- Email: [email protected]
Comments by Joseph P. Martino
T. Modis / Technological Forecasting & Social Change 74 (2007) 391–404
399
for business executives facing a major transition between products, technologies, or other fundamental change is not to strive for minimizing its impact but to plan for and anticipate a low-growth period comparable in duration to the high-growth period the just enjoyed. Proverbial wisdom has long claimed that there is goodness in every season. Some people may think that the most desirable climate is found on tropical islands like Mauritius and Seychelles. Not true! Our above argument based on harmony dictates a large and regular seasonal variation like that encountered in temperate climate, which has also been the cradle of most great civilizations. History is poor in significant cultures that emerged from the arctic or from the tropics, the former perhaps because of conditions hostile to life and the latter mostly because of lack of variation and motivation. Despite an idyllic setting, tropical islands are rather sterile. The expulsion of Adam and Eve from Paradise may have been, after all, the original blessing.
References [1] Thanks to one reviewer's comments I found out that Asimov has published some of these arguments in an article in 1956: I., [2] [3] [4] [5]
Asimov, The abnormality of being normal, Astounding Science Fiction, May 1956, vol. LVII, Number 3. T. Modis, A. Debecker, Chaos-like states can be expected before and after logistic growth, Technol. Forecast. Soc. Change 41 (1992). T. Modis, Fractal aspects of natural growth, Technol. Forecast. Soc. Change 47 (1994). H.B. Stewart, Recollecting the Future, Dow Jones-Irwin, Homewood, IL, 1989. T. Modis, Conquering Uncertainty, McGraw-Hill, New York, 1998.
Theodore Modis is the founder of Growth Dynamics, an organization specializing in strategic forecasting and management consulting (http://www.growth-dynamics.com). doi:10.1016/j.techfore.2006.07.003
Discussion of Modis article Comments by Joseph P. Martino The idea that to be “normal” is actually unusual is not a new one. Years ago Isaac Asimov wrote an article in ANALOG SCIENCE FICTION titled “The Abnormality of Being Normal.” The point was the same as Modis's. An entity may be “typical” in one respect, but demanding that it also be “typical” in another aspect adds a constraint. The more different aspects in which the entity is required to be “typical”, the fewer the entities that will qualify. It is important to note, however, that Modis uses the term “Normal”, as applied to the Gaussian distribution, improperly. It makes no sense to call a single item “normal” in a statistical sense. A population may be Normal with respect to a single characteristic, or a sample may be drawn from a Normal population (again with reference to a single characteristic), but a single individual drawn from a Normal population may have a value on that characteristic anywhere between minus infinity and plus infinity. It is not the individual that is Normal, but the distribution of the population. That is, a population (or sample) is “Normal” if the probability of being in a certain range of values conforms to the Normal probability distribution.
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T. Modis / Technological Forecasting & Social Change 74 (2007) 391–404
As for the validity of the Normal or Gaussian distribution, there is a joke among statisticians that everyone believes in the Normal distribution; the theoreticians because they believe it to be an experimental fact, and the experimenters because they believe it to be a mathematical theorem. In reality, there are many cases in experimental science where the distribution of values appears to have been drawn from a Normal distribution. However, there is no guarantee that every data set will turn out to have been drawn from a Normal distribution. Some experimental data appear to be drawn from other distributions (Binomial, Poisson, Exponential, and Hypergeometric are common ones). In a practical case, one has to apply a statistical test to verify whether a sample can be said to be drawn from a particular distribution. Mathematicians have not neglected the Logistic distribution, which as Modis points out, is unimodal and looks very much like the Normal. In fact, in estimating the probability distribution of certain types of events, such as probability of failure as a function of applied stress, the Logit curve (Logistic distribution) is often used in place of the Probit curve (Normal distribution), not because it is believed to be “better” in terms of representing reality, but because Logit is mathematically tractable while Probit is very difficult to handle. In reality, since the Logistic and the Normal do not differ by that much, it may be that the differences between Probit and Logit disappear in the experimental error. Modis's Eq. (2) is incorrect. This is NOT the differential equation equivalent to Eq. (1). Repeating Eq. (1): dX ¼ aX ðM −X Þ dt To get from this differential equation to the Logistic, after some manipulation Z Z dX ¼ a dX X ðL−X Þ one integrates to get: X ¼
L 1 þ ae−bt
(see [1], p. 678). The correct differential equation is: ðD X Þn ¼ aXn ðM −Xn Þ To get from this differential equation to a stairstep approximation to the Logistic, one sums: Xnþ1 ¼ Xn þ ðD X Þn Dt ¼ Xn þ aXn ðM −Xn ÞDt It has been a common mistake to think that because the differential equation for the Logistic resembles the equation for a certain type of chaotic behavior that the Logistic is chaotic. In fact, however, the chaotic behavior arises not from summing the differential equation, as in (2*′), but in iterating the chaotic equation, as in Modis's Eq. (2). That is, Xnþ1 ¼ aXn ðM −Xn Þ
T. Modis / Technological Forecasting & Social Change 74 (2007) 391–404
401
This misinterpretation is not unique to Modis. It has appeared several times in the literature. However, it is not correct. This is not to say that oscillations cannot occur at the overlap between one technological growth curve and a successor growth curve. There are examples in the literature of this happening. However, the phenomenon does not arise from any supposed chaotic behavior of the differential equation for the Logistic. Modis tries to argue that the Logistic is “natural” in some sense. However, it is not obviously any more “natural” than the Normal. One can make simple models of rabbits and grass that then follow a Logistic, but in the real world a model would have to include (at the very least) seasons of the year, predators, and competitors for grass. It is not clear that the population growth would be anything like a Logistic. It is clear that many phenomena in the real world do appear to produce a Normal population. The assumptions behind the Normal are not that hard to meet, so it would be expected that many phenomena would be approximately Normal (since the Normal goes to infinity in both directions, no physical phenomenon can ever be truly Normal). One might argue that the actual data would fit the Logistic as well as the Normal, but the assumptions behind the Logistic simply do not apply to the phenomena such as the “time to failure” of mechanical components, which appears to be Normal. Hence the Normal distribution is preferred because its assumptions “make sense” in terms of the actual phenomenon. As for Modis's remaining arguments, the problem of when to introduce a new product is a recurring one in industry. His argument, that one should plan the successor product while the predecessor product is still growing strongly, is undoubtedly correct. But as for timing that introduction correctly, surely the planner must take into account external economic conditions, the likely actions of competitors, and considerations such as physical life of the product (customers may have bought a second identical predecessor product when the first one wore out, and do not want to replace it prematurely with a successor product even if it is better), and even whether customers have finished paying for the predecessor product (a common consideration in airline purchases). I conclude that Modis's sine wave analysis is mathematically correct, but it is not clear whether it has anything to do with actual cases such as planning product introduction. Reference [1] Jeffrey Alan, Mathematics for Engineers and Scientists, 6/e, Chapman and Hall, New York, 2005.
Joseph P. Martino 905 S. Main Street, Sidney, OH 45365, USA E-mail address: [email protected]. 5 August 2006 doi:10.1016/j.techfore.2006.08.007
399
for business executives facing a major transition between products, technologies, or other fundamental change is not to strive for minimizing its impact but to plan for and anticipate a low-growth period comparable in duration to the high-growth period the just enjoyed. Proverbial wisdom has long claimed that there is goodness in every season. Some people may think that the most desirable climate is found on tropical islands like Mauritius and Seychelles. Not true! Our above argument based on harmony dictates a large and regular seasonal variation like that encountered in temperate climate, which has also been the cradle of most great civilizations. History is poor in significant cultures that emerged from the arctic or from the tropics, the former perhaps because of conditions hostile to life and the latter mostly because of lack of variation and motivation. Despite an idyllic setting, tropical islands are rather sterile. The expulsion of Adam and Eve from Paradise may have been, after all, the original blessing.
References [1] Thanks to one reviewer's comments I found out that Asimov has published some of these arguments in an article in 1956: I., [2] [3] [4] [5]
Asimov, The abnormality of being normal, Astounding Science Fiction, May 1956, vol. LVII, Number 3. T. Modis, A. Debecker, Chaos-like states can be expected before and after logistic growth, Technol. Forecast. Soc. Change 41 (1992). T. Modis, Fractal aspects of natural growth, Technol. Forecast. Soc. Change 47 (1994). H.B. Stewart, Recollecting the Future, Dow Jones-Irwin, Homewood, IL, 1989. T. Modis, Conquering Uncertainty, McGraw-Hill, New York, 1998.
Theodore Modis is the founder of Growth Dynamics, an organization specializing in strategic forecasting and management consulting (http://www.growth-dynamics.com). doi:10.1016/j.techfore.2006.07.003
Discussion of Modis article Comments by Joseph P. Martino The idea that to be “normal” is actually unusual is not a new one. Years ago Isaac Asimov wrote an article in ANALOG SCIENCE FICTION titled “The Abnormality of Being Normal.” The point was the same as Modis's. An entity may be “typical” in one respect, but demanding that it also be “typical” in another aspect adds a constraint. The more different aspects in which the entity is required to be “typical”, the fewer the entities that will qualify. It is important to note, however, that Modis uses the term “Normal”, as applied to the Gaussian distribution, improperly. It makes no sense to call a single item “normal” in a statistical sense. A population may be Normal with respect to a single characteristic, or a sample may be drawn from a Normal population (again with reference to a single characteristic), but a single individual drawn from a Normal population may have a value on that characteristic anywhere between minus infinity and plus infinity. It is not the individual that is Normal, but the distribution of the population. That is, a population (or sample) is “Normal” if the probability of being in a certain range of values conforms to the Normal probability distribution.
400
T. Modis / Technological Forecasting & Social Change 74 (2007) 391–404
As for the validity of the Normal or Gaussian distribution, there is a joke among statisticians that everyone believes in the Normal distribution; the theoreticians because they believe it to be an experimental fact, and the experimenters because they believe it to be a mathematical theorem. In reality, there are many cases in experimental science where the distribution of values appears to have been drawn from a Normal distribution. However, there is no guarantee that every data set will turn out to have been drawn from a Normal distribution. Some experimental data appear to be drawn from other distributions (Binomial, Poisson, Exponential, and Hypergeometric are common ones). In a practical case, one has to apply a statistical test to verify whether a sample can be said to be drawn from a particular distribution. Mathematicians have not neglected the Logistic distribution, which as Modis points out, is unimodal and looks very much like the Normal. In fact, in estimating the probability distribution of certain types of events, such as probability of failure as a function of applied stress, the Logit curve (Logistic distribution) is often used in place of the Probit curve (Normal distribution), not because it is believed to be “better” in terms of representing reality, but because Logit is mathematically tractable while Probit is very difficult to handle. In reality, since the Logistic and the Normal do not differ by that much, it may be that the differences between Probit and Logit disappear in the experimental error. Modis's Eq. (2) is incorrect. This is NOT the differential equation equivalent to Eq. (1). Repeating Eq. (1): dX ¼ aX ðM −X Þ dt To get from this differential equation to the Logistic, after some manipulation Z Z dX ¼ a dX X ðL−X Þ one integrates to get: X ¼
L 1 þ ae−bt
(see [1], p. 678). The correct differential equation is: ðD X Þn ¼ aXn ðM −Xn Þ To get from this differential equation to a stairstep approximation to the Logistic, one sums: Xnþ1 ¼ Xn þ ðD X Þn Dt ¼ Xn þ aXn ðM −Xn ÞDt It has been a common mistake to think that because the differential equation for the Logistic resembles the equation for a certain type of chaotic behavior that the Logistic is chaotic. In fact, however, the chaotic behavior arises not from summing the differential equation, as in (2*′), but in iterating the chaotic equation, as in Modis's Eq. (2). That is, Xnþ1 ¼ aXn ðM −Xn Þ
T. Modis / Technological Forecasting & Social Change 74 (2007) 391–404
401
This misinterpretation is not unique to Modis. It has appeared several times in the literature. However, it is not correct. This is not to say that oscillations cannot occur at the overlap between one technological growth curve and a successor growth curve. There are examples in the literature of this happening. However, the phenomenon does not arise from any supposed chaotic behavior of the differential equation for the Logistic. Modis tries to argue that the Logistic is “natural” in some sense. However, it is not obviously any more “natural” than the Normal. One can make simple models of rabbits and grass that then follow a Logistic, but in the real world a model would have to include (at the very least) seasons of the year, predators, and competitors for grass. It is not clear that the population growth would be anything like a Logistic. It is clear that many phenomena in the real world do appear to produce a Normal population. The assumptions behind the Normal are not that hard to meet, so it would be expected that many phenomena would be approximately Normal (since the Normal goes to infinity in both directions, no physical phenomenon can ever be truly Normal). One might argue that the actual data would fit the Logistic as well as the Normal, but the assumptions behind the Logistic simply do not apply to the phenomena such as the “time to failure” of mechanical components, which appears to be Normal. Hence the Normal distribution is preferred because its assumptions “make sense” in terms of the actual phenomenon. As for Modis's remaining arguments, the problem of when to introduce a new product is a recurring one in industry. His argument, that one should plan the successor product while the predecessor product is still growing strongly, is undoubtedly correct. But as for timing that introduction correctly, surely the planner must take into account external economic conditions, the likely actions of competitors, and considerations such as physical life of the product (customers may have bought a second identical predecessor product when the first one wore out, and do not want to replace it prematurely with a successor product even if it is better), and even whether customers have finished paying for the predecessor product (a common consideration in airline purchases). I conclude that Modis's sine wave analysis is mathematically correct, but it is not clear whether it has anything to do with actual cases such as planning product introduction. Reference [1] Jeffrey Alan, Mathematics for Engineers and Scientists, 6/e, Chapman and Hall, New York, 2005.
Joseph P. Martino 905 S. Main Street, Sidney, OH 45365, USA E-mail address: [email protected]. 5 August 2006 doi:10.1016/j.techfore.2006.08.007