data with no information regarding spatial and/or flux components. 5.1 Diffusion of a Bank Current Account We consider here the weekly stock diffusion of a special bank current account introduced by Cardine in north–east regions of Italy for small or medium size firms. Data are referred to the first 64 weeks starting from the origin of the service and are based on a modified counting unit (see Figure 1). By inspecting original data we observed a limited number of cancellations during the initial life of the service so that we suppose a limited effect of parameter r in this particular series. The perturbed BFG model is defined by Equation (23) and one exponential shock is embedded in intervention function, x(t) = 1 + c1 eb1 (t−a1 ) , in order to recover some structural change in economic and administrative facilities occurred in the first part of the cycle. Note that c1 represents the initial (positive or negative) value of the impulsive intervention, a1 denotes the initial intervention time and b1 , usually negative, describes the memory of the system, i.e., the “time” of permanence of intervention in the system. The proposed model is therefore the following Rt −D x(τ )dτ 0 1−e Rt w(t) = M + ε(t) (25) 1 1 −D 0 x(τ )dτ − e r2 r1 where r1 , r2 and D are referred to Equations (21) and (22). Stability and significance of the new model are controlled through a reduction testing based upon the elimination of the exit parameter r in BFG so that we obtain a GBM with

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Table 1 Cardine current account diffusion: north–east Italy. BFG vs GBM M

r

21872

0.0068 (-0.3806) (0.3941)

BFGe1

q

c1

12536 (8667) (16405) b1

0.0259 (-0.2594) (0.3112) 0.0235 (0.0163) (0,0306)

3.6759 (1.3916) (5.9602) 3.9556 (2.5898) (5.3215)

-0.1876 (-0.3267) (-0.0486) -0.2557 (-0.3549) (-0.1564)

GBMe1

BFGe1 GBMe1

m [M r2 ] 16577

p 0.0019 (-0.0138) (0.0176) 0.0036 (0.0025) (0.0047) a1 8.4878 (7.5938) (9.3818) 9.2770 (8.3809) (10.1730)

R2 0.99872 0.99858

( ) lower and upper 95% marginal linearized asymptotic confidence limits.

one exponential shock. If we examine Table 1 we note that global goodness-of-fit of perturbed model BFG with one exponential shock (BFGe1 for short) is quite good, R2 = 0.998719, but linearized asymptotic univariate confidence interval do not work properly in this situation probably for a quasi ill-conditioning due to the expected limited value of parameter r, and, surely, to the linearized version of marginal confidence limits. In these situations it is statistically convenient a simultaneous confidence regions testing or preferably, for simplicity reasons, a partially oriented test between nested models. This uncertainty may be reduced as far as the diffusion process reaches the central part of the cycle so that deflection, due to the limited withdrawals, may be properly recognized. In this case the distributed number of closed accounts is particularly small within the 64 weeks period. In order to test this problem we have implemented a reduced BFG, that generate a corresponding GBM, by setting parameter r to zero. Results depicted in Table (1) for the reduced GBM model, namely GBMe1, are more stable and conservative in the linearized marginal confidence limits. A more reliable way to study the significance of parameter r in BFG model may be performed by looking at the squared partial correlation which is equal to ˜ 2 (N − ˜ 2 = (R2 − R2 )/(1 − R2 ) = 0.09859 or equivalently to the ratio F = R R 1 1 2 ˜ 2 ) = 6.23. Parameter r in perturbed BFG results to be significant even k)/(1 − R if partially borderline. We observe that both models correctly identify the position and the nature of the intervention in the first part of the cycle with a very good marginal statistical significance. The asymptotic position of carrying capacity M r2 is not statistically stable in the first model. There is a certain departure between the estimates of M r2 and m within the two alternative representations, BFG and GBM. Nevertheless, parameter r is borderline significant and the upper confidence limit of m in GBMe1 is quite similar to estimated asymptotic behaviour of BFG that defines the limiting carrying capacity M r2 . Therefore, a conjoint evaluation may suggest a strong evidence in survival life-time (see Figures 1 and 2).


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Fig. 1 A current account diffusion: north–east Italy: BFG, zoom ; Data source: Cardine.

Fig. 2 A current account diffusion: north–east Italy: BFG; Data source: Cardine.

The asymptotic fraction of innovative adoptions are respectively Fp,r (t → +∞) = 28% and Fp (t → +∞) = 33%. The service, which is mainly sustained by institutional communication, may be expanded in the following periods and needs two years in order to reach 90% of market potential. A similar application of previous models within different economic homogeneous areas in Italy has supported the hypothesis of an imminent market’s saturation of that bank service in such areas.

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5.2 Italian Tricars Stock (1950–2002) We consider here the evolution of annual circulating Italian tricars stock starting from year 1950 to 2002. The sources of data are the ACI (Automobile Club d’Italia) archive combined with integrated information from ISTAT (Italian government statistical institute) archives. The tricars series exhibits some special features as we can see by inspecting Figure 3. We note that this kind of vehicles suffered a strong reduction during the period related to international oil crises supply (1973–1979) that induced a regional economic recession. We note a subsequent recovery of circulating tricars with a maximum located about 1990. Since that year, the Italian circulating tricars stock has begun to diminish, probably due to technology substitution effects. How can we model such a special “Automaton” submitted to one evident shock and to a degree of exit rate r that changes over time? We introduce in model (25) an exponential shock through intervention function x(t) = 1 + c1 eb1 (t−a1 ) . Besides, we consider a more flexible version of parameter r, namely a function r(t) = wtv , with w > 0 and v ≥ 0 in order to describe an increasing, now quite evident, exit rate. Obviously, we observe that r = w if v = 0. Note that function substitution, r = r(t), within model (25), which is obtained under a fixed assumption for r coefficient, corresponds approximately to a power function r˜(t) with similar parameters (w, ˜ v˜) within the Equation (20). Direct solution of Riccati Equation (20), under the modified time dependent exit rate, is not a simple matter. Fortunately, in this application, it is preferable a pragmatic choice of the final r(t) leaving the exact determination of r˜(t) unspecified. As a first step, we apply the modified model obtaining the results presented in Table 2, where, in particular, R12 = 0.981336. Table 2 Italian tricars stock (1950–2002); BFG + 1 exponential shock and variable exit rate

BFGe1rwv

BFGe1rwv

M

w

v

p

817233

1.4329E-6

2.5958

0.01954

q

c1

b1

a1

0.0521

-15.3858

-1.5954

26.84

R12 (SSE) 0.981336 (15.7465E9)

We can improve this preliminary robust dynamic estimate with a more flexible tool in order to absorb local interactions and autocorrelated deviations. We consider an ARMAX model (that is ARIMA(0,0,4)) with a regressor obtained as predicted series during the first step. The results are summarized in Table 3, where, in particular, R22 = 0.99011. We underline two major facts. M A components and first step predictor, denoted by P REbe1rwv, are statistically significant and the estimated deviance, SSE, is 8.3365E9 = (48 × 1.73367E8). We observe that the estimated coefficient


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Table 3 Italian tricars stock (1950–2002); ARIMA(0,0,4) + BFG + 1 exponential shock and variable exit rate Parameter MA(1) MA(2) MA(3) MA(4) PREbe1rwv Mean

Estimate -0.556999 -0.542280 -0.762372 -0.615259 0.9770480 8242.64

t -4.88289 -5.95655 -8.78945 -5.36366 24.71490 0.588253

P –value 0.000012 0.000000 0.000000 0.000002 0.000000 0.559121

Estimated white noise variance = 1.73677E8 with 48 degrees of freedom (R22 = 0.99011).

Fig. 3 Italian circulating tricars stock (1950-2002); BFG + 1 shock and variable exit rate; Data sources: ACI and ISTAT.

of predictor P REbe1rwv, 0.977048, is essentially equal to one. This confirms the optimal selection of first step modelling because a large change in its unitary level is a symptom depicting an imperfect identification of systematic mean evolution of modified and perturbed BFG model. If we examine the multiple partial squared ˜ 2 = (R2 − R2 )/(1 − R2 ) = correlation between the nested models we obtain R 2 1 1 ˜ 2 (N − k)/((1 − R ˜ 2 )s) = 0.47 and the corresponding F ratio is significant: F = R 7.1 (s is the number of incremental parameters between nested models). The strong reduction in residual variability highlights the significance of this second step analysis through ARMAX and we may appreciate the corresponding graphical aspects by inspecting Figure 3. The forecasted behaviour of this type of vehicles highlights an evident substitution effect and allows a quantitative precise assessment towards the end of a life cycle.

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6 Final Remarks and Discussion In this paper we have shown the existence of a clear link between a particular class of Cellular Automata and the aggregate differential approach based on a Riccati equation, whose general structure includes the Bass family models (BM, GBM). On the basis of this link, we have developed a new model, BFG characterized by Equation (13) which may be perturbed by an intervention function x(t). Note that Equation (13) is a representation of a possible weaker assumption in the spatial memory depth of the imitative pattern without the mean field approximation, i.e., ρ0 (t) = −rρ(t) + (p + Qh˜ σ (i, t)i)(1 − ρ(t)). With a plausible interpretation, we may consider a censoring effect in the mean value definition (4) that set to zeros the states very far from i due to the limited ray of interaction so that individual perception is based on a reduced level of σ(i, t), namely σ ˜ (i, t) = sσ(i, t), 0 < s < 1, due to a personal memory spatial depth. For this reason we may assume h˜ σ (i, t)i as a discounted function of ρ(t), i.e. h˜ σ (i, t)i = hsσ(i, t)i = shσ(i, t)i = sρ(t) so that Qh˜ σ (i, t)i = Qsρ(t) = qρ(t). Parameter Q denotes the pure imitative effect, and s represents a normalized spatial memory depth, i.e., a share of the neighboring pressure σ(i, t) or its limiting mean value ρ(t) under mean field approximation. Unfortunately, Equation (13) can not identify separately Q and s. Nevertheless, this is a minor problem in forecasting because parameter q jointly summarizes the pure imitative component and a simple version of local communication pattern. This remark about q decomposition may give some suggestions on the role of parameter’s s effect which may be partially compensated by external efforts on p by the means of general communication media. The obtained results may be of interest both for innovation theory and for managerial aspects related to forecasts of a specific innovation life cycle. From a theoretical point of view, the transition from individual (CA) to aggregate level (BFG) may represent a trial to face a problem that Ganguly et al. (2003) have called the inverse problem, from global to local mapping: “find a Cellular Automata rule that will have some preselected global properties”. Indeed, the estimated parameters p, q, r and M within our BFG model may be viewed both as global evolutive patterns that allow to determine, at aggregate level, the y(t) (or ρ(t)) by Equation (20) and as a possible direct tool for implementing the simulative experiments involved in the transition probability matrix relying on observed historical data. As we have seen constructing the CA, ρ(t) can be seen as a limiting behaviour of σ(i, t), whose value is necessary for the transition rule that defines the CA evolution. Clearly, the optimal solution would require the knowledge of the real σ(i, t), but this would mean to know the individual distribution p(n) (see Equation (1)). Because this information is generally not available, we believe that the CA transition rule could be constructed using an estimate of σ(i, t), ρ(t), inferred from a process historically observed, for which information is available. This procedure seems more reliable than one using direct simulations based on a transition rule, whose values are probably defined in an arbitrary manner. The simulation may be


17

better implemented once we are able to use a transition rule in which the values are justified from observed data. According to this perspective, we have focused our analysis on the identification of the four aggregate parameters (p, q, r, M ) simultaneously with the intervention function x(t) parameters. To connect theory and practice we have tested the goodness-of-fit of our model within two empirical applications showing a new way to face the problem of forecasting a diffusion process, considering at the same time the adoption parameters p and q and the exit rate r. Actually this represents a novelty in diffusion modellings because in general aggregate models like the Bass one do not consider “death” processes: they describe only the input components. In this sense, the BFG may constitute an improvement for forecasting aspects. Moreover, we may appreciate the descriptive role of the CA considered in this work, that has suggested a flexible and reasonable representation of diffusive phenomena, globally considered as a stock in time domain. The bank account case does not exhibit individual information on the adoption times. Only weekly aggregated data are available. We note that such a type of adoption process is characterized by a small, stable and statistically borderline exit rate r so that its evolution is quite similar to a GBM. Forecasting evaluations are very simple and do not require simulations, if we exclude the scenario opportunity offered by the function x(t) which allows the modulation of future interventions. In standard CA by Boccara and Fuk´s (1999) this feature is not available. Moreover, Riccati equation representation allows a clear interpretation by the means of the ratio q/p (see, for instance, Guseo 2004). In this case q/p = 13.6 highlights an ³asymptotic ´ fraction of innovative adoption equal to Fp,r (t → +∞) = qrp2 log 1 + pq = 28%. This is a useful suggestion for the management: in general, common products and service diffusions exhibit lower levels of this fraction so that in this case an intensive institutional communication effort would be the correct approach. The second application faces a different situation where we observe a strong effect of the exit rate r which may be no longer assumed constant for the whole life cycle. On the contrary, its effect becomes dominant reverting the monotonicity to a probably complete substitution of the tricars technology. The ratio q/p = 2.66 corresponds to an fraction of innovative adoption equal to Fp,r (t → ³ asymptotic ´

+∞) = q˜pr2 log 1 + pq = 36.7%, where r˜2 = 0, 99998 is determined under the hypothesis v = 0 in order to eliminate the direct effect of substitution. This high percentage confirms the substantial irreversibility of the observed descent trend because it highlights the role of innovative adoptions which reveal an active attitude of adopters to modify a preference on the basis of a new efficient opportunity. In this particular case the ARMAX improving approach, combined with a preliminary non-linear regressive identification, gives a more precise estimation and an accurate forecasting. At the same time it gives a robust corroboration of the per-

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turbed Riccati model, BFG. Computations in both cases do not require sophisticated techniques based on evolutionary methods, namely genetic algorithms and simulated annealing. The optimization procedure follows, at the first step, the well–known reproducible techniques of non–linear least squares (e.g. Levenberg–Marquardt). See, for instance, Seber and Wild (1989).

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