1. Selecting Appropriate Projections Input and Output Evaluation

2. Input Evaluation  Compares observed historical trend with the assumed trend line properties.

3. 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)

4. 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

5. 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

6. 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

7. A Parabola

8. 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

9. 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

10. 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

11. 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

12. Input Evaluation  Mean

13. Input Evaluation  Standard Deviation

14. Input Evaluation  Coefficient of Relative Variation

15. 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

16. Output Evaluation

17.  Mean Error (ME)  Mean Absolute Percentage Error (MAPE)

18. Output Evaluation  Mean Error

19. Output Evaluation  Mean Absolute Percentage Error

20. Output Evaluation  ME  Good for detecting estimation error or bias  Consistent over- or underestimation  MAPE  Evaluates total estimation error  “ Dimensionless ”  Good for any data

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

22. Math  Reciprocal  Logs  antilogs