or positively or negatively influencing forces

TECHNOLOGICAL

FC..ECASTING

AND SOCIAL CHANGE

30, 313-330

(1986)

Innovation Diffusion Models Expressing Asymmetry and/or Positively or Negatively Influencing Forces CHRISTOS

H. SKIADAS

ABSTRACT Two different characteristics of the innovation diffusion process, that is, the asymmetry and the appearance of positively or negatively influencing forces, are analyzed. Related diffusion models in use are presented and new generalized models are formulated. A five-parameter model is proposed in order to cover the above two different characteristics of innovation diffusion. The properties of this model and other related diffusion models are examined and the appropriate iterative nonlinear regression analysis technique is developed. Five comparative applications follow, predictions are made, as are comparisons between two generalized model&RM, and Von Bertalanffy-that express asymmetric diffusion behavior.

Introduction Various studies on innovation diffusion showed the need for models describing asymmetric behavior, a case which could not be satisfied by the popular Logistic [5, 9, 23, 301 model. For this purpose, the Gompertz and the Floyd [2, 10, 26, 341 models were used. The first shows an inflection point at l/e = 0.3679 of the upper limit and the second at one-third of the upper limit. However, it is in both cases assumed that the asymmetry occurs in a stable point below that of the logistic. There are more flexible models, e.g., the Generalized [3, 20, 22, 29, 301, Sharif-Kabir [36], Von Bertalanffy (Generalized Logistic) [19, 21, 33, 401, NSRL [7], mod. NSRL, GRMI and II [39], Weibull [35], and Lognormal [4] models, the inflection points of which cover a part or the whole interval between 0 and the upper limit. The main disadvantages of all these models are, first, that they do not include in a simple equation the three basic models that appear in several diffusion cases, that is, the Logistic, the Exponential [l], and the Coleman models [26], and, second, that they do not take into account negatively acting forces (or rejective forces). Only the Generalized includes positive forces influencing the diffusion process. The aim of this article is to propose a nonsymmetric model that includes Coleman, Exponential, and Logistic models and is based on GRM I and GRM II models, to

This work was done by the author during his collaboration with the Department of Chemical Engineering, Processes and Systems Engineering, of the National Technical University of Athens. This collaboration is still active. C. H. SKIADAS current research interests are in the areas of modeling the generation and diffusion of new technologies and innovations, and forecasting. Address reprint requests to Dr. Christos H. Skiadas, Londou 4, 106 81 Athens, Greece. 0 1986 by Elsevier Science Publishing

Co., Inc.

0040-1625/86/$03.50

314

C.H. SKIADAS

investigate the possible application of the Generalized model in order to express both positively and negatively acting forces, to introduce a general model that could express both asymmetry and positively or negatively acting forces, and finally to compare the nonsymmetric model proposed here with the Von Bertalanffy model.

The GRM, Model Consider the model expressed lnf -

u In (F - f)

by the equation:

= c, + bt,

o 2 0, b,F > 0,

(1)

where f is the number of adopters at time t, F is the total number of potential adopters, b is the diffusion parameter, and cr is a dimensionless parameter. When 0 =S u S 1, (1) is identical to the equation of the GRMI, model [39]. When u 3 1 eq. (1) can be transformed to the form: (l/u)

lnf -

In (F - f)

= c2 + (b/u) t,

(2)

which, by setting l/u = u* and b/u = b*, expresses the equation of GRMII, model [39]. The model expressed by eq. (1) and covering both GRMI, and GRMII, is here defined as “GRM,” model and is illustrated in Figure 1. At the bounds, u = 0 and u + x, the Exponential and the Coleman models result, whereas when u = 1 the Logistic model (Blackman/Fisher-Pry) is obtained. The GRM, model has a point of inflection varying from 0 to F when parameter u decreases from ~0 to 0. Another interesting property of parameter u is that it gives a measure of the asymmetry of the model (see Fig. 1).

Fig. 1. Illustration

of GRMl model.

\

/’

\

i

0

F/2

f

I

NONSYMMETRIC AND GENERALIZED DIFFUSION MODELS

315

Generalized

0

-9

F/2

f

F

Fig. 2. Illustration of generalized model.

The Generalized Model This model is expressed In

(f

+

g)

-

In

(F

by the equation:

-

f)

=

c3

+

br,

(3)

with g either positive or negative. Differentiation of eq. (3) with respect to t yields f+,f+g

(F - f)?

F+g

where j = dfldt. This equation can be found in Lawton and Lawton [ 161 and Lawrence and Lawton [ 151. The model is also summarized in the diffusion monograph by Mahajan and Peterson [ 191. Setting bgl(F + g) = a and bF/(F + g) = b* in eq. (4a) yields j- = a(F - f)

+ b* (F ; f)f.

This equation is identical to the differential equation of BasslMahajan-Schoeman model [3, 201, referred to as the “Generalized” model. The model expressed by (4a) is illustrated in Figure 2. It is a symmetric model. This can be easily demonstrated by substituting f with fk ? E, where fk = (F - g)/2 is the value off at the point of inflection in eq. (4a). Another form of the generalized model was proposed by Oliver [29]. This model has differential equation:

.i”= (W

(f- y) (k + y - f),

which, be setting g = -y

and F = k + y, becomes eq. (4a).

(4c)

316

C.H. SKIADAS

As regards eq. (4~) and parameter tractors in Spain [30], writes:

y, Oliver,

in his application

on the diffusion

of

In the context of the spread of an innovation [equation 4c] implies that there is some autonomous stimulus positive or negative, in addition to the existing level of the innovation within the country concerned.

Two examples on tractors diffusion provided by Oliver in Great Britain [29] and Spain [30] gave, respectively, positive and negative values for y, The coefficient g included in eq. (4a) is referred to in Lawrence and Lawton as the “effective number of prior product users” [ 151. In the Lawton and Lawton article [ 16) this parameter represents the “carriers” of the innovation. As they write: These carriers are not themselves adopters, they merely expose others to the practice and can, with a given probability, convert potential adopters into actual adopters. The disease analogy underlying this presentation is apparent, and just as diseases can be spread by means other than person-to person contact, so can innovations. In other cases, no (here 8) carriers may represent effects of dissemination by field agents, professors of education, provincial or state agencies, leaders in professional organizations, salesmen, and advertisers.

Clearly, these last assumptions would lead to a positive g. On the other hand, negatively influencing persons or “carriers” may also appear. Mahajan, Muller, and Kerin 1181write: Others still might actively spread unfavorable or an unsatisfactory use experience.

word-of-mouth

due to negative impressions

about a product

During the various applications of the Generalized model, parameter a and hence g were usually taken as positive. Here, by the introduction of eq. (3) it was assumed that g could be either positive or negative. This is in accordance with Oliver’s findings for tractors diffusion. Parameter g is not here assumed as to express positive of negative “carriers.” It is assumed that g is a measure, a sum of actions from several positively or negatively influencing sources, not always easily recognized, that are different from the influence of the adopters to potential adopters. It is clear that a negative g would not allow the diffusion process to start. This could be overcome by an action outside the diffusion system, such as central policy or promotional efforts. This would cause the innovation to spread to a value f0 > -g that would allow diffusion process to begin. In the very early stages of innovation diffusion the basic controlling mechanisms such as the diffusion agency network and the information network can play a crucial role to the individual adoption decision [ 131. In cases where a population harbors a strong negative attitude toward an innovation, a technological innovation, or a new technology, a conflict results between adopting and rejecting forces that may delay the diffusion process in its very early stages. In this case, the very early data are not smouth enough or may lead to an underestimation of the speed of the diffusion process and finally to poor forecasts. In some cases of technological substitution the elimination of the very early data (whenfis lower than a small fraction-usually lO%-of the upper limit F) is suggested [9, 35, 361. However, here, by introducing parameter g, the very early data might be in general retained. Extremely early data (mainly when g is negative) could be excluded in some cases, but this is a matter of the experience of the forecaster rather than to follow general rules.

NONSYMMETRIC AND GENERALIZED DIFFUSION MODELS

317

The GRMlg Model This is a five-parameter model with an equation arising from the eqs. (1) and (3) of GRM, and Generalized models, respectively:

ln (f + g) - u In (F - f) =

c

+

bt,

(3

where c = In CJO+ g) - u In (F - fO),fO is the value offat time t = 0, and u = 0 for the Generalized model and g = 0 for the GRM, model. The model expressed by eq. (5) is here defined as the GRM,, model and has its differential equation given by j _

(F - f) (f + 8)

b

(6)

F-f+uCf+g). The inflection

point of the model is obtained

at

F-g6

fk=

(7)

1+X&.

The GRM,, model has properties of both the models from which it was originated. a nonsymmetric model, as is the GRM, model, and it has parameter g expressing itively or negatively acting forces, as does the Generalized model.

It is pos-

Parameter Estimation of GRMI, Model Parameters b, fo, F, u, and g, say a,, i = 1, , . . . ,m, of the GRMr, model [eq. (5)] are estimated by iterative direct nonlinear least-squares [28, 311 by minimizing the sum of squared errors:

E, is the error term of the stochastic equation:

y,=ft+

5% Aai

j=, aa;

+ Ed,

t = 0,1,2,

. . . ,n (n + 1 observations),

(9)

where y, denotes provided data andf, is calculated by any suitable numerical method from eq. (5) for every t, given a set of starting values of the parameters. For smooth data, f0 can be set equal to y,, and F > y,. Usually u is set equal to unity and g equal to zero. A value of b between 0 and 1 could be sufficient. Another method is to estimate b, fo, and F, by means of the popular logistic model, and then use them, together with u = 1 and g = 0, as starting values. Sometimes good starting values can be obtained by investigating the v,,f) and (t,f) diagrams. A method of five points is also available for the original estimation of parameters. The first two terms in the right side of eq. (9) are the first terms of a Taylor expansion of the nonlinear functionfaround a given set of initial values of parameters ai and provide a linear approximation of the nonlinear function. dfJ&zi is the first derivative off, with respect to the parameter ai, and Aa; is a difference which is added to ai after each iteration

C.H. SKIADAS

318

until the value of Aa; converge to a sufficient small number. By using a damping factor p (usually varying between 0 and I) the values of parameters Ui are computed after each iteration from ar.k+

I

=

ai,k

+

PAa,.

(10)

The values of ‘Au, are computed from the set of m linear equations that result by taking the partial derivatives of eq. (8) [after substitution of E, from (9)] with respect to ai and setting them equal to zero:

where U, = y, - ft, i = 1,2, . . ,m. In our model, we set m = 5. In matrix notation, the system (11) may be written as V6 = d

(12)

where V is the m-order symmetric square matrix of the coefficients

of the linear equations:

aa, EC 12 a!

v=

[ and 6 and d are the vector of unknowns

and the vector of the constants,

6=

respectively.

d=

The standard error is given by

cu: .Y=

i

t

n+l-m’

When the iterative 2~: = EEL. Hence

procedure

=

O,l, . . . ,n

converges

sufficiently

(134

well

then,

ha, ---, 0

and

319

NONSYMMETRIC AND GENERALIZED DIFFUSION MODELS

XE:

I

s^-

t = O,l,

n+1-m’

(13b)

. .,n.

As in this last case, the residual sum of squared errors (SSE), the mean squared error (MSE), and the correlation coefficient R2 (also known as coefficient or determination) and the correlation coefficient corrected for degrees of freedom R* are, respectively,

MSE = n+l

-‘I nfl’

(1%

C&r” R*‘.l-

R2 =

1

-

c

(Y, -

(1

-

3’

Cl@

’

R2)L

Cl@)

n-m’ where t = O,l, . . .,n and j = Xyj(n + 1). Standard errors of parameters s,; are computed

SOi

=

SVZJ

3

i =

from

1, . . . ,m,

(17)

where b,.; are the diagonal elements of the inverse matrix V-’ of V, and the corresponding t statistics, say tai, are tai = J? ,

i = 1, .

(18)

. ,m.

hzi

The correlation matrix of estimates of parameters matrix with elements ci,i given by

Ui is a m-order symmetric

square

(19) All diagonal elements c;.i are equal to unity. The iterative nonlinear regression analysis method, presented in this section, can be applied to all GRM models as well as to other models. First, one must compute the function5 and its partial derivatives, $,,&zi, with respect to the parameters a,. For the specific case of the GRM, model, differentiation of eq. (5) with respect to the five parameters b, fo, F, u, and g easily provides, after rearrangement, the first partial derivatives off with respect to these parameters:

8 __=Aafo

1

G o+g

(2Ob)

320

C.H.SKIADAS

aft

4.h - fo) (F - fi) (F - .h) ’

-_=A

aF

GJOC)

2LAInF-f, au

(204

F-h’ fi

fo

-

ti +g> (fo + g) ’

GQ=9

where

A

cf,+ g) (F

=

&

- fr)

F-f,+u(fr+g)=b.

Nonsymmetric Models Related to GRMl The differential (I), is i =

b

f(F F-(1

equation

of the GRM, model, as provided by differentiation

-f) -a)f'

0

=su.

of eq.

(21)

As referred to in ref. 39, the GRM, model, which originated from GRMI, and GRMII,, is an alternative model of the NSRL and NSRL models, when parameter 6 of last two models is varying within the interval (0, 1). These two general models have the following equations:

f = b(F - f) (J’F)”

NSRL: OS6

(22) j = bf(1 - flF)”

mod. NSRL:

(23)

Another model that might have similar behavior to that of GRMr model is the Von Bertalanffy model. This model has differential equation:

Wa) The equation of this model, as derived by integration f =

F

[I

_

(1

_

fo”-“J)

,-bt]l/(l-6)

Another form of Von Bertalanffy’s and (24b), which take the form:

_f=

if[l

-

($“I.

-

(24b)

model is derived by setting 6 = 1 + + in eqs. (24a)

-16$,

F f = [l + [(Flf,)6

of eq. (24a), is

11 e-b’]“+

.

(254 Wb)

321

NONSYMMETRIC AND GENERALIZED DIFFUSION MODELS

The Von Bertalanffy model in the form of eqs. (24a) or (24b) can be found in refs. 19, 21, 29, and 40. The model in the form of eq. (25b) is referred as the “General Modified Exponential” [ 11, 12, 241 or as “Generalized Logistic” [33]. An interesting property of GRM, and Von Bertalanffy models arises when their differential equations, (21) and (25a), respectively, are transformed as follows: f

GRM,:

f = b [ 1 -

(l/u)

[F -

(1 - o)f] If .

Von Bertalanffy:

(27)

Both eqs. (26) and (27) represent Logistic model of the form:

two generalizations

of the differential

-$jj_f,

i=+

(28)

where the fixed upper limit F is replaced by a varying upper limit F(f). In the GRM, model the varying upper limit F(f) is a linear function of the form

F,(f) =

$[F -

F2(f)

=

F

0 f

(I

of adoptersf

(2%

(1 - u)fl,

whereas, in the Von Bertalanffy

equation of the

model F(f)

has the nonlinear

form:

-4d .

(30)

Both eqs. (29) and (30) of the varying upper limit are illustrated in Figure 3 in an C$Fcf>) diagram. When u = 1 or + = 1, Fcf, = F and yields the Logistic model, whereas when u > 1 or 0 < 4 < 1 the upper limit increases with f and when 0 < u < 1 or 4 > 1 the upper limit decreases with J When F(f) = f the diffusion will stop. The point A at which the diffusion will stop is the point at which the line F,(f) or the curve Fzff) meet the “Stop Process Line” F(f) = f(Fig. 3). Considering the function F,(j) of Von Bertalanffy model when 0 < + < 1, this function [eq. (30)] represents a diffusion process with an upper limit that is zero at the origin cf = 0) and increases with f, whereas when I < $ the value of F --, w when f + 0, which is unrealistic. The function F,(j) of the GRM, model does not start at zero whenf = 0 but at a level equal to F/u. This is quite realistic for some cases of innovation diffusion when information sources and advertisements formulate a first level of potential adopters before the diffusion process has begun. When this is not the case, the function F&J of Von Bertalanffy’s model, with 0 < + < 1, might be more appropriate. However, the above analysis and the illustration in Figure 3 have shown that the GRM, and Von Bertalanffy models have similar characteristics and might be used alter-

C.H. SKIADAS

Von

Von

Bertalanffy

,

Bertalan

F/c

0

F Fig. 3. The varying saturation

f

level.

natively. In that case the points of inflection of the two models may approach each other. These points of inflection, fk, have the following equations:

GRM,.

Von Bertalanffy:

fk=

fk=

F 1+x&

F

(1 + $)“Q

Equating fkfrom eqs. (31) and (32) results in eq. (33), which gives parameter function of parameter +: u = [(l + $)I’* -

112.

(31)

(32) u as a

(33)

It must also be noted that the illustration in Figure 1 for GRM, model also holds for the Von Bertalanffy model if, where 1 < u, this is replaced by 0 < 4 < 1 and where 0 < u < 1, this is replaced by 1 < $.

NONSYMMETRIC

AND GENERALIZED

DIFFUSION

323

MODELS

Illustrations Time-series data expressing the cumulative percentage of steel produced by the oxygen process in Spain, Japan, the United States, and Romania and the time-series expressing the cumulative percentage production of continuous casting steel in Austria [32] are used in the application that follows. In all cases, the five-parameter GRM,,, the four-parameter GRM, and Generalized, and the three-parameter Logistic models are studied comparatively. The best model is considered to be the one that shows low MSE and relatively small standard errors of estimated parameters. If two models show close values of MSE, then the model with the fewer number of parameters can be selected as the best. In all cases parameter estimates are based on the method presented above. SPAIN (1964-1976)

In the GRM,, model parameter u shows high standard error (Table 1) and the value of MSE = 1.0380 of this model is very close to the MSE = 1.0593 of the four-parameter Generalized model. Thus, the latter is the more appropriate for Spain. The MSE of GRM, and that of the Logistic are not comparable to the MSE of the other two models. Here the negative value of g = - 8.6470 is a crucial factor influencing the diffusion process. JAPAN (1958-1974)

In the GRM,, model, parameter g shows very high standard error (Table 1). The GRM, model (g = 0) is the most appropriate one for this country. The MSE = 2.4952 of this model is very close to 2.4910, which is the MSE of the five-parameter GRM,,. The Generalized and the Logistic models show higher values of MSE, 2.7384 and 4.2749, respectively. The diffusion process is primarily characterized by an asymmetry in the direction of the Coleman process, u = 2.7734. USA (1955-1975)

The values of MSE for the GRM,,, GRM, , and Generalized models are very close to each other (Table 2) but parameter g shows a high standard error in GRM,,, ruling out use of this last model. The GRM, and Generalized models show comparable MSEs, 0.8190 and 0.8331, respectively, with a slight superiority in favor of GRM,. Moreover,

TABLE 1 Parameter Estimates, MSE and Ez for the Diffusion of Oxygen Steel Technology in Spain and Japan (Standard Errors in Parentheses) Spain (1964-1976) Parameter

GRMI,

b

0.5404 (0.2335)

fo

10.1900

F

0 g MSE i?

(1.1042) 53.8504 (1.1456) 0.6947 (0.5386) - 7.5204 (3.3375) 1.0380 0.9935

“Data from ref. 32.

GRM, 0.2765 (0.0312) 9.1026 (0.7621) 53.0471 (0.8304) 0.1897 (0.0951) 0 1.2716 0.9930

Japan (1958-1974)

Generalized

Logistic

0.6582 (0.0650) 10.4082 (0.9238) 54.2618 (0.8891) 1

0.4473 (0.0382) 7.3326 (0.9265) 56.7972 (1.7173)

- 8.6470

(1.4723) 1.0593 0.9942

1 0 2.8964 0.9857

GRMI, 0.7357 (0.4202) 3.1836 (1.7086) 84.1005 (2.4589) 2.5055 (1.9641) 0.8313 (6.0334) 2.4910 0.9952

GRM, 0.7947 (0.1698) 3.3792 (0.9716) 84.2982 (2.0545) 2.7734 (0.9828) 0 2.4952 0.9956

Generalized

Logistic

0.3863 (0.0359) 2.2726 (1.4825) 82.1886 (1.3941)

0.4719 (0.0264) 5.7767 (0.7388) 80.4228 (1.2003)

I

1

10.0061 (4.7731) 2.7384 0.995 1

0 4.2749 0.9930

324

C.H. SKIADAS

Parameter

Estimates,

TABLE 2 MSE and g* for the Diffusion of Oxygen Steel Technology and Romania (Standard Errors in Parentheses)

United States (1955-1975) Parameter

GRMI,

b

0.4092 (0.0559) 0.7857 (0.5384) 58.9674 (1.2413) 0.7617 (0.2446) -0.4384 (0.6822) 0.7999 0.9977

fo F IJ &T MSE R=

in the United States

Romania (1968-1975)

GRM,

Generalized

Logistic

GRMI,

GRM,

Generalized

Logistic

0.3824 (0.0302) 0.4570 (0.1178) 58.6953 (1.0884) 0.6648 (0.1581) 0

0.4572 (0.0212) 1.0083 (0.4526) 59.8863 (0.8682) 1

0.4358 (0.0160) 0.3032 (0.0537) 60.4263 (0.8797) 1

1.1409 (0.4516) 6.1274 (0.6096) 37.0631 (0.4113) 0.6764 (0.4462) - 3.6732 (2.3012) 0.1419 0.9965

0.6997 (0.0768) 5.7773 (0.4771) 36.9036 (0.3663) 0.2899 (0.1105) 0

1.4430 (0.1114) 6.2446 (0.5220) 37.1828 (0.3434) 1

1.0944 (0.0949) 4.5859 (0.7082) 37.7489 (0.7399) 1

0.8190 0.9979

- 0.7824 (0.4920) 0.8331 0.9979

0 0.9489 0.9976

0.1865 0.9970

-4.7055 (0.8720) 0.1553 0.0074

0 0.8030 0.9901

“Data from ref. 32.

parameter u in GRM, shows a t statistic of 0.664810.1581 = 4.20 instead of an absolute value of 0.7824/0.4920 = 1.59 for the t statistic of g in the Generalized model. Thus, the GRM, model is preferred here. However, the Generalized model is not rejected. One can say that the diffusion process here is mainly influenced by the asymmetry (a = 0.6648), whereas there appears a slightly negative force (g = -0.7824). ROMANIA

(1968-1975)

The GRM,, model seems to be the most appropriate (Table 2), although the Generalized model shows an MSE = 0.1553 that is close the MSE = 0.1419 of GRMI,. The negative force and the asymmetry, expressed by a g = - 3.6732 and u = 0.6764 in GRM,,, both influence the diffusion process. The GRMr and Logistic models provide values for MSE that are higher than those of the other two models.

Parameter

Estimates,

TABLE 3 MSE, and Ez for the Diffusion of Continuous (Standard Errors in Parentheses)”

Casting Steel in Austria

Austria (1970-1980) Parameter b

fa F Is g MSE i? “Data from ref. 32.

GRMI,

GRM,

Generalized

Logistic

0.8670 (0.5125) 8.3652 (1.0108) 63.6389 (15.0808) 2.6260 (3.6511) - 7.9966 (1.5654) 0.8863 0.9924

0.2701 (0.0437) 6.1660 (0.8663) 52.8394 (5.4202) 0.1801 (0.2100) 0

0.5883 (0.0801) 7.9493 (0.8653) 56.3797 (3.1662) 1

0.3370 (0.0420) 5.5807 (0.8437) 74.2544 (11.2250) 1

- 6.8835 (1.2977) 0.9868 0.9930

0

1.9062 0.9865

2.5150 0.9848

325

NONSYMMETRIC AND GENERALIZED DIFFUSION MODELS AUSTRIA (1970-I 980)

The GRM,, model is not acceptable because parameter cr shows high standard error (Table 3). The Generalized model is the most appropriate, with MSE = 0.9868, which is close to the value of MSE = 0.8863 for GRMI,. The GRM, and the Logistic models both show a very high MSE-1.9062 and 2.5159, respectively. The process here is mainly influenced by a negative force, g = - 6.8835, for the Generalized model.

Predictions The three of the above cases with the longest time-series data, that is Spain, Japan and United States, were used for a 5-year prediction run. Parameters of five models (GRM,,, GRM,, Generalized, Logistic and Von Bertalanffy) were estimated for 5-year sorter time periods for each of the three countries and predictions for 5 years ahead were made. Von Bertalanffy’s model was applied in the form of eq. (32) with the same nonlinear regression analysis technique as for the other models. The residual sum of squared errors (SSE) between actual and predicted values was also computed. In Spain (Table 4) the best prediction is obtained from the Generalized model, in Japan (Table 5) the GRM, offers the best prediction, and in the United States the Generalized model gives the best prediction. The models providing the best prediction in

Parameter

Estimates,

TABLE 4 MSE, @, Predictions, and SSE of Predictions for the Diffusion of Oxygen Steel Technology in Spain (Standard Errors in Parentheses) Spain (196471) GRM,

Generalized

Logistic

Von Bertalanffv

0.2620 (0.0378) 9.2305 (0.6750) 46.1157 (5.8375) 0.0826

0.6481 (0.0588) 10.2691 (0.3710) 52.7951 (2.3255)

0.3193 (0.0432) 8.9451 (0.7150) 87.1976 (23.4654)

(1.2187) 9.2728 (0.6247) 48.8173 (8.1295)

c

1.2425 (0.3089) 10.5857 (0.2228) 73.8810 (13.6489) 5.8409

g

(3.8141) - 10.0762

(0.1458) 0

9

(0.4837) -

Parameter

GRM,.

b

h F

MSE is2

0.0279 0.9996

Year

Actual

1972 1973 1974 197s 1976

52.0 50.0 51.8 54.1 53.8

SSE

-

“Data from ref. 32.

I - 8.3345 (0.8079) -

0.5259 0.9952

0.0997 0.9991

1.2924

1

-

0

-

0.7809 0.9947

5.1526 (5.2689) 0.4699 0.9958

Predictions 49.5718 53.8786 57.4907 60.4914 62.9679

46.0173 46.1115 46.1155 46.1157 46.1157

47.9911 50.1052 51.3464 52.0254 52.3892

51.9013 58.3583 64.1585 69. IS29 73.2991

47.3642 48.3907 48.6978 48.7843 48.8082

178.2250

206.0248

23.1484

829.4072

86.8787

326

Parameter

C.H. SKIADAS

Estimates,

TABLE 5 MSE, R*, Predictions, and SSE of Predictions for the Diffusion of Oxygen Steel Technology in Japan (Standard Errors in Parentheses) Japan (1958-69)

GRM,

Generalized

Logistic

Von Bertalanffy

u

0.1946 (0.1270) 2.1469 (1.5562) 73.3605 (3.1486) 0.1929

0.7001 (0.1878) 3.7580 (1.1421) 81.1455 (6.4934) 2.0447

0.4495 (0.0629) 2.9228 (1.5262) 78.2861 (3.2772) I

0.5249 (0.0359) 4.9121 (0.7167) 75.8680 (2.1550) 1

0.3778 (0.1015) 3.5094 (1.8417) 80.1724 (5.1319) -

R

(0.2592) 23.9734

(1.2373) 0

5.4827

0

-

+

(23.4524) -

Parameter

GRM,,

b

fo F

MSE i?

-

2.1695 0.9946

Year

Actual

1970 1971 1972 1973 1974

73.8 19.9 19.4 80.4 82.7

SSE

-

(4.5866) -

2.5912 0.9945

2.3739 0.9950

3.0312 0.9944

0.3501 (0.4177) 2.4955 0.9947

Predictions 73.0069 73.2300 73.3127 73.3430 73.3541

75.6265 77.1873 78.3149 19.1254 79.7060

75.0070 76.1641 76.9199 77.4093 77.7246

73.9053 74.6944 75.1693 75.4531 75.6220

75.4716 76.9095 77.9165 78.6169 79.1019

219.3184

22.4610

55.2627

119.5789

30.0636

“Data from ref. 32.

Spain and Japan were the most appropriate models selected in the preceding application whereas, the Generalized model was also retained in the case of United States. The most crucial factor for the quality of predictions seems to be the estimation of parameter F [25]. In all three applications (Tables 4-6) the model that gave the best prediction provided a value for F closer to the final actual value than the other models. In two of the three cases, Spain and the United States, the standard error of the estimate of F was small in comparison to that of the other models. In the last two cases the estimated values of F for the five models vary considerably among each other. In some cases the introduction of a time-varying function for parameter F might improve the forecasts [37, 381. In other cases special methods and forecasting techniques have been proposed. Most of these techniques adapt the estimation of parameters of the models to special characteristics of the diffusion phenomenon or improve and even propose new models and estimation techniques [6, 8, 17, 27, 411. It appears, therefore, that predictions are more a matter of special methods and techniques than simply making extrapolations based on the model that best fits the data. Therefore, the predictions presented above give only little information for the “predictability” of the models. Some of these models might give better predictions in the future after the related forecasting techniques are improved, and a more integrated view of the factors influencing the

NONSYMMETRIC

Parameter

AND GENERALIZED

DIFFUSION

MODELS

321

TABLE 6 Estimates, MSE, E*‘, Predictions, and SSE of Predictions for the Diffusion of Oxygen Steel Technology in the United States (Standard Errors in Parentheses)” United States

(1955-70)

GRM,

Generalized

Logistic

Von Bertalanffy

0.3588 (0.0237) 0.5330 (0.0975) 52.4781 (4.0590) 0.4012

0.4345 (0.0256) 0.8609 (0.3172) 62.1506 (3.1907)

0.4059 (0.0171) 0.3935 (0.0623) 65.2259 (3.2430)

0.6311 (0.1511) 0.5669 (0.1100) 55.3120 (4.1969)

(T

0.3764 (0.0479) 0.6710 (0.3464) 53.7208 (5.4383) 0.4858

1

-

8

(0.2946) - 0.2242

(0.1774) 0

0

-

+

(0.5018) -

0.3580 0.9986

(0.5573) 0.2825 0.9988

Parameter

GRMI,

b

fo F

MSE R2

Actual

1971 1972 1973 1974 1975

52.2 54.7 55.6 54.7 60.1

SSE

- 0.5762 (0.3737) -

-

0.2788 0.9987

Year

1

0.3035 0.9987

0.2830 0.9988

1.8342

-

Predictions 50.2020 51.9785 52.8886 53.3307 53.5396

49.8811 51.3373 5 I .9965 52.2785 52.3960

51.6346 54.9044 57.2550 58.8889 59.9982

52.2185 55.9403 58.7297 60.7482 62.1721

50.4604 52.5748 53.8069 54.4967 54.8740

63.6642

94.8848

20.6578

52.2074

38.1107

‘Data from ref. 32

parameters of innovation and technological innovation diffusion [ 141, primarily parameter F, is obtained. In this study, emphasis has been placed on the explanatory ability of the models used, and for this purpose the mean squared error (MSE) of the estimates seems to be a good criterion. Comparisons Between the GRM, and Von Bertalanffy Models, and Between Their Parameters u and + In all cases studied above the Von Bertalanffy model as also applied. The results are included in Tables 4-7. In Table 8 the estimated values for parameters u of GRM, and 4 of Von Bertalanffy model are summarized, as well as the fraction of MSE of the two models. Parameter u is also calculated using eq. (33) and summarized in Table 8. This value of u for GRMl model was close to that calculated from eq. (33) except in one case, Spain (1964-76). The fraction of MSEs between the GRM, and Von Bertalanffy models in six cases was relatively close to 1. In one case, Spain (1964-7 l), the fraction was 1.1192 and in another case, Japan (1958-69), the fraction was 1.1972. The results give a slight superiority to the Von Bertalanffy model, but also confirm the similar behavior of the GRM, and Von Bertalanffy models, as was expected by the preceding analysis.

328

C.H. SKIADAS

TABLE 7 Parameter Estimates. MSE. and 2’ for Von Bertalanffv Model (Standard Errors in Parentheses)” Process

Oxygen

Parameter b fo F + MSE R=

Continuous Casting Steel

Steel

Spain ( 1964-76)

Japan (1958-74)

United States (1955-75)

1.0411 (0.2843) 9.2810 (0.7458) 53.3535 (0.8852) 4.1423 (1.4475) 1.1988 0.9934

0.3172 (0.0427) 3.0124 (3.2556) 83.5723 (1.7816) 0.1139 (0.2320) 2.5075 0.9955

0.5371 (0.0655) 0.5183 (0.1427) 58.9041 (1.0133) 1.4990 (0.3096) 0.8017 0.9979

Romania (1968-75)

Austria (1970-80)

1.9417 (0.3737) 5.8510 (0.4783) 36.985 1 (0.3653) 3.0700 (0.8537) 0.1783 0.997 1

0.9286 (0.6282) 6.2781 (0.8365) 54.2935 (6.1702) 3.6775 (2.8853) 1.7598 0.9876

“Data from ref. 32

Regarding Fcf), only Japan in both of the studied cases provided a value for parameter u higher than unity, which corresponds to values for parameter + lower than unity (Table 8). This means that only Japan shows a growing pattern for the varying upper limit F, as illustrated in Figure 3. In all other cases the varying saturation level follows a decreasing pattern, due to the restriction that 0 < u < 1 or 1 < +. Summary and Conclusions A nonsymmetric model, labeled GRM,, was formulated. This model gives a point of inflection that varies across the range between 0 and F and covers the three basic models-Coleman, Exponential, and Logistic. An analysis of the Generalized model showed that it may express negatively or positively influencing behavior. The equation of the Generalized model combined with the equation of GRM, gives the equation of a five-parameter model, named GRM Ig , that expresses both asymmetry and positively or negatively influencing forces. The properties of the GRM,, Generalized, and GRM,, models were examined and the used iterative nonlinear regression analysis technique developed. This technique seems to be appropriate for application of the above models as well as for application of the GRM models and other models, such as the Von Bertalanffy model, also studied here. This model is a nonsymmetric model that shows behavior similar to that of GRM, model. The Von Bertalanffy and the GRMl models TABLE 8 Comparison Between Parameters (r and 4 and MSE of GRM, and Von Bertalanffy Models 4) Country

@on Bertalanffy)

Spain (1964-76) Spain (1964-71) Japan (1958-74) Japan (1958-69) USA (1955-75) USA (1955-70) Romania ( 1968-75) Austria (1970-80)

4.1423 5.1526 0.1139 0.3501 1.4990 1.8342 3.0700 3.6775

(GEM,) 0.1897 0.0826 2.7734 2.0447 0.6648 0.4012 0.2899 0.1801

U = [(l + +)I’+ - 112 0.2351 0.1787 2.4903 1.8416 0.7094 0.5847 0.3360 0.2717

Fraction of MSE GRM,/Von Bertalanffy 1.0607 1.1192 0.9952 1.1972 1.0216 1.0018 1.0460 1.0832

NONSYMMETRIC

AND GENERALIZED

DIFFUSION

329

MODELS

were comparatively analyzed and the properties of their parameters cr and 4 investigated. It was shown that these two models are generalized forms of the Logistic model, and are formulated when the stable parameter F, expressing the upper limit of the diffusion process in the Logistic model, is replaced by functions that represent a varying upper limit F with respect to adopters f. Five comparative applications between GRM,,, GRM,, Generalized, and Logistic were performed, with the most appropriate model for each case selected, and the dynamics of the diffusion, expressed by parameters u and g, investigated. Predictions were made based on these four models as well as on the Von Bertalanffy model. Based on all above applications, comparisons between GRM, and Von Bertalanffy were made. It was shown that both models have similar behavior and might be used alternatively. Emphasis was primarily on the explanatory rather than the predictive behavior of the used models. It is hoped that the above models and methods will have useful applications to the forecasting area, but especially that they will lead to a better understanding of the mechanisms underlying the diffusion process. The author thanks Professor of Athens for his encouragement

N. Koumoutsos

of the National Technical

and useful suggestions

University

during the preparation

work. He is also grateful for the helpful comments of two anonymous

of this

referees.

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