Trend Projection

1
Trend Projection

• Questions from last day?
• Geometric series question

2
Geometric Series-Reprise

• We use a geometric curve for discrete time series
• Fundamental characteristic of many systems
• that increase at a constant rate of growth (r)
• in the absence of any constraints on growth
• A geometric series increases following the
  general form
• Lets try some examples for different values of b

3
Logarithmic Series

• http//www.ucl.ac.uk/Mathematics/geomath/level2/se
  ries/ser10.html

4
Review

• Experiential treatment of arithmetic and
  logarithmic scales
• Nottingham and Ruddington
• Hypothetical data
• Power Laws and logarithms
• Characteristics and graphs of two curves used in
  trend projections

5
Agenda

• Curve fitting
• Eyeball
• Criteria for fitting trend line to points
• Add 2 more curves to linear and geometric
• Why?
• To recognize properties for evaluation of series
• To understand how curves may be transformed
• To understand the effect of parameters

6
Eyeballing Exercise
7
Some Criteria for Fitting a Line

• Diagram courtesy of National Council of Teachers
  of Mathematics
• http//standards.nctm.org/document/eexamples/chap
  7/7.4/applet

8
The Least Squares Criterion

• The line which minimizes the sum of the squares
  of the errors.
• Demonstration courtesy of Erhan Erkut and Armann
  Ingolfsson (2000), "Let's Put the Squares in
  Least-Squares," INFORMS Transactions on
  Education, Vol. 1, No. 1,
• http//ite.pubs.informs.org/Vol1No1/ErkutIngolfsso
  n/leastsquares.xls

9
Linear curve

• 1st degree curve, for arithmetic series
• Y is an observed variable e,g, population
• X represents an observation period e.g. year
• a b are parameters, fixed for a given curve
• a is the Y intercept
• b is the slope or rate of change

10
Linear curve

• How does the curve look for various values of a
  and b?
• Linear transformation is not necessary!

11
Geometric Curve

• How does the curve look for various values of a
  and b?
• If b gt 1, Y increases without limit
• If b1, Ya
• If 0ltblt1, Y approaches zero
• If b is negative, Y oscillates

12
Geometric Curve

• Linear transformation How may we transform this
  curve?

13
Parabola

• How does the curve look for various values of a
  and b?
• a is Y intercept
• b is slope at Y intercept
• c controls shape of curve

14
Parabola

• Linear transformation How may we transform this
  curve?

15
Modified Exponential Curve

• How does the curve look for various values of a
  and b?
• ca is Y intercept
• If a lt 0, convex up
• If a gt 0, concave up
• If 0ltblt1, Y approaches asymptote
• If b gt 1 Y moves away from asymptote
• c is upper or lower bound

16
Modified Exponential Curve

• Linear transformation How may we transform this
  curve?
• If a is negative
• If a is positive

17
Gompertz Curve

• How does the curve look for various values of a
  and b?
• ca is Y intercept
• If log(a) lt 0, sigmoid
• If log(a) gt 0, concave up
• b affects direction of slope, depending on sign
  of a
• c is upper or lower bound

18
Gompertz Curve

• Linear transformation What function may we
  apply to transform this curve?
• If log(a) is negative
• If log(a) is positive

19
Logistic Curve

• How does the curve look for various values of a
  and b?
• 1/ca is Y intercept
• If 0ltblt1, positive sigmoid
• If bgt1, negative sigmoid
• 1/c is upper or lower bound

20
Logistic Curve

• Linear transformation What function may we
  apply to transform this curve?