The Growth Curve

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The Growth Curve

• Henry C. Co
• Technology and Operations Management,
• California Polytechnic and State University

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S-Curves

• The term growth curve (or S curve) represents
  a loose analogy between growth in performance of
  a technology and the growth in living organisms.
• Initially slow growth speeding up before slowing
  down to approacha a limit.
• Much research has been devoted to modeling growth
  processes, and there are many ways of doing this,
  including mechanistic models, time series,
  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 performance by chemical and physical
  laws that establish the maximum performance that
  can be obtained using a given principle of
  operation.
• Adoption of device using a different principle of
  operation means a transfer to a new growth curve.
  
• These patterns can be modeled using several
  mathematical functions.
• Applications
• Animal weights over time
• Growth of human populations
• Biomedical data
• Development of organisms.

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Example Animal Weights Over Time

• The 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 points for each cow on the
graph.
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A Typical Growth Curve

• The 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 Richards) curve is
  a widely used and flexible function for growth
  modeling.
• Where
• Y 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 Weights

• We 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 on technological
  performance, then extrapolating the growth curve
  beyond the range of the data to obtain an
  estimate of future performance.
• Assumptions
• The 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 formula correctly.
• The growth curve most frequently used by
  technological forecasters are
• the Pearl curve
• the Gompertz curve

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The Pearl Curve

• Named after U.S. demographer Raymond Pearl
• Pearl popularized its use for population
  forecasting.
• Also known as the logistics curve.
• The standard Pearl curve
• L is the upper limit to the growth of the
  variable y,
• e is the base of the natural logarithms
• t is time, and
• a and b are the coefficients obtained by fitting
  the curve to the data.

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• Properties of the Pearl curve
• Initial 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 tln(a)/b, when
  yL/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
  controlled independently.
• Changes in the coefficient a affect the location
  only they do not alter 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., the formula 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 by the difference between the data
  value and the upper limit (L-y), the transformed
  variable becomes a linear function of time t. The
  constant term is (ln a) and the slope is b.
• Once the coefficients a and b have been
  determined by regression analysis, they can be
  substituted back into the Pearl curve formula.
  The formula can then be extrapolated to future
  values of time by substituting the appropriate
  value for t.

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• 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 the Pearl curve, respectively.
• With an algebraic manipulation that is similar to
  the standard version of the Pearl curve
• The revised Pearl curve can be transformed into
  the following straight-line equation

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The Gompertz Curve

• Named after Benjamin Gompertz, an English actuary
  and mathematician.
• The standard Gompertz curve
• where
• y is the variable representing performance,
• L is the upper limit,
• e is the base of the natural logarithms
• t is time, and
• b and k are the coefficients obtained by fitting
  the curve to the data.

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• Properties of the Gompertz curve
• Like 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 occurs at 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 and the slope term is k.

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Gompertz Curve - Fitted Cow Data
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Pearl v. Gompertz Curves

• The slope of the Pearl curve involves y and
  (L-y), i.e., distance already come and distance
  yet to go to the upper limit.
• For large values of y, the 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.

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• Consider the growth in performance of a technical
  approach.
• Progress will be harder to achieve the closer the
  upper limit is approach.
• Is there any offsetting factor by which progress
  already achieved makes additional progress
  easier?
• If there is such an offsetting factor, then
  progress is a function 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 a function only of distance to go,
  and the Gompertz curve is the appropriate choice.

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Substitution Curves
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• Frequently, one is interested in forecasting the
  rate at which a new technology will be
  substituted for an older technology in a given
  application.
• Substitution of new technology for an older one
  often exhibits a growth curve.
• Initially, the older technology has the
  advantage. Initial rate of substitution is low.
• The older technology is well understood, its
  reliability is probably high, users have
  confidence in it, and both spare parts and
  technicians are readily available.
• The new technology is unknown and its reliability
  is uncertain spare parts are hard to obtain and
  skilled technicians are scarce.
• As the initial problems are solved, the rate of
  substitution increases.
• As the substitution becomes complete, however,
  there will remain a few applications for which
  the old technology is well suited. The rate of
  substitution slows, as the older technology
  becomes more and more difficult to replace.

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