Extrapolation Methods Summarized

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Extrapolation Technique Summarized

• The extrapolation technique (aka curve fitting)
  is a simplistic model that uses past gross
  population trends to project future population
  levels.
• The defining characteristics of trend
  extrapolation is that future values of any
  variable are determined solely by its historical
  values. (SLPP, p. 161 emphasis added)
• Basic Procedure 1) Identify overall past
  trend and fit proper curve THEN 2) Project
  future populations based upon your chosen curve
• We use a linear equation for most of these
  equations. A linear transformation is required to
  make projections for all but the Parabolic Curve.
• Advantages 1) Low data requirements 2) Very
  easy methodology 3) Therefore, low resource
  requirements
• Disadvantages 1) Uses only aggregate data
  2) Assumes that past trends will predict the
  future

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The Curves to Be Fit

• Linear Curve Plots a straight line based on the
  formula
• Y a bX
• Geometric Curve Plots a curve based upon a rate
  of compounding growth over discrete intervals via
  the formula Y aebX
• Parabolic (Polynomial) Curve A curve with one
  bend and a constantly changing slope. Formula Y
  a bX cX2
• Modified Exponential Curve An asymptotic
  growth curve that recognizes that a region will
  reach an upper limit of growth. It takes the
  form Y c abX
• Gompertz Curve Describes a growth pattern that
  is quite slow, increases for a time, and then
  tapers off as the population approaches a growth
  limit. Form Y c(a) exp (bX)
• Logistic Curve Similar to the Gompertz Curve,
  this is useful for describing phenomena that grow
  slowly at first, increase rapidly, and then slow
  with approach to a growth limit. Y (c
  abX)-1 Asymptotic Curve

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The Linear Curve (Y a bX)

• Fits a straight line to population data. The
  growth rate is assumed to be constant, with
  non-compounding incremental growth. Calculated
  exactly the same as using linear regression
  (least-squares criterion).
• Advantages --Simplest curve --Most widely
  used --Useful for slow or non-growth areas
• Disadvantages --Rarely appropriate to
  demographic data
• Example
• Y 55,000 6,000(X)
• In plain language, this equation tells us that
  for each year that passes, we can project an
  additional 6,000 people will be added to the
  population. So, in 10 years we would project
  60,000 more people using this equation.
• Evaluation Generally used as a staring point
  for curve fitting.

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Calculating the Linear Curve
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Basic Lesson 1 on Extrapolation

• The Linear Curve helps to illustrate a very basic
  principle of using the the extrapolation
  technique
• The choice of the Base Period can have a
  significant impact upon the projection generated.
• In our Leon County example, if we use a varying
  Base Year, we get the following results

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Basic Lesson 2 on Extrapolation

• The Linear Curve also helps to illustrate a
  second basic principle of using the the
  extrapolation technique
• Oftentimes analysts calibrate their model to
  fit the projection to the observed data.
• Calibration is very simply an adjustment that
  makes the projected population consistent with
  the launch year population.
• The calibration is calculated by subtracting the
  estimated population from the observed population
  in the launch year (Observed Estimated).
• In our Leon County example, the adjustment for
  BY1940 is
• Observed Pop 1980 87,000 Estimated Pop 1980
  82,500 Calibration 4,500
• Therefore the Calibrated Projections would be as
  follows
• Calibration is usually used with the Lin
  Regression technique, but can be used in others
  as well.

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The Geometric Curve (Y aebX)

• In this curve, a growth rate is assumed to be
  compounded at set intervals using a constant
  growth rate. To transform this equation into a
  linear equation, we use logarithms.
• Advantages --Assumes a constant rate of
  growth --Still simple to use
• Disadvantage --Does not take into account a
  growth limit
• Example
• Y 55,000 (1.00 0.06)X
• In plain language, this equation tells us that we
  have a 6 growth rate. After one year we project
  a population of 58,300. After 10 years we would
  project a population of 98,497.
• Evaluation Pretty good for short term
  fast-growing areas. However, over the long-run,
  this curve usually generates unrealistically high
  numbers.

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Calculating the Geometric Curve
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The Parabolic Curve (Y a bX cX2)

• Generally has a constantly changing slope and
  one bend. Very similar to the Linear Curve except
  for the additional parameter (c). Growing very
  quickly when c gt 0, declining quickly when c lt 0.
• Advantage --Models fast growing areas
• Disadvantages --Poor for long range
  projections (familiar refrain?) --No Growth
  Limit --More complex
• Example
• Y 43.46 8.78(X) 0.581(X2)
• When X0, Y 43.46. When X 6, Y 117.1
• Evaluation Exactly the same as the Geometric
  Curve good for fast growing areas, but poor over
  the long run.

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Modified Exponential Curve (Y c abX )

• The first of the Asymptotic Curves. Takes into
  account an upper or lower limit when computing
  projected values. The asymptote can be derived
  from local analysis or supplied by the model
  itself.
• Advantage --Growth limit is introduced --Bes
  t fitting growth limit
• Disadvantage --Much more complex
  calculations --Misleading Growth limit (high
  and low)
• Example
• Yc 114 - 64(0.75)X
• The growth limit is 114. The curve takes into
  account the number of time periods and as X gets
  larger the closer you get to the Growth limit.
  When X 0, Y 50 when X 2, Y 78, etc.
• Evaluation This curve largely depends upon the
  growth limit. If the limit is reasonable, then
  the curve can be a good one. Also, the ability to
  calculate the growth limit within the model is
  very useful.

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The Gompertz Curve (Y c(a) exp (bX))

• Describes a growth pattern that is initially
  quite slow, increases for a period and then
  tapers off. Like the Mod Exp curve, the upper
  limit can be assumed or derived by the model.
• Advantage --Reflects very common growth
  patterns
• Disadvantages --Getting even more
  complex --Misleading growth limit (limit can be
  high or low)
• Example
• log Yc 2.699 - 1.056(0.9221)X
• The equation itself is tough to understand. When
  X 0, Log Y 1.64, so Y 44.0 (via antilog
  calculation). Note Antilog of 2.699 is 500
  (the growth limit)
• Evaluation A very useful curve that can be
  fitted to all kinds of growth patterns. However,
  as with the previous curve, using an assumed
  growth limit can be problematic unless it is
  reasonable and makes sense for the case at hand.

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The Logistic Curve (Y (c abX)-1 )

• VERY similar to the Mod Exp and the Gompertz
  curves, except that we are taking the reciprocals
  of the observed values. A very popular curve.
• Advantages --Has proven to be a good
  projection tool --Considered a bit more stable
  than the Gompertz curve
• Disadvantages --Complex! --Hard to
  interpret the formula
• Example
• Yc-1 0.0020 0.217(0.8015)X
• Another difficult to interpret equation. When X
  0, Y 42.1. When X 6, Y 128.9. Note
  Reciprocal of .002 is 500 (GL)
• Evaluation Considered to be the best of the
  extrapolation curves. It reflects a well-known
  growth pattern. It is more stable than the
  Gompertz curve and it does not have a misleading
  growth limit.

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The Skill in Curve Fitting

• The Curve Fitting Procedure in more detail
• 1) Plot the data in a chart
• 2) Eyeball the data Identify and eliminate
  erroneous data Identify past population
  trends Eliminate curves that dont fit the data
• 3) Process the data using the chosen curves,
  Plot your results in charts
• 4) Calibrate the model if it seems approrpriate
• 5) Use quantitative procedures to identify
  best-fitting curves
• 6) Make your choice of forecast based upon a
  combination of quantitative and qualitative
  evaluations of the various projections
• Many issues affect how well curve fitting
• --Choice of the Base Period, including the Base
  Year
• --Selection of the Curve that best fits the
  data
• --Identification of the Best-Fitting Curve for
  that Curve type
• --Consider the possibility of a growth limit

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Whos in Charge Here?

• A planners task is to combine the most reliable
  information about the past with the most
  appropriate assumptions about the future to
  prepare the best possible forecast.
• These projection techniques are only quantitative
  procedures for for using limited information with
  the most appropriate assumptions about the future
  to predict the unknowable future.
• Planners often assume that the techniques are
  what make for good projections. This is simply
  not so!
• The Planner is the analyst and the computer,
  techniques, and data are the tools. I cannot
  emphasize enough that the Planner makes the
  decisions about appropriate and useful
  projections, not the computer and not the data.

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