Selecting Appropriate Projections

Input and Output Evaluation

Input Evaluation Compares observed historical trend with the

assumed trend line properties.Linear Population Projection, Leon County FL

0.0

20000.0

40000.0

60000.0

80000.0

100000.0

120000.0

140000.0

160000.0

1940 1945 1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010 2015

Linear

Observed Values

Input Evaluation Conceptual Question Being Asked:

Which type of curve best fits our observed historical trend?

We can “eyeball” (the art) We can employ comparative statistics

(the science)

Input Evaluation Linear Curve

Assumption: constant growth increments i.e., constant absolute change

Constant Growth Increments = “First Differences”

This is the “best fit” if the curve approximates a straight line

Input Evaluation Geometric Curve

Assumption: Growth increments for the logarithms of the geometric curve are equal to a constant value

Even more technically, these are the first differences of the logarithm of the observed values

That is, growth is exponential – the rates of change are constant

Input Evaluation Parabolic Curve

Assumption: Constant Second Differences (differences of the first difference)

This curve has a constantly changing slope, and one bend (given a sufficient number of observations i.e., it describes a parabola

A Parabola

Input Evaluation Modified Exponential Curve

Assumption: First differences decline or increase at a constant percentage

Assumption includes a limit, beyond which the curve will not exceed

Input Evaluation Gompertz Curve

Assumption: First differences in the logarithms of the dependent variable decline by a constant percentage

One of a family of “S” Curves

Input Evaluation Logistic Curve

Assumption: The first differences in the reciprocals of the observed values decline by a constant percentage.

“Reciprocal” = 1 / the observed value Curve is characterized by an “s” shape

Input Evaluation Compare the “Coefficient of Relative

Variation” (CRV) or CV Describes variation about the mean

value Variation = standard deviation Mean value = arithmetic mean (average) CRV is calculated to create a standardized

point of reference

Input Evaluation Mean

MZ iZ

Input Evaluation Standard Deviation

21

2

1

MZ

s i Z

Input Evaluation Coefficient of

Relative Variation

ZsCRV

Output Evaluation Compares the observed trend values

with the computed trend values Only for the period of the historical trend Assumes that if historical trend fits well,

the extrapolated trend will follow

Output EvaluationLinear Population Projection, Leon County FL

0.0

20000.0

40000.0

60000.0

80000.0

100000.0

120000.0

140000.0

160000.0

1940 1945 1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010 2015

Linear

Observed Values

Geometric Population Projection, Leon County FL

0.0

50000.0

100000.0

150000.0

200000.0

250000.0

300000.0

350000.0

400000.0

450000.0

1940 1945 1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010 2015

Geometric

Observed Values

Parabolic Population Projection, Leon County FL

0.0

50000.0

100000.0

150000.0

200000.0

250000.0

1940 1945 1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010 2015

Parabolic

Observed Values

Modified Exponential Population Projection (500,000 limit assumed), Leon County FL

0.0

20000.0

40000.0

60000.0

80000.0

100000.0

120000.0

140000.0

160000.0

1940 1945 1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010 2015

Modified Exponential (assumed)

Observed Values

Output Evaluation Mean Error (ME) Mean Absolute Percentage Error

(MAPE)

Output Evaluation

NYY

ME c

Mean Error

Output Evaluation

100

NYYY

MAPEc

Mean Absolute Percentage Error

Output Evaluation ME

Good for detecting estimation error or bias Consistent over- or underestimation

MAPE Evaluates total estimation error “Dimensionless”

Good for any data

Excel Formulas to Note =sum(x) =average(x) =stdev(x) =count(x) =concatenate(x,y)

Math Reciprocal

Logs

antilogs