selecting appropriate projections input and output evaluation
DESCRIPTION
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)TRANSCRIPT
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