S- Curve Framework

haizia

New member
The S-Curve Framework


The S-Curve emerged as a mathematical model and was afterwards applied to a variety of fields including physics, biology and economics. It describes for example the development of the embryo, the diffusion of viruses, the utility gained by people as the number of consumption choices increases, and so on.


In the innovation management field the S-Curve illustrates the introduction, growth and maturation of innovations as well as the technological cycles that most industries experience. In the early stages large amounts of money, effort and other resources are
expended on the new technology but small performance improvements are observed.

Then, as the knowledge about the technology accumulates, progress becomes more rapid.
As soon as major technical obstacles are overcome and the innovation reaches a certain adoption level an exponential growth will take place. During this phase relatively small increments of effort and resources will result in large performance gains. Finally, as the
technology starts to approach its physical limit, further pushing the performance becomes increasingly difficult, as the figure below shows (see Fig. 3.6. )
























Fig. 3.6 : The S-Curve

Consider the supercomputer industry, where the traditional architecture involved
single microprocessors. In the early stages of this technology a huge amount of money was
spent in research and development, and it required several years to produce the first
commercial prototype. Once the technology reached a certain level of development the
know-how and expertise behind supercomputers started to spread, boosting dramatically
the speed at which those systems evolved. After some time, however, microprocessors
started to yield lower and lower performance gains for a given time/effort span, suggesting
that the technology was close to its physical limit (based on the ability to squeeze transistors
in the silicon wafer). In order to solve the problem supercomputer producers adopted a
new architecture composed of many microprocessors working in parallel. This innovation
created a new S-curve, shifted to the right of the original one, with a higher performance
limit (based instead on the capacity to co-ordinate the work of the single processors – see
Fig. 3.7. ).

NOTES


Usually the S-curve is represented as the variation of performance in function of
the time/effort. Probably that is the most used metric because it is also the easiest to collect
data for. This fact does not imply, however, that performance is more accurate than the
other possible metrics, for instance the number of inventions, the level of the overall research,
or the profitability associated with the innovation.

One must be careful with the fact that different performance parameters tend to be
used over different phases of the innovation, as a result the outcomes may get mixed
together, or one parameter will end up influencing the outcome of another. Civil aircraft
provides a good example, on early stages of the industry fuel burn was a negligible parameter,
and all the emphasis was on the speed aircrafts could achieve and if they would thus be
able to get off the ground safely. Over the time, with the improvement of the aircrafts
almost everyone was able to reach the minimum speed and to take off, which made fuel
burn the main parameter for assessing performance of civil aircrafts.


Overall we can say that the S-Curve is a robust yet flexible framework to analyze
the introduction, growth and maturation of innovations and to understand the technological
cycles. The model also has plenty of empirical evidence, it was exhaustively studied within
many industries including semiconductors, telecommunications, hard drives, photocopiers,
jet engines and so on.


3.11.2. An S-Curve describing the growth of entities


There is a mathematical curve that is brought to mind by fads in general. It is the S-
shaped curve. This curve characterizes, or at least seems to characterize, a variety of
phenomena, including the demand for new toys. Its shape can most easily be explained by
imagining a few bacteria in a petri dish (see Fig. 3. 8.)).At first, the number of bacteria will

increase at a rapid exponential rate because of the rich nutrient broth and the ample space
in which to expand. Gradually, however, as the bacteria crowd each other, their rate of
increase slows and the number of bacteria stabilizes.






















Fig. 3.8


Interestingly, this curve {sometimes called the logistic curve) appears to describe
the growth of entities as disparate as Mozart’s symphony production, the rise of airline
traffic, new mainframe computer installations, and the building of Gothic cathedrals. If you
can’t think of more, the writer Cesare Marchetti and others have amassed a large number
of them and speculated that there is a kind of universal principle governing many natural
and human phenomena. What is especially provocative about the supposition is that in
many of their examples, there doesn’t seem to be anything analogous to the nutrients in the
petri dish — no resources whose depletion leads to an end to exponential growth and a
gradual leveling off.


But there is something that is being continuously depleted with time, and that is the
sense of novelty. Our natural proclivity to focus on the unusual, the dramatic, and the new
is strengthened immeasurably by newspapers and other media, but our interest wanes
quickly as well. We’re so fascinated by the sudden rise of the new celebrity, the spread of
the titillating rumor, and the increasingly frequent accounts of some exotic condition or lurid
crime that we forget the trivial fact that many phenomena have a limited life span. And
given this life span, it shouldn’t be too surprising if some phenomenon begins small, takes

off, and then gradually tapers off. It might be interesting to plot the cumulative number of
mentions that previously unknown persons, ideas, or fads receive in, say, the New York
Times to determine which ones generate S-curves.


The mathematics of the S-curve cannot be predictive of these phenomena without
more precise information. It may be nothing more than a suggestive mathematical metaphor.
Supplementing it with plausible empirical assumptions about the curve’s parameters and
arguments about its applicability, biologists have accurately predicted the growth of bacteria
in limited environments. Human population experts have also used it to forecast that the world’s population will level off at 11 to 12 billion.


We’ve all heard the television anchor with the authoritative voice intoning that such and such an index is declining (or rising) or seen the generic headline proclaiming things
getting worse (or better).At what point can we say, for example, that we are witnessing a
deterioration? Is it when the index is falling or when its ascent is slowing or its descent
quickening?Arguments can be adduced for the last two positions. (In calculus, the question
is whether the first derivative is negative — the index is falling — or whether the second
derivative is negative — either the ascent is slowing or the descent is quickening.) The
point is that even in this very simple case, a mere decline in the relevant index needn’t be a
cause for despair nor need a rise be a cause for joy.


The point on the graph of the S-curve where it switches from being concave up
(smiling) to concave down (frowning) is a critical one. This is where the growth, though still
positive, begins to slow. If the quantity indexed is something desirable, then, in a sense,
things start to get worse at this point. In another sense, they’re just getting better more
slowly. We must examine what the specific index in question is measuring to evaluate the
situation.


One last point about these indices: Situations in which the value of such an index is
rising most rapidly are often only superficially worrisome (or hopeful, depending upon
what is being measured). An example is the increase in the incidence of AIDS in rural
women over sixty years of age. If the baseline incidence is very low, a few new cases can
result in stories announcing a dramatic rise in the relevant index.


Rates of change, rates of rates of change, and the relations among them constitute
the bulk of the mathematical discipline of differential equations. It’s noteworthy that the

rudiments of the subject are implicit in seemingly straightforward news stories. Continuing

on this topic, however, will lead us beyond the point where the S-curve of interest in it
begins to level off.


3.11.3. Sigmoid Curve -The logistic curve


A sigmoid function is a mathematical function that produces a sigmoid curve - a
curve having an “S” shape. Often, sigmoid function refers to the special case of the logistic
function shown at right and defined by the formula


Members of the sigmoid family


In general, a sigmoid function is real-valued and differentiable, having either a non-
negative or non-positive first derivative and exactly one inflection point.


Besides the logistic function, sigmoid functions include the ordinary arc-tangent,
the hyperbolic tangent, and the error function, but also algebraic functions like . The integral
of any smooth, positive, “bump-shaped” function will be sigmoidal, thus the cumulative
distribution functions for many common probability distributions are sigmoidal.


The logistic sigmoid function is related to the hyperbolic tangent, e.g. by


Sigmoid functions in neural networks

Sigmoid functions are often used in neural networks to introduce nonlinearity in the
model and/or to clamp signals to within a specified range. A popular neural net element
computes a linear combination of its input signals, and applies a bounded sigmoid function
to the result; this model can be seen as a “smoothed” variant of the classical threshold
neuron.


A reason for its popularity in neural networks is because the sigmoid function
satisfies the differential equation

y’ = y(1 “ y).


The right hand side is a low order polynomial. Furthermore, the polynomial has
factors y and 1 “ y, both of which are simple to compute. Given y = sig(t) at a particular t,



the derivative of the sigmoid function at that t can be obtained by multiplying the two
factors together. These relationships result in simplified implementations of artificial neural
networks with artificial neurons.


Double sigmoid curve


The double sigmoid is a function similar to the sigmoid function with numerous
applications. Its general formula is:


where d is its centre and s is the steepness factor.


It is based on the Gaussian curve and graphically it is similar to two identical
sigmoids bonded together at the point x = d.


One of its applications is non-linear normalization of a sample, as it has the property
of eliminating outliers.


In this article I will introduce the S-curve framework, which is particularly useful
for analyzing technological cycles and predicting the introduction, adoption and maturation
of innovations
 
Top