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The Growth Curve

Henry C. CoTechnology and Operations Management,California Polytechnic and State University



S-Curves

The term “growth curve” (or “S curve”) representsa loose analogy between growth in performance of a technology and the growth in living organisms.

Initially slow growth speeding up before slowing down toapproacha a limit.

Much research has been devoted to modeling

growth processes, and there are many ways of doing this, including: mechanistic models, timeseries, stochastic differential equations etc.

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Many growth phenomena in nature show an “S” shaped pattern Any single technical approach is limited in its ultimate performanceby chemical and physical laws that establish the maximumperformance that can be obtained using a given principle of operation.

Adoption of device using a different principle of operation means atransfer to a new growth curve.These patterns can be modeled using several mathematicalfunctions.

Applications Animal weights over time;Growth of human populations;Biomedical data;Development of organisms.

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Animal Weights Over TimeThe weights of cows have been recorded every two weeks for 232 weeks,i.e. 116 observations in total

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We wish to fit a curve which best summarizes the distribution of the pointsfor each cow on the graph.

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A Typical Growth CurveThe lower asymptote is the starting level.The upper asymptote is the mature level.The point of inflexion is the point of maximum growth.The curve can be modeled in a number of ways.

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Generalized Logistic Curve

The generalized logistic (or Richard’s) curve is a widely usedand flexible function for growth modeling.

WhereY= weight, height, size etc;

X = time.

A controls the lower asymptote,

C controls the upper asymptote,

M controls the time of maximum growth,

B controls the growth rate, and

T controls where maximum growth occurs - nearer the lower or upper asymptote.

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Application to Cow WeightsWe will fit the generalized logistic curve to body weight data for each cow.Cow live weights have been recorded every two weeks for 232 weeks, i.e.116 observations in total.The curve is fitted separately for each cow, so that parameters can be

compared

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Using Growth Curves

Forecasting by growth curves involves fitting a growth curve to a set of data ontechnological performance, then extrapolating the growth curve beyond therange of the data to obtain an estimate of future performance.

AssumptionsThe upper limit to the growth curve is known.

The chosen growth curve to be fitted to the historical data is the correct one.

The historical data gives the coefficients of the chosen growth curve formulacorrectly.

The growth curve most frequently used by technological forecasters arethe Pearl curve

the Gompertz curve

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The Pearl Curve

Named after U.S. demographer Raymond PearlPearl popularized its use for population forecasting.

Also known as the “logistics curve.”

The standard Pearl curve:

L = upper limit to the growth of the variable y,

e = base of the natural logarithmst = timea,b = coefficients obtained by fitting the curve to the data.

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bt e L

y 1



Properties of the Pearl curveInitial value of zero at time = - and a value of L at time = + .

If the initial value is not zero, the initial value can be added as a

constant to the above formula.The inflection point occurs at t=ln(a)/b, when y=L/2.

The curve is symmetrical about this point, with the upper half being a reflection of the lower half.

The shape and location of the Pearl curve can be controlledindependently.

Changes in the coefficient a affect the location only; they do notalter the shape.

Changes in the coefficient b affect the shape only, they do not

alter the location.

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Fitting Pearl Curve to Data

It is customary to “straighten out” the curve first (i.e., theformula is transformed to a straight line):

The right-hand side is the equation of a straight line.

By taking the natural logarithm of the data value (y) divided bythe difference between the data value and the upper limit (L-y),the transformed variable becomes a linear function of time t. Theconstant term is – (ln a) and the slope is b.

Once the coefficients a and b have been determined byregression analysis, they can be substituted back into the Pearlcurve formula. The formula can then be extrapolated to futurevalues of time by substituting the appropriate value for t .

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bt a y L y

Y lnln



The revised Pearl curve:

Here again, L is the upper limit of the variable y, and t is the

time.The coefficients A and B control the location and shape of thePearl curve, respectively.

With an algebraic manipulation that is similar to the standardversion of the Pearl curve

The revised Pearl curve: can be transformed into thefollowing straight-line equation:

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Bt A

L y

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Bt A y L

yY log



The Gompertz Curve

Named after Benjamin Gompertz, an English actuary andmathematician.

The standard Gompertz curve:

wherey = variable representing performance,

L = upper limit,e = base of the natural logarithmst = timeb, k = coefficients obtained by fitting the curve to the data.

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kt be

Le y



Properties of the Gompertz curveLike the Pearl curve, the initial value is zero at time = - and a value of L at time = + .The curve is not symmetrical. The inflection point occursat t = (ln b )/k , when y = L /e .

By taking the logarithm of the Gompertz curve

twice, we obtain

When Y is regressed on t, the constant term is ln b andthe slope term is k .

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kt b y LY

lnlnln



Gompertz CurveFitted Cow Data

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Pearl v. Gompertz CurvesThe slope of the Pearl curve involves y and (L-y), i.e., distance alreadycome and distance yet to go to the upper limit.For large values of y, slope of the Gompertz curve involves only (L-y), i.e.,the Gompertz curve is a function only of distance to go to the upper limit.

17Comparison of the Pearl and Gompertz Curves

Curve Equation Slope

Pearl bt e

L y 1

L

y Lby

Gompertz kt be Le y for all values of y L ybky /lnkt be Le y Approximation

fory L/2

L ybk /



Consider the growth in performance of a technical approach.Progress will be harder to achieve the closer the upper limit isapproach.

Is there any offsetting factor by which progress alreadyachieved makes additional progress easier?

If there is such an offsetting factor, then progress is afunction of both distance to go and distance already come(thus the Pearl curve is the appropriate choice.).

If there is not any such offsetting factor, progress is afunction only of distance to go, and the Gompertz curve isthe appropriate choice.

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