The Weibull distribution as a general model for forecasting technological change

TECHNOLOGICAL

FORECASTING

AND SOCIAL CHANGE

18, 247-256

(1980)

The Weibull Distribution as a General Model for Forecasting Technological Change M. NAWAZ SHARIF and M. NAZRUL

ISLAM

ABSTRACT This paper presents a simple and widely applicable model for technological forecasting using the Weibull distribution function. It shows the generality of the Weibull function, which can accommodate different patterns of technological change. A method for estimating parameters and some applications are presented to illustrate the usefulness of the model.

Introduction Since the pioneering work of Mansfield [S], various models have been proposed to represent the time pattern of technological changes. These are now widely used for technological forecasting. Some of the well-known forecasting models available in the literature are the Blackman model [I], the Fisher-Pry model [2], the Floyd model [3], the Stapleton (normal distribution) model [lo], the Sharif-Kabir model [9], the Pearl curve [S], and the Gompertz curve [6]. Both the Blackman model and the Fisher-Pry model are actually Pearl curves, and all these models follow an S-shaped pattern. Experience has shown that technologies tend to follow a predictable pattern of growth that can reasonably be described by S-shaped curves. For practical purposes, the problem is to determine the steepness of the curve and the point at which inflection occurs. The inflection points of different models are shown in Figure 1. As can be observed from Figure 1 and Table 1, the selection of a model for forecasting technological change has an implicit relationship with the selection of the point of inflection of the S curve. Except for the Sharif-Kabir model (in which case the inflection point varies fromf = 0.333 to 0.5 wherefis the fraction of the upper limit achieved), all have a fixed inflection point. Though there is some flexibility in choosing the inflection point in the Sharif-Kabir model, there are two limitations: (i) The range is not very wide and (ii) the procedure for estimating the location of the inflection point within the short range is also complicated. Moreover, none of the above models has an inflection point beyond f = 0.5 or below f = 0.333. In addition to the fixed point of inflection, the Blackman, Fisher-Pry, and Stapleton models and Pearl curve have the additional constraint of being symmetrical about the inflection point. It is therefore not difficult to understand why each of these models shows remarkable success in describing certain situations and yet fails in other cases. In real-world situations the S curve can be symmetrical as well as nonsymmetrical, with the point of inflection in

M. NAWAZ SHARIF is Professor of Industrial and Systems Engineering and M. NAZRUL ISLAM is a doctoral student in the Division of Industrial Engineering and Management of the Asian Institute of Technology, Bangkok, Thailand. @M.

N. Sharif et al., 1981

M. N. SHARIF AND M. N. ISLAM

248

0.0 --

0.4=___________ ______ SHARIF 0.2

-__-0.6

___--- KABIR

-

_-----------

-0.5

_-_-_--

-4

-3

-2

-I

0

I

0.367

2

3

4

3

6

TIME

Fig. 1 Available models and their inflection points.

either the earlier or later phase of development. This paper presents a model capable of overcoming these limitations. The model is based on the Weibull distribution function. The following section describes the appropriateness of the Weibull function as a general model for technological forecasting. Later sections present a method for estimating model parameters and a few examples. Weibull

Distribution

as Model

The Weibull probability

density function

(p.d.f.)

is given by [7, 111

WEIBULL DISTRIBUTION

FOR TECHNOLOGICAL

CHANGE

249

TABLE 1 Available Models and Their Inflection Points Fraction of the upper limit when inflection occurs

Model

Expression

Blackman

In f F-f

Fisher-Pry

In f = C, + C,r 1 -f

0.500

Pearl curve

f=

0.500

= C, + C,r

0.500

F 1 + c/o

Stapleton

0.500

f = Fem',p 2'

Gompertz curve

In-+- f

F-f

In __ f F-f

Sharif-Kabir

f(r)

=

(~l(W)(tl~)‘“-“e-“‘~‘”

and its probability

distribution

F(t) = 1 - e-V/rrP

0.367

F F-f

= c, + C,r F

+ uF-f

for

= c, + c,r

t> 0

function (cummulative for

t > 0

0.500 for (T = 0 0.382 for u= 0.5 0.333 for o= 1

(1)

p.d.f.)

is given by (2)

where (Y > 0 is scale parameter, p > 0 is shape parameter and t is time. The curves forf(t) and F(t) for different values of o and p are shown in Figures 2 and 3. It can be seen that (Yand /? together determine the steepness of the curve, while /3 alone determines the shape of the curve. Note that the Pearl curve and the normal probability curve (Stapleton model) share the property that the location and the shape of the curve can be adjusted independently of one another. Depending on the value of p the Weibull curve becomes left skewed, symmetrical, or right skewed. The inflection points of the curve for specific values of /3 are shown in Table 2. In a way, p is similar to the delay coefficient in Sharif-Kabir model. Thus, the Weibull function effectively models technological forecasting for a wide variety of situations. The probability density function and the probability distribution function at any point of time may be interpreted as the rate of change of the technology and the fraction of the upper limit achieved respectively at that point of time. Based on the above discussions on the Weibull function, the following empirical model is proposed as a general model for technological forecasting: F(t) = L - Le-“‘*‘4

(3)

where I is time, L is the upper limit, F(t) is the fraction of the upper limit achieved at time t, and (Y, ,L?are constants. For technological substitution cases, the upper limit L is taken to be any desired value less than or equal to unity.

M. N. SHARIF AND M. N. ISLAM

250

0

I

2

3

Fig. 2 Weibull p.d.f. and distribution

4

5

6

7

t function for different values of cy (/3 = 3).

TABLE 2 Variations in the Weibull Curve

Value of p

Value off at Inflection Point

12.0000 2.2814

0.632 0.500

1.8424 1.6814

0.367 0.333

1 .oOOo

El

Remarks Right-skewed distribution Symmetrical distribution Blackman Fisher-Pry Pearl Stapleton Sharif-Kabir (a = 0) Left-skewed distribution Gompertz Floyd Sharif-Kabir (CT= 1) Exponential distribution

WEIBULL

DISTRIBUTION

FOR TECHNOLOGICAL

CHANGE

0.6 -

0.6

-

s LL

0

2

3

4

5

6

t

Fig. 3. Weibull p.d.f. and distribution

function for different values of p ((Y = 4).

Estimation of Parameters Equation (3) can also be written ln(ln&)=Pln(a)+Pln(~). Y=A+BX where

A =PIn(cz), B = P, X = In(r).

(4) (5)

M. N. SHARIF AND M. N. ISLAM

252

The value of constants can be obtained by fitting the line to the data by least-squares regression. However, one must be careful in evaluating the constants of (4) by the least-squares method. Note that the independent variable X is the logarithm of time, so that, unlike in other models, we cannot select any reference year as t = 0 for the purpose of regression analysis. Therefore, before fitting (4) to the data, one has to estimate the base year (i.e., the year at which F(t) = 0) where the time must equal zero, (because in the Weibull distribution, F(t) = 0 at t = 0 and F(t) approaches its limit as t approaches infinity). The base year can be estimated by trial and error with the correlation coefficient as the criteria. The following algorithm has been found useful:

500*

400 -

300 -

200 -

00

1690

I

1900

1910

1920

1930

I

I

1940

1950

1960

1970

1940

1950

1960

1970

Year

1900

1910

1920

1930

Year Fig. 4. Growth of telephone per 1000 people in the United States fitted to (a) the text model (r = 0.98356, base year 1867) and (h) the normal distribution model (r = 0.9777): o, historical data [6].

WEIBULL

DISTRIBUTION

FOR TECHNOLOGICAL

253

CHANGE

1. Choose the base year to be the year preceding the first available data year. 2. Fit (4) to the data with the base year chosen and compute the correlation coefficient . 3. Decrease the base year by 1 and perform step 2 with the new base year. If the correlation coefficient is less than the immediately preceding value, go to step 4. Otherwise repeat this step. 4. Select this as the correct base year for curve fitting. Illustrations To illustrate the model, one example from Stapleton’s (normal distribution) article and one from Fisher and F’ry were reworked. The resulting Weibull curves are shown in Figures 4 and 5 along with the normal distribution and Fisher-Pry curves. It can be seen that the model works perfectly in those cases.

1900

1920

1940

1960

1960

2000

2020

2040

2060

2040

2060

Year

1900

1920

1940

1960

1980

2000

2020

Year Fig. 5. Substkution of synthetic for natural fiber, fitted to (a) the text model (r = 0.9936, base year 1904) and (b) the Fisher-Pry model (r = 0.9852): o, historical data 141.

M. N. SHARIF AND M N. ISLAM

254

0.6 -

0.6 -

‘21 0.4 -

0.2 YEAR OF FORECAST 0.0

1930

1935

1940

1945

1950

1955

1960

1965

1970

Year

Fig. 6. Substitution data [4].

of diesel and electric for steam locomotives

in the United States: o, historical

In order to compare this model with Sharif-Kabir model, two examples from Sharif and Kabir were reworked. The base years were estimated by the procedure given in the previous section. All data points corresponding to F(t) < 8% of limit were ignored. Forecasts were made at the time when F(t) reached 30% of its limit. The resulting curves are shown in Figures 6 and 7. Table 3 further compares the performance of the proposed model with that of Sharif-Kabir model. It can be seen from Figures 6 and 7 that the historical data confirm the long-range predictions of the model. The correlation coefficients and effective lifespan (ELS), the time for F(t) to go from 10% and 90% of its limit, are also comparable to those of the Sharif-Kabir model. The ELS in cases 1 and 2 are 11 and 55 years, respectively. For case

0.6 -

0.6 t tL

YEAR OF FORECAST

I885

1895

1905

1915

1925

1935

1943

1955

1965

Year Fig. 7. Substitution

of metal for wood in the U.S. merchant marine market; o, historical data [6].

WEIBULL

DISTRIBUTION

FOR TECHNOLOGICAL

CHANGE

255

TABLE 3 Model Parameters for Comparison Sharif-Kabir

Proposed model Cases

I 2

yea

Correlation coefficient

ELS years

1918 1848

0.99434 0.99528

11 50

Base F(r)” 0.46 0.30

F(r)” 0.5 0.4

Case 1: Substitution of diesel and electric locomotives for steam locomotives Case 2: Substitution of metal for wood in the U.S. Merchant Marine. “At the time of forecast.

model

Correlation coefficient 0.99377 0.99951

ELS years 1.5 55

in the United States

1 the superiority of the proposed model may be caused by its flexibility locating the inflection point. For case 2, as can be observed in Figure 7, there was some disruption in the last stage of the actual substitution process. To demonstrate its ability to handle problems other than substitution, the proposed model was also applied to the present growth of dwellings with electric power in the United States. The upper limit chosen in this case was 100. (This case is taken from Martin0 [6].) The predictions of the model along with the historical data points are presented in Figure 8. Here, too, the data confirm the long-range predictions of the model. Conclusion From the examples presented it can be seen that the proposed model based on the Weibull distribution can be used effectively to predict technological growth as well as technological substitution. Estimation of the base year can easily be done with the help of an iterative procedure. Moreover, the flexibility of the model, in terms of accommodating all possible variations of the S curve, makes it a simple but versatile tool for forecasting technological change.

1900

1910

1920

1930

1940

1950

1960

1970

1980

Year Fig. 8. Growth of U.S. dwellings, r = 0.99728, base year 1881).

in percent, with electric power: o, historical data [6] (limit of 100,

M. N. SHARIF AND M. N. ISLAM

256

References I. Blackman, A.W., Jr., A Mathematical Model for Trend Forecasts, Techno/. Forecast. Sot. Change 3, 441-452 (1972). 2. Fisher, J.C., and Pry, R.H., A Simple Substitution Model of Technological Change, Technol. Forecasr. Sot. Change 3, 75-88 (1971). 3. Floyd, A., Trend Forecasting: A Methodology for Figure of Merit, Technological Forecusringfor Industry md Go~~ernment: Methods md Applictitions (J. Bright, ed.), Prentice-Hall, Englewood Cliffs, N.J. 1962, pp. 95-109. 4. HutoriculStutistics ofthe UnitedStates Colonial Times IO 1978, A Statistical Abstract Supplement Prepared by the U.S. Department of Commerce, Bureau of the Census. 5. Mansfield, E., Technological Change and the Rate of Imitation, Economerricu 29, 741-765 (October 1961). 6. Martino, J.P., Technological Forecusringfi,r Decision Making. American Elsevier, New York, 1975. 7. Mode, E.B., E/ements ofProbubifity md Sturistica, Prentice-Hall, Englewood Cliffs, N.J., 1966. 8. Pearl, R., The Biology qf PopulationGrowth, Knopf, New York, 1925. 9. Sharif, M.N., and Kabir, C., A Generalized Model for Forecasting Technological Substitution, Technol. Forecast.

Sot. Chunge

8, 353-364

(1976).

IO. Stapleton, E., The Normal Distribution as a Model of Technological Substitution, Technol. Forecost. Change 8, 325-334 (1976). I I. Walpole, R.E., and Myers, R.H., Probabiliry und Srutistics,for Engineers and Scientisrs, Macmillan, York. 1972. Pcceired 22 September

1980; revised 14 Nowmber

1980

Sot.

New